From 76806676383a8bbbbb61accc44576213322f1665 Mon Sep 17 00:00:00 2001
From: Daniel Wang <dan@taiko.xyz>
Date: Thu, 1 Aug 2024 14:56:57 +0800
Subject: [PATCH 01/17] more

---
 .../protocol/contracts/L2/Lib1559Math.sol     | 32 +++++-------------
 packages/protocol/test/L2/Lib1559Math.t.sol   | 33 +++++++++++++++++++
 2 files changed, 41 insertions(+), 24 deletions(-)

diff --git a/packages/protocol/contracts/L2/Lib1559Math.sol b/packages/protocol/contracts/L2/Lib1559Math.sol
index 1ee53ef118d..ae4c19d2b3e 100644
--- a/packages/protocol/contracts/L2/Lib1559Math.sol
+++ b/packages/protocol/contracts/L2/Lib1559Math.sol
@@ -43,34 +43,18 @@ library Lib1559Math {
 
     /// @dev eth_qty(excess_gas_issued) / (TARGET * ADJUSTMENT_QUOTIENT)
     /// @param _gasExcess The gas excess value
-    /// @param _adjustmentFactor The product of gasTarget and adjustmentQuotient
-    function basefee(
-        uint256 _gasExcess,
-        uint256 _adjustmentFactor
-    )
-        internal
-        pure
-        returns (uint256)
-    {
-        if (_adjustmentFactor == 0) {
-            revert EIP1559_INVALID_PARAMS();
-        }
-        return _ethQty(_gasExcess, _adjustmentFactor) / LibFixedPointMath.SCALING_FACTOR;
+    /// @param _target The product of gasTarget and adjustmentQuotient
+    function basefee(uint256 _gasExcess, uint256 _target) internal pure returns (uint256) {
+        if (_target == 0) revert EIP1559_INVALID_PARAMS();
+        return ethQty(_gasExcess, _target) / _target;
     }
 
-    /// @dev exp(gas_qty / TARGET / ADJUSTMENT_QUOTIENT)
-    function _ethQty(
-        uint256 _gasExcess,
-        uint256 _adjustmentFactor
-    )
-        private
-        pure
-        returns (uint256)
-    {
-        uint256 input = _gasExcess * LibFixedPointMath.SCALING_FACTOR / _adjustmentFactor;
+    /// @dev exp(_gasExcess / _target)
+    function ethQty(uint256 _gasExcess, uint256 _target) internal pure returns (uint256) {
+        uint256 input = _gasExcess * LibFixedPointMath.SCALING_FACTOR / _target;
         if (input > LibFixedPointMath.MAX_EXP_INPUT) {
             input = LibFixedPointMath.MAX_EXP_INPUT;
         }
-        return uint256(LibFixedPointMath.exp(int256(input)));
+        return uint256(LibFixedPointMath.exp(int256(input))) / LibFixedPointMath.SCALING_FACTOR;
     }
 }
diff --git a/packages/protocol/test/L2/Lib1559Math.t.sol b/packages/protocol/test/L2/Lib1559Math.t.sol
index a4d275d26be..b471ac9a5d8 100644
--- a/packages/protocol/test/L2/Lib1559Math.t.sol
+++ b/packages/protocol/test/L2/Lib1559Math.t.sol
@@ -6,8 +6,41 @@ import "../TaikoTest.sol";
 contract TestLib1559Math is TaikoTest {
     using LibMath for uint256;
 
+    uint256 public constant gips = 2_000_000; // gas issuance per second
+    uint256 public constant quotient = 4;
+    uint256 public constant target = gips * quotient;
+
+    function test_ethQty() external {
+        assertEq(Lib1559Math.ethQty(0, 60_000_000 / 8), 1);
+        assertEq(Lib1559Math.ethQty(60_000_000, 60_000_000 / 8), 2980);
+        assertEq(Lib1559Math.ethQty(60_000_000 * 2, 60_000_000 / 8), 8_886_110);
+        assertEq(Lib1559Math.ethQty(60_000_000 * 3, 60_000_000 / 8), 26_489_122_129);
+        assertEq(Lib1559Math.ethQty(60_000_000 * 4, 60_000_000 / 8), 78_962_960_182_680);
+        assertEq(
+            Lib1559Math.ethQty(60_000_000 * 10, 60_000_000 / 8),
+            55_406_223_843_935_100_525_863_115_942_268_902
+        );
+        assertEq(
+            Lib1559Math.ethQty(60_000_000 * 100, 60_000_000 / 8),
+            57_896_044_618_658_097_650_144_101_621_524_338_577_433_870_140_581_303_254_786
+        );
+    }
+
+    function test_basefee() external {
+        uint256 basefee;
+        for (uint256 i; basefee <= 5000;) {
+            // uint 0.01 gwei
+            basefee = Lib1559Math.basefee(i * gips, target) / 10_000_000;
+            if (basefee != 0) {
+                console2.log("basefee (uint 0.01gwei) after", i, "seconds:", basefee);
+            }
+            i += 12;
+        }
+    }
+
     function test_eip1559_math() external pure {
         LibL2Config.Config memory config = LibL2Config.get();
+
         uint256 adjustmentFactor = config.gasTargetPerL1Block * config.basefeeAdjustmentQuotient;
 
         uint256 baseFee;

From d207123345fc6eb3a53f785e85cd97e8cc0f2253 Mon Sep 17 00:00:00 2001
From: Daniel Wang <dan@taiko.xyz>
Date: Thu, 1 Aug 2024 15:37:10 +0800
Subject: [PATCH 02/17] improve basefee impl

---
 packages/protocol/contracts/L1/TaikoData.sol  |  4 +-
 packages/protocol/contracts/L1/TaikoL1.sol    |  2 +-
 .../protocol/contracts/L1/libs/LibData.sol    |  2 +-
 .../contracts/L1/libs/LibProposing.sol        |  2 +-
 .../protocol/contracts/L2/Lib1559Math.sol     |  6 +-
 packages/protocol/contracts/L2/TaikoL2.sol    | 68 +++++++++++--------
 .../protocol/contracts/hekla/HeklaTaikoL1.sol |  2 +-
 .../contracts/mainnet/MainnetTaikoL1.sol      |  2 +-
 packages/protocol/test/L2/Lib1559Math.t.sol   | 68 +++++++------------
 .../test/L2/TaikoL2EIP1559Configurable.sol    |  2 +-
 10 files changed, 75 insertions(+), 83 deletions(-)

diff --git a/packages/protocol/contracts/L1/TaikoData.sol b/packages/protocol/contracts/L1/TaikoData.sol
index e2cae944c1a..8b08b699a14 100644
--- a/packages/protocol/contracts/L1/TaikoData.sol
+++ b/packages/protocol/contracts/L1/TaikoData.sol
@@ -41,7 +41,7 @@ library TaikoData {
         // ---------------------------------------------------------------------
         uint8 basefeeAdjustmentQuotient;
         uint8 basefeeSharingPctg;
-        uint32 gasTargetPerL1Block;
+        uint32 gasIssuancePerSecond;
         // ---------------------------------------------------------------------
         // Group 6: Others
         // ---------------------------------------------------------------------
@@ -126,7 +126,7 @@ library TaikoData {
         uint8 blobIndex;
         uint8 basefeeAdjustmentQuotient;
         uint8 basefeeSharingPctg;
-        uint32 gasTargetPerL1Block;
+        uint32 gasIssuancePerSecond;
     }
 
     /// @dev Struct representing transition to be proven.
diff --git a/packages/protocol/contracts/L1/TaikoL1.sol b/packages/protocol/contracts/L1/TaikoL1.sol
index e4d70136e6c..e590bb1851b 100644
--- a/packages/protocol/contracts/L1/TaikoL1.sol
+++ b/packages/protocol/contracts/L1/TaikoL1.sol
@@ -287,7 +287,7 @@ contract TaikoL1 is EssentialContract, ITaikoL1, TaikoEvents {
             maxAnchorHeightOffset: 64,
             basefeeAdjustmentQuotient: 8,
             basefeeSharingPctg: 75,
-            gasTargetPerL1Block: 60_000_000,
+            gasIssuancePerSecond: 2_000_000,
             ontakeForkHeight: 374_400 // = 7200 * 52
          });
     }
diff --git a/packages/protocol/contracts/L1/libs/LibData.sol b/packages/protocol/contracts/L1/libs/LibData.sol
index d8ea53b7432..f497b7fa0ac 100644
--- a/packages/protocol/contracts/L1/libs/LibData.sol
+++ b/packages/protocol/contracts/L1/libs/LibData.sol
@@ -78,7 +78,7 @@ library LibData {
             blobIndex: 0,
             basefeeAdjustmentQuotient: 0,
             basefeeSharingPctg: 0,
-            gasTargetPerL1Block: 0
+            gasIssuancePerSecond: 0
         });
     }
 }
diff --git a/packages/protocol/contracts/L1/libs/LibProposing.sol b/packages/protocol/contracts/L1/libs/LibProposing.sol
index 0e8b2c893cd..14afcf4f5cd 100644
--- a/packages/protocol/contracts/L1/libs/LibProposing.sol
+++ b/packages/protocol/contracts/L1/libs/LibProposing.sol
@@ -176,7 +176,7 @@ library LibProposing {
                 blobIndex: local.params.blobIndex,
                 basefeeAdjustmentQuotient: _config.basefeeAdjustmentQuotient,
                 basefeeSharingPctg: _config.basefeeSharingPctg,
-                gasTargetPerL1Block: _config.gasTargetPerL1Block
+                gasIssuancePerSecond: _config.gasIssuancePerSecond
             });
         }
 
