2d:null 2e:["$","title",null,{"children":"Symbolic Computation with mathjs, Tagged Templates, and ObservableHQ | Adam Thomas | Observable"}] 33:T8fb,I've been reading a lot of scientific papers and struggling to grok the formulas. I have gotten used to being able to open up the inspection panel on an ObservableHQ notebook and fiddle with parameters and even whole functions. I wanted to be able to do the same for the papers I was reading but PDFs are notoriously hard to edit and the executing ones are usually to be avoided. I figured the next best thing was to implement the formulas in javascript here on ObservableHQ, but I kept finding bugs introduced by my bad transcriptions from ${tex`\LaTeX{}`} styled formulas to imperative programs. I wanted to be able to verify that my implementation matched the source material through visual inspection. I've also noticed a number of notebooks here on ObservableHQ which introduce and describe formulas styled using the *tex* template tag builtin, but the implementation is typically hidden off in an attic section at the bottom of the page. I hope that those kinds of notebooks start to pick up the executable formula concept because, rather selfishly, it's easier for me to learn if I'm not trying to hold the description, the formula and the implementation in my head at the same time. In an effort to provide a way to have both the nicely rendered expression and the implementation kept in lock-step, this notebook implements the template tag: λ. The tag takes a mathjs compatible expression in backticks; and returns a *document* built from the ${tex`\LaTeX{}`} representation, with the value set to either a compiled function, or the root node of the mathjs syntax tree. This allows for a compact representation of function expressions composed of the expression intermixed with sub expressions (using the $\{\} form). When the sub expression is a variable or function assignment expression tagged with λ, the sub expressions' name (taken from the assignment node in the parsed expression) will be rendered in place, but the sub expressions' value (as determined by the ObservableHQ runtime) is used to generate the dependant cell's value. The end result is (afaict) well behaving reactive functions that allow visual inspection of the implementation and the results.34:Tca7,chart = f => { const debug = true; const svg = d3.create('svg').attr("width",width).attr("height",height); const x = d3.scaleLinear().domain([-4,4]).range([margin,width-margin]); const y = d3.scaleLinear().domain([-3,3]).range([height-margin,margin]); const clipID = DOM.uid('clip'); svg.append('defs').append('clipPath') .attr('id',clipID.id).append('rect') .attr('x',margin) .attr('y',margin) .attr('width',width-margin*2) .attr('height',height-margin*2); svg.append('g') .attr('class','axisLeft') .attr('transform',`translate(${margin},0)`) .call(d3.axisLeft(y)); svg.append('g') .attr('class','axisBottom') .attr('transform',`translate(0,${(height-margin)})`) .call(d3.axisBottom(x)); svg.append('g') .style('clip-path',clipID) .attr('class','grid') .selectAll('line') .data([[0,undefined],[undefined,0]]) .join('line') .style('stroke','lightgrey'); svg.append('g') .style('clip-path',clipID) .attr('class','lines') .style('fill','none') .selectAll('path') .data( mathjs.isComplex(f(0)) ? [λ`g(x) = re(${f}(x))`.value, λ`g(x) = im(${f}(x))`.value] : [f] ) .join('path') .style('stroke',(d,i) => d3.schemeCategory10[i]); svg.append('g') .style('clip-path',clipID) .attr('class','markers') .selectAll('g') .data(mathjs.isComplex(f(0))?