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

Percentage Accurate: 98.0% → 98.5%

Time: 23.2s

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: 98.0% 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: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin^{-1} \left(\frac{x \cdot \left(\sqrt{t} \cdot 0.05555555555555555\right)}{y \cdot z}\right)\\ \left(0.25 \cdot \left(\pi \cdot \pi\right) - {t\_1}^{2}\right) \cdot \left(\frac{1}{\mathsf{fma}\left(\pi, 0.5, t\_1\right)} \cdot 0.3333333333333333\right) \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (asin (/ (* x (* (sqrt t) 0.05555555555555555)) (* y z)))))
   (*
    (- (* 0.25 (* PI PI)) (pow t_1 2.0))
    (* (/ 1.0 (fma PI 0.5 t_1)) 0.3333333333333333))))

C

double code(double x, double y, double z, double t) {
	double t_1 = asin(((x * (sqrt(t) * 0.05555555555555555)) / (y * z)));
	return ((0.25 * (((double) M_PI) * ((double) M_PI))) - pow(t_1, 2.0)) * ((1.0 / fma(((double) M_PI), 0.5, t_1)) * 0.3333333333333333);
}

Julia

function code(x, y, z, t)
	t_1 = asin(Float64(Float64(x * Float64(sqrt(t) * 0.05555555555555555)) / Float64(y * z)))
	return Float64(Float64(Float64(0.25 * Float64(pi * pi)) - (t_1 ^ 2.0)) * Float64(Float64(1.0 / fma(pi, 0.5, t_1)) * 0.3333333333333333))
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[ArcSin[N[(N[(x * N[(N[Sqrt[t], $MachinePrecision] * 0.05555555555555555), $MachinePrecision]), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(0.25 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] - N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(Pi * 0.5 + t$95$1), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 97.7%

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

4. Applied rewrites 98.1%

   \[\leadsto \color{blue}{\frac{\left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.125 - {\sin^{-1} \left(\frac{3 \cdot \left(x \cdot \sqrt{t}\right)}{y \cdot \left(z \cdot 54\right)}\right)}^{3}\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(\sin^{-1} \left(\frac{3 \cdot \left(x \cdot \sqrt{t}\right)}{y \cdot \left(z \cdot 54\right)}\right), \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\frac{3 \cdot \left(x \cdot \sqrt{t}\right)}{y \cdot \left(z \cdot 54\right)}\right)\right), \left(\pi \cdot \pi\right) \cdot 0.25\right)}} \]

6. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\left(\cos^{-1} \left(\frac{x \cdot \left(\sqrt{t} \cdot 0.05555555555555555\right)}{y \cdot z}\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\sin^{-1} \left(\frac{x \cdot \left(\sqrt{t} \cdot 0.05555555555555555\right)}{y \cdot z}\right), \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\frac{x \cdot \left(\sqrt{t} \cdot 0.05555555555555555\right)}{y \cdot z}\right)\right), -0.25 \cdot \left(\pi \cdot \pi\right)\right)}\right) \cdot \mathsf{fma}\left(\sin^{-1} \left(\frac{x \cdot \left(\sqrt{t} \cdot 0.05555555555555555\right)}{y \cdot z}\right), \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\frac{x \cdot \left(\sqrt{t} \cdot 0.05555555555555555\right)}{y \cdot z}\right)\right), -0.25 \cdot \left(\pi \cdot \pi\right)\right)} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

8. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\left(0.25 \cdot \left(\pi \cdot \pi\right) - {\sin^{-1} \left(\frac{x \cdot \left(\sqrt{t} \cdot 0.05555555555555555\right)}{y \cdot z}\right)}^{2}\right) \cdot \left(\frac{1}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\frac{x \cdot \left(\sqrt{t} \cdot 0.05555555555555555\right)}{y \cdot z}\right)\right)} \cdot 0.3333333333333333\right)} \]
9. Add Preprocessing

Alternative 2: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot \left(\sqrt{t} \cdot 0.05555555555555555\right)}{y \cdot z}\\ \left(0.25 \cdot \left(\pi \cdot \pi\right) - {\sin^{-1} t\_1}^{2}\right) \cdot \frac{1}{\left(\pi - \cos^{-1} t\_1\right) \cdot 3} \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x (* (sqrt t) 0.05555555555555555)) (* y z))))
   (*
    (- (* 0.25 (* PI PI)) (pow (asin t_1) 2.0))
    (/ 1.0 (* (- PI (acos t_1)) 3.0)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * (sqrt(t) * 0.05555555555555555)) / (y * z);
	return ((0.25 * (((double) M_PI) * ((double) M_PI))) - pow(asin(t_1), 2.0)) * (1.0 / ((((double) M_PI) - acos(t_1)) * 3.0));
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * (Math.sqrt(t) * 0.05555555555555555)) / (y * z);
	return ((0.25 * (Math.PI * Math.PI)) - Math.pow(Math.asin(t_1), 2.0)) * (1.0 / ((Math.PI - Math.acos(t_1)) * 3.0));
}

