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endDigits: number[];\n} {\n const startDigits = decompose(start, base, numDigits);\n const endDigits = decompose(end, base, numDigits);\n\n const prefixDigits: number[] = [];\n for (let i = 0; i < startDigits.length; i++) {\n if (startDigits[i] === endDigits[i]) {\n prefixDigits.push(startDigits[i]);\n } else break;\n }\n\n return {\n prefixDigits,\n startDigits: startDigits.splice(prefixDigits.length),\n endDigits: endDigits.splice(prefixDigits.length),\n };\n}\n\nexport function frontGroupings(digits: number[], base: number): number[][] {\n const digitsReversed = Array.from(digits).reverse();\n const nonZeroDigits = dropUntil(\n zipWithIndex(digitsReversed),\n (a) => a[0] !== 0,\n );\n\n if (nonZeroDigits.length === 0) {\n return [[0]];\n }\n const fromFront = nonZeroDigits\n .filter((_, i) => i !== nonZeroDigits.length - 1)\n .flatMap(([d, i]) => {\n const fixedDigits = Array.from(digits);\n fixedDigits.length = fixedDigits.length - (i + 1);\n\n const result: number[][] = [];\n for (let n = d + 1; n < base; n++) {\n result.push([...Array.from(fixedDigits), n]);\n }\n return result;\n });\n\n return [nonZeroDigits.map((a) => a[0]).reverse(), ...fromFront];\n}\n\nexport function backGroupings(digits: number[], base: number): number[][] {\n const digitsReversed = Array.from(digits).reverse();\n const nonMaxDigits = dropUntil(\n zipWithIndex(digitsReversed),\n (a) => a[0] !== base - 1,\n );\n\n if (nonMaxDigits.length === 0) {\n return [[base - 1]];\n }\n\n const fromBack = nonMaxDigits\n .filter((_, i) => i !== nonMaxDigits.length - 1)\n .flatMap(([d, i]) => {\n const fixedDigits = Array.from(digits);\n fixedDigits.length = fixedDigits.length - (i + 1);\n\n const result: number[][] = [];\n for (let n = d - 1; n >= 0; n--) {\n result.push([...Array.from(fixedDigits), n]);\n }\n\n return result;\n });\n\n return [...fromBack.reverse(), nonMaxDigits.map((a) => a[0]).reverse()];\n}\n\nexport function middleGroupings(\n firstDigitStart: number,\n firstDigitEnd: number,\n): number[][] {\n const result = [];\n while (++firstDigitStart < firstDigitEnd) {\n result.push([firstDigitStart]);\n }\n return result;\n}\n\nexport function groupByIgnoringDigits(\n start: bigint,\n end: bigint,\n base: number,\n numDigits: number,\n): number[][] {\n const { prefixDigits, startDigits, endDigits } = separatePrefix(\n start,\n end,\n base,\n numDigits,\n );\n\n if (\n start === end ||\n (startDigits.every((n) => n === 0) &&\n endDigits.every((n) => n === base - 1) &&\n prefixDigits.length !== 0)\n ) {\n return [prefixDigits];\n } else if (prefixDigits.length === numDigits - 1) {\n const result = [];\n for (\n let i = startDigits[startDigits.length - 1];\n i <= endDigits[endDigits.length - 1];\n i++\n ) {\n result.push([...prefixDigits, i]);\n }\n\n return result;\n } else {\n const front = frontGroupings(startDigits, base);\n const middle = middleGroupings(startDigits[0], endDigits[0]);\n const back = backGroupings(endDigits, base);\n\n const groupings = [...front, ...middle, ...back];\n\n return groupings.map((g) => [...prefixDigits, ...g]);\n }\n}\n\nexport interface RoundingInterval {\n beginInterval: bigint;\n roundingMod: bigint;\n}\n\nexport type CETPayout = {\n indexFrom: bigint;\n indexTo: bigint;\n payout: bigint;\n};\n\n/**\n * Performs optimized evaluation and rounding for strictly monotonic hyperbolas on intervals (from, to)\n * e.g. hyperbolas with the form b = c = 0\n *\n * The next start of a payout range is determined by finding the outcome at the next mid-rounding payout.\n * Uses an inverse function of the hyperbola to find the outcome.\n *\n * Optimizes rounding from O(to - from) to O(totalCollateral / rounding)\n *\n *\n * Evaluates and rounds a payout_function equivalent to:\n *\n * payout_function_v0\n * num_pieces: 1\n * endpoint_0: from\n * endpoint_payout_0: fromPayout\n * extra_precision_0: 0\n * payout_curve_piece: HyperbolaPayoutCurve\n * endpoint_1: to\n * endpoint_payout_1: toPayout\n */\nexport function splitIntoRanges(\n from: bigint,\n to: bigint,\n fromPayout: bigint, // endpoint_payout\n toPayout: bigint, // endpoint_payout\n totalCollateral: bigint,\n curve: HyperbolaPayoutCurve | PolynomialPayoutCurve,\n roundingIntervals: RoundingInterval[],\n): CETPayout[] {\n if (to - from <= 0) {\n throw new Error('`to` must be strictly greater than `from`');\n }\n\n const reversedIntervals = [...roundingIntervals].reverse();\n\n const getRoundingForOutcome = (outcome: bigint): [bigint, number] => {\n const roundingIndex = reversedIntervals.findIndex(\n (interval) => interval.beginInterval <= outcome,\n );\n\n return [\n roundingIndex !