[x => mathjs.re(f(x)),x => mathjs.im(f(x))]:[f]) .join('g') .style('fill',(d,i) => d3.schemeCategory10[i]); const ddf = f.ast ? derivative(f,2) : () => NaN; const zoom = d3.zoom().on('zoom', function(e) { const node = d3.select(this); const xprime = e.transform.rescaleX(x); const yprime = e.transform.rescaleY(y); node.select('.axisLeft').call(d3.axisLeft(yprime)); node.select('.axisBottom').call(d3.axisBottom(xprime)); const sampler = f => simplifyjs( xprime .ticks(500) .reduce( (acc, chg, i, s) => (i === s.length -1 || Math.pow(acc.at(-1).x-chg,2)+Math.pow(acc.at(-1).y-f(chg),2) > Math.pow(0.001,2)) ? [...acc,{x:chg,y:f(chg)}] : [...acc.slice(0,-1), {x:chg,y:f(chg)}], [{x:xprime.domain()[0],y:f(xprime.domain()[0])}] ), .001, true ); const linegen = f => d3.line() .x(d=>xprime(d.x)) .y(d=>yprime(d.y)) .defined(d=>!isNaN(d.y)) .curve(d3.curveLinear) (sampler(f)) node.select('.lines').selectAll('path') .attr('d',linegen) if(debug) { node.select('.markers') .selectAll('g') .selectAll('path') .data(sampler) .join('path') .attr('transform',d => `translate(${xprime(d.x)},${yprime(d.y)})`) .attr('d',d3.symbol(d3.symbolCircle,4)); } node.select('.grid').selectAll('line') .attr('x1',(d,i)=>d[0]+0==d[0]?xprime.range()[0]:xprime(d[1])) .attr('x2',(d,i)=>d[0]+0==d[0]?xprime.range()[1]:xprime(d[1])) .attr('y1',(d,i)=>d[1]+0==d[1]?yprime.range()[0]:yprime(d[0])) .attr('y2',(d,i)=>d[1]+0==d[1]?yprime.range()[1]:yprime(d[0])) }); return svg .call(zoom) .call(zoom.transform, d3.zoomIdentity) .node() }35:[] 31:{"id":"bc4aa80d5a647313","slug":"mathjs-template-tag","title":"Symbolic Computation with mathjs, Tagged Templates, and ObservableHQ","thumbnail":"ecb6bbc663a13956563cf28eb18c7bcf123e6387380b0bffa99491bc99ecda91","access_level":"public","create_time":"$D2020-06-26T04:52:48.956Z","update_time":"$D2023-12-04T07:52:12.920Z","view_time":null,"first_public_time":"$D2021-05-26T23:48:13.959Z","has_unpublished_edits":null,"fork":false,"stars":0,"starred":false,"owner":{"type":"team","id":"797bce4a19d1218c","login":"devdsp","name":"Adam Thomas","avatar_url":"https://avatars-next.observableusercontent.com/086762d85c863b3bffce6f78bf73bc973d9cb8f1a1ae706be18bb72dcc97da1f","tier":"starter_2024","bio":""},"version":8266,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"ef340ccc893d0793","login":"devdsp","name":"Adam Thomas","avatar_url":"https://avatars-next.observableusercontent.com/086762d85c863b3bffce6f78bf73bc973d9cb8f1a1ae706be18bb72dcc97da1f","tier":null,"bio":""}],"members":[{"type":"individual","id":"ef340ccc893d0793","login":"devdsp","name":"Adam Thomas","avatar_url":"https://avatars-next.observableusercontent.com/086762d85c863b3bffce6f78bf73bc973d9cb8f1a1ae706be18bb72dcc97da1f","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"titling":"auto","title":"Symbolic Computation with mathjs, Tagged Templates, and ObservableHQ","cells":[{"id":0,"value":"# Symbolic Computation with mathjs, Tagged Templates, and ObservableHQ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":724,"value":"$33","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2581,"value":"## Gaussian\nBelow is an example of the two parameter form (expected value: μ, and standard deviation: σ) of the Gaussian function. Note that a cell's name is how it's referenced in another template expression (mean), but the symbol from within that expression (μ) is rendered in the view of both the original definition and the dependant expression.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":2580,"value":"viewof mean=λ`μ=0`","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2579,"value":"viewof stddev=λ`σ=.5`","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1146,"value":"viewof gaussian = λ`ϕ(x) = 1/${stddev}sqrt(2pi) * e^(- (x-${mean})^2 / 2${stddev}^2)`","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2577,"value":"The value of the *viewof gaussian* cell is a regular javascript function.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":2576,"value":"gaussian","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2575,"value":"Therefore, the gaussian function can be called like any other javascript function.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":2574,"value":"gaussian(.2)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2573,"value":"Changing the mean or the standard deviation in the referenced cells will automatically regenerate the gaussian function and be observable in the results.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":2572,"value":"Plotting the function over a range of values works as expected; seen here using the Plot system builtin of the ObservableHQ standard library.