{
 "generated": "2026-10-28",
 "description": "Terminal precision: each scenario is N identical steps run through each library in its native representation for several inputs, against the same loop in exact arithmetic (mpmath, 300 digits), twice: from the exact parameters (digits) and from the parameters as the library received them (digitsFormat). See methodology.",
 "versions": {
  "bl127": "6c2eeaf",
  "huff": "sepolia:0xfae694D0c2c44181791F838c54Ed64C3151FfE30@11857616",
  "solady": "v0.1.26",
  "prb-math": "v4.2.0",
  "abdk": "v3.2",
  "balancer": "balancer-v3-monorepo@2b9ce4b1b8ad (LogExpMath.sol, vendored)",
  "aave": "aave-v3-origin@8305565ae342 (MathUtils, transcribed)",
  "morpho": "morpho-blue@8e26ca6a8dbc (MathLib, transcribed)",
  "fraxlend": "fraxlend@2bed49d4dcc6 (VariableInterestRate V2, transcribed)",
  "euler": "evk-periphery@af436991e135 (IRMAdaptiveCurve, ExpLib, transcribed)",
  "maker": "dss pot.sol rpow via solady v0.1.26",
  "solmate": "solmate@89365b8 (no tag: v6 predates SignedWadMath)",
  "openzeppelin": "openzeppelin-contracts v5.7.0 (Math, SignedMath, vendored)",
  "uniswap": "v4-core v4.0.0 (FullMath, vendored)",
  "snekmate": "v0.1.2 (math module, vyper 0.4.3)",
  "forge": "forge Version: 1.7.1"
 },
 "ladder": [
  1,
  2,
  5,
  10,
  20,
  50,
  100,
  365,
  1000,
  10000
 ],
 "ladderLog10": {
  "1": "0.0",
  "2": "0.30103",
  "5": "0.69897",
  "10": "1.0",
  "20": "1.30103",
  "50": "1.69897",
  "100": "2.0",
  "365": "2.56229",
  "1000": "3.0",
  "10000": "4.0"
 },
 "forms": [
  {
   "id": "bl127",
   "label": "BL127 (deployed, staticcall)",
   "scale": "2**128"
  },
  {
   "id": "bl127lib",
   "label": "BL127Lib (inline)",
   "scale": "2**128"
  },
  {
   "id": "abdk",
   "label": "ABDKMath64x64",
   "scale": "2**64"
  },
  {
   "id": "solady",
   "label": "Solady FixedPointMathLib",
   "scale": "10**18"
  },
  {
   "id": "prb",
   "label": "PRBMath SD59x18",
   "scale": "10**18"
  },
  {
   "id": "quad",
   "label": "ABDKMathQuad (binary128)",
   "scale": "binary128"
  },
  {
   "id": "balancer",
   "label": "Balancer LogExpMath",
   "scale": "10**18"
  },
  {
   "id": "aave",
   "label": "Aave v3 MathUtils (RAY)",
   "scale": "10**27"
  },
  {
   "id": "morpho",
   "label": "Morpho Blue MathLib",
   "scale": "10**18"
  },
  {
   "id": "maker",
   "label": "Maker rpow (RAY)",
   "scale": "10**27"
  },
  {
   "id": "bl19",
   "label": "BL19 (float, inline)",
   "scale": "bl19"
  },
  {
   "id": "solmate",
   "label": "Solmate SignedWadMath / FixedPointMathLib",
   "scale": "10**18"
  },
  {
   "id": "oz",
   "label": "OpenZeppelin Math (mulDiv, sqrt)",
   "scale": "10**18"
  },
  {
   "id": "uniswap",
   "label": "Uniswap v4 FullMath (mulDiv)",
   "scale": "10**18"
  },
  {
   "id": "prbud",
   "label": "PRBMath UD60x18",
   "scale": "10**18"
  },
  {
   "id": "snekmate",
   "label": "snekmate math (Vyper)",
   "scale": "10**18"
  },
  {
   "id": "fraxlend",
   "label": "FraxLend VariableInterestRate V2",
   "scale": "10**18"
  },
  {
   "id": "euler",
   "label": "Euler v2 IRMAdaptiveCurve",
   "scale": "10**18"
  }
 ],
 "digitsCap": 100,
 "axisMax": 80,
 "methodology": {
  "truth": "For every scenario, input and N the same loop is run in exact real arithmetic with mpmath at 300 decimal digits (scripts/ladder/ladder.py), from the exact rational parameters in scenarios.json. Closed forms are used where the loop has one (compounding, the cumulative product, the geometric mean, the round trip, per-second and continuous accrual, the option walk, the decay and the adaptive rate); the bonding curve, the amortising loan and the Black-Scholes chain are stepped.",
