2b:null 2c:["$","title",null,{"children":"Ordering floats and approximate bit difference | Edward Misback | Observable"}] 31:Tc38,// for choosing the ticks on a plot over all floating point values // Adapted from code from Herbie: https://github.com/herbie-fp/herbie/blob/main/src/web/plot.rkt function choose_ticks_clientside (min, max) { function first_power10(min, max) { let value = max < 0 ? - (10 ** Math.ceil(Math.log(-max)/ Math.log(10))) : 10 ** (Math.floor(Math.log(max) / Math.log(10))) return value <= min ? false : value } function clamp(x, lo, hi) { return Math.min(hi, Math.max(x, lo)) } function choose_between(min, max, number) { // ; Returns a given number of ticks, roughly evenly spaced, between min and max // ; For any tick, n divisions below max, the tick is an ordinal corresponding to: // ; (a) a power of 10 between n and (n + ε) divisions below max where ε is some tolerance, or // ; (b) a value, n divisions below max let sub_range = Math.round((max - min) / (1 + number)) let near = (x, n) => (x <= n) && (Math.abs((x - n) / sub_range) <= .2) // <= tolerance return [...Array(number)].map((_, i) => i + 1).map(itr => { let power10 = first_power10( ordinal_to_real(clamp(max - ((itr + 1) * sub_range), min, max)), ordinal_to_real(clamp(max - (itr * sub_range), min, max)) ) return power10 && near(real_to_ordinal(power10), max - (itr * sub_range)) ? real_to_ordinal(power10) : max - (itr - sub_range) }) } function pick_spaced_ordinals(necessary, min, max, number) { // NOTE use of mathjs bignumber arithmetic is required in this function! let sub_range = math.divide(math.bignumber(math.subtract(math.bignumber(max), math.bignumber(min))), math.bignumber(number)) // size of a division on the ordinal range let necessary_star = (function loop(necessary) { return necessary.length < 2 ? necessary : math.smaller(math.subtract(necessary[1], necessary[0]), sub_range) ? loop(necessary.slice(1)) : [necessary[0], ...loop(necessary.slice(1))] })(necessary) // filter out necessary points that are too close let all = (function loop(necessary, min_star, start) { if (start >= number) { return [] } if (necessary.length == 0) { return choose_between(min_star, max, number - start) } let idx = false for (let i=0; i math.subtract(math.bignumber(a), math.bignumber(b))) } function choose_ticks (min, max) { let tick_count = 13 let necessary = [min, -1.0, 0, 1.0, max].filter(v => (min <= v) && (v <= max) && (min <= max)).map(v => real_to_ordinal(v)) let major_ticks = pick_spaced_ordinals(necessary, real_to_ordinal(min), real_to_ordinal(max), tick_count).map(v => ordinal_to_real(v)) return major_ticks } return choose_ticks(min, max) }32:[] 2f:{"id":"85d80ca248f27a84","slug":"ordering-floats","title":"Ordering floats and approximate bit difference","thumbnail":"9407303eb3d32eac00f4537cdae95d95cac4e503c77e625aa53712ea95cf370f","access_level":"public","create_time":"$D2022-01-19T05:14:03.028Z","update_time":"$D2022-09-15T17:40:39.441Z","view_time":null,"first_public_time":"$D2022-01-19T05:49:39.673Z","fork":false,"stars":1,"starred":false,"owner":{"type":"team","id":"5508584884fd137b","login":"elmisback","name":"Edward Misback","avatar_url":"https://avatars-next.observableusercontent.com/9a42c09ac3fb39913eeacae071d5ba669ac49bf75c82372d0d717ec0acb59024","tier":"starter_2024"},"version":585,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"e03d30b651790cf0","login":"elmisback","name":"Edward Misback","avatar_url":"https://avatars-next.observableusercontent.com/9a42c09ac3fb39913eeacae071d5ba669ac49bf75c82372d0d717ec0acb59024","tier":null}],"members":[{"type":"individual","id":"e03d30b651790cf0","login":"elmisback","name":"Edward Misback","avatar_url":"https://avatars-next.observableusercontent.com/9a42c09ac3fb39913eeacae071d5ba669ac49bf75c82372d0d717ec0acb59024","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"subscribed":null,"forks":3,"collection":null,"type":"notebook","body":{"title":"Ordering floats and approximate bit difference","cells":[{"id":0,"value":"# Ordering floats and approximate bit difference","pinned":true,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":280,"value":"md`NOTE: the performance of the code below can be improved. ordinal(x) calls to_signed_int(x) three times when it should be able to call it just once (cache first result, and cache mbn(-0.0)). See the notes below for updating to non-deprecated calls.`","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":130,"value":"md`The following table shows an ordinal mapping for Float64 \\`-MAX_VALUE\\`, \\`-MAX_VALUE/2\\`, \\`-1.0\\`, \\`-1/2\\`, \\`-MIN_VALUE\\`, \\`-0.0\\`, \\`0.0\\`, etc.:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":89,"value":"md`${[-Number.MAX_VALUE, \n -Number.MAX_VALUE/2, \n -1, \n -.5, \n -Number.MIN_VALUE, \"-0.0\", 0, Number.MIN_VALUE, .5, 1, Number.MAX_VALUE/2, Number.MAX_VALUE].map(v => `').join('')}

Ordinal	Float64
` +ordinal(v).toString() + '	' + v + '

`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":149,"value":"viewof example_float = Inputs.number({value: -1.57e-13, label: \"example_float\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":146,"value":"example_ordinal = ordinal(example_float).toString()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":374,"value":"9218868437227405311 // NOTE there are too many ordinals to represent as normal JS Numbers!","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":302,"value":"ordinal(Number.MAX_VALUE).toString()","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":334,"value":"math.subtract(mbn(-0.0), math.subtract(mbn(-0.0), mbn(42))).toString()","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":337,"value":"mbn(42).toString()","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":339,"value":"math.subtract(mbn(-0.0), mbn(42)).toString()","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":321,"value":"Number.MIN_VALUE *2","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":136,"value":"md`These are some examples of numbers I'm getting for the bit difference when I take the log2 of differences over this order.