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

Percentage Accurate: 98.0% → 99.6%

Time: 18.1s

Alternatives: 2

Speedup: 0.5×

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: 98.0% 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: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot {\left({\cos^{-1} \left(\left(0.05555555555555555 \cdot x\right) \cdot \frac{\sqrt{t}}{z \cdot y}\right)}^{0.25}\right)}^{4} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (*
  0.3333333333333333
  (pow
   (pow (acos (* (* 0.05555555555555555 x) (/ (sqrt t) (* z y)))) 0.25)
   4.0)))

C

double code(double x, double y, double z, double t) {
	return 0.3333333333333333 * pow(pow(acos(((0.05555555555555555 * x) * (sqrt(t) / (z * y)))), 0.25), 4.0);
}

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(((0.05555555555555555d0 * x) * (sqrt(t) / (z * y)))) ** 0.25d0) ** 4.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return 0.3333333333333333 * Math.pow(Math.pow(Math.acos(((0.05555555555555555 * x) * (Math.sqrt(t) / (z * y)))), 0.25), 4.0);
}

Python

def code(x, y, z, t):
	return 0.3333333333333333 * math.pow(math.pow(math.acos(((0.05555555555555555 * x) * (math.sqrt(t) / (z * y)))), 0.25), 4.0)

Julia

function code(x, y, z, t)
	return Float64(0.3333333333333333 * ((acos(Float64(Float64(0.05555555555555555 * x) * Float64(sqrt(t) / Float64(z * y)))) ^ 0.25) ^ 4.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 0.3333333333333333 * ((acos(((0.05555555555555555 * x) * (sqrt(t) / (z * y)))) ^ 0.25) ^ 4.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(0.3333333333333333 * N[Power[N[Power[N[ArcCos[N[(N[(0.05555555555555555 * x), $MachinePrecision] * N[(N[Sqrt[t], $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.25], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.0%

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{3}\right), \color{blue}{\cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)}\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \cos^{-1} \color{blue}{\left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)}\right) \]

   3. acos-lowering-acos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\right)\right) \]

   4. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\left(\frac{\left(3 \cdot \frac{x}{y \cdot 27}\right) \cdot \sqrt{t}}{z \cdot 2}\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\left(\frac{\left(3 \cdot \frac{x}{y \cdot 27}\right) \cdot \sqrt{t}}{2 \cdot z}\right)\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\left(\frac{\frac{3 \cdot x}{y \cdot 27} \cdot \sqrt{t}}{2 \cdot z}\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\left(\frac{\frac{3 \cdot x}{27 \cdot y} \cdot \sqrt{t}}{2 \cdot z}\right)\right)\right) \]

   8. times-frac N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\left(\frac{\left(\frac{3}{27} \cdot \frac{x}{y}\right) \cdot \sqrt{t}}{2 \cdot z}\right)\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\left(\frac{\frac{3}{27} \cdot \left(\frac{x}{y} \cdot \sqrt{t}\right)}{2 \cdot z}\right)\right)\right) \]

   10. times-frac N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\left(\frac{\frac{3}{27}}{2} \cdot \frac{\frac{x}{y} \cdot \sqrt{t}}{z}\right)\right)\right) \]

   11. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\left(\frac{\frac{3}{27}}{2} \cdot \left(\frac{x}{y} \cdot \frac{\sqrt{t}}{z}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{3}{27}}{2}\right), \left(\frac{x}{y} \cdot \frac{\sqrt{t}}{z}\right)\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{9}}{2}\right), \left(\frac{x}{y} \cdot \frac{\sqrt{t}}{z}\right)\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\mathsf{*.f64}\left(\frac{1}{18}, \left(\frac{x}{y} \cdot \frac{\sqrt{t}}{z}\right)\right)\right)\right) \]

   15. associate-*l/ N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\mathsf{*.f64}\left(\frac{1}{18}, \left(\frac{x \cdot \frac{\sqrt{t}}{z}}{y}\right)\right)\right)\right) \]

