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

Percentage Accurate: 98.1% → 99.5%

Time: 15.9s

Alternatives: 5

Speedup: 0.7×

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: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin^{-1} \left(x \cdot \left(0.05555555555555555 \cdot \frac{\sqrt{t}}{y \cdot z}\right)\right)\\ \frac{{\pi}^{2} \cdot -0.027777777777777776 - {t_1}^{2} \cdot -0.1111111111111111}{t_1 \cdot -0.3333333333333333 + \pi \cdot -0.16666666666666666} \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (asin (* x (* 0.05555555555555555 (/ (sqrt t) (* y z)))))))
   (/
    (-
     (* (pow PI 2.0) -0.027777777777777776)
     (* (pow t_1 2.0) -0.1111111111111111))
    (+ (* t_1 -0.3333333333333333) (* PI -0.16666666666666666)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = asin((x * (0.05555555555555555 * (sqrt(t) / (y * z)))));
	return ((pow(((double) M_PI), 2.0) * -0.027777777777777776) - (pow(t_1, 2.0) * -0.1111111111111111)) / ((t_1 * -0.3333333333333333) + (((double) M_PI) * -0.16666666666666666));
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.asin((x * (0.05555555555555555 * (Math.sqrt(t) / (y * z)))));
	return ((Math.pow(Math.PI, 2.0) * -0.027777777777777776) - (Math.pow(t_1, 2.0) * -0.1111111111111111)) / ((t_1 * -0.3333333333333333) + (Math.PI * -0.16666666666666666));
}

Python

def code(x, y, z, t):
	t_1 = math.asin((x * (0.05555555555555555 * (math.sqrt(t) / (y * z)))))
	return ((math.pow(math.pi, 2.0) * -0.027777777777777776) - (math.pow(t_1, 2.0) * -0.1111111111111111)) / ((t_1 * -0.3333333333333333) + (math.pi * -0.16666666666666666))

Julia

function code(x, y, z, t)
	t_1 = asin(Float64(x * Float64(0.05555555555555555 * Float64(sqrt(t) / Float64(y * z)))))
	return Float64(Float64(Float64((pi ^ 2.0) * -0.027777777777777776) - Float64((t_1 ^ 2.0) * -0.1111111111111111)) / Float64(Float64(t_1 * -0.3333333333333333) + Float64(pi * -0.16666666666666666)))
end

MATLAB

function tmp = code(x, y, z, t)
	t_1 = asin((x * (0.05555555555555555 * (sqrt(t) / (y * z)))));
	tmp = (((pi ^ 2.0) * -0.027777777777777776) - ((t_1 ^ 2.0) * -0.1111111111111111)) / ((t_1 * -0.3333333333333333) + (pi * -0.16666666666666666));
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[ArcSin[N[(x * N[(0.05555555555555555 * N[(N[Sqrt[t], $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[Power[Pi, 2.0], $MachinePrecision] * -0.027777777777777776), $MachinePrecision] - N[(N[Power[t$95$1, 2.0], $MachinePrecision] * -0.1111111111111111), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 * -0.3333333333333333), $MachinePrecision] + N[(Pi * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 97.3%

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

   1. metadata-eval 97.3%

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

   2. *-commutative 97.3%

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

   3. times-frac 97.3%

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

   4. associate-*l* 97.3%

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

   5. associate-/l/ 98.4%

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

   6. *-commutative 98.4%

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

   7. associate-*r* 98.4%

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

   8. *-commutative 98.4%

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

   9. associate-/l/ 98.1%

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

   10. associate-*l* 98.1%

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

   11. times-frac 98.1%

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

   12. *-commutative 98.1%

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

3. Simplified 98.1%

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

   1. add-sqr-sqrt_binary64 98.1%

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

5. Applied rewrite-once 98.1%

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

7. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\frac{\left(-{\pi}^{2} \cdot 0.027777777777777776\right) - {\sin^{-1} \left(x \cdot \left(\sqrt{t} \cdot \frac{0.05555555555555555}{z \cdot y}\right)\right)}^{2} \cdot -0.1111111111111111}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(x \cdot \left(\sqrt{t} \cdot \frac{0.05555555555555555}{z \cdot y}\right)\right)\right) \cdot -0.3333333333333333}} \]
8. Step-by-step derivation

   1. distribute-rgt-neg-in 99.9%

      \[\leadsto \frac{\color{blue}{{\pi}^{2} \cdot \left(-0.027777777777777776\right)} - {\sin^{-1} \left(x \cdot \left(\sqrt{t} \cdot \frac{0.05555555555555555}{z \cdot y}\right)\right)}^{2} \cdot -0.1111111111111111}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(x \cdot \left(\sqrt{t} \cdot \frac{0.05555555555555555}{z \cdot y}\right)\right)\right) \cdot -0.3333333333333333} \]

