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

Percentage Accurate: 98.0% → 99.4%

Time: 24.6s

Alternatives: 4

Speedup: 0.4×

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: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{0.05555555555555555 \cdot \frac{x}{y}}}\right)\\ \left(0.3333333333333333 \cdot \frac{1}{\frac{\pi}{2} + t\_1}\right) \cdot \left(\frac{\pi \cdot \pi}{4} - {t\_1}^{2}\right) \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (asin (/ (sqrt t) (/ z (* 0.05555555555555555 (/ x y)))))))
   (*
    (* 0.3333333333333333 (/ 1.0 (+ (/ PI 2.0) t_1)))
    (- (/ (* PI PI) 4.0) (pow t_1 2.0)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = asin((sqrt(t) / (z / (0.05555555555555555 * (x / y)))));
	return (0.3333333333333333 * (1.0 / ((((double) M_PI) / 2.0) + t_1))) * (((((double) M_PI) * ((double) M_PI)) / 4.0) - pow(t_1, 2.0));
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.asin((Math.sqrt(t) / (z / (0.05555555555555555 * (x / y)))));
	return (0.3333333333333333 * (1.0 / ((Math.PI / 2.0) + t_1))) * (((Math.PI * Math.PI) / 4.0) - Math.pow(t_1, 2.0));
}

Python

def code(x, y, z, t):
	t_1 = math.asin((math.sqrt(t) / (z / (0.05555555555555555 * (x / y)))))
	return (0.3333333333333333 * (1.0 / ((math.pi / 2.0) + t_1))) * (((math.pi * math.pi) / 4.0) - math.pow(t_1, 2.0))

Julia

function code(x, y, z, t)
	t_1 = asin(Float64(sqrt(t) / Float64(z / Float64(0.05555555555555555 * Float64(x / y)))))
	return Float64(Float64(0.3333333333333333 * Float64(1.0 / Float64(Float64(pi / 2.0) + t_1))) * Float64(Float64(Float64(pi * pi) / 4.0) - (t_1 ^ 2.0)))
end

MATLAB

function tmp = code(x, y, z, t)
	t_1 = asin((sqrt(t) / (z / (0.05555555555555555 * (x / y)))));
	tmp = (0.3333333333333333 * (1.0 / ((pi / 2.0) + t_1))) * (((pi * pi) / 4.0) - (t_1 ^ 2.0));
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[ArcSin[N[(N[Sqrt[t], $MachinePrecision] / N[(z / N[(0.05555555555555555 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(0.3333333333333333 * N[(1.0 / N[(N[(Pi / 2.0), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(Pi * Pi), $MachinePrecision] / 4.0), $MachinePrecision] - N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 98.1%

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

   1. *-commutative N/A

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

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

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

4. Applied egg-rr 98.1%

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

6. Applied egg-rr 98.1%

   \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\pi}{2} + \sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{0.05555555555555555}{\frac{y}{x}}}}\right)}{\frac{\pi \cdot \pi}{4} - {\sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{0.05555555555555555}{\frac{y}{x}}}}\right)}^{2}}}} \cdot 0.3333333333333333 \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{1}{3} \cdot \color{blue}{\frac{1}{\frac{\frac{\mathsf{PI}\left(\right)}{2} + \sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{\frac{1}{18}}{\frac{y}{x}}}}\right)}{\frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{4} - {\sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{\frac{1}{18}}{\frac{y}{x}}}}\right)}^{2}}}} \]

   2. associate-/r/ N/A

      \[\leadsto \frac{1}{3} \cdot \left(\frac{1}{\frac{\mathsf{PI}\left(\right)}{2} + \sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{\frac{1}{18}}{\frac{y}{x}}}}\right)} \cdot \color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{4} - {\sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{\frac{1}{18}}{\frac{y}{x}}}}\right)}^{2}\right)}\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(\frac{1}{3} \cdot \frac{1}{\frac{\mathsf{PI}\left(\right)}{2} + \sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{\frac{1}{18}}{\frac{y}{x}}}}\right)}\right) \cdot \color{blue}{\left(\frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{4} - {\sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{\frac{1}{18}}{\frac{y}{x}}}}\right)}^{2}\right)} \]