diff --git a/packages/protocol/contracts/L2/Lib1559Math.sol b/packages/protocol/contracts/L2/Lib1559Math.sol
index ae4c19d2b3e..6504b759617 100644
--- a/packages/protocol/contracts/L2/Lib1559Math.sol
+++ b/packages/protocol/contracts/L2/Lib1559Math.sol
@@ -36,9 +36,6 @@ library Lib1559Math {
         // block, however, this block's gas used will affect the next
         // block's base fee.
         basefee_ = basefee(gasExcess_, uint256(_adjustmentQuotient) * _gasTarget);
-
-        // Always make sure basefee is nonzero, this is required by the node.
-        if (basefee_ == 0) basefee_ = 1;
     }
 
     /// @dev eth_qty(excess_gas_issued) / (TARGET * ADJUSTMENT_QUOTIENT)
@@ -46,7 +43,8 @@ library Lib1559Math {
     /// @param _target The product of gasTarget and adjustmentQuotient
     function basefee(uint256 _gasExcess, uint256 _target) internal pure returns (uint256) {
         if (_target == 0) revert EIP1559_INVALID_PARAMS();
-        return ethQty(_gasExcess, _target) / _target;
+        uint256 fee = ethQty(_gasExcess, _target) / _target;
+        return fee == 0 ? 1 : fee;
     }
 
     /// @dev exp(_gasExcess / _target)
diff --git a/packages/protocol/contracts/L2/TaikoL2.sol b/packages/protocol/contracts/L2/TaikoL2.sol
index 1fcfe68709c..b8cc1015b05 100644
--- a/packages/protocol/contracts/L2/TaikoL2.sol
+++ b/packages/protocol/contracts/L2/TaikoL2.sol
@@ -35,12 +35,12 @@ contract TaikoL2 is EssentialContract {
 
     /// @notice The gas excess value used to calculate the base fee.
     /// @dev Slot 3.
-    uint64 public gasExcess;
+    uint64 public parentGasExcess;
 
     /// @notice The last synced L1 block height.
     uint64 public lastSyncedBlock;
     uint64 private parentTimestamp;
-    uint64 private __deprecated2; // was __currentBlockTimestamp
+    uint64 private parentGasTarget;
 
     /// @notice The L1's chain ID.
     uint64 public l1ChainId;
@@ -49,8 +49,8 @@ contract TaikoL2 is EssentialContract {
 
     /// @notice Emitted when the latest L1 block details are anchored to L2.
     /// @param parentHash The hash of the parent block.
-    /// @param gasExcess The gas excess value used to calculate the base fee.
-    event Anchored(bytes32 parentHash, uint64 gasExcess);
+    /// @param parentGasExcess The gas excess value used to calculate the base fee.
+    event Anchored(bytes32 parentHash, uint64 parentGasExcess);
 
     error L2_BASEFEE_MISMATCH();
     error L2_FORK_ERROR();
@@ -65,12 +65,12 @@ contract TaikoL2 is EssentialContract {
     /// @param _owner The owner of this contract. msg.sender will be used if this value is zero.
     /// @param _rollupAddressManager The address of the {AddressManager} contract.
     /// @param _l1ChainId The ID of the base layer.
-    /// @param _gasExcess The initial gasExcess.
+    /// @param _parentGasExcess The initial parentGasExcess.
     function init(
         address _owner,
         address _rollupAddressManager,
         uint64 _l1ChainId,
-        uint64 _gasExcess
+        uint64 _parentGasExcess
     )
         external
         initializer
@@ -95,11 +95,16 @@ contract TaikoL2 is EssentialContract {
         }
 
         l1ChainId = _l1ChainId;
-        gasExcess = _gasExcess;
-        parentTimestamp = uint64(block.timestamp);
+        parentGasExcess = _parentGasExcess;
         (publicInputHash,) = _calcPublicInputHash(block.number);
     }
 
+    function init2() external reinitializer(2) {
+        parentGasExcess = 0;
+        parentTimestamp = uint64(block.timestamp);
+        parentGasTarget = 0;
+    }
+
     /// @notice Anchors the latest L1 block details to L2 for cross-layer
     /// message verification.
     /// @dev This function can be called freely as the golden touch private key is publicly known,
@@ -134,7 +139,7 @@ contract TaikoL2 is EssentialContract {
         uint64 _anchorBlockId,
         bytes32 _anchorStateRoot,
         uint32 _parentGasUsed,
-        uint32 _gasTargetPerL1Block,
+        uint32 _gasIssuancePerSecond,
         uint8 _basefeeAdjustmentQuotient
     )
         external
@@ -145,7 +150,7 @@ contract TaikoL2 is EssentialContract {
             _anchorBlockId,
             _anchorStateRoot,
             _parentGasUsed,
-            _gasTargetPerL1Block,
+            _gasIssuancePerSecond,
             _basefeeAdjustmentQuotient
         );
     }
@@ -177,21 +182,21 @@ contract TaikoL2 is EssentialContract {
     /// @param _anchorBlockId The synced L1 height in the next Taiko block
     /// @param _parentGasUsed Gas used in the parent block.
     /// @return basefee_ The calculated EIP-1559 base fee per gas.
-    /// @return gasExcess_ The new gasExcess value.
+    /// @return parentGasExcess_ The new parentGasExcess value.
     function getBasefee(
         uint64 _anchorBlockId,
         uint32 _parentGasUsed
     )
         public
         view
-        returns (uint256 basefee_, uint64 gasExcess_)
+        returns (uint256 basefee_, uint64 parentGasExcess_)
     {
         LibL2Config.Config memory config = getConfig();
 
-        (basefee_, gasExcess_) = Lib1559Math.calc1559BaseFee(
+        (basefee_, parentGasExcess_) = Lib1559Math.calc1559BaseFee(
             config.gasTargetPerL1Block,
             config.basefeeAdjustmentQuotient,
-            gasExcess,
+            parentGasExcess,
             uint64(_anchorBlockId - lastSyncedBlock) * config.gasTargetPerL1Block,
             _parentGasUsed
         );
@@ -225,29 +230,29 @@ contract TaikoL2 is EssentialContract {
 
     /// @notice Calculates the basefee and the new gas excess value based on parent gas used and gas
     /// excess.
-    /// @param _gasTargetPerL1Block The gas target for L2 based on each L1 block.
+    /// @param _gasIssuancePerSecond The gas target for L2 per second.
     /// @param _blocktime The time between this block and the parent block.
     /// @param _adjustmentQuotient The gas adjustment quotient.
-    /// @param _gasExcess The current gas excess value.
+    /// @param _parentGasExcess The current gas excess value.
     /// @param _parentGasUsed Total gas used by the parent block.
     /// @return basefee_ Next block's base fee.
-    /// @return gasExcess_ The new gas excess value.
+    /// @return parentGasExcess_ The new gas excess value.
     function calculateBaseFee(
-        uint32 _gasTargetPerL1Block,
+        uint32 _gasIssuancePerSecond,
         uint64 _blocktime,
         uint8 _adjustmentQuotient,
-        uint64 _gasExcess,
+        uint64 _parentGasExcess,
         uint32 _parentGasUsed
     )
         public
         pure
-        returns (uint256 basefee_, uint64 gasExcess_)
+        returns (uint256 basefee_, uint64 parentGasExcess_)
     {
         return Lib1559Math.calc1559BaseFee(
-            _gasTargetPerL1Block,
+            _gasIssuancePerSecond,
             _adjustmentQuotient,
-            _gasExcess,
-            _blocktime * _gasTargetPerL1Block / 12,
+            _parentGasExcess,
+            _blocktime * _gasIssuancePerSecond,
             _parentGasUsed
         );
     }
@@ -256,7 +261,7 @@ contract TaikoL2 is EssentialContract {
         uint64 _anchorBlockId,
         bytes32 _anchorStateRoot,
         uint32 _parentGasUsed,
-        uint32 _gasTargetPerL1Block,
+        uint32 _gasIssuancePerSecond,
         uint8 _basefeeAdjustmentQuotient
     )
         private
@@ -276,14 +281,22 @@ contract TaikoL2 is EssentialContract {
             _calcPublicInputHash(parentId);
         if (publicInputHash != currentPublicInputHash) revert L2_PUBLIC_INPUT_HASH_MISMATCH();
 
