2f:T5ad,viewof value = { const el = document.createElement("div"); el.style.display = "flex"; el.value = 0n; el.append(createBitsInput(1, {color: "aqua", label: "sign"})); el.append(createBitsInput(q, {color: "lightgreen", label: "exponent (biased)"})); el.append(createBitsInput(p - 1, {color: "salmon", label: "mantissa"})); const bitInputs = el.querySelectorAll(".bit-input"); for (let i = 0; i < p + q; i++) { const mask = 1n << BigInt(p + q - 1 - i); bitInputs[i].addEventListener("change", (e) => { el.value = (el.value & ~mask) | (bitInputs[i].checked ? mask : 0n); el.dispatchEvent(new Event("input", {bubbles: true})); }) } const update = () => { for (let i = 0; i < p + q; i++) { const mask = 1n << BigInt(p + q - 1 - i); bitInputs[i].checked = (el.value & mask) != 0; } }; update(); el.addEventListener("input", (e) => { if (!e.target.classList.contains("bit-input")) update(); }); return el; function createBitsInput(nBits, {color, label}) { const group = htl.html`

${label}

`; const inputs = document.createElement("div"); group.append(inputs); for (let i = 0; i < nBits; i++) { const mask = (1n << BigInt(nBits - 1 - i)); const input = htl.html``; inputs.append(input); } return group; } }30:T550,{ const MAX_SERIES_TERMS = 6; const leading_bit = kind == SUBNORMAL ? 0 : 1; const s = sign ? "-" : ""; if (kind == Infinity) return tex`\text{value} = ${s} \infty`; if (typeof kind == "number" && isNaN(kind)) return tex`\text{value} = \text{NaN}`; // express the mantissa as a series of fractions let m_series = [kind == SUBNORMAL ? 0 : 1]; for (let i = 0; i < p; i++) { let mask = (1n<= integer_part) break; lo_pow = hi_pow + 1; hi_pow = hi_pow * 2; } // binary search for the logarithm, within the range [lo, hi] let pow, est; while (true) { pow = Math.floor((lo_pow + hi_pow) / 2); est = BigInt(1) << BigInt(pow); let below = integer_part < est; let above = integer_part >= est * 2n; if (below) { hi_pow = pow - 1; } else if (above) { lo_pow = pow + 1; } else { break; } } if (inverse) { // for numbers less than 1, we need to negate the power. in addition, we must round down. if (num.inverse().equals(est)) return -pow; return -pow - 1; } return pow; }32:Ta33,/** convert a decimal number to a float, rounded towards zero */ function toFloat(str) { // handle infinity if (str == "∞" || str == "-∞") { let sign = str == "-∞"; // manually construct an infinity float return (BigInt(sign) << BigInt(p + q - 1)) | ((1n << BigInt(p + q - 1)) - 1n) & ~((1n << BigInt(p - 1)) - 1n); } // convert to a big rational using the fraction.js library, catching errors and returning NaN if they occur let num; try { num = new Fraction(str); } catch { // NaN return (1n << BigInt(p + q - 1)) - 1n; } // determine the sign of the number, then proceed through conversion with absolute value let sign = num.s == -1n ? 1n : 0n; num = num.abs(); // fast case for zero if (num.equals(0)) return 0n; // now, we'll convert from a arbitrary-precision rational number `num` to // a floating point number. // our goal is to create a float that gets as close to `num` as possible, without going over, // first by calculating the exponent and then by proceeding step by step through the mantissa bits. // as we go, we'll track the current value of our result in float_val. let float_val; // begin by determining the ideal exponent (ignoring subnormals for now) let e = frac_log2(num); // we have to special case for sub-normals... // e_biased will be set to the biased exponent // place_val will be set to the value of the "hidden bit", and then div by 2 for each mantissa bit let place_val, e_biased; if (e >= e_min) { // this is a normal number e_biased = e + e_bias; float_val = place_val = new Fraction(2n).pow(e); } else { // this is a subnormal, manually override e_biased and start with zero