   16. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\mathsf{*.f64}\left(\frac{1}{18}, \left(\frac{\frac{x \cdot \sqrt{t}}{z}}{y}\right)\right)\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\mathsf{*.f64}\left(\frac{1}{18}, \left(\frac{\frac{\sqrt{t} \cdot x}{z}}{y}\right)\right)\right)\right) \]

   18. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\mathsf{*.f64}\left(\frac{1}{18}, \left(\frac{\sqrt{t} \cdot \frac{x}{z}}{y}\right)\right)\right)\right) \]

3. Simplified 98.1%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \cos^{-1} \left(0.05555555555555555 \cdot \frac{\frac{x \cdot \sqrt{t}}{z}}{y}\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\left(\frac{\frac{x \cdot \sqrt{t}}{z}}{y} \cdot \frac{1}{18}\right)\right)\right) \]

   2. div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\left(\left(\frac{x \cdot \sqrt{t}}{z} \cdot \frac{1}{y}\right) \cdot \frac{1}{18}\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\left(\frac{x \cdot \sqrt{t}}{z} \cdot \left(\frac{1}{y} \cdot \frac{1}{18}\right)\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\left(\frac{\sqrt{t} \cdot x}{z} \cdot \left(\frac{1}{y} \cdot \frac{1}{18}\right)\right)\right)\right) \]

   5. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\left(\left(\sqrt{t} \cdot \frac{x}{z}\right) \cdot \left(\frac{1}{y} \cdot \frac{1}{18}\right)\right)\right)\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\left(\sqrt{t} \cdot \left(\frac{x}{z} \cdot \left(\frac{1}{y} \cdot \frac{1}{18}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{t}\right), \left(\frac{x}{z} \cdot \left(\frac{1}{y} \cdot \frac{1}{18}\right)\right)\right)\right)\right) \]

   8. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(t\right), \left(\frac{x}{z} \cdot \left(\frac{1}{y} \cdot \frac{1}{18}\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(t\right), \mathsf{*.f64}\left(\left(\frac{x}{z}\right), \left(\frac{1}{y} \cdot \frac{1}{18}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, z\right), \left(\frac{1}{y} \cdot \frac{1}{18}\right)\right)\right)\right)\right) \]

   11. associate-*l/ N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, z\right), \left(\frac{1 \cdot \frac{1}{18}}{y}\right)\right)\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, z\right), \left(\frac{\frac{1}{18}}{y}\right)\right)\right)\right)\right) \]

   13. /-lowering-/.f64 98.1%

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(t\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{/.f64}\left(\frac{1}{18}, y\right)\right)\right)\right)\right) \]

6. Applied egg-rr 98.1%

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

8. Applied egg-rr 95.8%

   \[\leadsto 0.3333333333333333 \cdot \color{blue}{{\left({\cos^{-1} \left(\frac{0.05555555555555555}{\frac{y}{\frac{x}{\frac{z}{\sqrt{t}}}}}\right)}^{0.5}\right)}^{2}} \]
9. Step-by-step derivation

   1. sqr-pow N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({\cos^{-1} \left(\frac{\frac{1}{18}}{\frac{y}{\frac{x}{\frac{z}{\sqrt{t}}}}}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\cos^{-1} \left(\frac{\frac{1}{18}}{\frac{y}{\frac{x}{\frac{z}{\sqrt{t}}}}}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{2}\right)\right) \]

   2. unpow-prod-down N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({\cos^{-1} \left(\frac{\frac{1}{18}}{\frac{y}{\frac{x}{\frac{z}{\sqrt{t}}}}}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{2} \cdot \color{blue}{{\left({\cos^{-1} \left(\frac{\frac{1}{18}}{\frac{y}{\frac{x}{\frac{z}{\sqrt{t}}}}}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{2}}\right)\right) \]

   3. pow-prod-up N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({\cos^{-1} \left(\frac{\frac{1}{18}}{\frac{y}{\frac{x}{\frac{z}{\sqrt{t}}}}}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{\color{blue}{\left(2 + 2\right)}}\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\left({\cos^{-1} \left(\frac{\frac{1}{18}}{\frac{y}{\frac{x}{\frac{z}{\sqrt{t}}}}}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{4}\right)\right) \]