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":1157,"value":"Plot.plot({\n grid:true,\n y:{label:'ϕ(x)',nice:true},\n x:{label:\"x\",nice:true},\n marks: [\n Plot.line(d3.range(-2,2,.01),{ x:d=>d, y:d=>gaussian(d) }) \n // Note for the point free purists: I hear you... but the returned functions are \n // specific about the number of paramaters passed to them (as they should be); \n // the (any,integer,array)->any signature of the mapping function means the\n // arrow function which discards the index and array paramaters is required.\n ]\n})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2570,"value":"## Multiquadric\n### but with the parameters folded in as constants","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":2569,"value":"Another thing to note is that constant expressions that don't result in a named assignment node get passed straight through to the rendering function. It still works, but it loses some of the expressiveness of the earlier examples.\n\nHere's an example with an expression based cell as well as a λ tagged template which evaluates to a constant.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":7669,"value":"4/2","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2568,"value":"ϵ = 2*2","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2567,"value":"viewof mu = λ`.1*${ϵ}/10`","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1187,"value":"viewof multiquadric = λ`ϕ(t) = sqrt(1+ ${ϵ}^2 norm(t-${mu})^2)`","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1190,"value":"Plot.plot({\n grid:true,\n y:{label:'ϕ(x)',nice:true},\n x:{label:\"x\"},\n marks: [\n Plot.ruleY([0]),\n Plot.line(d3.range(-1,1.01,.01),{ x:d=>d, y:d=>multiquadric(d) })\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2562,"value":"## Curried functions\n### and first class functions\nBoth work out of the box though it's rather verbose on the rendering side.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":2561,"value":"viewof curried = λ`a(p) = b(fn) = c(bias) = fn(p)+bias`","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2560,"value":"curried(.01)(gaussian)(1)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2559,"value":"## Complex functions\n### like the Morlet wavelet\nYup, we've got those.\n\n...and just look at this compact form with the editor panels closed! There's no other hidden function implementing these formulas that has to be translated and maintained. \n\nOpen up the σ cell, make a change, and watch the graph below update when the cell is run.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":1086,"value":"viewof σ = λ`σ=7`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":6,"value":"viewof c = λ`c(σ) = (1+ e^-σ^2 - 2e^-(3/4 σ^2))^-(1/2)`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":39,"value":"viewof k = λ`k(σ) = e^-(1/2 σ^2)`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":57,"value":"viewof Ψ = λ`Ψ(t) = ${c}(${σ}) pi^-(1/4) e^-(1/2 t^2) (e^(i ${σ} t) - ${k}(${σ}))`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":761,"value":"chart( Ψ );","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":7913,"value":"function controlPoints(x) {\n var i,\n n = x.length - 1,\n m,\n a = new Array(n),\n b = new Array(n),\n r = new Array(n);\n a[0] = 0, b[0] = 2, r[0] = x[0] + 2 * x[1];\n for (i = 1; i < n - 1; ++i) a[i] = 1, b[i] = 4, r[i] = 4 * x[i] + 2 * x[i + 1];\n a[n - 1] = 2, b[n - 1] = 7, r[n - 1] = 8 * x[n - 1] + x[n];\n for (i = 1; i < n; ++i) m = a[i] / b[i - 1], b[i] -= m, r[i] -= m * r[i - 1];\n a[n - 1] = r[n - 1] / b[n - 1];\n for (i = n - 2; i >= 0; --i) a[i] = (r[i] - a[i + 1]) / b[i];\n b[n - 1] = (x[n] + a[n - 1]) / 2;\n for (i = 0; i < n - 1; ++i) b[i] = 2 * x[i + 1] - a[i + 1];\n return [a, b];\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":7925,"value":"DeCasteljauStep = (p) => (t) => p.map(v => d3.pairs(v, (a,b) => d3.interpolate(a,b)(t)));","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":7956,"value":"DeCasteljau = p => t => d3.transpose(DeCasteljauStep(DeCasteljauStep(DeCasteljauStep(p)(t))(t))(t))[0]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":137,"value":"$34","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2545,"value":"## Conclusion\nThere's still work I'd like to do to support the ubiquitous Σ notation and I'm currently reading a lot of formulas with infima which feels further off. It's also not particularly robust. However, as a proof of concept, I'm pleased with it.