  "rounding": "Inputs are floored on the grids and rounded to nearest on the floats: each library receives the parameters floored into its own representation (2^-128, 2^-64, 10^-18 or 10^-27 units) or, for ABDKMathQuad and BL19, the nearest float (scripts/ladder/inputs.json). truthFormat is the exact loop started from those parameters, so digitsFormat measures the library's arithmetic alone, while digits measures what a protocol sees: the format's rounding of the inputs plus the arithmetic. One exception: Morpho accrues from the per-second rate, rateAnnual / 31536000 floored to WAD, while truthFormat is built from the annual rate the other libraries receive, so Morpho's digitsFormat still includes its own input's rounding. Likewise in the adaptive rate, FraxLend and Euler compute from their own inputs (utilisation, step, half-life or speed) while truthFormat is built from rate0 and x, so their digitsFormat includes the rounding of those inputs (Euler's speed floored to WAD per second, FraxLend's half-life to whole seconds).",
  "inputs": "The parameters are short decimals (0.1/365, 10000, 0.05) as a protocol's governance would set them. A short decimal can sit exactly on the 10^-18 and 10^-27 grids, so WAD and RAY libraries often receive their inputs exactly while the binary formats round them; this favours WAD and RAY in digits (against exact inputs) and is visible as the gap between digits and digitsFormat. The decay scenario's eps are reciprocals of time constants in seconds (1/1036800 for twelve days), which no grid holds exactly.",
  "powers": "Integer powers are each library's own idiom, not one algorithm: BL127 and BL19 use compound (a correctly rounded (1 + r)^n that never forms a rounded 1 + r), Maker uses rpow (binary exponentiation in RAY), the WAD and 64.64 libraries and binary128 use exp(n ln(1 + r)) or their pow. Per-second accrual and compound-pow measure those choices as much as the formats.",
  "digits": "digits = floor(-log10 |v - t| / |t|), the number of correct leading significant digits by relative error, an integer; a result equal to the truth is exact (exact = true, digits = 100, the one cap on this field) and is drawn as a marker, never as a number. digits1 is the same quantity floored to one decimal instead of to an integer. It is not a string comparison: a value can be correct to 36 digits while its 37th printed digit happens to agree with the truth, or while a carry makes the printed prefix differ earlier.",
  "precision": "No format can be right to more digits than it holds at the result's magnitude: prec is 237 log10 2 = 71.3 for BL19 and 113 log10 2 = 34.0 for binary128 at any magnitude, and -log10(ulp / |t|) on the fixed grids, with ulp 2^-128 (BL127), 2^-64 (64.64), 10^-18 (WAD) or 10^-27 (RAY): 38.5 for BL127 at 1, 42.5 at 10^4. A cell may land closer to the truth than that by luck; digitsCapped = min(digits1, prec), and prec for an exact cell, is what the charts plot and the summaries aggregate, so no median or band exceeds the format's precision.",
  "summary": "Each scenario's median, minimum and maximum at each N are taken over the inputs that every library with any completed input at that N completed, so every library's median is over the same inputs. Each library's completed count is reported, and a library that completed fewer than all inputs is shown as k/n. Where no input is common to all of them, a revert counts as 0 digits over all inputs instead (basis revertsAsZero); the dataset records the basis and the inputs per N.",
  "interest": "For the compounding and accrual scenarios the digits are also measured on the interest, b - principal against t - principal (interestDigits1, interestCapped, with the format's precision taken against the interest's magnitude). Relative digits of a balance flatter a format when the interest is a small part of it: after one day the interest is about 10^-4 of the balance.",
  "wei": "For the deep-tail option, absErrWei is |v - t| in wei of the strike at 18 decimals ($100 = 10^20 wei). Past d1 = -8 the true price falls below one wei and every library's error with it, so relative digits there describe the format, not money.",
  "headline": "The Results page shows a single N = 10,000 per scenario; the ladder page shows every N.",