+        // Check if the gas settings has changed
+        uint64 newGasTarget = uint64(_gasIssuancePerSecond) * _basefeeAdjustmentQuotient;
+        if (newGasTarget < parentGasTarget) {
+            // Reset parentGasExcess to avoid sudden increase of basefee. See
+            // test_change_of_quotient_and_gips in Lib1559Math.t.sol.
+            parentGasExcess = 0;
+        }
+
         // Verify the base fee per gas is correct
         (uint256 basefee, uint64 newGasExcess) = block.number < ontakeForkHeight()
             ? getBasefee(_anchorBlockId, _parentGasUsed)
             : calculateBaseFee(
-                _gasTargetPerL1Block,
+                _gasIssuancePerSecond,
                 uint64(block.timestamp - parentTimestamp),
                 _basefeeAdjustmentQuotient,
-                gasExcess,
+                parentGasExcess,
                 _parentGasUsed
             );
 
@@ -305,8 +318,9 @@ contract TaikoL2 is EssentialContract {
         l2Hashes[parentId] = parentHash;
 
         publicInputHash = newPublicInputHash;
-        gasExcess = newGasExcess;
+        parentGasExcess = newGasExcess;
         parentTimestamp = uint64(block.timestamp);
+        parentGasTarget = newGasExcess;
 
         emit Anchored(parentHash, newGasExcess);
     }
diff --git a/packages/protocol/contracts/hekla/HeklaTaikoL1.sol b/packages/protocol/contracts/hekla/HeklaTaikoL1.sol
index 1e8e8c01361..eb2ce129066 100644
--- a/packages/protocol/contracts/hekla/HeklaTaikoL1.sol
+++ b/packages/protocol/contracts/hekla/HeklaTaikoL1.sol
@@ -22,7 +22,7 @@ contract HeklaTaikoL1 is TaikoL1 {
             maxAnchorHeightOffset: 64,
             basefeeAdjustmentQuotient: 8,
             basefeeSharingPctg: 75,
-            gasTargetPerL1Block: 60_000_000,
+            gasIssuancePerSecond: 2_000_000,
             ontakeForkHeight: 720_000 // = 7200 * 100
          });
     }
diff --git a/packages/protocol/contracts/mainnet/MainnetTaikoL1.sol b/packages/protocol/contracts/mainnet/MainnetTaikoL1.sol
index fb09b4b3853..6e8600d3613 100644
--- a/packages/protocol/contracts/mainnet/MainnetTaikoL1.sol
+++ b/packages/protocol/contracts/mainnet/MainnetTaikoL1.sol
@@ -32,7 +32,7 @@ contract MainnetTaikoL1 is TaikoL1 {
             maxAnchorHeightOffset: 64,
             basefeeAdjustmentQuotient: 8,
             basefeeSharingPctg: 75,
-            gasTargetPerL1Block: 60_000_000,
+            gasIssuancePerSecond: 2_000_000,
             ontakeForkHeight: 374_400 // = 7200 * 52
          });
     }
diff --git a/packages/protocol/test/L2/Lib1559Math.t.sol b/packages/protocol/test/L2/Lib1559Math.t.sol
index b471ac9a5d8..fbdd0af7015 100644
--- a/packages/protocol/test/L2/Lib1559Math.t.sol
+++ b/packages/protocol/test/L2/Lib1559Math.t.sol
@@ -6,66 +6,46 @@ import "../TaikoTest.sol";
 contract TestLib1559Math is TaikoTest {
     using LibMath for uint256;
 
-    uint256 public constant gips = 2_000_000; // gas issuance per second
-    uint256 public constant quotient = 4;
-    uint256 public constant target = gips * quotient;
-
     function test_ethQty() external {
-        assertEq(Lib1559Math.ethQty(0, 60_000_000 / 8), 1);
-        assertEq(Lib1559Math.ethQty(60_000_000, 60_000_000 / 8), 2980);
-        assertEq(Lib1559Math.ethQty(60_000_000 * 2, 60_000_000 / 8), 8_886_110);
-        assertEq(Lib1559Math.ethQty(60_000_000 * 3, 60_000_000 / 8), 26_489_122_129);
-        assertEq(Lib1559Math.ethQty(60_000_000 * 4, 60_000_000 / 8), 78_962_960_182_680);
-        assertEq(
-            Lib1559Math.ethQty(60_000_000 * 10, 60_000_000 / 8),
-            55_406_223_843_935_100_525_863_115_942_268_902
-        );
-        assertEq(
-            Lib1559Math.ethQty(60_000_000 * 100, 60_000_000 / 8),
-            57_896_044_618_658_097_650_144_101_621_524_338_577_433_870_140_581_303_254_786
-        );
+        assertEq(Lib1559Math.ethQty(0, 60_000_000 * 8), 1);
+        assertEq(Lib1559Math.ethQty(60_000_000, 60_000_000 * 8), 1);
+        assertEq(Lib1559Math.ethQty(60_000_000 * 100, 60_000_000 * 8), 268_337);
+        assertEq(Lib1559Math.ethQty(60_000_000 * 200, 60_000_000 * 8), 72_004_899_337);
     }
 
     function test_basefee() external {
         uint256 basefee;
         for (uint256 i; basefee <= 5000;) {
             // uint 0.01 gwei
-            basefee = Lib1559Math.basefee(i * gips, target) / 10_000_000;
+            basefee = Lib1559Math.basefee(i * 2_000_000, 2_000_000 * 4) / 10_000_000;
             if (basefee != 0) {
                 console2.log("basefee (uint 0.01gwei) after", i, "seconds:", basefee);
             }
-            i += 12;
+            i += 1;
         }
     }
 
-    function test_eip1559_math() external pure {
-        LibL2Config.Config memory config = LibL2Config.get();
-
-        uint256 adjustmentFactor = config.gasTargetPerL1Block * config.basefeeAdjustmentQuotient;
-
-        uint256 baseFee;
-        uint256 i;
-        uint256 target = 0.01 gwei;
+    function test_change_of_quotient_and_gips() public {
+        uint256 excess = 150 * 2_000_000;
 
-        for (uint256 k; k < 5; ++k) {
-            for (; baseFee < target; ++i) {
-                baseFee = Lib1559Math.basefee(config.gasTargetPerL1Block * i, adjustmentFactor);
-            }
-            console2.log("base fee:", baseFee);
-            console2.log("    gasExcess:", config.gasTargetPerL1Block * i);
-            console2.log("    i:", i);
-            target *= 10;
-        }
-    }
+        // uint 0.01 gwei
+        uint256 basefee = Lib1559Math.basefee(excess, 2_000_000 * 4) / 10_000_000;
+        console2.log("basefee (uint 0.01gwei) with quotient = 4: ", basefee);
 
-    function test_eip1559_math_max() external pure {
-        LibL2Config.Config memory config = LibL2Config.get();
-        uint256 adjustmentFactor = config.gasTargetPerL1Block * config.basefeeAdjustmentQuotient;
+        // basefee will decrease if gas issued per second or quotient increased.
+        basefee = Lib1559Math.basefee(excess, 2_000_000 * 8) / 10_000_000;
+        console2.log("basefee (uint 0.01gwei) with quotient = 8: ", basefee);
+        basefee = Lib1559Math.basefee(excess, 4_000_000 * 4) / 10_000_000;
+        console2.log("basefee (uint 0.01gwei) with gips = 4_000_000: ", basefee);
 
-        uint256 gasExcess = type(uint64).max;
-        uint256 baseFee = Lib1559Math.basefee(gasExcess, adjustmentFactor);
+        // basefee will increase if gas issued per second or quotient decreased.
+        basefee = Lib1559Math.basefee(excess, 2_000_000 * 2) / 10_000_000;
+        console2.log("basefee (uint 0.01gwei) with quotient = 2: ", basefee);
+        basefee = Lib1559Math.basefee(excess, 1_000_000 * 4) / 10_000_000;
+        console2.log("basefee (uint 0.01gwei) with gips = 1_000_000: ", basefee);
 
-        console2.log("base fee (gwei):", baseFee / 1 gwei);
-        console2.log("    gasExcess:", gasExcess);
+        // basefee will be the same if gas issued per second * quotient decreased remains same
+        basefee = Lib1559Math.basefee(excess, 4_000_000 * 2) / 10_000_000;
+        console2.log("basefee (uint 0.01gwei) ", basefee);
     }
 }
diff --git a/packages/protocol/test/L2/TaikoL2EIP1559Configurable.sol b/packages/protocol/test/L2/TaikoL2EIP1559Configurable.sol
index e8402ce58f4..75b2cc0529d 100644
--- a/packages/protocol/test/L2/TaikoL2EIP1559Configurable.sol
+++ b/packages/protocol/test/L2/TaikoL2EIP1559Configurable.sol
@@ -34,7 +34,7 @@ contract TaikoL2EIP1559Configurable is TaikoL2 {
         if (_newConfig.basefeeAdjustmentQuotient == 0) revert L2_INVALID_CONFIG();
 
         customConfig = _newConfig;
-        gasExcess = _newGasExcess;
+        parentGasExcess = _newGasExcess;
 
         emit ConfigAndExcessChanged(_newConfig, _newGasExcess);
     }

From 219602f30019e9041db0d2740364121b2537c519 Mon Sep 17 00:00:00 2001
From: Daniel Wang <dan@taiko.xyz>
Date: Thu, 1 Aug 2024 15:37:50 +0800
Subject: [PATCH 03/17] Update Lib1559Math.t.sol