   2. metadata-eval 99.9%

      \[\leadsto \frac{{\pi}^{2} \cdot \color{blue}{-0.027777777777777776} - {\sin^{-1} \left(x \cdot \left(\sqrt{t} \cdot \frac{0.05555555555555555}{z \cdot y}\right)\right)}^{2} \cdot -0.1111111111111111}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(x \cdot \left(\sqrt{t} \cdot \frac{0.05555555555555555}{z \cdot y}\right)\right)\right) \cdot -0.3333333333333333} \]

   3. *-commutative 99.9%

      \[\leadsto \frac{{\pi}^{2} \cdot -0.027777777777777776 - {\sin^{-1} \left(x \cdot \color{blue}{\left(\frac{0.05555555555555555}{z \cdot y} \cdot \sqrt{t}\right)}\right)}^{2} \cdot -0.1111111111111111}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(x \cdot \left(\sqrt{t} \cdot \frac{0.05555555555555555}{z \cdot y}\right)\right)\right) \cdot -0.3333333333333333} \]

   4. associate-*l/ 99.9%

      \[\leadsto \frac{{\pi}^{2} \cdot -0.027777777777777776 - {\sin^{-1} \left(x \cdot \color{blue}{\frac{0.05555555555555555 \cdot \sqrt{t}}{z \cdot y}}\right)}^{2} \cdot -0.1111111111111111}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(x \cdot \left(\sqrt{t} \cdot \frac{0.05555555555555555}{z \cdot y}\right)\right)\right) \cdot -0.3333333333333333} \]

   5. associate-*r/ 99.9%

      \[\leadsto \frac{{\pi}^{2} \cdot -0.027777777777777776 - {\sin^{-1} \left(x \cdot \color{blue}{\left(0.05555555555555555 \cdot \frac{\sqrt{t}}{z \cdot y}\right)}\right)}^{2} \cdot -0.1111111111111111}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(x \cdot \left(\sqrt{t} \cdot \frac{0.05555555555555555}{z \cdot y}\right)\right)\right) \cdot -0.3333333333333333} \]

   6. *-commutative 99.9%

      \[\leadsto \frac{{\pi}^{2} \cdot -0.027777777777777776 - {\sin^{-1} \left(x \cdot \left(0.05555555555555555 \cdot \frac{\sqrt{t}}{\color{blue}{y \cdot z}}\right)\right)}^{2} \cdot -0.1111111111111111}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(x \cdot \left(\sqrt{t} \cdot \frac{0.05555555555555555}{z \cdot y}\right)\right)\right) \cdot -0.3333333333333333} \]

9. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{{\pi}^{2} \cdot -0.027777777777777776 - {\sin^{-1} \left(x \cdot \left(0.05555555555555555 \cdot \frac{\sqrt{t}}{y \cdot z}\right)\right)}^{2} \cdot -0.1111111111111111}{\mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(x \cdot \left(0.05555555555555555 \cdot \frac{\sqrt{t}}{y \cdot z}\right)\right)\right) \cdot -0.3333333333333333}} \]
10. Step-by-step derivation

    1. *-commutative 99.9%

       \[\leadsto \frac{{\pi}^{2} \cdot -0.027777777777777776 - {\sin^{-1} \left(x \cdot \left(0.05555555555555555 \cdot \frac{\sqrt{t}}{y \cdot z}\right)\right)}^{2} \cdot -0.1111111111111111}{\color{blue}{-0.3333333333333333 \cdot \mathsf{fma}\left(\pi, 0.5, \sin^{-1} \left(x \cdot \left(0.05555555555555555 \cdot \frac{\sqrt{t}}{y \cdot z}\right)\right)\right)}} \]

    2. fma-udef 99.9%

       \[\leadsto \frac{{\pi}^{2} \cdot -0.027777777777777776 - {\sin^{-1} \left(x \cdot \left(0.05555555555555555 \cdot \frac{\sqrt{t}}{y \cdot z}\right)\right)}^{2} \cdot -0.1111111111111111}{-0.3333333333333333 \cdot \color{blue}{\left(\pi \cdot 0.5 + \sin^{-1} \left(x \cdot \left(0.05555555555555555 \cdot \frac{\sqrt{t}}{y \cdot z}\right)\right)\right)}} \]