  "agree": "agree is the count of leading significant digits of the printed value (42 significant digits) that are literally equal to the printed truth. The demo page colours the rest of the value with it. It is a reading aid, not a measurement: a carry across 0.999\u2026 / 1.000\u2026 gives 0.",
  "gas": "gasTotal is measured inside the adapter around the scenario function (the parameter reads and the loop), after the runner has warmed the adapter and the BL127 object, so the call into the adapter and the ABI decoding are excluded. The BL127 (deployed) form includes one staticcall per op, and so does snekmate, a Vyper library that a Solidity adapter can only call; BL127Lib and the other libraries are internal calls. Aave, Morpho and Maker answer only the accrual scenarios their models describe, FraxLend and Euler only the adaptive rate, and OpenZeppelin and Uniswap FullMath, which have no exp, ln or real power (and FullMath no root), only the scenarios built from mul, div and their roots; each reports Unsupported elsewhere, which the tables and charts leave empty (a library with no model is not a revert). gasPerStep = gasTotal / N. Reverted cells carry no gas.",
  "live": "LadderRunner.run(scenario, form, n, p) on Sepolia executes any cell with one eth_call and returns (ok, raw, reason, gasPerStep, gasTotal). scripts/ladder/livecap.py sends every cell's floored parameters p and compares the returned raw word with the committed one bit for bit (or the revert selector for a reverted cell). liveMaxN is the largest N per form the public RPC completed within its gas limit, measured by scripts/ladder/livecap.py.",
  "reproduce": "make ladder regenerates ladder.json and ladder.md from the Foundry harness (test/ladder/Ladder.t.sol) and this script; make ladder-check fails CI when the committed dataset differs. Versions of every library and of forge are recorded in the dataset."
 },
 "scenarios": [
  {
   "id": "compound",
   "index": 0,
   "demo": true,
   "title": "Daily compound interest",
   "step": "b = b * (1 + r)",
   "result": "balance after N days",
   "why": "Per-block or per-day accrual in every lending protocol: one mul per step, N steps.",
   "order": [
    "principal",
    "rate"
   ],
   "derived": {},
   "ops": {
    "bl127": [
     "mul"
    ],
    "bl127lib": [
     "mul"
    ],
    "abdk": [
     "mul"
    ],
    "solady": [
     "sMulWad"
    ],
    "prb": [
     "mul"
    ],
    "quad": [
     "mul"
    ],
    "balancer": [
     "mulDown"
    ],
    "aave": [
     "unsupported"
    ],
    "morpho": [
     "unsupported"
    ],
    "maker": [
     "rpow",
     "rmul"
    ],
    "bl19": [
     "mul"
    ],
    "solmate": [
     "wadMul"
    ],
    "oz": [
     "mulDiv(a, b, 1e18)"
    ],
    "uniswap": [
     "mulDiv(a, b, 1e18)"
    ],
    "prbud": [
     "mul"
    ],
    "snekmate": [
     "mul_div(a, b, 1e18)"
    ],
    "fraxlend": [
     "unsupported"
    ],
    "euler": [
     "unsupported"
    ]
   },
   "inputs": [
    {
     "name": "reference",
     "reference": true,
     "why": "$10,000 at 10% APR, the headline cell",
     "params": {
      "principal": "10000",
      "rate": "0.1/365"
     },
     "resolved": {
      "principal": "10000.0",
      "rate": "0.000273972602739726027397260273972602739726027"
     },
     "truth": {
      "1": "10002.7397260273972602739726027397260273973",
      "2": "10005.4802026646650403452805404391067742541",
      "5": "10013.7061382924386210524857642965155424808",
      "10": "10027.4310624075663873328585783270357249574",
      "20": "10054.9373711336135548503369243222913751056",
      "50": "10137.9098421416552959332725454243516547326",
      "100": "10277.721596739264220164675884991779643531",
      "365": "11051.5578161626437393801159668054461177946",
      "1000": "13151.2942008121758666272801482974151894245",
      "10000": "154769.34069144112806190764064318746174622"
     },
     "truthFormat": {
      "2**128": {
       "1": "10002.7397260273972602739726027397260273831",
       "2": "10005.4802026646650403452805404391067742258",