---
 packages/protocol/test/L2/Lib1559Math.t.sol | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/packages/protocol/test/L2/Lib1559Math.t.sol b/packages/protocol/test/L2/Lib1559Math.t.sol
index fbdd0af7015..cfa55a0a74d 100644
--- a/packages/protocol/test/L2/Lib1559Math.t.sol
+++ b/packages/protocol/test/L2/Lib1559Math.t.sol
@@ -13,7 +13,7 @@ contract TestLib1559Math is TaikoTest {
         assertEq(Lib1559Math.ethQty(60_000_000 * 200, 60_000_000 * 8), 72_004_899_337);
     }
 
-    function test_basefee() external {
+    function test_basefee() external pure {
         uint256 basefee;
         for (uint256 i; basefee <= 5000;) {
             // uint 0.01 gwei
@@ -25,7 +25,7 @@ contract TestLib1559Math is TaikoTest {
         }
     }
 
-    function test_change_of_quotient_and_gips() public {
+    function test_change_of_quotient_and_gips() public pure {
         uint256 excess = 150 * 2_000_000;
 
         // uint 0.01 gwei

From f1a6cd9629ef3229b3e3ca81ddbdfbca16cf0c1a Mon Sep 17 00:00:00 2001
From: Daniel Wang <dan@taiko.xyz>
Date: Thu, 1 Aug 2024 15:49:54 +0800
Subject: [PATCH 04/17] more

---
 packages/protocol/contracts/L1/TaikoL1.sol             | 2 +-
 packages/protocol/contracts/hekla/HeklaTaikoL1.sol     | 2 +-
 packages/protocol/contracts/mainnet/MainnetTaikoL1.sol | 2 +-
 3 files changed, 3 insertions(+), 3 deletions(-)

diff --git a/packages/protocol/contracts/L1/TaikoL1.sol b/packages/protocol/contracts/L1/TaikoL1.sol
index e590bb1851b..96a5651eddb 100644
--- a/packages/protocol/contracts/L1/TaikoL1.sol
+++ b/packages/protocol/contracts/L1/TaikoL1.sol
@@ -287,7 +287,7 @@ contract TaikoL1 is EssentialContract, ITaikoL1, TaikoEvents {
             maxAnchorHeightOffset: 64,
             basefeeAdjustmentQuotient: 8,
             basefeeSharingPctg: 75,
-            gasIssuancePerSecond: 2_000_000,
+            gasIssuancePerSecond: 5_000_000,
             ontakeForkHeight: 374_400 // = 7200 * 52
          });
     }
diff --git a/packages/protocol/contracts/hekla/HeklaTaikoL1.sol b/packages/protocol/contracts/hekla/HeklaTaikoL1.sol
index eb2ce129066..32299e5d74d 100644
--- a/packages/protocol/contracts/hekla/HeklaTaikoL1.sol
+++ b/packages/protocol/contracts/hekla/HeklaTaikoL1.sol
@@ -22,7 +22,7 @@ contract HeklaTaikoL1 is TaikoL1 {
             maxAnchorHeightOffset: 64,
             basefeeAdjustmentQuotient: 8,
             basefeeSharingPctg: 75,
-            gasIssuancePerSecond: 2_000_000,
+            gasIssuancePerSecond: 5_000_000,
             ontakeForkHeight: 720_000 // = 7200 * 100
          });
     }
diff --git a/packages/protocol/contracts/mainnet/MainnetTaikoL1.sol b/packages/protocol/contracts/mainnet/MainnetTaikoL1.sol
index 6e8600d3613..c2ff8c232c8 100644
--- a/packages/protocol/contracts/mainnet/MainnetTaikoL1.sol
+++ b/packages/protocol/contracts/mainnet/MainnetTaikoL1.sol
@@ -32,7 +32,7 @@ contract MainnetTaikoL1 is TaikoL1 {
             maxAnchorHeightOffset: 64,
             basefeeAdjustmentQuotient: 8,
             basefeeSharingPctg: 75,
-            gasIssuancePerSecond: 2_000_000,
+            gasIssuancePerSecond: 5_000_000,
             ontakeForkHeight: 374_400 // = 7200 * 52
          });
     }

From f5849109be82037217fa930999cc477bd1dbf1e9 Mon Sep 17 00:00:00 2001
From: dantaik <dantaik@users.noreply.github.com>
Date: Thu, 1 Aug 2024 07:51:23 +0000
Subject: [PATCH 05/17] forge fmt & update contract layout table

---
 packages/protocol/contract_layout.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/packages/protocol/contract_layout.md b/packages/protocol/contract_layout.md
index 26860eabd3a..bec4136538e 100644
--- a/packages/protocol/contract_layout.md
+++ b/packages/protocol/contract_layout.md
@@ -35,10 +35,10 @@
 | __gap           | uint256[49]                 | 202  | 0      | 1568  | contracts/L2/TaikoL2.sol:TaikoL2 |
 | l2Hashes        | mapping(uint256 => bytes32) | 251  | 0      | 32    | contracts/L2/TaikoL2.sol:TaikoL2 |
 | publicInputHash | bytes32                     | 252  | 0      | 32    | contracts/L2/TaikoL2.sol:TaikoL2 |
-| gasExcess       | uint64                      | 253  | 0      | 8     | contracts/L2/TaikoL2.sol:TaikoL2 |
+| parentGasExcess | uint64                      | 253  | 0      | 8     | contracts/L2/TaikoL2.sol:TaikoL2 |
 | lastSyncedBlock | uint64                      | 253  | 8      | 8     | contracts/L2/TaikoL2.sol:TaikoL2 |
 | parentTimestamp | uint64                      | 253  | 16     | 8     | contracts/L2/TaikoL2.sol:TaikoL2 |
-| __deprecated2   | uint64                      | 253  | 24     | 8     | contracts/L2/TaikoL2.sol:TaikoL2 |
+| parentGasTarget | uint64                      | 253  | 24     | 8     | contracts/L2/TaikoL2.sol:TaikoL2 |
 | l1ChainId       | uint64                      | 254  | 0      | 8     | contracts/L2/TaikoL2.sol:TaikoL2 |
 | __gap           | uint256[46]                 | 255  | 0      | 1472  | contracts/L2/TaikoL2.sol:TaikoL2 |
 

From 53770a703925edb39d872ef098b800b166de9ebd Mon Sep 17 00:00:00 2001
From: Daniel Wang <dan@taiko.xyz>
Date: Thu, 1 Aug 2024 15:57:07 +0800
Subject: [PATCH 06/17] Update Lib1559Math.t.sol

---
 packages/protocol/test/L2/Lib1559Math.t.sol | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/packages/protocol/test/L2/Lib1559Math.t.sol b/packages/protocol/test/L2/Lib1559Math.t.sol
index cfa55a0a74d..83e5ae4f7f9 100644
--- a/packages/protocol/test/L2/Lib1559Math.t.sol
+++ b/packages/protocol/test/L2/Lib1559Math.t.sol
@@ -32,19 +32,19 @@ contract TestLib1559Math is TaikoTest {
         uint256 basefee = Lib1559Math.basefee(excess, 2_000_000 * 4) / 10_000_000;
         console2.log("basefee (uint 0.01gwei) with quotient = 4: ", basefee);
 
-        // basefee will decrease if gas issued per second or quotient increased.
+        /// basefee will decrease if (gas_issued_per_second * quotient) increases.
         basefee = Lib1559Math.basefee(excess, 2_000_000 * 8) / 10_000_000;
         console2.log("basefee (uint 0.01gwei) with quotient = 8: ", basefee);
         basefee = Lib1559Math.basefee(excess, 4_000_000 * 4) / 10_000_000;
         console2.log("basefee (uint 0.01gwei) with gips = 4_000_000: ", basefee);
 
-        // basefee will increase if gas issued per second or quotient decreased.
+        // basefee will increase if (gas_issued_per_second * quotient) decreases.
         basefee = Lib1559Math.basefee(excess, 2_000_000 * 2) / 10_000_000;
         console2.log("basefee (uint 0.01gwei) with quotient = 2: ", basefee);
         basefee = Lib1559Math.basefee(excess, 1_000_000 * 4) / 10_000_000;
         console2.log("basefee (uint 0.01gwei) with gips = 1_000_000: ", basefee);
 
