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

Percentage Accurate: 97.9% → 99.5%

Time: 20.2s

Alternatives: 6

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: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \sqrt{\mathsf{PI}\left(\right)}\\ t_2 := 0.05555555555555555 \cdot \frac{\frac{x}{y}}{z}\\ t_3 := \sin^{-1} \left(\sqrt{t} \cdot t\_2\right)\\ {3}^{-1} \cdot \frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(t\_2 \cdot \sqrt{t}\right)}^{3}}{\mathsf{fma}\left(\frac{\mathsf{PI}\left(\right)}{4} \cdot t\_1, t\_1, \left(t\_3 - \frac{\mathsf{PI}\left(\right)}{-2}\right) \cdot t\_3\right)} \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (PI)))
        (t_2 (* 0.05555555555555555 (/ (/ x y) z)))
        (t_3 (asin (* (sqrt t) t_2))))
   (*
    (pow 3.0 -1.0)
    (/
     (- (pow (/ (PI) 2.0) 3.0) (pow (asin (* t_2 (sqrt t))) 3.0))
     (fma (* (/ (PI) 4.0) t_1) t_1 (* (- t_3 (/ (PI) -2.0)) t_3))))))

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. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 97.4

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

5. Applied rewrites 97.4%

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

   1. lift-acos.f64 N/A

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

   2. acos-asin N/A

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} - \sin^{-1} \left(\left(\frac{x}{z \cdot y} \cdot \frac{1}{18}\right) \cdot \sqrt{t}\right)\right)} \]

   3. lift-PI.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \left(\frac{\color{blue}{\mathsf{PI}\left(\right)}}{2} - \sin^{-1} \left(\left(\frac{x}{z \cdot y} \cdot \frac{1}{18}\right) \cdot \sqrt{t}\right)\right) \]

   4. lift-/.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \left(\color{blue}{\frac{\mathsf{PI}\left(\right)}{2}} - \sin^{-1} \left(\left(\frac{x}{z \cdot y} \cdot \frac{1}{18}\right) \cdot \sqrt{t}\right)\right) \]

   5. flip3-- N/A

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\left(\frac{x}{z \cdot y} \cdot \frac{1}{18}\right) \cdot \sqrt{t}\right)}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} \left(\left(\frac{x}{z \cdot y} \cdot \frac{1}{18}\right) \cdot \sqrt{t}\right) \cdot \sin^{-1} \left(\left(\frac{x}{z \cdot y} \cdot \frac{1}{18}\right) \cdot \sqrt{t}\right) + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} \left(\left(\frac{x}{z \cdot y} \cdot \frac{1}{18}\right) \cdot \sqrt{t}\right)\right)}} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\left(\frac{x}{z \cdot y} \cdot \frac{1}{18}\right) \cdot \sqrt{t}\right)}^{3}}{\frac{\mathsf{PI}\left(\right)}{2} \cdot \frac{\mathsf{PI}\left(\right)}{2} + \left(\sin^{-1} \left(\left(\frac{x}{z \cdot y} \cdot \frac{1}{18}\right) \cdot \sqrt{t}\right) \cdot \sin^{-1} \left(\left(\frac{x}{z \cdot y} \cdot \frac{1}{18}\right) \cdot \sqrt{t}\right) + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} \left(\left(\frac{x}{z \cdot y} \cdot \frac{1}{18}\right) \cdot \sqrt{t}\right)\right)}} \]

7. Applied rewrites 98.0%

   \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\left(0.05555555555555555 \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)}^{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2} + \left({\sin^{-1} \left(\left(0.05555555555555555 \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)}^{2} + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} \left(\left(0.05555555555555555 \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)\right)}} \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\left(\frac{1}{18} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)}^{3}}{\color{blue}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2} + \left({\sin^{-1} \left(\left(\frac{1}{18} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)}^{2} + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} \left(\left(\frac{1}{18} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)\right)}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\left(\frac{1}{18} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)}^{3}}{\color{blue}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}} + \left({\sin^{-1} \left(\left(\frac{1}{18} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)}^{2} + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} \left(\left(\frac{1}{18} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)\right)} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\left(\frac{1}{18} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)}^{3}}{{\color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}}^{2} + \left({\sin^{-1} \left(\left(\frac{1}{18} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)}^{2} + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} \left(\left(\frac{1}{18} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)\right)} \]

   4. lift-PI.f64 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\left(\frac{1}{18} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)}^{3}}{{\left(\frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)}^{2} + \left({\sin^{-1} \left(\left(\frac{1}{18} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)}^{2} + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} \left(\left(\frac{1}{18} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)\right)} \]