       "5": "10013.7061382924386210524857642965155424101",
       "10": "10027.4310624075663873328585783270357248157",
       "20": "10054.9373711336135548503369243222913748213",
       "50": "10137.9098421416552959332725454243516540161",
       "100": "10277.7215967392642201646758849917796420783",
       "365": "11051.5578161626437393801159668054461120931",
       "1000": "13151.2942008121758666272801482974151708361",
       "10000": "154769.34069144112806190764064318745955867"
      },
      "2**64": {
       "1": "10002.7397260273972600785191700012433102529",
       "2": "10005.4802026646650399542665771907777877954",
       "5": "10013.7061382924386200741471826822447632445",
       "10": "10027.4310624075663853734995663192074778207",
       "20": "10054.937371133613550920869440439192010187",
       "50": "10137.9098421416552860285397736420984834779",
       "100": "10277.7215967392642000820183148249192217963",
       "365": "11051.5578161626436605593414014599397698577",
       "1000": "13151.294200812175609651124924142183760655",
       "10000": "154769.340691441097819994175573140474179894"
      },
      "10**18": {
       "1": "10002.73972602739726",
       "2": "10005.48020266466503979718521298555076",
       "5": "10013.7061382924386196811209138045354520256",
       "10": "10027.4310624075663845863696540850290456649",
       "20": "10054.9373711336135493422912540217635458895",
       "50": "10137.9098421416552820495285819326298125931",
       "100": "10277.7215967392641920142469703402753810544",
       "365": "11051.5578161626436288948077526349463981174",
       "1000": "13151.2942008121755064165378636582564191694",
       "10000": "154769.340691441085670962504669178459129233"
      },
      "10**27": {
       "1": "10002.7397260273972602739726",
       "2": "10005.4802026646650403452805349581534997185",
       "5": "10013.706138292438621052485750582867037561",
       "10": "10027.4310624075663873328585508621464825374",
       "20": "10054.9373711336135548503368692418346721003",
       "50": "10137.9098421416552959332724065869120198153",
       "100": "10277.721596739264220164675603487490497016",
       "365": "11051.5578161626437393801148619523639760896",
       "1000": "13151.2942008121758666272765461899923430328",
       "10000": "154769.340691441128061907216733736102006072"
      },
      "binary128": {
       "1": "10002.7397260273972602739726027397260273243",
       "2": "10005.4802026646650403452805404391067741082",
       "5": "10013.7061382924386210524857642965155421159",
       "10": "10027.4310624075663873328585783270357242265",
       "20": "10054.9373711336135548503369243222913736397",
       "50": "10137.9098421416552959332725454243516510376",
       "100": "10277.7215967392642201646758849917796360393",
       "365": "11051.557816162643739380115966805446088391",
       "1000": "13151.2942008121758666272801482974150935609",
       "10000": "154769.340691441128061907640643187450464637"
      },
      "bl19": {
       "1": "10002.7397260273972602739726027397260273973",
       "2": "10005.4802026646650403452805404391067742541",
       "5": "10013.7061382924386210524857642965155424808",
       "10": "10027.4310624075663873328585783270357249574",
       "20": "10054.9373711336135548503369243222913751056",
       "50": "10137.9098421416552959332725454243516547326",
       "100": "10277.721596739264220164675884991779643531",
       "365": "11051.5578161626437393801159668054461177946",
       "1000": "13151.2942008121758666272801482974151894245",
       "10000": "154769.34069144112806190764064318746174622"
      }
     },
     "cells": {
      "bl127": {
       "1": {
        "value": "10002.7397260273972602739726027397260273831",
        "digits": 38,
        "digitsFormat": 100,
        "exact": false,
        "agree": 39,
        "errUlps": "-4810.96",
        "relErr": "-1.41343e-39",
        "prec": "42.5",
        "digits1": "38.8",
        "digitsCapped": "38.8",
        "digitsFormatCapped": "42.5",