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":2544,"value":"and here's the template function:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":2543,"value":"λ = (...args) => {\n const syntax = d3.reduce(args[0],(a,c,i) => `${a} ${args[i].name ?? args[i]} ${c}`);\n const scope = new Map(d3.merge(args.slice(1).map(d => [...(d.scope?.entries()??[])])));\n const ast = mathjs.parse(syntax);\n const evaled = mathjs.evaluate(syntax,scope);\n const name = ast.name ? typeof ast.name === 'function' ? ast.name() : ast.name : syntax;\n return Object.assign(\n tex`${ast.toTex()}`,\n {\n value: Object.assign(\n typeof evaled === 'function' \n ? evaled \n : name \n ? {name} \n : {name:syntax},\n {scope,ast}\n )\n }\n )\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":7362,"value":"interval = [-4,4]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":7701,"value":"viewof curvature = λ`d(f′,f′′) = k(x) = abs(f′′(x))/(1+f′(x)^2)^(3/2)`","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":7774,"value":"initialPoints = f => transform => transform.rescaleX(d3.scaleLinear().domain([-4,4])).ticks(100).map(x=>[x,f(x)])","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":7777,"value":"initialPoints(x => Ψ(x).re)(d3.zoomIdentity).filter(([x],i,s) => {\n const k = curvature(Ψprime,Ψprimeprime)(x).re;\n return Math.abs(k) < 0.01 || Math.abs(k) > .5 || i==0 || i == s.length-1\n})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":7783,"value":"viewof Ψk = λ`Ψk(${Ψ.ast.params[0]}) = abs(${Ψprimeprime}(${Ψ.ast.params[0]}))/(1+${Ψprime}(${Ψ.ast.params[0]})^2)^(3/2)`","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":7874,"value":"viewof dk = derivative(Ψk)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":3834,"value":"Plot.plot({\n x:{domain: interval },\n y:{domain:[-2,2]},\n grid: true,\n height: width,\n width: width,\n marks:[\n Plot.ruleY([0],{stroke:'grey'}),\n Plot.lineX(\n d3.scaleLinear().domain(interval).ticks(300),\n {y: x => Ψ(x).re, stroke:d3.schemeCategory10[0]}\n ),\n Plot.line(\n refined,\n {stroke: d3.schemeCategory10[2]}\n ),\n ],\n})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":8193,"value":"{\n const k = 1;\n const x = d3.scaleLinear().domain([-4,4]).ticks(1000);\n const y = d3.map(x,x => Ψ(x).re);\n const kcosine = d3.range(k,x.length-k)\n .map(i => [mathjs.subtract([x[i-k],y[i-k]],[x[i],y[i]]),mathjs.subtract([x[i],y[i]],[x[i+k],y[i+k]])])\n .map(([l1,l2]) => (mathjs.dot(l1,l2)/(mathjs.hypot(l1)*mathjs.hypot(l2))));\n\n const mean = d3.mean(kcosine);\n const stdv = d3.deviation(kcosine);\n return x.filter((_,i,s) => i === 0 || i === s.length-1 || (kcosine[i-k]-mean)/stdv < .27);\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":7914,"value":"refined = {\n const f = x=>Ψ(x).re\n const x = d3.scaleLinear().domain([-4,4]).ticks(5);\n const y = x.map(f);\n const xy = d3.transpose([x,y]);\n const subStep = p => d3.scaleLinear().ticks(5).map(t => DeCasteljau(p)(t));\n const resample = f => P => (d3.map(d3.transpose(P)[0], x => [x,f(x)]));\n\n const refine = (f,xy) => \n d3.map(\n d3.zip(\n d3.pairs(xy),\n d3.transpose(d3.map(d3.transpose(xy),v => d3.transpose(controlPoints(v))))\n ),\n ([o,tp]) => subStep(d3.transpose(d3.sort(d3.merge([o,d3.transpose(tp)]),([x]) => x)))\n ) \n .flatMap(P => Math.sqrt(d3.mean(d3.map(P,([x,y]) => (y-f(x))**2))) > .005 ? refine(f,resample(f)(P)) : P)\n .reduce((acc,chg) => acc.at(-1) ? acc.at(-1)[0] !