    3. metadata-eval 99.9%

       \[\leadsto \frac{{\pi}^{2} \cdot -0.027777777777777776 - {\sin^{-1} \left(x \cdot \left(0.05555555555555555 \cdot \frac{\sqrt{t}}{y \cdot z}\right)\right)}^{2} \cdot -0.1111111111111111}{-0.3333333333333333 \cdot \left(\pi \cdot \color{blue}{\frac{1}{2}} + \sin^{-1} \left(x \cdot \left(0.05555555555555555 \cdot \frac{\sqrt{t}}{y \cdot z}\right)\right)\right)} \]

    4. div-inv 99.9%

       \[\leadsto \frac{{\pi}^{2} \cdot -0.027777777777777776 - {\sin^{-1} \left(x \cdot \left(0.05555555555555555 \cdot \frac{\sqrt{t}}{y \cdot z}\right)\right)}^{2} \cdot -0.1111111111111111}{-0.3333333333333333 \cdot \left(\color{blue}{\frac{\pi}{2}} + \sin^{-1} \left(x \cdot \left(0.05555555555555555 \cdot \frac{\sqrt{t}}{y \cdot z}\right)\right)\right)} \]

    5. distribute-rgt-in 99.9%

       \[\leadsto \frac{{\pi}^{2} \cdot -0.027777777777777776 - {\sin^{-1} \left(x \cdot \left(0.05555555555555555 \cdot \frac{\sqrt{t}}{y \cdot z}\right)\right)}^{2} \cdot -0.1111111111111111}{\color{blue}{\frac{\pi}{2} \cdot -0.3333333333333333 + \sin^{-1} \left(x \cdot \left(0.05555555555555555 \cdot \frac{\sqrt{t}}{y \cdot z}\right)\right) \cdot -0.3333333333333333}} \]

    6. +-commutative 99.9%

       \[\leadsto \frac{{\pi}^{2} \cdot -0.027777777777777776 - {\sin^{-1} \left(x \cdot \left(0.05555555555555555 \cdot \frac{\sqrt{t}}{y \cdot z}\right)\right)}^{2} \cdot -0.1111111111111111}{\color{blue}{\sin^{-1} \left(x \cdot \left(0.05555555555555555 \cdot \frac{\sqrt{t}}{y \cdot z}\right)\right) \cdot -0.3333333333333333 + \frac{\pi}{2} \cdot -0.3333333333333333}} \]

    7. div-inv 99.9%

       \[\leadsto \frac{{\pi}^{2} \cdot -0.027777777777777776 - {\sin^{-1} \left(x \cdot \left(0.05555555555555555 \cdot \frac{\sqrt{t}}{y \cdot z}\right)\right)}^{2} \cdot -0.1111111111111111}{\sin^{-1} \left(x \cdot \left(0.05555555555555555 \cdot \frac{\sqrt{t}}{y \cdot z}\right)\right) \cdot -0.3333333333333333 + \color{blue}{\left(\pi \cdot \frac{1}{2}\right)} \cdot -0.3333333333333333} \]

    8. metadata-eval 99.9%

       \[\leadsto \frac{{\pi}^{2} \cdot -0.027777777777777776 - {\sin^{-1} \left(x \cdot \left(0.05555555555555555 \cdot \frac{\sqrt{t}}{y \cdot z}\right)\right)}^{2} \cdot -0.1111111111111111}{\sin^{-1} \left(x \cdot \left(0.05555555555555555 \cdot \frac{\sqrt{t}}{y \cdot z}\right)\right) \cdot -0.3333333333333333 + \left(\pi \cdot \color{blue}{0.5}\right) \cdot -0.3333333333333333} \]

    9. associate-*l* 99.9%

       \[\leadsto \frac{{\pi}^{2} \cdot -0.027777777777777776 - {\sin^{-1} \left(x \cdot \left(0.05555555555555555 \cdot \frac{\sqrt{t}}{y \cdot z}\right)\right)}^{2} \cdot -0.1111111111111111}{\sin^{-1} \left(x \cdot \left(0.05555555555555555 \cdot \frac{\sqrt{t}}{y \cdot z}\right)\right) \cdot -0.3333333333333333 + \color{blue}{\pi \cdot \left(0.5 \cdot -0.3333333333333333\right)}} \]

    10. metadata-eval 99.9%

        \[\leadsto \frac{{\pi}^{2} \cdot -0.027777777777777776 - {\sin^{-1} \left(x \cdot \left(0.05555555555555555 \cdot \frac{\sqrt{t}}{y \cdot z}\right)\right)}^{2} \cdot -0.1111111111111111}{\sin^{-1} \left(x \cdot \left(0.05555555555555555 \cdot \frac{\sqrt{t}}{y \cdot z}\right)\right) \cdot -0.3333333333333333 + \pi \cdot \color{blue}{-0.16666666666666666}} \]