   5. add-sqr-sqrt N/A

      \[\leadsto \frac{1}{3} \cdot \frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\left(\frac{1}{18} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)}^{3}}{{\left(\frac{\color{blue}{\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}}}{2}\right)}^{2} + \left({\sin^{-1} \left(\left(\frac{1}{18} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)}^{2} + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} \left(\left(\frac{1}{18} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)\right)} \]

   6. associate-/l* N/A

      \[\leadsto \frac{1}{3} \cdot \frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\left(\frac{1}{18} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)}^{3}}{{\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{\sqrt{\mathsf{PI}\left(\right)}}{2}\right)}}^{2} + \left({\sin^{-1} \left(\left(\frac{1}{18} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)}^{2} + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} \left(\left(\frac{1}{18} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)\right)} \]

   7. unpow-prod-down N/A

      \[\leadsto \frac{1}{3} \cdot \frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\left(\frac{1}{18} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)}^{3}}{\color{blue}{{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{2} \cdot {\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{2}\right)}^{2}} + \left({\sin^{-1} \left(\left(\frac{1}{18} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)}^{2} + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} \left(\left(\frac{1}{18} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)\right)} \]

   8. pow2 N/A

      \[\leadsto \frac{1}{3} \cdot \frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\left(\frac{1}{18} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)}^{3}}{\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \cdot {\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{2}\right)}^{2} + \left({\sin^{-1} \left(\left(\frac{1}{18} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)}^{2} + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} \left(\left(\frac{1}{18} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)\right)} \]

   9. add-sqr-sqrt N/A

      \[\leadsto \frac{1}{3} \cdot \frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\left(\frac{1}{18} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)}^{3}}{\color{blue}{\mathsf{PI}\left(\right)} \cdot {\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{2}\right)}^{2} + \left({\sin^{-1} \left(\left(\frac{1}{18} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)}^{2} + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} \left(\left(\frac{1}{18} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)\right)} \]

   10. lift-PI.f64 N/A

       \[\leadsto \frac{1}{3} \cdot \frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\left(\frac{1}{18} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)}^{3}}{\color{blue}{\mathsf{PI}\left(\right)} \cdot {\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{2}\right)}^{2} + \left({\sin^{-1} \left(\left(\frac{1}{18} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)}^{2} + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} \left(\left(\frac{1}{18} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)\right)} \]

9. Applied rewrites 99.5%

   \[\leadsto \frac{1}{3} \cdot \frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\left(0.05555555555555555 \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)}^{3}}{\color{blue}{\mathsf{fma}\left(\mathsf{PI}\left(\right), {\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{2}\right)}^{2}, \sin^{-1} \left(\left(\frac{\frac{x}{y}}{z} \cdot 0.05555555555555555\right) \cdot \sqrt{t}\right) \cdot \left(\sin^{-1} \left(\left(\frac{\frac{x}{y}}{z} \cdot 0.05555555555555555\right) \cdot \sqrt{t}\right) - \frac{\mathsf{PI}\left(\right)}{-2}\right)\right)}} \]
10. Step-by-step derivation

    1. lift-fma.f64 N/A

       \[\leadsto \frac{1}{3} \cdot \frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\left(\frac{1}{18} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)}^{3}}{\color{blue}{\mathsf{PI}\left(\right) \cdot {\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{2}\right)}^{2} + \sin^{-1} \left(\left(\frac{\frac{x}{y}}{z} \cdot \frac{1}{18}\right) \cdot \sqrt{t}\right) \cdot \left(\sin^{-1} \left(\left(\frac{\frac{x}{y}}{z} \cdot \frac{1}{18}\right) \cdot \sqrt{t}\right) - \frac{\mathsf{PI}\left(\right)}{-2}\right)}} \]

    2. *-commutative N/A

       \[\leadsto \frac{1}{3} \cdot \frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\left(\frac{1}{18} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)}^{3}}{\color{blue}{{\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{2}\right)}^{2} \cdot \mathsf{PI}\left(\right)} + \sin^{-1} \left(\left(\frac{\frac{x}{y}}{z} \cdot \frac{1}{18}\right) \cdot \sqrt{t}\right) \cdot \left(\sin^{-1} \left(\left(\frac{\frac{x}{y}}{z} \cdot \frac{1}{18}\right) \cdot \sqrt{t}\right) - \frac{\mathsf{PI}\left(\right)}{-2}\right)} \]

