2c:null 2d:["$","title",null,{"children":"Floating Point Calculator | Ben Aubin | Observable"}] 32: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; } }33: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; }35: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; }36:[] 30:{"id":"098eb38425b144ea","slug":"floating-point","title":"Floating Point Calculator","thumbnail":"9952c022608d4042dca15c69049cf7d08cda0aa7706e3803ce58939856710155","access_level":"public","create_time":"$D2022-05-14T20:05:35.170Z","update_time":"$D2023-04-28T17:52:28.476Z","view_time":null,"first_public_time":"$D2022-05-14T22:31:00.788Z","has_unpublished_edits":null,"fork":false,"stars":16,"starred":false,"owner":{"type":"team","id":"47b6fada5d996fda","login":"benaubin","name":"Ben Aubin","avatar_url":"https://avatars-next.observableusercontent.com/1bc40c046ba75a59de56327d538bfcce5374b1f23270e4a8b58ef897f585ae6d","tier":"starter_2024"},"version":2968,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"dccffcacdd858684","login":"benaubin","name":"Ben Aubin","avatar_url":"https://avatars-next.observableusercontent.com/1bc40c046ba75a59de56327d538bfcce5374b1f23270e4a8b58ef897f585ae6d","tier":null}],"members":[{"type":"individual","id":"dccffcacdd858684","login":"benaubin","name":"Ben Aubin","avatar_url":"https://avatars-next.observableusercontent.com/1bc40c046ba75a59de56327d538bfcce5374b1f23270e4a8b58ef897f585ae6d","tier":null,"role":"editor"}],"comments":[{"id":"91730ddb83f0c2c5","reply_to_id":null,"content":"This is really cool! I'm going to add \"fork, add support for posit style floating point numbers, and come up with some way of comparing them\" to my todo list. Feel free to beat me to it though ;)\n\nhttps://posithub.org/","create_time":"$D2022-05-17T11:29:26.164Z","update_time":null,"resolved":true,"user":{"type":"individual","id":"d36034ddfb454bdd","login":"jobleonard","name":"Job van der Zwan","avatar_url":"https://avatars-next.observableusercontent.com/808373580aa09723ceabb9cb31d01ea6db287280b59eaa63618e259f5d28dd71","tier":null}},{"id":"b8d8812c5f811ad3","reply_to_id":"91730ddb83f0c2c5","content":"Very cool! I'll keep an eye out for a merge, also, as a note, I plan to implement binary search to speed up toFloat","create_time":"$D2022-05-17T13:06:17.704Z","update_time":null,"resolved":true,"user":{"type":"individual","id":"dccffcacdd858684","login":"benaubin","name":"Ben Aubin","avatar_url":"https://avatars-next.observableusercontent.com/1bc40c046ba75a59de56327d538bfcce5374b1f23270e4a8b58ef897f585ae6d","tier":null}}],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":32,"collection":null,"type":"notebook","body":{"titling":"auto","title":"Floating Point Calculator","cells":[{"id":0,"value":"# Floating Point Calculator","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":282,"value":"Calculate [IEEE-754](https://en.wikipedia.org/wiki/IEEE_754) style floating point numbers with arbitrary precision (`p`) and range (`q`).","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":3,"value":"viewof p = Inputs.number([2, Infinity], {value: 24, step: 1, label: \"precision\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":13,"value":"viewof q = Inputs.number([2, 14], {value: 8, step: 1, label: \"exponent bits\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":506,"value":"Enter as decimal aproximation, hex, or click to modify the binary digits","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":393,"value":"{\n const toString = v => `0x${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","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"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","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":44,"value":"$32","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2421,"value":"Inputs.bind(Inputs.button([\n [\"+\", (v) => v + 1n],\n [\"-\", (v) => v - 1n]\n], {label: \"step\"}), viewof value)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":200,"value":"$33","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":778,"value":"Inputs.bind(Inputs.button([\n [\"0\", () => 0n],\n [\"smallest normal\", () => 1n << BigInt(p-1)],\n [\"1\", () => toFloat(\"1\")],\n [\"π\", () => {\n const hex_digits = Math.ceil((p+1)/2);\n const fixed = BBPbigint(0n, hex_digits);\n const mantissa = BigInt(\"0b\" + fixed.toString(2).slice(1, p));\n let exp = (BigInt(e_bias+1) << BigInt(p - 1));\n return exp | mantissa;\n }],\n [\"max\", () => ((1n << BigInt(p + q - 1)) - 1n - (1n << BigInt(p - 1))) ],\n [\"∞\", () => toFloat(\"∞\")],\n]), viewof value)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1137,"value":"## Deconstructed Representation","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":502,"value":"sign = Number(value >> BigInt(q + p - 1)) & 1","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":501,"value":"exp_biased = Number((value >> BigInt(p - 1)) & ((1n << BigInt(q)) - 1n))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":500,"value":"mantissa = value & ((1n << BigInt(p - 1)) - 1n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":179,"value":"E = exp_biased == 0 ? 1 - e_bias : exp_biased - e_bias","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"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","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1694,"value":"leading_bit = kind == SUBNORMAL ? 0 : 1;","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1670,"value":"significand = Fraction(mantissa, 1n << BigInt(p-1)).add(leading_bit).mul(-2*sign+1);","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1069,"value":"rational = (E >= 0) ? significand.mul(1n << BigInt(E)) : significand.div(1n << BigInt(-E))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2390,"value":"viewof next_largest = {\n let val = toFrac(value + 1n);\n let el = tex`\\text{next\\_largest} = ${val ? val.toLatex(true) : `\\\\text{NaN}`}`;\n el.value = val;\n return el;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2414,"value":"viewof next_smaller = {\n let val = toFrac(value - 1n);\n let el = tex`\\text{next\\_smaller} = ${val ? val.toLatex(true) : `\\\\text{NaN}`}`;\n el.value = val;\n return el;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1088,"value":"## bias/min/max","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":27,"value":"e_bias = pow2(q-1) - 1","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1082,"value":"e_min = 1-e_bias","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1080,"value":"e_max = e_bias","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":724,"value":"## implementation details","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":131,"value":"","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":2209,"value":"$34","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":785,"value":"$35","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2360,"value":"/// Convert a floating point number to a Fraction.js rational\nfunction toFrac(float) {\n // extract the relevant bits of the floating point number\n let sign = Number(float >> BigInt(q + p - 1)) & 1;\n let signum = BigInt(-2*sign+1); // positive sign becomes 1, negative sign becomes -1\n let exp_biased = Number((float >> BigInt(p - 1)) & ((1n << BigInt(q)) - 1n));\n let mantissa = float & ((1n << BigInt(p - 1)) - 1n);\n\n if (exp_biased == pow2(q) - 1) return null; // test for infinity/NaN\n let E = Math.max(exp_biased - e_bias, e_min); // debias the exponent\n \n // calculate the significand, the numerator is the mantissa, denominator is 2^(p-1)\n let significand = Fraction(mantissa * signum, 1n << BigInt(p-1));\n // hidden bit trick\n if (exp_biased != 0) significand = significand.add(1);\n \n let scalar = 1n << BigInt(Math.abs(E)); \n return (E >= 0) ? significand.mul(scalar) : significand.div(scalar);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":684,"value":"function pow2(e) { return 1 << e }","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":703,"value":"SUBNORMAL = \"SUBNORMAL\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":718,"value":"NORMALIZED = \"NORMALIZED\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2727,"value":"import {BBPbigint} from \"@rreusser/computing-with-the-bailey-borwein-plouffe-formula\"\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1595,"value":"Fraction = (await import(\"https://cdn.skypack.dev/pin/fraction.js@v4.2.0-4T2HgYMCUUjn2ZtoMwJX/mode=imports,min/unoptimized/bigfraction.js\")).default","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1706,"value":"{\n aprox_input.value = kind == SUBNORMAL || kind == NORMALIZED ? rational :\n kind == Infinity ? ( (sign == 1 ? \"-∞\" : \"∞\") ) : \"NaN\";\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W36","stdlib":"1"}} 31:"$30:body"