`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":20,"value":"bit_difference(Number.MAX_VALUE, -Number.MAX_VALUE).toString()","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":48,"value":"bit_difference(0, 1).toString()","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":120,"value":"bit_difference(0, 1e-320).toString()","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":25,"value":"// ULP difference, note -0.0 is treated as equal to 0.0\nulp_difference(Number.MIN_VALUE, -Number.MIN_VALUE).toString()","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":171,"value":"md`### Code`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":283,"value":"mbn(\"-9218868437227405311\")","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":5,"value":"mbn = x => math.bignumber(to_signed_int(x).toString())","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":387,"value":"mbn_neg_0 = mbn(-0.0)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":3,"value":"// TODO name consistently with below\nto_signed_int = float64 => {\n const buffer = new ArrayBuffer(8)\n const view = new DataView(buffer)\n view.setFloat64(0, float64)\n return view.getBigInt64(0)\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":383,"value":"// TODO \"real\" should be \"float64\"\nfunction real_from_signed_int (signed) {\n const buffer = new ArrayBuffer(8)\n const view = new DataView(buffer)\n view.setBigInt64(0, signed)\n return view.getFloat64(0)\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":389,"value":"// TODO \"real\" should be \"float64\"\nfunction real_to_ordinal(real) {\n let signed = to_signed_int(real)\n let mbn = math.bignumber(signed.toString())\n return signed >= 0 ? mbn : math.subtract(mbn_neg_0, mbn)\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":391,"value":"function ordinal_to_real(ordinal) {\n return ordinal >=0 ? real_from_signed_int(BigInt(ordinal)) : real_from_signed_int(BigInt(math.subtract(mbn_neg_0, math.bignumber(ordinal))))\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":9,"value":"// TODO: (Deprecated, use real/float64_to_ordinal instead)\nordinal = x => to_signed_int(x) >= 0 ? mbn(x) : math.subtract(mbn(-0.0), mbn(x)) // because order of signed negatives is reversed","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":11,"value":"// TODO: ordinal is deprecated\nulp_difference = (x, y) => math.add(math.abs(math.subtract(ordinal(x), ordinal(y))), 1)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":15,"value":"bit_difference = (x, y) => math.log2(ulp_difference(x, y).toString()) // toString will truncate, but this is okay","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":275,"value":"{\n const start = performance.now()\n ;[...new Array(8000)].map(e => ulp_difference(e, 2))\n return performance.now() - start\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":184,"value":"{\n const start = performance.now()\n ;[...new Array(8000)].map(e => math.log2(ulp_difference(3, 4).toString()))\n return performance.now() - start\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":7,"value":"math = require('mathjs')","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":411,"value":"\t9223372036854775808 < 10748314587012925997 // 4652007308841189376","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":487,"value":"points = [[-1e+308,5442597155835700],[-1e+206,1534620040627248120],[-1e+103,3075435233412954392],[\"-1\",4616189618054758400],[\"-1e-102\",6142370034567588547],[\"-1e-205\",7683126141587503524],[\"0\",9223372036854775808],[\"1e-206\",10748314587012925997],[\"1e-103\",12289385714708364181],[\"1\",13830554455654793216],[\"1e+102\",15356310588812194163],[\"1e+205\",16897476253732898419],[\"1e+308\",18440533523121328391]]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":490,"value":"Promise.all(points.map(p => herbie_ordinal_to_real(Number(p[1]))))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":547,"value":"BigInt(mbn_neg_0)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":446,"value":"async function herbie_ordinal_to_real(ordinal) {\n const math = await (import('https://cdn.skypack.dev/mathjs'))\n // (approximate) -- using mathjs\n function real_from_signed_int (signed) {\n const buffer = new ArrayBuffer(8)\n const view = new DataView(buffer)\n view.setBigInt64(0, signed)\n return view.getFloat64(0)\n }\n function to_signed_int (float64) {\n const buffer = new ArrayBuffer(8)\n const view = new DataView(buffer)\n view.setFloat64(0, float64)\n return view.getBigInt64(0)\n }\n \n const straight_conversion = real_from_signed_int(BigInt(ordinal))\n const zero = math.bignumber(to_signed_int(-0.0).toString())\n return - straight_conversion < 0 ? - real_from_signed_int(math.subtract(zero, math.bignumber(ordinal))) : - straight_conversion\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":433,"value":"my_ordinal_to_real(9218868437227405311)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":429,"value":"my_ordinal_to_real(16897476253732898419)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":227,"value":"ordinal_to_real(9223372036854775808)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":431,"value":"ordinal_to_real(13830554455654793216)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":463,"value":"real_from_signed_int(\"6142370034567588547\")","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":249,"value":"id = '@elmisback/ordering-floats'","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":223,"value":"import {chart} with {id} from '@observablehq/notebook-visualizer'","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":405,"value":"$31","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"}],"files":"$W32","stdlib":"1","titling":"auto"}} 30:"$2f:body"