-        // basefee will be the same if gas issued per second * quotient decreased remains same
+        /// basefee will remain the same if (gas_issued_per_second * quotient) remains the same.
         basefee = Lib1559Math.basefee(excess, 4_000_000 * 2) / 10_000_000;
         console2.log("basefee (uint 0.01gwei) ", basefee);
     }

From 75f6c4f9822555c77f0c79ce1f92984eae916ed8 Mon Sep 17 00:00:00 2001
From: Daniel Wang <dan@taiko.xyz>
Date: Thu, 1 Aug 2024 16:03:51 +0800
Subject: [PATCH 07/17] Update TaikoL2.sol

---
 packages/protocol/contracts/L2/TaikoL2.sol | 10 +++++-----
 1 file changed, 5 insertions(+), 5 deletions(-)

diff --git a/packages/protocol/contracts/L2/TaikoL2.sol b/packages/protocol/contracts/L2/TaikoL2.sol
index b8cc1015b05..74afe2b22c5 100644
--- a/packages/protocol/contracts/L2/TaikoL2.sol
+++ b/packages/protocol/contracts/L2/TaikoL2.sol
@@ -65,12 +65,12 @@ contract TaikoL2 is EssentialContract {
     /// @param _owner The owner of this contract. msg.sender will be used if this value is zero.
     /// @param _rollupAddressManager The address of the {AddressManager} contract.
     /// @param _l1ChainId The ID of the base layer.
-    /// @param _parentGasExcess The initial parentGasExcess.
+    /// @param _initialGasExcess The initial parentGasExcess.
     function init(
         address _owner,
         address _rollupAddressManager,
         uint64 _l1ChainId,
-        uint64 _parentGasExcess
+        uint64 _initialGasExcess
     )
         external
         initializer
@@ -95,12 +95,12 @@ contract TaikoL2 is EssentialContract {
         }
 
         l1ChainId = _l1ChainId;
-        parentGasExcess = _parentGasExcess;
+        parentGasExcess = _initialGasExcess;
         (publicInputHash,) = _calcPublicInputHash(block.number);
     }
 
-    function init2() external reinitializer(2) {
-        parentGasExcess = 0;
+    function init2(uint64 _initialGasExcess) external reinitializer(2) {
+        parentGasExcess = _initialGasExcess;
         parentTimestamp = uint64(block.timestamp);
         parentGasTarget = 0;
     }

From c2c21b8b0185cc0235666ec36135a6becb004ce3 Mon Sep 17 00:00:00 2001
From: Daniel Wang <dan@taiko.xyz>
Date: Thu, 1 Aug 2024 16:11:00 +0800
Subject: [PATCH 08/17] more

---
 packages/protocol/contracts/L2/TaikoL2.sol  | 5 +++++
 packages/protocol/test/L2/Lib1559Math.t.sol | 2 +-
 2 files changed, 6 insertions(+), 1 deletion(-)

diff --git a/packages/protocol/contracts/L2/TaikoL2.sol b/packages/protocol/contracts/L2/TaikoL2.sol
index 74afe2b22c5..68c10a446a9 100644
--- a/packages/protocol/contracts/L2/TaikoL2.sol
+++ b/packages/protocol/contracts/L2/TaikoL2.sol
@@ -99,6 +99,11 @@ contract TaikoL2 is EssentialContract {
         (publicInputHash,) = _calcPublicInputHash(block.number);
     }
 
+    /// @dev Reinitialize some state variables.
+    /// We may want to init the basefee to a default value using one of the following values.
+    /// - _initialGasExcess = 274*5_000_000 => basefee =0.01 gwei
+    /// - _initialGasExcess = 282*5_000_000 => basefee =0.05 gwei
+    /// - _initialGasExcess = 288*5_000_000 => basefee =0.1 gwei
     function init2(uint64 _initialGasExcess) external reinitializer(2) {
         parentGasExcess = _initialGasExcess;
         parentTimestamp = uint64(block.timestamp);
diff --git a/packages/protocol/test/L2/Lib1559Math.t.sol b/packages/protocol/test/L2/Lib1559Math.t.sol
index 83e5ae4f7f9..b845726d9a2 100644
--- a/packages/protocol/test/L2/Lib1559Math.t.sol
+++ b/packages/protocol/test/L2/Lib1559Math.t.sol
@@ -17,7 +17,7 @@ contract TestLib1559Math is TaikoTest {
         uint256 basefee;
         for (uint256 i; basefee <= 5000;) {
             // uint 0.01 gwei
-            basefee = Lib1559Math.basefee(i * 2_000_000, 2_000_000 * 4) / 10_000_000;
+            basefee = Lib1559Math.basefee(i * 5_000_000, 5_000_000 * 8) / 10_000_000;
             if (basefee != 0) {
                 console2.log("basefee (uint 0.01gwei) after", i, "seconds:", basefee);
             }

From e3f75f87f8b3bb91afef346943db7dd0738cfca8 Mon Sep 17 00:00:00 2001
From: Daniel Wang <dan@taiko.xyz>
Date: Thu, 1 Aug 2024 16:11:28 +0800
Subject: [PATCH 09/17] Update TaikoL2.sol

---
 packages/protocol/contracts/L2/TaikoL2.sol | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/packages/protocol/contracts/L2/TaikoL2.sol b/packages/protocol/contracts/L2/TaikoL2.sol
index 68c10a446a9..409208477ef 100644
--- a/packages/protocol/contracts/L2/TaikoL2.sol
+++ b/packages/protocol/contracts/L2/TaikoL2.sol
@@ -104,7 +104,7 @@ contract TaikoL2 is EssentialContract {
     /// - _initialGasExcess = 274*5_000_000 => basefee =0.01 gwei
     /// - _initialGasExcess = 282*5_000_000 => basefee =0.05 gwei
     /// - _initialGasExcess = 288*5_000_000 => basefee =0.1 gwei
-    function init2(uint64 _initialGasExcess) external reinitializer(2) {
+    function init2(uint64 _initialGasExcess) external onlyOwner reinitializer(2) {
         parentGasExcess = _initialGasExcess;
         parentTimestamp = uint64(block.timestamp);
         parentGasTarget = 0;

From 8fe29e0f349dc3d4014d13345fdc444510998e56 Mon Sep 17 00:00:00 2001
From: Daniel Wang <dan@taiko.xyz>
Date: Thu, 1 Aug 2024 16:21:51 +0800
Subject: [PATCH 10/17] Update taikoL2.ts

---
 packages/protocol/utils/generate_genesis/taikoL2.ts | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/packages/protocol/utils/generate_genesis/taikoL2.ts b/packages/protocol/utils/generate_genesis/taikoL2.ts
index 9e631d35f6c..30d33fedd4f 100644
--- a/packages/protocol/utils/generate_genesis/taikoL2.ts
+++ b/packages/protocol/utils/generate_genesis/taikoL2.ts
@@ -515,7 +515,7 @@ async function generateContractConfigs(
                 // TaikoL2 => CrossChainOwned
                 l1ChainId,
                 // TaikoL2
-                gasExcess: param1559.gasExcess,
+                parentGasExcess: param1559.gasExcess,
                 publicInputHash: `${ethers.utils.solidityKeccak256(
                     ["bytes32[256]"],
                     [

From 51bf4df8c36b4d484427a278626a30d5af7b0881 Mon Sep 17 00:00:00 2001
From: Daniel Wang <99078276+dantaik@users.noreply.github.com>
Date: Fri, 2 Aug 2024 12:42:24 +0800
Subject: [PATCH 11/17] Update packages/protocol/contracts/L2/TaikoL2.sol

Co-authored-by: Brecht Devos <Brechtp.Devos@gmail.com>
---
 packages/protocol/contracts/L2/TaikoL2.sol | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/packages/protocol/contracts/L2/TaikoL2.sol b/packages/protocol/contracts/L2/TaikoL2.sol
index 409208477ef..8080297d68b 100644
--- a/packages/protocol/contracts/L2/TaikoL2.sol
+++ b/packages/protocol/contracts/L2/TaikoL2.sol
@@ -325,7 +325,7 @@ contract TaikoL2 is EssentialContract {
         publicInputHash = newPublicInputHash;
         parentGasExcess = newGasExcess;
         parentTimestamp = uint64(block.timestamp);
-        parentGasTarget = newGasExcess;
+        parentGasTarget = newGasTarget;
 
         emit Anchored(parentHash, newGasExcess);
     }

From 2fb9aa13d67a87d515cf67bea2768871f7a18a54 Mon Sep 17 00:00:00 2001
From: Daniel Wang <dan@taiko.xyz>
Date: Fri, 2 Aug 2024 12:22:43 +0800
Subject: [PATCH 12/17] more