e_biased = 0; place_val = new Fraction(2n).pow(e_min); float_val = new Fraction(0); } // now that we have the exponent and sign, we begin to construct the float let float = (BigInt(sign) << BigInt(p + q - 1)) | (BigInt(e_biased) << BigInt(p - 1)); // okay, now for every bit in the mantissa (from MSb to LSb), try to increase float_val by the approprate place value // if adding place_val would make float_val > num, we can't set that bit. // if place_val <= num, set the mantissa bit at the appropriate position for (let i = p - 1; i >= 0; i--) { place_val = place_val.div(2); // try to increase float_val by the approprate place value let tmp = float_val.add(place_val); let cmp = num.compare(tmp); if (cmp >= 0) { float_val = tmp; float |= (1n << BigInt(i - 1)); } if (cmp == 0) return float; } return float; }33:[] 2d:{"title":"Floating Point Calculator","cells":[{"id":0,"value":"# Floating Point Calculator","pinned":false,"hidden":false,"mode":"md"},{"id":282,"value":"Calculate [IEEE-754](https://en.wikipedia.org/wiki/IEEE_754) style floating point numbers with arbitrary precision (`p`) and range (`q`).","pinned":false,"hidden":false,"mode":"md"},{"id":3,"value":"viewof p = Inputs.number([2, Infinity], {value: 24, step: 1, label: \"precision\"})","pinned":false,"hidden":false,"mode":"ojs"},{"id":13,"value":"viewof q = Inputs.number([2, 14], {value: 8, step: 1, label: \"exponent bits\"})","pinned":false,"hidden":false,"mode":"ojs"},{"id":506,"value":"Enter as decimal aproximation, hex, or click to modify the binary digits","pinned":false,"hidden":false,"mode":"md"},{"id":393,"value":"{\n const toString = v => `0x${v.toString(16).toUpperCase()}`;\n const view = viewof value;\n const input = Inputs.text({\n label: \"hex\",\n pattern: \"0x[A-F0-9]+\",\n value: toString(view.value),\n });\n input.querySelector(\"input\").style.fontFamily = \"monospace\";\n \n const onInput = () => input.value = toString(view.value);\n view.addEventListener(\"input\", onInput);\n invalidation.then(() => view.removeEventListener(\"input\", onInput));\n\n input.oninput = () => {\n view.value = BigInt(input.value);\n view.dispatchEvent(new Event(\"input\"));\n };\n return input;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":987,"value":"aprox_input = {\n const input = Inputs.text({\n label: \"decimal value ≈\"\n });\n \n input.onchange = () => {\n (viewof value).value = toFloat(input.value);\n (viewof value).dispatchEvent(new Event(\"input\"));\n };\n return input;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":44,"value":"$2f","pinned":false,"hidden":false,"mode":"ojs"},{"id":2421,"value":"Inputs.bind(Inputs.button([\n [\"+\", (v) => v + 1n],\n [\"-\", (v) => v - 1n]\n], {label: \"step\"}), viewof value)","pinned":false,"hidden":false,"mode":"ojs"},{"id":200,"value":"$30","pinned":false,"hidden":false,"mode":"ojs"},{"id":778,"value":"Inputs.bind(Inputs.button([\n [\"0\", () => 0n],\n [\"smallest normal\", () => 1n << BigInt(p-1)],\n [\"1\", () => toFloat(\"1\")],\n [\"π\", () => {\n const hex_digits = Math.ceil((p+1)/2);\n const fixed = BBPbigint(0n, hex_digits);\n const mantissa = BigInt(\"0b\" + fixed.toString(2).slice(1, p));\n let exp = (BigInt(e_bias+1) << BigInt(p - 1));\n return exp | mantissa;\n }],\n [\"max\", () => ((1n << BigInt(p + q - 1)) - 1n - (1n << BigInt(p - 1))) ],\n [\"∞\", () => toFloat(\"∞\")],\n]), viewof value)","pinned":false,"hidden":false,"mode":"ojs"},{"id":1137,"value":"## Deconstructed Representation","pinned":false,"hidden":false,"mode":"md"},{"id":502,"value":"sign = Number(value >> BigInt(q + p - 1)) & 1","pinned":false,"hidden":false,"mode":"ojs"},{"id":501,"value":"exp_biased = Number((value >> BigInt(p - 1)) & ((1n << BigInt(q)) - 1n))","pinned":false,"hidden":false,"mode":"ojs"},{"id":500,"value":"mantissa = value & ((1n << BigInt(p - 1)) - 1n)","pinned":false,"hidden":false,"mode":"ojs"},{"id":179,"value":"E = exp_biased == 0 ? 