        "interestDigits1": "35.2",
        "interestCapped": "35.2",
        "ulps": "-4.81e+3",
        "raw": "3403755949666702274259673128036673260340000",
        "gas": 1335,
        "gasPerStep": 1335
       },
       "2": {
        "value": "10005.4802026646650403452805404391067742258",
        "digits": 38,
        "digitsFormat": 42,
        "exact": false,
        "agree": 39,
        "errUlps": "-9625.4",
        "relErr": "-2.8271e-39",
        "prec": "42.5",
        "digits1": "38.5",
        "digitsFormat1": "42.6",
        "digitsCapped": "38.5",
        "digitsFormatCapped": "42.5",
        "interestDigits1": "35.2",
        "interestCapped": "35.2",
        "ulps": "-9.63e+3",
        "raw": "3404688485543323288581388106975861389995554",
        "gas": 2144,
        "gasPerStep": 1072
       },
       "5": {
        "value": "10013.7061382924386210524857642965155424101",
        "digits": 38,
        "digitsFormat": 42,
        "exact": false,
        "agree": 39,
        "errUlps": "-24083.1",
        "relErr": "-7.06769e-39",
        "prec": "42.5",
        "digits1": "38.1",
        "digitsFormat1": "42.2",
        "digitsCapped": "38.1",
        "digitsFormatCapped": "42.2",
        "interestDigits1": "35.2",
        "interestCapped": "35.2",
        "ulps": "-2.41e+4",
        "raw": "3407487626388881358454318285018398702905994",
        "gas": 4571,
        "gasPerStep": 914
       },
       "10": {
        "value": "10027.4310624075663873328585783270357248157",
        "digits": 37,
        "digitsFormat": 41,
        "exact": false,
        "agree": 38,
        "errUlps": "-48232.5",
        "relErr": "-1.41355e-38",
        "prec": "42.5",
        "digits1": "37.8",
        "digitsFormat1": "41.9",
        "digitsCapped": "37.8",
        "digitsFormatCapped": "41.9",
        "interestDigits1": "35.2",
        "interestCapped": "35.2",
        "ulps": "-4.82e+4",
        "raw": "3412157976052587301682235746989530010598495",
        "gas": 8616,
        "gasPerStep": 861
       },
       "20": {
        "value": "10054.9373711336135548503369243222913748213",
        "digits": 37,
        "digitsFormat": 41,
        "exact": false,
        "agree": 37,
        "errUlps": "-96731.2",
        "relErr": "-2.82714e-38",
        "prec": "42.5",
        "digits1": "37.5",
        "digitsFormat1": "41.5",
        "digitsCapped": "37.5",
        "digitsFormatCapped": "41.5",
        "interestDigits1": "35.2",
        "interestCapped": "35.2",
        "ulps": "-9.67e+4",
        "raw": "3421517887891144695366396716445441388188205",
        "gas": 16707,
        "gasPerStep": 835
       },
       "50": {
        "value": "10137.909842141655295933272545424351654016",
        "digits": 37,
        "digitsFormat": 41,
        "exact": false,
        "agree": 38,
        "errUlps": "-243824.0",
        "relErr": "-7.06787e-38",
        "prec": "42.5",
        "digits1": "37.1",
        "digitsFormat1": "41.1",
        "digitsCapped": "37.1",
        "digitsFormatCapped": "41.1",
        "interestDigits1": "35.2",
        "interestCapped": "35.2",
        "ulps": "-2.44e+5",
        "raw": "3449751956715040084008762809673487784914014",
        "gas": 40982,
        "gasPerStep": 819
       },
       "100": {
        "value": "10277.7215967392642201646758849917796420782",
        "digits": 36,
        "digitsFormat": 40,
        "exact": false,
        "agree": 37,
        "errUlps": "-494375.0",
        "relErr": "-1.41358e-37",
        "prec": "42.5",
        "digits1": "36.8",
        "digitsFormat1": "40.8",
        "digitsCapped": "36.8",
        "digitsFormatCapped": "40.8",
        "interestDigits1": "35.2",
        "interestCapped": "35.2",
        "ulps": "-4.94e+5",
        "raw": "3497327431492883849142179832619559394729276",
        "gas": 81447,
        "gasPerStep": 814
       },
       "365": {
        "value": "11051.5578161626437393801159668054461120925",
        "digits": 36,
