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

Percentage Accurate: 97.9% → 98.4%

Time: 36.4s

Alternatives: 4

Speedup: 1.2×

Specification

Math

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

FPCore

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

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)));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (1.0d0 / 3.0d0) * acos((((3.0d0 * (x / (y * 27.0d0))) / (z * 2.0d0)) * sqrt(t)))
end function

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)));
}

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)))

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = (1.0 / 3.0) * acos((((3.0 * (x / (y * 27.0))) / (z * 2.0)) * sqrt(t)));
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]

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

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)));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (1.0d0 / 3.0d0) * acos((((3.0d0 * (x / (y * 27.0d0))) / (z * 2.0d0)) * sqrt(t)))
end function

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)));
}

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)))

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = (1.0 / 3.0) * acos((((3.0 * (x / (y * 27.0))) / (z * 2.0)) * sqrt(t)));
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]

Alternative 1: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin^{-1} \left(\frac{x \cdot \sqrt{t}}{18 \cdot \left(z \cdot y\right)}\right)\\ 0.3333333333333333 \cdot \frac{\left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.0625 - {t\_1}^{4}}{\mathsf{fma}\left(\pi, 0.5, t\_1\right) \cdot \mathsf{fma}\left(\pi, \pi \cdot 0.25, {t\_1}^{2}\right)} \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (asin (/ (* x (sqrt t)) (* 18.0 (* z y))))))
   (*
    0.3333333333333333
    (/
     (- (* (* (* PI PI) (* PI PI)) 0.0625) (pow t_1 4.0))
     (* (fma PI 0.5 t_1) (fma PI (* PI 0.25) (pow t_1 2.0)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = asin(((x * sqrt(t)) / (18.0 * (z * y))));
	return 0.3333333333333333 * (((((((double) M_PI) * ((double) M_PI)) * (((double) M_PI) * ((double) M_PI))) * 0.0625) - pow(t_1, 4.0)) / (fma(((double) M_PI), 0.5, t_1) * fma(((double) M_PI), (((double) M_PI) * 0.25), pow(t_1, 2.0))));
}

Julia

function code(x, y, z, t)
	t_1 = asin(Float64(Float64(x * sqrt(t)) / Float64(18.0 * Float64(z * y))))
	return Float64(0.3333333333333333 * Float64(Float64(Float64(Float64(Float64(pi * pi) * Float64(pi * pi)) * 0.0625) - (t_1 ^ 4.0)) / Float64(fma(pi, 0.5, t_1) * fma(pi, Float64(pi * 0.25), (t_1 ^ 2.0)))))
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[ArcSin[N[(N[(x * N[Sqrt[t], $MachinePrecision]), $MachinePrecision] / N[(18.0 * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(0.3333333333333333 * N[(N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * 0.0625), $MachinePrecision] - N[Power[t$95$1, 4.0], $MachinePrecision]), $MachinePrecision] / N[(N[(Pi * 0.5 + t$95$1), $MachinePrecision] * N[(Pi * N[(Pi * 0.25), $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 98.5%

   \[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. metadata-eval 98.5

      \[\leadsto \color{blue}{0.3333333333333333} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \]

4. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{0.3333333333333333} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \]
5. Step-by-step derivation

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. *-commutative N/A

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

   5. times-frac N/A

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

   6. metadata-eval N/A

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \left(\left(\frac{\color{blue}{\frac{1}{9}}}{2} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right) \]

   7. metadata-eval N/A

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \left(\left(\color{blue}{\frac{1}{18}} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right) \]

   8. associate-/r* N/A

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \left(\left(\frac{1}{18} \cdot \color{blue}{\frac{x}{y \cdot z}}\right) \cdot \sqrt{t}\right) \]

   9. associate-*r/ N/A

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \left(\color{blue}{\frac{\frac{1}{18} \cdot x}{y \cdot z}} \cdot \sqrt{t}\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \frac{1}{3} \cdot \cos^{-1} \left(\color{blue}{\frac{\frac{1}{18} \cdot x}{y \cdot z}} \cdot \sqrt{t}\right) \]

   11. *-commutative N/A

       \[\leadsto \frac{1}{3} \cdot \cos^{-1} \left(\frac{\color{blue}{x \cdot \frac{1}{18}}}{y \cdot z} \cdot \sqrt{t}\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \frac{1}{3} \cdot \cos^{-1} \left(\frac{\color{blue}{x \cdot \frac{1}{18}}}{y \cdot z} \cdot \sqrt{t}\right) \]