== chg[0] ? [...acc,chg] : acc : [chg], []);\n \n return refine(f,xy);\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":3496,"value":"viewof Ψprimeprime = derivative(Ψ,2);","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":4009,"value":"viewof Ψprime = derivative(Ψ);","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":6480,"value":"expand = (f) => {\n let ast = f.ast;\n do {\n ast = ast.transform(\n node => node.fn && f.scope.has(node.fn.name) \n ? mathjs.resolve(\n f.scope.get(node.fn.name).ast.expr.clone(),\n Object.fromEntries(d3.merge([\n f.scope.get(node.fn.name).ast.params\n .filter(k=>!f.scope.has(k))\n .map((k,i)=>[k,node.args[i]]),\n f.scope.get(node.fn.name).scope.entries(),\n ]))\n )\n : node\n );\n } while(ast.filter(node => node.type === 'FunctionNode' && f.scope.has(node.fn.name)).length);\n \n ast = ast.transform(node=>node.type === 'FunctionAssignmentNode' ? node : mathjs.simplify(node));\n \n return Object.assign(\n tex`${ast.toTex()}`,\n {value:Object.assign(ast.evaluate(f.scope),{ast,scope:f.scope})}\n );\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":5360,"value":"derivative = (f,order=1) => {\n const ast = new mathjs.FunctionAssignmentNode(\n getLagrangesNotation(f.name,order),\n f.ast.params,\n d3.range(order).reduce(\n (a) => mathjs.derivative(a,f.ast.params[0]),\n f.ast\n .transform(\n (node,path,parent) => parent && node.isFunctionNode \n ? f.scope.get(node.name)?.ast?.expr ?? node \n : node\n )\n )\n );\n const df = Object.assign(ast.evaluate(f.scope),{scope:f.scope,ast});\n return Object.assign(\n tex`${ast.toTex()}`,\n {value: df}\n );\n};","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2542,"value":"## Attic\ndon't worry, there aren't any re-implementations down here, I promise.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":2540,"value":"margin = 50","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2539,"value":"height = width*3/4","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1843,"value":"x = d3.scaleLinear().domain([-1,1]).range([50,width-50]);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":1841,"value":"y = d3.scaleLinear().domain([-1,1]).range([width-50,50]);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":7475,"value":"getLagrangesNotation = {\n const superscriptlut = ({\n \"⁰\":0,\"¹\":1,\"²\":2,\"³\":3,\"⁴\":4,\"⁵\":5,\"⁶\":6,\"⁷\":7,\"⁸\":8,\"⁹\":9,\n 0:\"⁰\",1:\"¹\",2:\"²\",3:\"³\",4:\"⁴\",5:\"⁵\",6:\"⁶\",7:\"⁷\",8:\"⁸\",9:\"⁹\",\n '′':1, '″':2, '‴':3, '⁗':4,\n });\n \n const makedervivsuper = (order) => order > 0 && order < 5 \n ? [,'′','″','‴','⁗'][order] \n : `⁽${[...order.toString()].map(c=>superscriptlut[c]).join(\"\")}⁾`;\n \n const derivreg = /(?.*?)(⁽(?[²³¹⁰⁴-⁹]+)⁾|(?[′-‴⁗])|)$/u\n return (name,order=1) => {\n const groups = derivreg.exec(name)?.groups;\n const current = (groups?.order !== undefined \n ? Number([...groups.order].map(c => superscriptlut[c]).join(\"\"))\n : groups?.primes !== undefined \n ? superscriptlut[groups.primes]\n : 0\n );\n return groups?.name + makedervivsuper(current+order)\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":130,"value":"d3 = require(\"d3\",\"d3-hexbin\");","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":227,"value":"mathjs = require('mathjs').then(obj => {\n const isAlphaOriginal = obj.parse.isAlpha\n obj.parse.isAlpha = function (c, cPrev, cNext) {\n return isAlphaOriginal(c, cPrev, cNext) || [\"⁰\",\"¹\",\"²\",\"³\",\"⁴\",\"⁵\",\"⁶\",\"⁷\",\"⁸\",\"⁹\",'′','″','‴','⁗'].includes(c)\n }\n return obj;\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":3161,"value":"simplifyjs = require(\"https://cdn.jsdelivr.net/npm/simplify-js@1.2.3/simplify.min.js\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":2533,"value":"P.S. After all the faffing about I did trying to hook in to the current notebook's runtime or initialising a second runtime just for the mathjs expressions engine, all so I could get incremental/reactive computation - the implementation using the tagged templates feature of javascript feels really underwhelming! Though there is a certain *elegance* to it...","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"}],"files":"$W35","stdlib":"1"}} 32:"$31:body"