1 - e_bias : exp_biased - e_bias","pinned":false,"hidden":false,"mode":"ojs"},{"id":702,"value":"kind = \n exp_biased == 0 ? SUBNORMAL :\n exp_biased == pow2(q) - 1 ? (mantissa == 0 ? Infinity : NaN) :\n NORMALIZED;","pinned":false,"hidden":false,"mode":"ojs"},{"id":1694,"value":"leading_bit = kind == SUBNORMAL ? 0 : 1;","pinned":true,"hidden":false,"mode":"ojs"},{"id":1670,"value":"significand = Fraction(mantissa, 1n << BigInt(p-1)).add(leading_bit).mul(-2*sign+1);","pinned":true,"hidden":false,"mode":"ojs"},{"id":1069,"value":"rational = (E >= 0) ? significand.mul(1n << BigInt(E)) : significand.div(1n << BigInt(-E))","pinned":true,"hidden":false,"mode":"ojs"},{"id":2390,"value":"viewof next_largest = {\n let val = toFrac(value + 1n);\n let el = tex`\\text{next\\_largest} = ${val ? val.toLatex(true) : `\\\\text{NaN}`}`;\n el.value = val;\n return el;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":2414,"value":"viewof next_smaller = {\n let val = toFrac(value - 1n);\n let el = tex`\\text{next\\_smaller} = ${val ? val.toLatex(true) : `\\\\text{NaN}`}`;\n el.value = val;\n return el;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1088,"value":"## bias/min/max","pinned":false,"hidden":false,"mode":"md"},{"id":27,"value":"e_bias = pow2(q-1) - 1","pinned":false,"hidden":false,"mode":"ojs"},{"id":1082,"value":"e_min = 1-e_bias","pinned":false,"hidden":false,"mode":"ojs"},{"id":1080,"value":"e_max = e_bias","pinned":false,"hidden":false,"mode":"ojs"},{"id":724,"value":"## implementation details","pinned":false,"hidden":false,"mode":"md"},{"id":131,"value":"","pinned":false,"hidden":false,"mode":"html"},{"id":2209,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":785,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":2360,"value":"/// Convert a floating point number to a Fraction.js rational\nfunction toFrac(float) {\n // extract the relevant bits of the floating point number\n let sign = Number(float >> BigInt(q + p - 1)) & 1;\n let signum = BigInt(-2*sign+1); // positive sign becomes 1, negative sign becomes -1\n let exp_biased = Number((float >> BigInt(p - 1)) & ((1n << BigInt(q)) - 1n));\n let mantissa = float & ((1n << BigInt(p - 1)) - 1n);\n\n if (exp_biased == pow2(q) - 1) return null; // test for infinity/NaN\n let E = Math.max(exp_biased - e_bias, e_min); // debias the exponent\n \n // calculate the significand, the numerator is the mantissa, denominator is 2^(p-1)\n let significand = Fraction(mantissa * signum, 1n << BigInt(p-1));\n // hidden bit trick\n if (exp_biased != 0) significand = significand.add(1);\n \n let scalar = 1n << BigInt(Math.abs(E)); \n return (E >= 0) ? significand.mul(scalar) : significand.div(scalar);\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":684,"value":"function pow2(e) { return 1 << e }","pinned":false,"hidden":false,"mode":"ojs"},{"id":703,"value":"SUBNORMAL = \"SUBNORMAL\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":718,"value":"NORMALIZED = \"NORMALIZED\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":2727,"value":"import {BBPbigint} from \"@rreusser/computing-with-the-bailey-borwein-plouffe-formula\"\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":1595,"value":"Fraction = (await import(\"https://cdn.skypack.dev/pin/fraction.js@v4.2.0-4T2HgYMCUUjn2ZtoMwJX/mode=imports,min/unoptimized/bigfraction.js\")).default","pinned":true,"hidden":false,"mode":"ojs"},{"id":1706,"value":"{\n aprox_input.value = kind == SUBNORMAL || kind == NORMALIZED ? rational :\n kind == Infinity ? ( (sign == 1 ? \"-∞\" : \"∞\") ) : \"NaN\";\n}","pinned":true,"hidden":false,"mode":"ojs"}],"files":"$W33","stdlib":"1","titling":"auto"}