    3. rem-square-sqrt N/A

       \[\leadsto \frac{1}{3} \cdot \frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\left(\frac{1}{18} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)}^{3}}{{\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{2}\right)}^{2} \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} + \sin^{-1} \left(\left(\frac{\frac{x}{y}}{z} \cdot \frac{1}{18}\right) \cdot \sqrt{t}\right) \cdot \left(\sin^{-1} \left(\left(\frac{\frac{x}{y}}{z} \cdot \frac{1}{18}\right) \cdot \sqrt{t}\right) - \frac{\mathsf{PI}\left(\right)}{-2}\right)} \]

    4. lift-sqrt.f64 N/A

       \[\leadsto \frac{1}{3} \cdot \frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\left(\frac{1}{18} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)}^{3}}{{\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{2}\right)}^{2} \cdot \left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) + \sin^{-1} \left(\left(\frac{\frac{x}{y}}{z} \cdot \frac{1}{18}\right) \cdot \sqrt{t}\right) \cdot \left(\sin^{-1} \left(\left(\frac{\frac{x}{y}}{z} \cdot \frac{1}{18}\right) \cdot \sqrt{t}\right) - \frac{\mathsf{PI}\left(\right)}{-2}\right)} \]

    5. lift-sqrt.f64 N/A

       \[\leadsto \frac{1}{3} \cdot \frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\left(\frac{1}{18} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)}^{3}}{{\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{2}\right)}^{2} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right) + \sin^{-1} \left(\left(\frac{\frac{x}{y}}{z} \cdot \frac{1}{18}\right) \cdot \sqrt{t}\right) \cdot \left(\sin^{-1} \left(\left(\frac{\frac{x}{y}}{z} \cdot \frac{1}{18}\right) \cdot \sqrt{t}\right) - \frac{\mathsf{PI}\left(\right)}{-2}\right)} \]

    6. associate-*r* N/A

       \[\leadsto \frac{1}{3} \cdot \frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\left(\frac{1}{18} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)}^{3}}{\color{blue}{\left({\left(\frac{\sqrt{\mathsf{PI}\left(\right)}}{2}\right)}^{2} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} + \sin^{-1} \left(\left(\frac{\frac{x}{y}}{z} \cdot \frac{1}{18}\right) \cdot \sqrt{t}\right) \cdot \left(\sin^{-1} \left(\left(\frac{\frac{x}{y}}{z} \cdot \frac{1}{18}\right) \cdot \sqrt{t}\right) - \frac{\mathsf{PI}\left(\right)}{-2}\right)} \]

11. Applied rewrites 99.5%

    \[\leadsto \frac{1}{3} \cdot \frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\left(0.05555555555555555 \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)}^{3}}{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{PI}\left(\right)}{4} \cdot \sqrt{\mathsf{PI}\left(\right)}, \sqrt{\mathsf{PI}\left(\right)}, \left(\sin^{-1} \left(\sqrt{t} \cdot \left(0.05555555555555555 \cdot \frac{\frac{x}{y}}{z}\right)\right) - \frac{\mathsf{PI}\left(\right)}{-2}\right) \cdot \sin^{-1} \left(\sqrt{t} \cdot \left(0.05555555555555555 \cdot \frac{\frac{x}{y}}{z}\right)\right)\right)}} \]

12. Final simplification 99.5%

    \[\leadsto {3}^{-1} \cdot \frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\left(0.05555555555555555 \cdot \frac{\frac{x}{y}}{z}\right) \cdot \sqrt{t}\right)}^{3}}{\mathsf{fma}\left(\frac{\mathsf{PI}\left(\right)}{4} \cdot \sqrt{\mathsf{PI}\left(\right)}, \sqrt{\mathsf{PI}\left(\right)}, \left(\sin^{-1} \left(\sqrt{t} \cdot \left(0.05555555555555555 \cdot \frac{\frac{x}{y}}{z}\right)\right) - \frac{\mathsf{PI}\left(\right)}{-2}\right) \cdot \sin^{-1} \left(\sqrt{t} \cdot \left(0.05555555555555555 \cdot \frac{\frac{x}{y}}{z}\right)\right)\right)} \]
13. Add Preprocessing

Alternative 2: 97.9% accurate, 0.7× speedup

Math

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

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (* (pow 3.0 -1.0) (acos (* (/ (* 3.0 (/ x (* y 27.0))) (* z 2.0)) (sqrt t)))))

C

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

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
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 ** (-1.0d0)) * acos((((3.0d0 * (x / (y * 27.0d0))) / (z * 2.0d0)) * sqrt(t)))
end function