        "digitsFormat": 40,
        "exact": false,
        "agree": 37,
        "errUlps": "-1.94032e+6",
        "relErr": "-5.15953e-37",
        "prec": "42.5",
        "digits1": "36.2",
        "digitsFormat1": "40.2",
        "digitsCapped": "36.2",
        "digitsFormatCapped": "40.2",
        "interestDigits1": "35.2",
        "interestCapped": "35.2",
        "ulps": "-1.94e+6",
        "raw": "3760650251847422126544826182474085878843312",
        "gas": 296079,
        "gasPerStep": 811
       },
       "1000": {
        "value": "13151.2942008121758666272801482974151708344",
        "digits": 35,
        "digitsFormat": 39,
        "exact": false,
        "agree": 36,
        "errUlps": "-6.32587e+6",
        "relErr": "-1.41355e-36",
        "prec": "42.6",
        "digits1": "35.8",
        "digitsFormat1": "39.8",
        "digitsCapped": "35.8",
        "digitsFormatCapped": "39.8",
        "interestDigits1": "35.2",
        "interestCapped": "35.2",
        "ulps": "-6.33e+6",
        "raw": "4475153518725978999354785710825927528088900",
        "gas": 811502,
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     ]
    },
    "50": {
     "basis": "common",
     "inputs": [
      "reference",
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      "tiny-high",
      "mid-1pct",
      "mid-5pct",
      "mid-50pct",
      "large",
      "huge",
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      "fraction"
     ]
    },
    "100": {
     "basis": "common",
     "inputs": [
      "reference",
      "tiny-low",
      "tiny-high",
      "mid-1pct",
      "mid-5pct",
      "mid-50pct",
      "large",
      "huge",
      "huge-high",
      "fraction"
     ]
    },
    "365": {
     "basis": "common",
     "inputs": [
      "reference",
      "tiny-low",
      "tiny-high",
      "mid-1pct",
      "mid-5pct",
      "mid-50pct",
      "large",
      "huge",
      "huge-high",
      "fraction"
     ]
    },
    "1000": {
     "basis": "common",
     "inputs": [
      "reference",
      "tiny-low",
      "tiny-high",
      "mid-1pct",
      "mid-5pct",
      "mid-50pct",
      "large",
      "huge",
      "huge-high",
      "fraction"
     ]
    },
    "10000": {
     "basis": "common",
     "inputs": [
      "reference",
      "tiny-low",
      "tiny-high",
      "mid-1pct",
      "mid-5pct",
      "mid-50pct",
      "large",
      "huge",
      "huge-high",
      "fraction"
     ]
    }
   },
   "liveMaxN": {
    "bl127": 10000,
    "bl19": 10000
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  },
  {
   "id": "compound-pow",
   "index": 1,
   "demo": true,
   "title": "Compound interest in one pow",
   "step": "b0 * (1 + r)^N",
   "result": "the same balance, computed with one real-exponent pow",
   "why": "The clever way to skip the loop. pow is exp(N ln(1+r)) in every library, so its precision is bounded by ln and exp, and the error grows with N without any loop.",
   "order": [
    "principal",
    "rate"
   ],
   "derived": {},
   "ops": {
    "bl127": [
     "pow",
     "mul"
    ],
    "bl127lib": [
     "pow",
     "mul"
    ],
    "abdk": [
     "exp",
     "mul",
     "ln (pow = exp(y ln x))"
    ],
    "solady": [
     "powWad",
     "sMulWad"
    ],
    "prb": [
     "pow",
     "mul"
    ],
    "quad": [
     "exp(y ln x)",
     "mul"
    ],
    "balancer": [
     "pow",
     "mulDown"
    ],
    "aave": [
     "unsupported"
    ],
    "morpho": [
     "unsupported"
    ],
    "maker": [
     "unsupported"
    ],
    "bl19": [
     "pow",
     "mul"
    ],
    "solmate": [
     "wadPow",
     "wadMul"
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    "oz": [
     "unsupported"
    ],
    "uniswap": [
     "unsupported"
    ],
    "prbud": [
     "pow",
     "mul"
    ],
    "snekmate": [
     "wad_exp(wad_ln(x) y)",
     "mul_div(a, b, 1e18)"
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    "fraxlend": [
     "unsupported"
    ],
    "euler": [
     "unsupported"
    ]
   },