---
 .../protocol/contracts/L2/Lib1559Math.sol     | 21 ++++-
 packages/protocol/contracts/L2/TaikoL2.sol    |  7 +-
 .../thirdparty/solmate/LibFixedPointMath.sol  | 78 +++++++++++++++++++
 packages/protocol/test/L2/Lib1559Math.t.sol   | 52 ++++++++-----
 4 files changed, 134 insertions(+), 24 deletions(-)

diff --git a/packages/protocol/contracts/L2/Lib1559Math.sol b/packages/protocol/contracts/L2/Lib1559Math.sol
index 6504b759617..d901dd24072 100644
--- a/packages/protocol/contracts/L2/Lib1559Math.sol
+++ b/packages/protocol/contracts/L2/Lib1559Math.sol
@@ -38,6 +38,25 @@ library Lib1559Math {
         basefee_ = basefee(gasExcess_, uint256(_adjustmentQuotient) * _gasTarget);
     }
 
+    function adjustExcess(
+        uint64 _gasExcess,
+        uint64 _gasTarget,
+        uint64 _newGasTarget
+    )
+        internal
+        pure
+        returns (uint64)
+    {
+        int256 ratio = int256(LibFixedPointMath.SCALING_FACTOR * _newGasTarget / _gasTarget);
+        int256 newGasExcess = (
+            LibFixedPointMath.ln(ratio) * int256(uint256(_newGasTarget))
+                + ratio * int256(uint256(_gasExcess))
+        ) / int256(LibFixedPointMath.SCALING_FACTOR);
+
+        if (newGasExcess < 0) newGasExcess = 0;
+        return uint64(uint256(newGasExcess).min(type(uint64).max));
+    }
+
     /// @dev eth_qty(excess_gas_issued) / (TARGET * ADJUSTMENT_QUOTIENT)
     /// @param _gasExcess The gas excess value
     /// @param _target The product of gasTarget and adjustmentQuotient
@@ -49,7 +68,7 @@ library Lib1559Math {
 
     /// @dev exp(_gasExcess / _target)
     function ethQty(uint256 _gasExcess, uint256 _target) internal pure returns (uint256) {
-        uint256 input = _gasExcess * LibFixedPointMath.SCALING_FACTOR / _target;
+        uint256 input = LibFixedPointMath.SCALING_FACTOR * _gasExcess / _target;
         if (input > LibFixedPointMath.MAX_EXP_INPUT) {
             input = LibFixedPointMath.MAX_EXP_INPUT;
         }
diff --git a/packages/protocol/contracts/L2/TaikoL2.sol b/packages/protocol/contracts/L2/TaikoL2.sol
index 8080297d68b..ebb9562a74f 100644
--- a/packages/protocol/contracts/L2/TaikoL2.sol
+++ b/packages/protocol/contracts/L2/TaikoL2.sol
@@ -289,9 +289,10 @@ contract TaikoL2 is EssentialContract {
         // Check if the gas settings has changed
         uint64 newGasTarget = uint64(_gasIssuancePerSecond) * _basefeeAdjustmentQuotient;
         if (newGasTarget < parentGasTarget) {
-            // Reset parentGasExcess to avoid sudden increase of basefee. See
-            // test_change_of_quotient_and_gips in Lib1559Math.t.sol.
-            parentGasExcess = 0;
+            // adjust parentGasExcess to keep the basefee unchanged. Note that due to math
+            // calculation precision, the basefee may change slightly.
+            parentGasExcess =
+                Lib1559Math.adjustExcess(parentGasExcess, parentGasTarget, newGasTarget);
         }
 
         // Verify the base fee per gas is correct
diff --git a/packages/protocol/contracts/thirdparty/solmate/LibFixedPointMath.sol b/packages/protocol/contracts/thirdparty/solmate/LibFixedPointMath.sol
index 2ad599363c2..df7fd59d005 100644
--- a/packages/protocol/contracts/thirdparty/solmate/LibFixedPointMath.sol
+++ b/packages/protocol/contracts/thirdparty/solmate/LibFixedPointMath.sol
@@ -79,4 +79,82 @@ library LibFixedPointMath {
             );
         }
     }
+
+    function ln(int256 x) internal pure returns (int256 r) {
+        unchecked {
+            require(x > 0, "UNDEFINED");
+
+            // We want to convert x from 10**18 fixed point to 2**96 fixed point.
+            // We do this by multiplying by 2**96 / 10**18. But since
+            // ln(x * C) = ln(x) + ln(C), we can simply do nothing here
+            // and add ln(2**96 / 10**18) at the end.
+
+            // Reduce range of x to (1, 2) * 2**96
+            // ln(2^k * x) = k * ln(2) + ln(x)
+            int256 k = int256(log2(uint256(x))) - 96;
+            x <<= uint256(159 - k);
+            x = int256(uint256(x) >> 159);
+
+            // Evaluate using a (8, 8)-term rational approximation.
+            // p is made monic, we will multiply by a scale factor later.
+            int256 p = x + 3_273_285_459_638_523_848_632_254_066_296;
+            p = ((p * x) >> 96) + 24_828_157_081_833_163_892_658_089_445_524;
+            p = ((p * x) >> 96) + 43_456_485_725_739_037_958_740_375_743_393;
+            p = ((p * x) >> 96) - 11_111_509_109_440_967_052_023_855_526_967;
+            p = ((p * x) >> 96) - 45_023_709_667_254_063_763_336_534_515_857;
+            p = ((p * x) >> 96) - 14_706_773_417_378_608_786_704_636_184_526;
+            p = p * x - (795_164_235_651_350_426_258_249_787_498 << 96);
+
+            // We leave p in 2**192 basis so we don't need to scale it back up for the division.
+            // q is monic by convention.
+            int256 q = x + 5_573_035_233_440_673_466_300_451_813_936;
+            q = ((q * x) >> 96) + 71_694_874_799_317_883_764_090_561_454_958;
+            q = ((q * x) >> 96) + 283_447_036_172_924_575_727_196_451_306_956;
+            q = ((q * x) >> 96) + 401_686_690_394_027_663_651_624_208_769_553;
+            q = ((q * x) >> 96) + 204_048_457_590_392_012_362_485_061_816_622;
+            q = ((q * x) >> 96) + 31_853_899_698_501_571_402_653_359_427_138;
+            q = ((q * x) >> 96) + 909_429_971_244_387_300_277_376_558_375;
+            assembly {
+                // Div in assembly because solidity adds a zero check despite the unchecked.
+                // The q polynomial is known not to have zeros in the domain.
+                // No scaling required because p is already 2**96 too large.
+                r := sdiv(p, q)
+            }
+
+            // r is in the range (0, 0.125) * 2**96
+
+            // Finalization, we need to:
+            // * multiply by the scale factor s = 5.549…
+            // * add ln(2**96 / 10**18)
+            // * add k * ln(2)
+            // * multiply by 10**18 / 2**96 = 5**18 >> 78
+
+            // mul s * 5e18 * 2**96, base is now 5**18 * 2**192
+            r *= 1_677_202_110_996_718_588_342_820_967_067_443_963_516_166;
+            // add ln(2) * k * 5e18 * 2**192
+            r +=
+            16_597_577_552_685_614_221_487_285_958_193_947_469_193_820_559_219_878_177_908_093_499_208_371
+                * k;
+            // add ln(2**96 / 10**18) * 5e18 * 2**192
+            r +=
+                600_920_179_829_731_861_736_702_779_321_621_459_595_472_258_049_074_101_567_377_883_020_018_308;
+            // base conversion: mul 2**18 / 2**192
+            r >>= 174;
+        }
+    }
+
+    function log2(uint256 x) internal pure returns (uint256 r) {
+        require(x > 0, "UNDEFINED");
+
+        assembly {
+            r := shl(7, lt(0xffffffffffffffffffffffffffffffff, x))
+            r := or(r, shl(6, lt(0xffffffffffffffff, shr(r, x))))
+            r := or(r, shl(5, lt(0xffffffff, shr(r, x))))
+            r := or(r, shl(4, lt(0xffff, shr(r, x))))
+            r := or(r, shl(3, lt(0xff, shr(r, x))))
+            r := or(r, shl(2, lt(0xf, shr(r, x))))
+            r := or(r, shl(1, lt(0x3, shr(r, x))))
+            r := or(r, lt(0x1, shr(r, x)))
+        }
+    }
 }
diff --git a/packages/protocol/test/L2/Lib1559Math.t.sol b/packages/protocol/test/L2/Lib1559Math.t.sol
index b845726d9a2..89db181b7ab 100644
--- a/packages/protocol/test/L2/Lib1559Math.t.sol
+++ b/packages/protocol/test/L2/Lib1559Math.t.sol
@@ -25,27 +25,39 @@ contract TestLib1559Math is TaikoTest {
         }
     }
 
-    function test_change_of_quotient_and_gips() public pure {
-        uint256 excess = 150 * 2_000_000;
+    function test_change_of_quotient_and_gips() public {
+        uint64 excess = 150 * 2_000_000;
+        uint64 target = 4 * 2_000_000;
+        uint256 unit = 10_000_000; // 0.01 gwei
 