== -1\n ? reversedIntervals[roundingIndex].roundingMod\n : BigInt(1),\n roundingIndex,\n ];\n };\n\n const clamp = (val: bigint) =>\n BigIntMath.clamp(BigInt(0), val, totalCollateral);\n\n const totalCollateralBN = new BigNumber(totalCollateral.toString());\n const clampBN = (val: BigNumber) =>\n BigNumber.max(0, BigNumber.min(val, totalCollateralBN));\n\n const result: CETPayout[] = [];\n\n // outcome = endpoint_0\n result.push({\n payout: fromPayout,\n indexFrom: from,\n indexTo: from,\n });\n\n // In the case of a constant payout curve (fromPayout === toPayout), we can skip the range evaluation\n if (fromPayout === toPayout) {\n result.push({\n payout: toPayout,\n indexFrom: from,\n indexTo: to,\n });\n // outcome = endpoint_1\n result.push({\n payout: toPayout,\n indexFrom: to,\n indexTo: to,\n });\n // merge neighbouring ranges with same payout\n return mergePayouts(result);\n }\n\n let currentOutcome = from + BigInt(1);\n\n // iterate over entire range of outcomes from [from, to]\n while (currentOutcome < to) {\n const [rounding, roundingIndex] = getRoundingForOutcome(currentOutcome);\n\n // either the next rounding interval, or the end of the range\n const nextFirstRoundingOutcome =\n reversedIntervals[roundingIndex - 1]?.beginInterval || to;\n\n // temporary variable to hold the current payout\n let currentPayout = new BigNumber(\n roundPayout(\n clampBN(curve.getPayout(currentOutcome)),\n rounding,\n ).toString(),\n );\n\n let currentMidRoundedOutcome = currentOutcome;\n\n const isAscending = curve\n .getPayout(nextFirstRoundingOutcome)\n .gt(currentPayout);\n\n // Add loop counter to prevent infinite loops\n let loopCounter = 0;\n const maxIterations = Number(to - from) + 1000; // Allow some extra iterations\n\n while (currentMidRoundedOutcome < nextFirstRoundingOutcome) {\n // Prevent infinite loops\n if (++loopCounter > maxIterations) {\n // Breaking out of potential infinite loop - add remaining range and exit\n result.push({\n payout: clamp(toBigInt(currentPayout)),\n indexFrom: currentMidRoundedOutcome,\n indexTo: to - BigInt(1),\n });\n currentOutcome = to;\n break;\n }\n\n const nextRoundedPayout = currentPayout\n .integerValue()\n .plus(isAscending ? Number(rounding) : -Number(rounding));\n\n const nextRoundedPayoutBigInt = toBigInt(nextRoundedPayout);\n\n const nextMidRoundedPayout = currentPayout.plus(\n isAscending ? Number(rounding) / 2 : -Number(rounding) / 2,\n );\n\n let nextMidRoundedOutcome = curve.getOutcomeForPayout(\n nextMidRoundedPayout,\n );\n\n // Handle invalid outcomes from getOutcomeForPayout\n if (nextMidRoundedOutcome < 0) {\n // If getOutcomeForPayout returns invalid value, advance manually\n nextMidRoundedOutcome = currentMidRoundedOutcome + BigInt(1);\n }\n\n if (\n (!isAscending &&\n nextMidRoundedOutcome >= 0 &&\n curve.getPayout(nextMidRoundedOutcome).lt(nextMidRoundedPayout)) ||\n (isAscending &&\n nextMidRoundedOutcome >= 0 &&\n curve.getPayout(nextMidRoundedOutcome).gte(nextMidRoundedPayout))\n ) {\n nextMidRoundedOutcome = nextMidRoundedOutcome - BigInt(1);\n // Ensure we don't go negative\n if (nextMidRoundedOutcome < 0) {\n nextMidRoundedOutcome = currentMidRoundedOutcome;\n }\n }\n\n const nextOutcome = curve.getOutcomeForPayout(nextRoundedPayout);\n\n if (nextMidRoundedOutcome >= nextFirstRoundingOutcome) {\n result.push({\n payout: clamp(toBigInt(currentPayout)),\n indexFrom: currentMidRoundedOutcome,\n indexTo: nextFirstRoundingOutcome - BigInt(1),\n });\n\n currentOutcome = nextFirstRoundingOutcome;\n break;\n }\n\n if (\n nextRoundedPayoutBigInt > totalCollateral ||\n nextRoundedPayoutBigInt < 0 ||\n nextOutcome >= to ||\n nextOutcome < 0 // undefined on curve\n ) {\n if (nextMidRoundedOutcome < from || nextMidRoundedOutcome > to) {\n result.push({\n payout: clamp(toBigInt(curren