Java

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

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return math.pow(3.0, -1.0) * math.acos((((3.0 * (x / (y * 27.0))) / (z * 2.0)) * math.sqrt(t)))

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64((3.0 ^ -1.0) * acos(Float64(Float64(Float64(3.0 * Float64(x / Float64(y * 27.0))) / Float64(z * 2.0)) * sqrt(t))))
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = (3.0 ^ -1.0) * acos((((3.0 * (x / (y * 27.0))) / (z * 2.0)) * sqrt(t)));
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[Power[3.0, -1.0], $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]

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. Add Preprocessing

3. Final simplification 98.0%

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

Alternative 3: 97.3% accurate, 1.1× speedup

Math

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

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (*
  (acos (* (* (/ (/ (sqrt t) z) y) x) 0.05555555555555555))
  0.3333333333333333))

C

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

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
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(((((sqrt(t) / z) / y) * x) * 0.05555555555555555d0)) * 0.3333333333333333d0
end function

Java

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

Python

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

Julia

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

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = acos(((((sqrt(t) / z) / y) * x) * 0.05555555555555555)) * 0.3333333333333333;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[ArcCos[N[(N[(N[(N[(N[Sqrt[t], $MachinePrecision] / z), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision] * 0.05555555555555555), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $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. 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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 97.1%

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

Alternative 4: 98.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \cos^{-1} \left(\sqrt{t} \cdot \frac{-3 \cdot x}{\left(-54 \cdot z\right) \cdot y}\right) \cdot 0.3333333333333333 \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (* (acos (* (sqrt t) (/ (* -3.0 x) (* (* -54.0 z) y)))) 0.3333333333333333))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return acos((sqrt(t) * ((-3.0 * x) / ((-54.0 * z) * y)))) * 0.3333333333333333;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
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((sqrt(t) * (((-3.0d0) * x) / (((-54.0d0) * z) * y)))) * 0.3333333333333333d0
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return Math.acos((Math.sqrt(t) * ((-3.0 * x) / ((-54.0 * z) * y)))) * 0.3333333333333333;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return math.acos((math.sqrt(t) * ((-3.0 * x) / ((-54.0 * z) * y)))) * 0.3333333333333333

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(acos(Float64(sqrt(t) * Float64(Float64(-3.0 * x) / Float64(Float64(-54.0 * z) * y)))) * 0.3333333333333333)
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = acos((sqrt(t) * ((-3.0 * x) / ((-54.0 * z) * y)))) * 0.3333333333333333;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[ArcCos[N[(N[Sqrt[t], $MachinePrecision] * N[(N[(-3.0 * x), $MachinePrecision] / N[(N[(-54.0 * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $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. 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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 97.1%

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

7. Applied rewrites 97.3%

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

9. Applied rewrites 97.5%

   \[\leadsto \cos^{-1} \left(\sqrt{t} \cdot \frac{-3 \cdot x}{\left(-54 \cdot z\right) \cdot y}\right) \cdot 0.3333333333333333 \]
10. Add Preprocessing

Alternative 5: 98.1% accurate, 1.2× speedup

Math

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

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (*
  (acos (* x (/ (* 0.05555555555555555 (sqrt t)) (* z y))))
  0.3333333333333333))

C

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

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
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 * ((0.05555555555555555d0 * sqrt(t)) / (z * y)))) * 0.3333333333333333d0
end function

Java

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

Python

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

Julia

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

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = acos((x * ((0.05555555555555555 * sqrt(t)) / (z * y)))) * 0.3333333333333333;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[ArcCos[N[(x * N[(N[(0.05555555555555555 * N[Sqrt[t], $MachinePrecision]), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $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. 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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 97.1%

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

7. Applied rewrites 97.3%

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

9. Applied rewrites 97.4%

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

Alternative 6: 98.0% accurate, 1.2× speedup

Math

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

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (*
  (acos (* x (* (sqrt t) (/ 0.05555555555555555 (* z y)))))
  0.3333333333333333))

C

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

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
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 * (sqrt(t) * (0.05555555555555555d0 / (z * y))))) * 0.3333333333333333d0
end function

Java

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

Python

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

Julia

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

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = acos((x * (sqrt(t) * (0.05555555555555555 / (z * y))))) * 0.3333333333333333;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[ArcCos[N[(x * N[(N[Sqrt[t], $MachinePrecision] * N[(0.05555555555555555 / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $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. 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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 97.1%

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

7. Applied rewrites 97.3%

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

Developer Target 1: 98.1% 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 2024337 
(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)))))