   5. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(\left({\cos^{-1} \left(\frac{\frac{1}{18}}{\frac{y}{\frac{x}{\frac{z}{\sqrt{t}}}}}\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right), \color{blue}{4}\right)\right) \]

10. Applied egg-rr 98.1%

    \[\leadsto 0.3333333333333333 \cdot \color{blue}{{\left({\cos^{-1} \left(\frac{\frac{0.05555555555555555}{\frac{z \cdot {t}^{-0.5}}{x}}}{y}\right)}^{0.25}\right)}^{4}} \]
11. Step-by-step derivation

    1. associate-/r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{acos.f64}\left(\left(\frac{\frac{\frac{1}{18}}{z \cdot {t}^{\frac{-1}{2}}} \cdot x}{y}\right)\right), \frac{1}{4}\right), 4\right)\right) \]

    2. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{acos.f64}\left(\left(\frac{\frac{1}{18}}{z \cdot {t}^{\frac{-1}{2}}} \cdot \frac{x}{y}\right)\right), \frac{1}{4}\right), 4\right)\right) \]

    3. associate-/l/ N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{acos.f64}\left(\left(\frac{\frac{\frac{1}{18}}{{t}^{\frac{-1}{2}}}}{z} \cdot \frac{x}{y}\right)\right), \frac{1}{4}\right), 4\right)\right) \]

    4. frac-times N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{acos.f64}\left(\left(\frac{\frac{\frac{1}{18}}{{t}^{\frac{-1}{2}}} \cdot x}{z \cdot y}\right)\right), \frac{1}{4}\right), 4\right)\right) \]

    5. div-inv N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{acos.f64}\left(\left(\frac{\left(\frac{1}{18} \cdot \frac{1}{{t}^{\frac{-1}{2}}}\right) \cdot x}{z \cdot y}\right)\right), \frac{1}{4}\right), 4\right)\right) \]

    6. pow-flip N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{acos.f64}\left(\left(\frac{\left(\frac{1}{18} \cdot {t}^{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)}\right) \cdot x}{z \cdot y}\right)\right), \frac{1}{4}\right), 4\right)\right) \]

    7. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{acos.f64}\left(\left(\frac{\left(\frac{1}{18} \cdot {t}^{\frac{1}{2}}\right) \cdot x}{z \cdot y}\right)\right), \frac{1}{4}\right), 4\right)\right) \]

    8. pow1/2 N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{acos.f64}\left(\left(\frac{\left(\frac{1}{18} \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right)\right), \frac{1}{4}\right), 4\right)\right) \]

    9. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{acos.f64}\left(\left(\frac{\frac{1}{18} \cdot \left(\sqrt{t} \cdot x\right)}{z \cdot y}\right)\right), \frac{1}{4}\right), 4\right)\right) \]

    10. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{acos.f64}\left(\left(\frac{\frac{1}{18} \cdot \left(\sqrt{t} \cdot x\right)}{y \cdot z}\right)\right), \frac{1}{4}\right), 4\right)\right) \]

    11. associate-*r/ N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{acos.f64}\left(\left(\frac{1}{18} \cdot \frac{\sqrt{t} \cdot x}{y \cdot z}\right)\right), \frac{1}{4}\right), 4\right)\right) \]

    12. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{acos.f64}\left(\left(\frac{1}{18} \cdot \frac{x \cdot \sqrt{t}}{y \cdot z}\right)\right), \frac{1}{4}\right), 4\right)\right) \]

    13. associate-/l* N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{acos.f64}\left(\left(\frac{1}{18} \cdot \left(x \cdot \frac{\sqrt{t}}{y \cdot z}\right)\right)\right), \frac{1}{4}\right), 4\right)\right) \]

    14. associate-*r* N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{acos.f64}\left(\left(\left(\frac{1}{18} \cdot x\right) \cdot \frac{\sqrt{t}}{y \cdot z}\right)\right), \frac{1}{4}\right), 4\right)\right) \]

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

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{18} \cdot x\right), \left(\frac{\sqrt{t}}{y \cdot z}\right)\right)\right), \frac{1}{4}\right), 4\right)\right) \]

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

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{18}, x\right), \left(\frac{\sqrt{t}}{y \cdot z}\right)\right)\right), \frac{1}{4}\right), 4\right)\right) \]