Python

def code(x, y, z, t):
	t_1 = (x * (math.sqrt(t) * 0.05555555555555555)) / (y * z)
	return ((0.25 * (math.pi * math.pi)) - math.pow(math.asin(t_1), 2.0)) * (1.0 / ((math.pi - math.acos(t_1)) * 3.0))

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * Float64(sqrt(t) * 0.05555555555555555)) / Float64(y * z))
	return Float64(Float64(Float64(0.25 * Float64(pi * pi)) - (asin(t_1) ^ 2.0)) * Float64(1.0 / Float64(Float64(pi - acos(t_1)) * 3.0)))
end

MATLAB

function tmp = code(x, y, z, t)
	t_1 = (x * (sqrt(t) * 0.05555555555555555)) / (y * z);
	tmp = ((0.25 * (pi * pi)) - (asin(t_1) ^ 2.0)) * (1.0 / ((pi - acos(t_1)) * 3.0));
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[(N[Sqrt[t], $MachinePrecision] * 0.05555555555555555), $MachinePrecision]), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(0.25 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] - N[Power[N[ArcSin[t$95$1], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(Pi - N[ArcCos[t$95$1], $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 97.7%

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

4. Applied rewrites 98.1%

   \[\leadsto \color{blue}{\frac{\left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot 0.125 - {\sin^{-1} \left(\frac{3 \cdot \left(x \cdot \sqrt{t}\right)}{y \cdot \left(z \cdot 54\right)}\right)}^{3}\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(\sin^{-1} \left(\frac{3 \cdot \left(x \cdot \sqrt{t}\right)}{y \cdot \left(z \cdot 54\right)}\right), \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\frac{3 \cdot \left(x \cdot \sqrt{t}\right)}{y \cdot \left(z \cdot 54\right)}\right)\right), \left(\pi \cdot \pi\right) \cdot 0.25\right)}} \]

6. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\left(\cos^{-1} \left(\frac{x \cdot \left(\sqrt{t} \cdot 0.05555555555555555\right)}{y \cdot z}\right) \cdot \frac{0.3333333333333333}{\mathsf{fma}\left(\sin^{-1} \left(\frac{x \cdot \left(\sqrt{t} \cdot 0.05555555555555555\right)}{y \cdot z}\right), \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\frac{x \cdot \left(\sqrt{t} \cdot 0.05555555555555555\right)}{y \cdot z}\right)\right), -0.25 \cdot \left(\pi \cdot \pi\right)\right)}\right) \cdot \mathsf{fma}\left(\sin^{-1} \left(\frac{x \cdot \left(\sqrt{t} \cdot 0.05555555555555555\right)}{y \cdot z}\right), \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\frac{x \cdot \left(\sqrt{t} \cdot 0.05555555555555555\right)}{y \cdot z}\right)\right), -0.25 \cdot \left(\pi \cdot \pi\right)\right)} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

8. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\left(0.25 \cdot \left(\pi \cdot \pi\right) - {\sin^{-1} \left(\frac{x \cdot \left(\sqrt{t} \cdot 0.05555555555555555\right)}{y \cdot z}\right)}^{2}\right) \cdot \left(\frac{1}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\frac{x \cdot \left(\sqrt{t} \cdot 0.05555555555555555\right)}{y \cdot z}\right)\right)} \cdot 0.3333333333333333\right)} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. metadata-eval N/A

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

   5. clear-num N/A

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

   6. lower-/.f64 N/A

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

   7. div-inv N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f64 99.6

      \[\leadsto \left(0.25 \cdot \left(\pi \cdot \pi\right) - {\sin^{-1} \left(\frac{x \cdot \left(\sqrt{t} \cdot 0.05555555555555555\right)}{y \cdot z}\right)}^{2}\right) \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(\frac{x \cdot \left(\sqrt{t} \cdot 0.05555555555555555\right)}{y \cdot z}\right)\right) \cdot 3}} \]

10. Applied rewrites 99.6%

    \[\leadsto \left(0.25 \cdot \left(\pi \cdot \pi\right) - {\sin^{-1} \left(\frac{x \cdot \left(\sqrt{t} \cdot 0.05555555555555555\right)}{y \cdot z}\right)}^{2}\right) \cdot \color{blue}{\frac{1}{\left(\pi - \cos^{-1} \left(\frac{x \cdot \left(\sqrt{t} \cdot 0.05555555555555555\right)}{y \cdot z}\right)\right) \cdot 3}} \]
11. Add Preprocessing

Alternative 3: 98.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 98.5

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

5. Applied rewrites 98.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

7. Applied rewrites 98.5%

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

8. Final simplification 98.5%

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

Alternative 4: 97.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

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

   4. associate-*r/ N/A

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

   5. lower-/.f64 N/A

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

   6. associate-*r* 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. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-*.f64 98.1

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

5. Applied rewrites 98.1%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \cos^{-1} \left(\frac{0.05555555555555555 \cdot \left(x \cdot \sqrt{t}\right)}{y \cdot z}\right)} \]
6. 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 2024232 
(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)))))