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

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

8. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{\frac{\pi}{2} + \sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{0.05555555555555555 \cdot \frac{x}{y}}}\right)}\right) \cdot \left(\frac{\pi \cdot \pi}{4} - {\sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{0.05555555555555555 \cdot \frac{x}{y}}}\right)}^{2}\right)} \]
9. Add Preprocessing

Alternative 2: 97.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (/
  0.3333333333333333
  (/ 1.0 (acos (/ (sqrt t) (/ z (* 0.05555555555555555 (/ x y))))))))

C

double code(double x, double y, double z, double t) {
	return 0.3333333333333333 / (1.0 / acos((sqrt(t) / (z / (0.05555555555555555 * (x / 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 / (1.0d0 / acos((sqrt(t) / (z / (0.05555555555555555d0 * (x / y))))))
end function

Java

public static double code(double x, double y, double z, double t) {
	return 0.3333333333333333 / (1.0 / Math.acos((Math.sqrt(t) / (z / (0.05555555555555555 * (x / y))))));
}

Python

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

Julia

function code(x, y, z, t)
	return Float64(0.3333333333333333 / Float64(1.0 / acos(Float64(sqrt(t) / Float64(z / Float64(0.05555555555555555 * Float64(x / y)))))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 0.3333333333333333 / (1.0 / acos((sqrt(t) / (z / (0.05555555555555555 * (x / y))))));
end

Wolfram

code[x_, y_, z_, t_] := N[(0.3333333333333333 / N[(1.0 / N[ArcCos[N[(N[Sqrt[t], $MachinePrecision] / N[(z / N[(0.05555555555555555 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.1%

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

   1. *-commutative N/A

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

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

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

4. Applied egg-rr 98.1%

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

6. Applied egg-rr 98.1%

   \[\leadsto \color{blue}{\frac{1}{\frac{\frac{\pi}{2} + \sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{0.05555555555555555}{\frac{y}{x}}}}\right)}{\frac{\pi \cdot \pi}{4} - {\sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{0.05555555555555555}{\frac{y}{x}}}}\right)}^{2}}}} \cdot 0.3333333333333333 \]
7. Step-by-step derivation

   1. associate-*l/ N/A

      \[\leadsto \frac{1 \cdot \frac{1}{3}}{\color{blue}{\frac{\frac{\mathsf{PI}\left(\right)}{2} + \sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{\frac{1}{18}}{\frac{y}{x}}}}\right)}{\frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{4} - {\sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{\frac{1}{18}}{\frac{y}{x}}}}\right)}^{2}}}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{3}}{\frac{\color{blue}{\frac{\mathsf{PI}\left(\right)}{2} + \sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{\frac{1}{18}}{\frac{y}{x}}}}\right)}}{\frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{4} - {\sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{\frac{1}{18}}{\frac{y}{x}}}}\right)}^{2}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \color{blue}{\left(\frac{\frac{\mathsf{PI}\left(\right)}{2} + \sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{\frac{1}{18}}{\frac{y}{x}}}}\right)}{\frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{4} - {\sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{\frac{1}{18}}{\frac{y}{x}}}}\right)}^{2}}\right)}\right) \]

   4. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \left(\frac{1}{\color{blue}{\frac{\frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{4} - {\sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{\frac{1}{18}}{\frac{y}{x}}}}\right)}^{2}}{\frac{\mathsf{PI}\left(\right)}{2} + \sin^{-1} \left(\frac{\sqrt{t}}{\frac{z}{\frac{\frac{1}{18}}{\frac{y}{x}}}}\right)}}}\right)\right) \]

8. Applied egg-rr 98.1%

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

Alternative 3: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 0.3333333333333333d0 * acos((sqrt(t) / (z * (18.0d0 / (x / y)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.1%

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

   1. *-commutative N/A

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

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

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

4. Applied egg-rr 98.1%

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

5. Final simplification 98.1%

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

Alternative 4: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.1%

   \[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \]
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. *-lowering-*.f64 N/A

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

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

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

   3. associate-*r* N/A

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

   4. associate-/r* N/A

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

   5. associate-*r/ N/A

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

   6. /-lowering-/.f64 N/A

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

   7. associate-*r/ N/A

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

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

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

   9. associate-*r* N/A

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

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

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

   11. *-commutative N/A

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

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

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

   13. sqrt-lowering-sqrt.f64 97.7%

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

5. Simplified 97.7%

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

6. Final simplification 97.7%

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

Developer Target 1: 97.9% 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 2024191 
(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)))))