    17. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{18}, x\right), \mathsf{/.f64}\left(\left(\sqrt{t}\right), \left(y \cdot z\right)\right)\right)\right), \frac{1}{4}\right), 4\right)\right) \]

    18. sqrt-lowering-sqrt.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{18}, x\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(t\right), \left(y \cdot z\right)\right)\right)\right), \frac{1}{4}\right), 4\right)\right) \]

    19. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{18}, x\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(t\right), \left(z \cdot y\right)\right)\right)\right), \frac{1}{4}\right), 4\right)\right) \]

    20. *-lowering-*.f64 99.9%

        \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{18}, x\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(t\right), \mathsf{*.f64}\left(z, y\right)\right)\right)\right), \frac{1}{4}\right), 4\right)\right) \]

12. Applied egg-rr 99.9%

    \[\leadsto 0.3333333333333333 \cdot {\left({\cos^{-1} \color{blue}{\left(\left(0.05555555555555555 \cdot x\right) \cdot \frac{\sqrt{t}}{z \cdot y}\right)}}^{0.25}\right)}^{4} \]
13. Add Preprocessing

Alternative 2: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return 0.3333333333333333 * acos((0.05555555555555555 * (((x * sqrt(t)) / 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((0.05555555555555555d0 * (((x * sqrt(t)) / z) / y)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{3}\right), \color{blue}{\cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)}\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \cos^{-1} \color{blue}{\left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)}\right) \]

   3. acos-lowering-acos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\right)\right) \]

   4. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\left(\frac{\left(3 \cdot \frac{x}{y \cdot 27}\right) \cdot \sqrt{t}}{z \cdot 2}\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\left(\frac{\left(3 \cdot \frac{x}{y \cdot 27}\right) \cdot \sqrt{t}}{2 \cdot z}\right)\right)\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\left(\frac{\frac{3 \cdot x}{y \cdot 27} \cdot \sqrt{t}}{2 \cdot z}\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\left(\frac{\frac{3 \cdot x}{27 \cdot y} \cdot \sqrt{t}}{2 \cdot z}\right)\right)\right) \]

   8. times-frac N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\left(\frac{\left(\frac{3}{27} \cdot \frac{x}{y}\right) \cdot \sqrt{t}}{2 \cdot z}\right)\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\left(\frac{\frac{3}{27} \cdot \left(\frac{x}{y} \cdot \sqrt{t}\right)}{2 \cdot z}\right)\right)\right) \]

   10. times-frac N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\left(\frac{\frac{3}{27}}{2} \cdot \frac{\frac{x}{y} \cdot \sqrt{t}}{z}\right)\right)\right) \]

   11. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\left(\frac{\frac{3}{27}}{2} \cdot \left(\frac{x}{y} \cdot \frac{\sqrt{t}}{z}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{3}{27}}{2}\right), \left(\frac{x}{y} \cdot \frac{\sqrt{t}}{z}\right)\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{9}}{2}\right), \left(\frac{x}{y} \cdot \frac{\sqrt{t}}{z}\right)\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\mathsf{*.f64}\left(\frac{1}{18}, \left(\frac{x}{y} \cdot \frac{\sqrt{t}}{z}\right)\right)\right)\right) \]

   15. associate-*l/ N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\mathsf{*.f64}\left(\frac{1}{18}, \left(\frac{x \cdot \frac{\sqrt{t}}{z}}{y}\right)\right)\right)\right) \]

   16. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\mathsf{*.f64}\left(\frac{1}{18}, \left(\frac{\frac{x \cdot \sqrt{t}}{z}}{y}\right)\right)\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\mathsf{*.f64}\left(\frac{1}{18}, \left(\frac{\frac{\sqrt{t} \cdot x}{z}}{y}\right)\right)\right)\right) \]

   18. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{acos.f64}\left(\mathsf{*.f64}\left(\frac{1}{18}, \left(\frac{\sqrt{t} \cdot \frac{x}{z}}{y}\right)\right)\right)\right) \]

3. Simplified 98.1%

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

Developer Target 1: 98.2% 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 2024158 
(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)))))