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

Percentage Accurate: 98.1% → 98.8%

Time: 30.7s

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: 98.1% 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: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{3}{\cos^{-1} \left(\frac{\frac{x \cdot \sqrt{t}}{y \cdot 18}}{z}\right)}\right)}^{-1} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (pow (/ 3.0 (acos (/ (/ (* x (sqrt t)) (* y 18.0)) z))) -1.0))

C

double code(double x, double y, double z, double t) {
	return pow((3.0 / acos((((x * sqrt(t)) / (y * 18.0)) / z))), -1.0);
}

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 = (3.0d0 / acos((((x * sqrt(t)) / (y * 18.0d0)) / z))) ** (-1.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return Math.pow((3.0 / Math.acos((((x * Math.sqrt(t)) / (y * 18.0)) / z))), -1.0);
}

Python

def code(x, y, z, t):
	return math.pow((3.0 / math.acos((((x * math.sqrt(t)) / (y * 18.0)) / z))), -1.0)

Julia

function code(x, y, z, t)
	return Float64(3.0 / acos(Float64(Float64(Float64(x * sqrt(t)) / Float64(y * 18.0)) / z))) ^ -1.0
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (3.0 / acos((((x * sqrt(t)) / (y * 18.0)) / z))) ^ -1.0;
end

Wolfram

code[x_, y_, z_, t_] := N[Power[N[(3.0 / N[ArcCos[N[(N[(N[(x * N[Sqrt[t], $MachinePrecision]), $MachinePrecision] / N[(y * 18.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 97.8%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-acos.f64 N/A

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

   3. associate-*r/ N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-*.f64 96.5

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

5. Simplified 96.5%

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

7. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{\frac{\left(\left(\pi \cdot \pi\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 0.0625\right) - {\sin^{-1} \left(\frac{x \cdot \sqrt{t}}{\left(y \cdot 18\right) \cdot z}\right)}^{4}\right) \cdot 1}{\mathsf{fma}\left(\pi, \pi \cdot 0.25, {\sin^{-1} \left(\frac{x \cdot \sqrt{t}}{\left(y \cdot 18\right) \cdot z}\right)}^{2}\right) \cdot \left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\frac{x \cdot \sqrt{t}}{\left(y \cdot 18\right) \cdot z}\right)\right) \cdot 3\right)}} \]

9. Applied egg-rr 98.0%

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

    1. lift-sqrt.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. associate-*r* N/A

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

    4. lift-*.f64 N/A

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

    5. associate-/r* N/A

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

    6. lower-/.f64 N/A

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

    7. lower-/.f64 98.7

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

11. Applied egg-rr 98.7%

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

Alternative 2: 99.7% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (pow (/ 3.0 (acos (* (sqrt t) (/ (* x 0.05555555555555555) (* y z))))) -1.0))

C

double code(double x, double y, double z, double t) {
	return pow((3.0 / acos((sqrt(t) * ((x * 0.05555555555555555) / (y * z))))), -1.0);
}

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 = (3.0d0 / acos((sqrt(t) * ((x * 0.05555555555555555d0) / (y * z))))) ** (-1.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return Math.pow((3.0 / Math.acos((Math.sqrt(t) * ((x * 0.05555555555555555) / (y * z))))), -1.0);
}

Python

def code(x, y, z, t):
	return math.pow((3.0 / math.acos((math.sqrt(t) * ((x * 0.05555555555555555) / (y * z))))), -1.0)

Julia

function code(x, y, z, t)
	return Float64(3.0 / acos(Float64(sqrt(t) * Float64(Float64(x * 0.05555555555555555) / Float64(y * z))))) ^ -1.0
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (3.0 / acos((sqrt(t) * ((x * 0.05555555555555555) / (y * z))))) ^ -1.0;
end

Wolfram

code[x_, y_, z_, t_] := N[Power[N[(3.0 / N[ArcCos[N[(N[Sqrt[t], $MachinePrecision] * N[(N[(x * 0.05555555555555555), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 97.8%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-acos.f64 N/A

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

   3. associate-*r/ N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-*.f64 96.5

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

5. Simplified 96.5%

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

7. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{\frac{\left(\left(\pi \cdot \pi\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 0.0625\right) - {\sin^{-1} \left(\frac{x \cdot \sqrt{t}}{\left(y \cdot 18\right) \cdot z}\right)}^{4}\right) \cdot 1}{\mathsf{fma}\left(\pi, \pi \cdot 0.25, {\sin^{-1} \left(\frac{x \cdot \sqrt{t}}{\left(y \cdot 18\right) \cdot z}\right)}^{2}\right) \cdot \left(\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\frac{x \cdot \sqrt{t}}{\left(y \cdot 18\right) \cdot z}\right)\right) \cdot 3\right)}} \]

9. Applied egg-rr 98.0%

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

    1. lift-sqrt.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. *-commutative N/A

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

    4. lift-*.f64 N/A

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

    5. associate-/l* N/A

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

    6. *-commutative N/A

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

    7. lower-*.f64 N/A

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

    8. lift-*.f64 N/A

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

    9. *-commutative N/A

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

    10. lift-*.f64 N/A

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

    11. associate-*l* N/A

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

    12. associate-/r* N/A

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

    13. *-commutative N/A

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

    14. lower-/.f64 N/A

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

    15. div-inv N/A

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

    16. metadata-eval N/A

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

    17. metadata-eval N/A

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

    18. lower-*.f64 N/A

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

    19. metadata-eval N/A

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

    20. lower-*.f64 98.5

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

11. Applied egg-rr 98.5%

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

12. Final simplification 98.5%

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

Alternative 3: 98.1% 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 97.8%

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

   1. metadata-eval 97.8

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

4. Applied egg-rr 97.8%

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

5. Final simplification 97.8%

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

Alternative 4: 98.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \cos^{-1} \left(\sqrt{t} \cdot \frac{x}{\left(y \cdot 18\right) \cdot z}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* 0.3333333333333333 (acos (* (sqrt t) (/ x (* (* y 18.0) z))))))

C

double code(double x, double y, double z, double t) {
	return 0.3333333333333333 * acos((sqrt(t) * (x / ((y * 18.0) * z))));
}

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 / ((y * 18.0d0) * z))))
end function

Java

public static double code(double x, double y, double z, double t) {
	return 0.3333333333333333 * Math.acos((Math.sqrt(t) * (x / ((y * 18.0) * z))));
}

Python

def code(x, y, z, t):
	return 0.3333333333333333 * math.acos((math.sqrt(t) * (x / ((y * 18.0) * z))))

Julia

function code(x, y, z, t)
	return Float64(0.3333333333333333 * acos(Float64(sqrt(t) * Float64(x / Float64(Float64(y * 18.0) * z)))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 0.3333333333333333 * acos((sqrt(t) * (x / ((y * 18.0) * z))));
end

Wolfram

code[x_, y_, z_, t_] := N[(0.3333333333333333 * N[ArcCos[N[(N[Sqrt[t], $MachinePrecision] * N[(x / N[(N[(y * 18.0), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.8%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-acos.f64 N/A

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

   3. associate-*r/ N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-*.f64 96.5

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

5. Simplified 96.5%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l/ N/A

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

   7. lift-/.f64 N/A

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

   8. associate-*l* N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 97.0

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

7. Applied egg-rr 97.0%

   \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \color{blue}{\left(\frac{x}{\left(y \cdot 18\right) \cdot z} \cdot \sqrt{t}\right)} \]

8. Final simplification 97.0%

   \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(\sqrt{t} \cdot \frac{x}{\left(y \cdot 18\right) \cdot z}\right) \]
9. 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 2024208 
(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)))))