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

Average Error: 1.4 → 0.4

Time: 6.9s

Precision: binary64

Cost: 26432

\[ \begin{array}{c}[y, z] = \mathsf{sort}([y, z])\\ \end{array} \]

Math

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

↓

\[e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(\left(0.05555555555555555 \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)\right)} + -1 \]

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
 (+
  (exp
   (log1p
    (*
     0.3333333333333333
     (acos (* (* 0.05555555555555555 (/ (/ x y) z)) (sqrt t))))))
  -1.0))

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 exp(log1p((0.3333333333333333 * acos(((0.05555555555555555 * ((x / y) / z)) * sqrt(t)))))) + -1.0;
}

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.exp(Math.log1p((0.3333333333333333 * Math.acos(((0.05555555555555555 * ((x / y) / z)) * Math.sqrt(t)))))) + -1.0;
}

Python

def code(x, y, z, t):
	return (1.0 / 3.0) * math.acos((((3.0 * (x / (y * 27.0))) / (z * 2.0)) * math.sqrt(t)))

↓

def code(x, y, z, t):
	return math.exp(math.log1p((0.3333333333333333 * math.acos(((0.05555555555555555 * ((x / y) / z)) * math.sqrt(t)))))) + -1.0

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(exp(log1p(Float64(0.3333333333333333 * acos(Float64(Float64(0.05555555555555555 * Float64(Float64(x / y) / z)) * sqrt(t)))))) + -1.0)
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[Exp[N[Log[1 + N[(0.3333333333333333 * N[ArcCos[N[(N[(0.05555555555555555 * N[(N[(x / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]

Target

Original	1.4
Target	1.2
Herbie	0.4

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

Derivation

1. Initial program 1.4

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

2. Simplified 1.4

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

   [Start] 1.4	\[ \frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \]
   metadata-eval [=>] 1.4	\[ \color{blue}{0.3333333333333333} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \]
   associate-*r/ [=>] 1.4	\[ 0.3333333333333333 \cdot \cos^{-1} \left(\frac{\color{blue}{\frac{3 \cdot x}{y \cdot 27}}}{z \cdot 2} \cdot \sqrt{t}\right) \]
   *-commutative [=>] 1.4	\[ 0.3333333333333333 \cdot \cos^{-1} \left(\frac{\frac{3 \cdot x}{\color{blue}{27 \cdot y}}}{z \cdot 2} \cdot \sqrt{t}\right) \]
   times-frac [=>] 1.4	\[ 0.3333333333333333 \cdot \cos^{-1} \left(\frac{\color{blue}{\frac{3}{27} \cdot \frac{x}{y}}}{z \cdot 2} \cdot \sqrt{t}\right) \]
   *-commutative [=>] 1.4	\[ 0.3333333333333333 \cdot \cos^{-1} \left(\frac{\frac{3}{27} \cdot \frac{x}{y}}{\color{blue}{2 \cdot z}} \cdot \sqrt{t}\right) \]
   times-frac [=>] 1.4	\[ 0.3333333333333333 \cdot \cos^{-1} \left(\color{blue}{\left(\frac{\frac{3}{27}}{2} \cdot \frac{\frac{x}{y}}{z}\right)} \cdot \sqrt{t}\right) \]
   metadata-eval [=>] 1.4	\[ 0.3333333333333333 \cdot \cos^{-1} \left(\left(\frac{\color{blue}{0.1111111111111111}}{2} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right) \]
   metadata-eval [=>] 1.4	\[ 0.3333333333333333 \cdot \cos^{-1} \left(\left(\color{blue}{0.05555555555555555} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right) \]

3. Applied egg-rr 0.4

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(\left(0.05555555555555555 \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)\right)} - 1} \]

4. Final simplification 0.4

   \[\leadsto e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(\left(0.05555555555555555 \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)\right)} + -1 \]

Alternatives

Alternative 1
Error	1.4
Cost	13504

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

Reproduce

herbie shell --seed 2023031 
(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)))))