   13. *-lowering-*.f64 98.0

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

6. Applied egg-rr 98.0%

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

8. Applied egg-rr 98.4%

   \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{\left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.0625 - {\sin^{-1} \left(\frac{x \cdot \sqrt{t}}{\left(z \cdot y\right) \cdot 18}\right)}^{4}}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\frac{x \cdot \sqrt{t}}{\left(z \cdot y\right) \cdot 18}\right)\right) \cdot \mathsf{fma}\left(\pi, \pi \cdot 0.25, {\sin^{-1} \left(\frac{x \cdot \sqrt{t}}{\left(z \cdot y\right) \cdot 18}\right)}^{2}\right)}} \]

9. Final simplification 98.4%

   \[\leadsto 0.3333333333333333 \cdot \frac{\left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.0625 - {\sin^{-1} \left(\frac{x \cdot \sqrt{t}}{18 \cdot \left(z \cdot y\right)}\right)}^{4}}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\frac{x \cdot \sqrt{t}}{18 \cdot \left(z \cdot y\right)}\right)\right) \cdot \mathsf{fma}\left(\pi, \pi \cdot 0.25, {\sin^{-1} \left(\frac{x \cdot \sqrt{t}}{18 \cdot \left(z \cdot y\right)}\right)}^{2}\right)} \]
10. Add Preprocessing

Alternative 2: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin^{-1} \left(\frac{x \cdot \sqrt{t}}{18 \cdot \left(z \cdot y\right)}\right)\\ \left(\pi \cdot \left(\pi \cdot 0.25\right) - {t\_1}^{2}\right) \cdot \left(0.3333333333333333 \cdot \frac{1}{\mathsf{fma}\left(\pi, 0.5, t\_1\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (asin (/ (* x (sqrt t)) (* 18.0 (* z y))))))
   (*
    (- (* PI (* PI 0.25)) (pow t_1 2.0))
    (* 0.3333333333333333 (/ 1.0 (fma PI 0.5 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = asin(((x * sqrt(t)) / (18.0 * (z * y))));
	return ((((double) M_PI) * (((double) M_PI) * 0.25)) - pow(t_1, 2.0)) * (0.3333333333333333 * (1.0 / fma(((double) M_PI), 0.5, t_1)));
}

Julia

function code(x, y, z, t)
	t_1 = asin(Float64(Float64(x * sqrt(t)) / Float64(18.0 * Float64(z * y))))
	return Float64(Float64(Float64(pi * Float64(pi * 0.25)) - (t_1 ^ 2.0)) * Float64(0.3333333333333333 * Float64(1.0 / fma(pi, 0.5, t_1))))
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[ArcSin[N[(N[(x * N[Sqrt[t], $MachinePrecision]), $MachinePrecision] / N[(18.0 * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(Pi * N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision] - N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 * N[(1.0 / N[(Pi * 0.5 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 98.5%

   \[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. metadata-eval 98.5

      \[\leadsto \color{blue}{0.3333333333333333} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \]

4. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{0.3333333333333333} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \]

6. Applied egg-rr 96.6%

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

8. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\left(\pi \cdot \left(\pi \cdot 0.25\right) - {\sin^{-1} \left(\frac{x \cdot \sqrt{t}}{\left(y \cdot z\right) \cdot 18}\right)}^{2}\right) \cdot \left(0.3333333333333333 \cdot \frac{1}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\frac{x \cdot \sqrt{t}}{\left(y \cdot z\right) \cdot 18}\right)\right)}\right)} \]

9. Final simplification 98.4%

   \[\leadsto \left(\pi \cdot \left(\pi \cdot 0.25\right) - {\sin^{-1} \left(\frac{x \cdot \sqrt{t}}{18 \cdot \left(z \cdot y\right)}\right)}^{2}\right) \cdot \left(0.3333333333333333 \cdot \frac{1}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\frac{x \cdot \sqrt{t}}{18 \cdot \left(z \cdot y\right)}\right)\right)}\right) \]
10. Add Preprocessing

Alternative 3: 97.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \cos^{-1} \left(\sqrt{t} \cdot \frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (*
  0.3333333333333333
  (acos (* (sqrt t) (/ (* 3.0 (/ x (* y 27.0))) (* z 2.0))))))