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     "reference": true,
     "why": "$10,000 at 10% APR, the headline cell",
     "params": {
      "principal": "10000",
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   "summaryBasis": {
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   },
   "liveMaxN": {
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  },
  {
   "id": "amortise",
   "index": 4,
   "demo": true,
   "title": "Amortising loan",
   "step": "b = b * (1 + r) - payment",
   "result": "outstanding balance after N daily payments",
   "why": "Subtracting nearly equal numbers. Every loan's term is past the ladder's end, so the balance stays positive on every cell; the term-10,001 loans retire one step past N = 10,000, where the balance is one payment and relative error is amplified thousands of times.",
   "order": [
    "principal",
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   ],
   "derived": {
    "payment": "principal * rate / (1 - (1 + rate)^(-term))"
   },
   "ops": {
    "bl127": [
     "mul",
     "sub"
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    "bl127lib": [
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     "mul",
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    "quad": [
     "mul",
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    "balancer": [
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    "aave": [
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    "morpho": [
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    "maker": [
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    "bl19": [
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    "solmate": [
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    "oz": [
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    ],
    "uniswap": [
     "mulDiv(a, b, 1e18)",
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    "snekmate": [
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    "fraxlend": [
     "unsupported"
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    "euler": [
     "unsupported"
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     "reference": true,
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      "rate": "0.05/365",
      "term": "10001"
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     "resolved": {
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      },
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    {
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     "params": {
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     "resolved": {
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     ]
    },
    "365": {
     "basis": "common",
     "inputs": [
      "five-pct",
      "unit"
     ]
    },
    "1000": {
     "basis": "common",
     "inputs": [
      "five-pct"
     ]
    },
    "10000": {
     "basis": "common",
     "inputs": [
      "reference",
      "five-pct",
      "unit",
      "million",
      "fifty-pct"
     ]
    }
   },
   "liveMaxN": {
    "bl127": 10000,
    "bl19": 10000
   }
  },
  {
   "id": "accrual-per-second",
   "index": 9,
   "demo": true,
   "title": "Per-second accrual",
   "step": "b = b0 * (1 + r/31536000)^(N * 86400)",
   "result": "balance after N days of per-second compounding",
   "why": "How Maker's jug and pot accrue: the rate per second, compounded every second, N days at a time. The rate is 1.6e-9 at 5% APR, which an 18-decimal grid holds to ten digits before any arithmetic; BL127 and BL19 compute the power with compound, which never forms a rounded 1 + r; the WAD and 64.64 libraries as exp(n ln(1 + r)), Maker as rpow.",
   "order": [
    "principal",
    "rateSec"
   ],
   "derived": {},
   "ops": {
    "bl127": [
     "compound",
     "mul"
    ],
    "bl127lib": [
     "compound",
     "mul"
    ],
    "abdk": [
     "ln",
     "mul",
     "exp"
    ],
    "solady": [
     "powWad",
     "sMulWad"
    ],
    "prb": [
     "pow",