         // uint 0.01 gwei
-        uint256 basefee = Lib1559Math.basefee(excess, 2_000_000 * 4) / 10_000_000;
-        console2.log("basefee (uint 0.01gwei) with quotient = 4: ", basefee);
-
-        /// basefee will decrease if (gas_issued_per_second * quotient) increases.
-        basefee = Lib1559Math.basefee(excess, 2_000_000 * 8) / 10_000_000;
-        console2.log("basefee (uint 0.01gwei) with quotient = 8: ", basefee);
-        basefee = Lib1559Math.basefee(excess, 4_000_000 * 4) / 10_000_000;
-        console2.log("basefee (uint 0.01gwei) with gips = 4_000_000: ", basefee);
-
-        // basefee will increase if (gas_issued_per_second * quotient) decreases.
-        basefee = Lib1559Math.basefee(excess, 2_000_000 * 2) / 10_000_000;
-        console2.log("basefee (uint 0.01gwei) with quotient = 2: ", basefee);
-        basefee = Lib1559Math.basefee(excess, 1_000_000 * 4) / 10_000_000;
-        console2.log("basefee (uint 0.01gwei) with gips = 1_000_000: ", basefee);
-
-        /// basefee will remain the same if (gas_issued_per_second * quotient) remains the same.
-        basefee = Lib1559Math.basefee(excess, 4_000_000 * 2) / 10_000_000;
-        console2.log("basefee (uint 0.01gwei) ", basefee);
+        uint256 baselineBasefee = Lib1559Math.basefee(excess, target) / unit;
+        console2.log("baseline basefee: ", baselineBasefee);
+
+        uint256 basefee = Lib1559Math.basefee(excess, target * 2) / unit;
+        console2.log("basefee will decrease if target increases:", basefee);
+
+        basefee = Lib1559Math.basefee(excess, target / 2) / unit;
+        console2.log("basefee will increase if target decreases:", basefee);
+
+        console2.log("maintain basefee when target increases");
+        {
+            uint64 newTarget = 5 * 2_000_000;
+            uint64 newExcess = Lib1559Math.adjustExcess(excess, target, newTarget);
+            basefee = Lib1559Math.basefee(newExcess, newTarget) / unit;
+            console2.log("new gas excess: ", newExcess);
+            console2.log("basefee: ", basefee);
+            assertEq(baselineBasefee, basefee);
+        }
+
+        console2.log("maintain basefee when target decreases");
+        {
+            uint64 newTarget = 3 * 2_000_000;
+            uint64 newExcess = Lib1559Math.adjustExcess(excess, target, newTarget);
+            basefee = Lib1559Math.basefee(newExcess, newTarget) / unit;
+            console2.log("new gas excess: ", newExcess);
+            console2.log("basefee: ", basefee);
+            assertEq(baselineBasefee, basefee);
+        }
     }
 }

From 7b56994c04a67d7d3d05522715573a9a564b2fc2 Mon Sep 17 00:00:00 2001
From: Daniel Wang <dan@taiko.xyz>
Date: Fri, 2 Aug 2024 12:40:36 +0800
Subject: [PATCH 13/17] Update Lib1559Math.sol

---
 .../protocol/contracts/L2/Lib1559Math.sol     | 29 ++++++++++++++-----
 1 file changed, 22 insertions(+), 7 deletions(-)

diff --git a/packages/protocol/contracts/L2/Lib1559Math.sol b/packages/protocol/contracts/L2/Lib1559Math.sol
index d901dd24072..d646e555ca1 100644
--- a/packages/protocol/contracts/L2/Lib1559Math.sol
+++ b/packages/protocol/contracts/L2/Lib1559Math.sol
@@ -38,6 +38,8 @@ library Lib1559Math {
         basefee_ = basefee(gasExcess_, uint256(_adjustmentQuotient) * _gasTarget);
     }
 
+    /// @dev Returns the new gas excess that will keep the basefee the same.
+    /// `_newGasTarget * ln(_newGasTarget / _target) + _gasExcess * _newGasTarget / _target`
     function adjustExcess(
         uint64 _gasExcess,
         uint64 _gasTarget,
@@ -47,14 +49,27 @@ library Lib1559Math {
         pure
         returns (uint64)
     {
-        int256 ratio = int256(LibFixedPointMath.SCALING_FACTOR * _newGasTarget / _gasTarget);
-        int256 newGasExcess = (
-            LibFixedPointMath.ln(ratio) * int256(uint256(_newGasTarget))
-                + ratio * int256(uint256(_gasExcess))
-        ) / int256(LibFixedPointMath.SCALING_FACTOR);
+        if (_gasTarget == 0) revert EIP1559_INVALID_PARAMS();
 
-        if (newGasExcess < 0) newGasExcess = 0;
-        return uint64(uint256(newGasExcess).min(type(uint64).max));
+        uint256 f = LibFixedPointMath.SCALING_FACTOR;
+        uint256 ratio;
+        unchecked {
+            ratio = f * _newGasTarget / _gasTarget;
+        }
+
+        if (ratio > type(int256).max) revert EIP1559_INVALID_PARAMS();
+
+        int256 lnRatio = LibFixedPointMath.ln(int256(ratio));
+
+        uint256 newGasExcess;
+        assembly {
+            newGasExcess := sdiv(add(mul(lnRatio, _newGasTarget), mul(ratio, _gasExcess)), f)
+            switch gt(newGasExcess, 0)
+            case 1 { newGasExcess := shr(192, shl(192, newGasExcess)) }
+            default { newGasExcess := 0 }
+        }
+
+        return uint64(newGasExcess);
     }
 
     /// @dev eth_qty(excess_gas_issued) / (TARGET * ADJUSTMENT_QUOTIENT)

From 679f8a316be4ec8591558f5acbe0e4cd6dcf388f Mon Sep 17 00:00:00 2001
From: Daniel Wang <dan@taiko.xyz>
Date: Fri, 2 Aug 2024 12:45:31 +0800
Subject: [PATCH 14/17] Update TaikoL2.sol

---
 packages/protocol/contracts/L2/TaikoL2.sol | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/packages/protocol/contracts/L2/TaikoL2.sol b/packages/protocol/contracts/L2/TaikoL2.sol
index ebb9562a74f..009c7824dd1 100644
--- a/packages/protocol/contracts/L2/TaikoL2.sol
+++ b/packages/protocol/contracts/L2/TaikoL2.sol
@@ -288,7 +288,7 @@ contract TaikoL2 is EssentialContract {
 
         // Check if the gas settings has changed
         uint64 newGasTarget = uint64(_gasIssuancePerSecond) * _basefeeAdjustmentQuotient;
-        if (newGasTarget < parentGasTarget) {
+        if (newGasTarget != parentGasTarget) {
             // adjust parentGasExcess to keep the basefee unchanged. Note that due to math
             // calculation precision, the basefee may change slightly.
             parentGasExcess =

From 55dbace871209bfdb2a5f6a1bf8d786bc1d59f53 Mon Sep 17 00:00:00 2001
From: Daniel Wang <dan@taiko.xyz>
Date: Fri, 2 Aug 2024 14:37:17 +0800
Subject: [PATCH 15/17] Update Lib1559Math.sol

---
 packages/protocol/contracts/L2/Lib1559Math.sol | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/packages/protocol/contracts/L2/Lib1559Math.sol b/packages/protocol/contracts/L2/Lib1559Math.sol
index d646e555ca1..51165a2df72 100644
--- a/packages/protocol/contracts/L2/Lib1559Math.sol
+++ b/packages/protocol/contracts/L2/Lib1559Math.sol
@@ -57,7 +57,7 @@ library Lib1559Math {
             ratio = f * _newGasTarget / _gasTarget;
         }
 
-        if (ratio > type(int256).max) revert EIP1559_INVALID_PARAMS();
+        if (ratio > uint256(type(int256).max)) revert EIP1559_INVALID_PARAMS();
 
         int256 lnRatio = LibFixedPointMath.ln(int256(ratio));
 

From baf39b10ab95440a9dfad1cac98a6b6fe66db146 Mon Sep 17 00:00:00 2001
From: Daniel Wang <dan@taiko.xyz>
Date: Fri, 2 Aug 2024 14:41:06 +0800
Subject: [PATCH 16/17] Update TaikoL2.sol

---
 packages/protocol/contracts/L2/TaikoL2.sol | 12 ++++++------
 1 file changed, 6 insertions(+), 6 deletions(-)

diff --git a/packages/protocol/contracts/L2/TaikoL2.sol b/packages/protocol/contracts/L2/TaikoL2.sol
index 009c7824dd1..e20782887e3 100644
--- a/packages/protocol/contracts/L2/TaikoL2.sol
+++ b/packages/protocol/contracts/L2/TaikoL2.sol
@@ -287,8 +287,9 @@ contract TaikoL2 is EssentialContract {
         if (publicInputHash != currentPublicInputHash) revert L2_PUBLIC_INPUT_HASH_MISMATCH();
 