C

double code(double x, double y, double z, double t) {
	return 0.3333333333333333 * acos((sqrt(t) * ((3.0 * (x / (y * 27.0))) / (z * 2.0))));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 0.3333333333333333d0 * acos((sqrt(t) * ((3.0d0 * (x / (y * 27.0d0))) / (z * 2.0d0))))
end function

Java

public static double code(double x, double y, double z, double t) {
	return 0.3333333333333333 * Math.acos((Math.sqrt(t) * ((3.0 * (x / (y * 27.0))) / (z * 2.0))));
}

Python

def code(x, y, z, t):
	return 0.3333333333333333 * math.acos((math.sqrt(t) * ((3.0 * (x / (y * 27.0))) / (z * 2.0))))

Julia

function code(x, y, z, t)
	return Float64(0.3333333333333333 * acos(Float64(sqrt(t) * Float64(Float64(3.0 * Float64(x / Float64(y * 27.0))) / Float64(z * 2.0)))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 0.3333333333333333 * acos((sqrt(t) * ((3.0 * (x / (y * 27.0))) / (z * 2.0))));
end

Wolfram

code[x_, y_, z_, t_] := N[(0.3333333333333333 * N[ArcCos[N[(N[Sqrt[t], $MachinePrecision] * N[(N[(3.0 * N[(x / N[(y * 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.5%

   \[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. metadata-eval 98.5

      \[\leadsto \color{blue}{0.3333333333333333} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \]

4. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{0.3333333333333333} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \]

5. Final simplification 98.5%

   \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(\sqrt{t} \cdot \frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2}\right) \]
6. Add Preprocessing

Alternative 4: 98.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 0.3333333333333333d0 * acos((sqrt(t) * ((x * 0.05555555555555555d0) / (z * y))))
end function

Java

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

Python

def code(x, y, z, t):
	return 0.3333333333333333 * math.acos((math.sqrt(t) * ((x * 0.05555555555555555) / (z * y))))

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = 0.3333333333333333 * acos((sqrt(t) * ((x * 0.05555555555555555) / (z * y))));
end

Wolfram

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

Derivation

1. Initial program 98.5%

   \[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. metadata-eval 98.5

      \[\leadsto \color{blue}{0.3333333333333333} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \]

4. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{0.3333333333333333} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \]
5. Step-by-step derivation

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. *-commutative N/A

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

   5. times-frac N/A

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

   6. metadata-eval N/A

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \left(\left(\frac{\color{blue}{\frac{1}{9}}}{2} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right) \]

   7. metadata-eval N/A

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \left(\left(\color{blue}{\frac{1}{18}} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right) \]

   8. associate-/r* N/A

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \left(\left(\frac{1}{18} \cdot \color{blue}{\frac{x}{y \cdot z}}\right) \cdot \sqrt{t}\right) \]

   9. associate-*r/ N/A

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \left(\color{blue}{\frac{\frac{1}{18} \cdot x}{y \cdot z}} \cdot \sqrt{t}\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \frac{1}{3} \cdot \cos^{-1} \left(\color{blue}{\frac{\frac{1}{18} \cdot x}{y \cdot z}} \cdot \sqrt{t}\right) \]

   11. *-commutative N/A

       \[\leadsto \frac{1}{3} \cdot \cos^{-1} \left(\frac{\color{blue}{x \cdot \frac{1}{18}}}{y \cdot z} \cdot \sqrt{t}\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \frac{1}{3} \cdot \cos^{-1} \left(\frac{\color{blue}{x \cdot \frac{1}{18}}}{y \cdot z} \cdot \sqrt{t}\right) \]

   13. *-lowering-*.f64 98.0

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

6. Applied egg-rr 98.0%

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

7. Final simplification 98.0%

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

Developer Target 1: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = acos((((x / 27.0d0) / (y * z)) * (sqrt(t) / (2.0d0 / 3.0d0)))) / 3.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = acos((((x / 27.0) / (y * z)) * (sqrt(t) / (2.0 / 3.0)))) / 3.0;
end

Wolfram

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

Reproduce

herbie shell --seed 2024198 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, D"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (acos (* (/ (/ x 27) (* y z)) (/ (sqrt t) (/ 2 3)))) 3))

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