     "mul"
    ],
    "quad": [
     "ln",
     "mul",
     "exp"
    ],
    "balancer": [
     "pow",
     "mulDown"
    ],
    "aave": [
     "unsupported"
    ],
    "morpho": [
     "unsupported"
    ],
    "maker": [
     "rpow",
     "rmul"
    ],
    "bl19": [
     "compound",
     "mul"
    ],
    "solmate": [
     "wadPow",
     "wadMul"
    ],
    "oz": [
     "unsupported"
    ],
    "uniswap": [
     "unsupported"
    ],
    "prbud": [
     "pow",
     "mul"
    ],
    "snekmate": [
     "wad_exp(wad_ln(x) y)",
     "mul_div(a, b, 1e18)"
    ],
    "fraxlend": [
     "unsupported"
    ],
    "euler": [
     "unsupported"
    ]
   },
   "inputs": [
    {
     "name": "reference",
     "reference": true,
     "why": "$10,000 at 5% APR, the lending protocol's steady state",
     "params": {
      "principal": "10000",
      "rateSec": "0.05/31536000"
     },
     "resolved": {
      "principal": "10000.0",
      "rateSec": "0.00000000158548959918822932521562658548959918822933"
     },
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  {
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   "index": 11,
   "demo": true,
   "title": "An option walking out of the money",
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   "result": "the Black-Scholes call price after N down-ticks of the spot",
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    "bl19": 10000
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  },
  {
   "id": "decay",
   "index": 12,
   "demo": true,
   "title": "Decay, the share released",
   "step": "D = D e^(-eps); result 1 - D",
   "result": "the share released after N steps of decay",
   "why": "Vesting, TWAP weights and exponential moving averages multiply by e^(-dt/tau) every step. The factor is a hair below one, and on an 18-decimal grid eps and e^(-eps) both sit on a grid of spacing 1e-18, so the factor's distance from one keeps about 18 + log10(eps) digits, and the released share with it: six digits at eps = 1.06e-12, three at 1.06e-15, one at 1.06e-17, and below 1e-18 nothing is released at all. Each eps is dt/tau for a time constant in seconds, which no grid holds exactly; a power of ten sits on the WAD grid and flatters it (at eps = 1e-17 exactly, Solady kept 17 digits).",
   "order": [
    "eps"
   ],
   "derived": {},
   "ops": {
    "bl127": [
     "exp",
     "mul",
     "sub"
    ],
    "bl127lib": [
     "exp",
     "mul",
     "sub"
    ],
    "abdk": [
     "exp",
     "mul",
     "sub"
    ],
    "solady": [
     "expWad",
     "sMulWad"
    ],
    "prb": [
     "exp",
     "mul"
    ],
    "quad": [
     "exp",
     "mul",
     "sub"
    ],
    "balancer": [
     "exp",
     "mulDown"
    ],
    "aave": [
     "unsupported"
    ],
    "morpho": [
     "unsupported"
    ],
    "maker": [
     "unsupported"
    ],
    "bl19": [
     "exp",
     "mul",
     "sub"
    ],
    "solmate": [
     "wadExp",
     "wadMul"
    ],
    "oz": [
     "unsupported"
    ],
    "uniswap": [
     "unsupported"
    ],
    "prbud": [
     "exp",
     "mul"
    ],
    "snekmate": [
     "wad_exp",
     "mul_div(a, b, 1e18)"
    ],
    "fraxlend": [
     "unsupported"
    ],
    "euler": [
     "unsupported"
    ]
   },
   "inputs": [
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     "name": "reference",
     "reference": true,
     "why": "a one-second step against a twelve-day time constant: eps = 9.6e-7",
     "params": {
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 "liveNote": "liveMaxN is the largest N on the ladder that the public Sepolia RPC's eth_call completed for every input of that form. Above it the runner reports a bare revert because the adapter ran out of gas under the RPC's eth_call limits (calls needing more than about 40M gas, or a few seconds of execution, fail); cells above it are checked from your own node. Anyone with their own node can run every cell.",
 "liveChecked": 1913,
 "liveMismatches": 0,
 "liveBlock": 12017732,
 "liveDate": "2026-10-29"
}