Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, D

Average Error: 1.2 → 0.3

Time: 1.1min

Precision: binary64

Cost: 26368

Math

\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \]

↓

\[\sqrt[3]{{\cos^{-1} \left(\frac{\sqrt{t} \cdot \frac{x}{y}}{\frac{z}{0.05555555555555555}}\right)}^{3}} \cdot 0.3333333333333333 \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (/ 1.0 3.0) (acos (* (/ (* 3.0 (/ x (* y 27.0))) (* z 2.0)) (sqrt t)))))

↓

(FPCore (x y z t)
 :precision binary64
 (*
  (cbrt (pow (acos (/ (* (sqrt t) (/ x y)) (/ z 0.05555555555555555))) 3.0))
  0.3333333333333333))

C

double code(double x, double y, double z, double t) {
	return (1.0 / 3.0) * acos((((3.0 * (x / (y * 27.0))) / (z * 2.0)) * sqrt(t)));
}

↓

double code(double x, double y, double z, double t) {
	return cbrt(pow(acos(((sqrt(t) * (x / y)) / (z / 0.05555555555555555))), 3.0)) * 0.3333333333333333;
}

Java

public static double code(double x, double y, double z, double t) {
	return (1.0 / 3.0) * Math.acos((((3.0 * (x / (y * 27.0))) / (z * 2.0)) * Math.sqrt(t)));
}

↓

public static double code(double x, double y, double z, double t) {
	return Math.cbrt(Math.pow(Math.acos(((Math.sqrt(t) * (x / y)) / (z / 0.05555555555555555))), 3.0)) * 0.3333333333333333;
}

Julia

function code(x, y, z, t)
	return Float64(Float64(1.0 / 3.0) * acos(Float64(Float64(Float64(3.0 * Float64(x / Float64(y * 27.0))) / Float64(z * 2.0)) * sqrt(t))))
end

↓

function code(x, y, z, t)
	return Float64(cbrt((acos(Float64(Float64(sqrt(t) * Float64(x / y)) / Float64(z / 0.05555555555555555))) ^ 3.0)) * 0.3333333333333333)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(1.0 / 3.0), $MachinePrecision] * N[ArcCos[N[(N[(N[(3.0 * N[(x / N[(y * 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * 2.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := N[(N[Power[N[Power[N[ArcCos[N[(N[(N[Sqrt[t], $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] / N[(z / 0.05555555555555555), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Target

Original	1.2
Target	1.2
Herbie	0.3

\[\frac{\cos^{-1} \left(\frac{\frac{x}{27}}{y \cdot z} \cdot \frac{\sqrt{t}}{\frac{2}{3}}\right)}{3} \]

Derivation

1. Initial program 1.2

   \[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \]

2. Simplified 1.2

   \[\leadsto \color{blue}{\cos^{-1} \left(\frac{3}{z} \cdot \left(\frac{\frac{x}{y}}{54} \cdot \sqrt{t}\right)\right) \cdot 0.3333333333333333} \]
   Proof

3. Applied egg-rr 1.8

   \[\leadsto \color{blue}{\sqrt[3]{\cos^{-1} \left(\frac{3 \cdot x}{z \cdot \left(54 \cdot y\right)} \cdot \sqrt{t}\right)} \cdot \left(\sqrt[3]{{\cos^{-1} \left(\frac{3 \cdot x}{z \cdot \left(54 \cdot y\right)} \cdot \sqrt{t}\right)}^{2}} \cdot 0.3333333333333333\right)} \]

4. Simplified 0.3

   \[\leadsto \color{blue}{\sqrt[3]{{\cos^{-1} \left(\frac{\sqrt{t} \cdot \frac{x}{y}}{\frac{z}{0.05555555555555555}}\right)}^{3}} \cdot 0.3333333333333333} \]
   Proof

Alternatives

Alternative 1
Error	1.9
Cost	13504

\[\cos^{-1} \left(\frac{0.05555555555555555 \cdot \left(x \cdot \sqrt{t}\right)}{y \cdot z}\right) \cdot 0.3333333333333333 \]

Alternative 2
Error	1.2
Cost	13504

\[\cos^{-1} \left(\frac{\sqrt{t} \cdot \frac{x}{y}}{\frac{z}{0.05555555555555555}}\right) \cdot 0.3333333333333333 \]

Alternative 3
Error	1.2
Cost	13504

\[\frac{\cos^{-1} \left(\frac{x}{\frac{z \cdot y}{0.05555555555555555}} \cdot \sqrt{t}\right)}{3} \]

Reproduce

herbie shell --seed 2023033 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, D"
  :precision binary64

  :herbie-target
  (/ (acos (* (/ (/ x 27.0) (* y z)) (/ (sqrt t) (/ 2.0 3.0)))) 3.0)

  (* (/ 1.0 3.0) (acos (* (/ (* 3.0 (/ x (* y 27.0))) (* z 2.0)) (sqrt t)))))