         // Check if the gas settings has changed
+        bool postFork = block.number >= ontakeForkHeight();
         uint64 newGasTarget = uint64(_gasIssuancePerSecond) * _basefeeAdjustmentQuotient;
-        if (newGasTarget != parentGasTarget) {
+        if (postFork && newGasTarget != parentGasTarget) {
             // adjust parentGasExcess to keep the basefee unchanged. Note that due to math
             // calculation precision, the basefee may change slightly.
             parentGasExcess =
@@ -296,15 +297,15 @@ contract TaikoL2 is EssentialContract {
         }
 
         // Verify the base fee per gas is correct
-        (uint256 basefee, uint64 newGasExcess) = block.number < ontakeForkHeight()
-            ? getBasefee(_anchorBlockId, _parentGasUsed)
-            : calculateBaseFee(
+        (uint256 basefee, uint64 newGasExcess) = postFork
+            ? calculateBaseFee(
                 _gasIssuancePerSecond,
                 uint64(block.timestamp - parentTimestamp),
                 _basefeeAdjustmentQuotient,
                 parentGasExcess,
                 _parentGasUsed
-            );
+            )
+            : getBasefee(_anchorBlockId, _parentGasUsed);
 
         if (!skipFeeCheck() && block.basefee != basefee) revert L2_BASEFEE_MISMATCH();
 
@@ -319,7 +320,6 @@ contract TaikoL2 is EssentialContract {
         }
 
         // Update state variables
-
         bytes32 parentHash = blockhash(parentId);
         l2Hashes[parentId] = parentHash;
 

From 585e5fee9b83ed64039783dbca0e8453ea807996 Mon Sep 17 00:00:00 2001
From: Daniel Wang <dan@taiko.xyz>
Date: Mon, 5 Aug 2024 12:27:46 +0800
Subject: [PATCH 17/17] improve

---
 .../protocol/contracts/L2/Lib1559Math.sol     | 35 +++++++------------
 packages/protocol/contracts/L2/TaikoL2.sol    |  6 ++--
 packages/protocol/test/L2/Lib1559Math.t.sol   |  2 ++
 3 files changed, 17 insertions(+), 26 deletions(-)

diff --git a/packages/protocol/contracts/L2/Lib1559Math.sol b/packages/protocol/contracts/L2/Lib1559Math.sol
index 51165a2df72..aac64054f24 100644
--- a/packages/protocol/contracts/L2/Lib1559Math.sol
+++ b/packages/protocol/contracts/L2/Lib1559Math.sol
@@ -15,8 +15,7 @@ library Lib1559Math {
     error EIP1559_INVALID_PARAMS();
 
     function calc1559BaseFee(
-        uint32 _gasTarget,
-        uint8 _adjustmentQuotient,
+        uint256 _gasTarget,
         uint64 _gasExcess,
         uint64 _gasIssuance,
         uint32 _parentGasUsed
@@ -35,7 +34,7 @@ library Lib1559Math {
         // bonding curve, regardless the actual amount of gas used by this
         // block, however, this block's gas used will affect the next
         // block's base fee.
-        basefee_ = basefee(gasExcess_, uint256(_adjustmentQuotient) * _gasTarget);
+        basefee_ = basefee(gasExcess_, _gasTarget);
     }
 
     /// @dev Returns the new gas excess that will keep the basefee the same.
@@ -52,38 +51,30 @@ library Lib1559Math {
         if (_gasTarget == 0) revert EIP1559_INVALID_PARAMS();
 
         uint256 f = LibFixedPointMath.SCALING_FACTOR;
-        uint256 ratio;
-        unchecked {
-            ratio = f * _newGasTarget / _gasTarget;
-        }
-
+        uint256 ratio = f * _newGasTarget / _gasTarget;
         if (ratio > uint256(type(int256).max)) revert EIP1559_INVALID_PARAMS();
 
-        int256 lnRatio = LibFixedPointMath.ln(int256(ratio));
+        int256 lnRatio = LibFixedPointMath.ln(int256(ratio)); // may be negative
 
         uint256 newGasExcess;
         assembly {
             newGasExcess := sdiv(add(mul(lnRatio, _newGasTarget), mul(ratio, _gasExcess)), f)
-            switch gt(newGasExcess, 0)
-            case 1 { newGasExcess := shr(192, shl(192, newGasExcess)) }
-            default { newGasExcess := 0 }
         }
 
-        return uint64(newGasExcess);
+        return uint64(newGasExcess.min(type(uint64).max));
     }
 
-    /// @dev eth_qty(excess_gas_issued) / (TARGET * ADJUSTMENT_QUOTIENT)
-    /// @param _gasExcess The gas excess value
-    /// @param _target The product of gasTarget and adjustmentQuotient
-    function basefee(uint256 _gasExcess, uint256 _target) internal pure returns (uint256) {
-        if (_target == 0) revert EIP1559_INVALID_PARAMS();
-        uint256 fee = ethQty(_gasExcess, _target) / _target;
+    /// @dev exp(_gasExcess / _gasTarget) / _gasTarget
+    function basefee(uint256 _gasExcess, uint256 _gasTarget) internal pure returns (uint256) {
+        uint256 fee = ethQty(_gasExcess, _gasTarget) / _gasTarget;
         return fee == 0 ? 1 : fee;
     }
 
-    /// @dev exp(_gasExcess / _target)
-    function ethQty(uint256 _gasExcess, uint256 _target) internal pure returns (uint256) {
-        uint256 input = LibFixedPointMath.SCALING_FACTOR * _gasExcess / _target;
+    /// @dev exp(_gasExcess / _gasTarget)
+    function ethQty(uint256 _gasExcess, uint256 _gasTarget) internal pure returns (uint256) {
+        if (_gasTarget == 0) revert EIP1559_INVALID_PARAMS();
+
+        uint256 input = LibFixedPointMath.SCALING_FACTOR * _gasExcess / _gasTarget;
         if (input > LibFixedPointMath.MAX_EXP_INPUT) {
             input = LibFixedPointMath.MAX_EXP_INPUT;
         }
diff --git a/packages/protocol/contracts/L2/TaikoL2.sol b/packages/protocol/contracts/L2/TaikoL2.sol
index e20782887e3..68b73e48733 100644
--- a/packages/protocol/contracts/L2/TaikoL2.sol
+++ b/packages/protocol/contracts/L2/TaikoL2.sol
@@ -199,8 +199,7 @@ contract TaikoL2 is EssentialContract {
         LibL2Config.Config memory config = getConfig();
 
         (basefee_, parentGasExcess_) = Lib1559Math.calc1559BaseFee(
-            config.gasTargetPerL1Block,
-            config.basefeeAdjustmentQuotient,
+            uint256(config.gasTargetPerL1Block) * config.basefeeAdjustmentQuotient,
             parentGasExcess,
             uint64(_anchorBlockId - lastSyncedBlock) * config.gasTargetPerL1Block,
             _parentGasUsed
@@ -254,8 +253,7 @@ contract TaikoL2 is EssentialContract {
         returns (uint256 basefee_, uint64 parentGasExcess_)
     {
         return Lib1559Math.calc1559BaseFee(
-            _gasIssuancePerSecond,
-            _adjustmentQuotient,
+            uint256(_gasIssuancePerSecond) * _adjustmentQuotient,
             _parentGasExcess,
             _blocktime * _gasIssuancePerSecond,
             _parentGasUsed
diff --git a/packages/protocol/test/L2/Lib1559Math.t.sol b/packages/protocol/test/L2/Lib1559Math.t.sol
index 89db181b7ab..e05e9ddf104 100644
--- a/packages/protocol/test/L2/Lib1559Math.t.sol
+++ b/packages/protocol/test/L2/Lib1559Math.t.sol
@@ -45,6 +45,7 @@ contract TestLib1559Math is TaikoTest {
             uint64 newTarget = 5 * 2_000_000;
             uint64 newExcess = Lib1559Math.adjustExcess(excess, target, newTarget);
             basefee = Lib1559Math.basefee(newExcess, newTarget) / unit;
+            console2.log("old gas excess: ", excess);
             console2.log("new gas excess: ", newExcess);
             console2.log("basefee: ", basefee);
             assertEq(baselineBasefee, basefee);
@@ -55,6 +56,7 @@ contract TestLib1559Math is TaikoTest {
             uint64 newTarget = 3 * 2_000_000;
             uint64 newExcess = Lib1559Math.adjustExcess(excess, target, newTarget);
             basefee = Lib1559Math.basefee(newExcess, newTarget) / unit;
+            console2.log("old gas excess: ", excess);
             console2.log("new gas excess: ", newExcess);
             console2.log("basefee: ", basefee);
             assertEq(baselineBasefee, basefee);