11. Applied egg-rr 99.9%

    \[\leadsto \frac{{\pi}^{2} \cdot -0.027777777777777776 - {\sin^{-1} \left(x \cdot \left(0.05555555555555555 \cdot \frac{\sqrt{t}}{y \cdot z}\right)\right)}^{2} \cdot -0.1111111111111111}{\color{blue}{\sin^{-1} \left(x \cdot \left(0.05555555555555555 \cdot \frac{\sqrt{t}}{y \cdot z}\right)\right) \cdot -0.3333333333333333 + \pi \cdot -0.16666666666666666}} \]

12. Final simplification 99.9%

    \[\leadsto \frac{{\pi}^{2} \cdot -0.027777777777777776 - {\sin^{-1} \left(x \cdot \left(0.05555555555555555 \cdot \frac{\sqrt{t}}{y \cdot z}\right)\right)}^{2} \cdot -0.1111111111111111}{\sin^{-1} \left(x \cdot \left(0.05555555555555555 \cdot \frac{\sqrt{t}}{y \cdot z}\right)\right) \cdot -0.3333333333333333 + \pi \cdot -0.16666666666666666} \]

Alternative 2: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ {\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{3}{\cos^{-1} \left(x \cdot \left(0.05555555555555555 \cdot \frac{\sqrt{t}}{y \cdot z}\right)\right)}\right)\right)\right)}^{-1} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (pow
  (expm1
   (log1p (/ 3.0 (acos (* x (* 0.05555555555555555 (/ (sqrt t) (* y z))))))))
  -1.0))

C

double code(double x, double y, double z, double t) {
	return pow(expm1(log1p((3.0 / acos((x * (0.05555555555555555 * (sqrt(t) / (y * z)))))))), -1.0);
}

Java

public static double code(double x, double y, double z, double t) {
	return Math.pow(Math.expm1(Math.log1p((3.0 / Math.acos((x * (0.05555555555555555 * (Math.sqrt(t) / (y * z)))))))), -1.0);
}

Python

def code(x, y, z, t):
	return math.pow(math.expm1(math.log1p((3.0 / math.acos((x * (0.05555555555555555 * (math.sqrt(t) / (y * z)))))))), -1.0)

Julia

function code(x, y, z, t)
	return expm1(log1p(Float64(3.0 / acos(Float64(x * Float64(0.05555555555555555 * Float64(sqrt(t) / Float64(y * z)))))))) ^ -1.0
end

Wolfram

code[x_, y_, z_, t_] := N[Power[N[(Exp[N[Log[1 + N[(3.0 / N[ArcCos[N[(x * N[(0.05555555555555555 * N[(N[Sqrt[t], $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 97.3%

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

   1. metadata-eval 97.3%

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

   2. *-commutative 97.3%

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

   3. times-frac 97.3%

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

   4. associate-*l* 97.3%

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

   5. associate-/l/ 98.4%

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

   6. *-commutative 98.4%

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

   7. associate-*r* 98.4%

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

   8. *-commutative 98.4%

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

   9. associate-/l/ 98.1%

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

   10. associate-*l* 98.1%

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

   11. times-frac 98.1%

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

   12. *-commutative 98.1%

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

3. Simplified 98.1%

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

   1. add-sqr-sqrt_binary64 98.1%

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

5. Applied rewrite-once 98.1%

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

7. Applied egg-rr 98.4%

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

   1. neg-mul-1 98.4%

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

   2. associate-/r* 98.4%

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

   3. metadata-eval 98.4%

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

   4. associate-*r/ 98.4%

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

   5. metadata-eval 98.4%

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

   6. *-commutative 98.4%

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

   7. associate-*l/ 98.4%

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

   8. associate-*r/ 98.4%

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

   9. *-commutative 98.4%

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

9. Simplified 98.4%

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

    1. clear-num 98.4%

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

    2. inv-pow 99.9%

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

    3. associate-/l/ 99.9%

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

    4. associate-/r* 99.9%

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

    5. metadata-eval 99.9%

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

11. Applied egg-rr 99.9%

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

    1. expm1-log1p-u_binary64 99.9%

       \[\leadsto \color{blue}{{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{3}{\cos^{-1} \left(x \cdot \left(0.05555555555555555 \cdot \frac{\sqrt{t}}{y \cdot z}\right)\right)}\right)\right)\right)}^{-1}} \]

13. Applied rewrite-once 99.9%

    \[\leadsto {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{3}{\cos^{-1} \left(x \cdot \left(0.05555555555555555 \cdot \frac{\sqrt{t}}{y \cdot z}\right)\right)}\right)\right)\right)}}^{-1} \]

14. Final simplification 99.9%

    \[\leadsto {\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{3}{\cos^{-1} \left(x \cdot \left(0.05555555555555555 \cdot \frac{\sqrt{t}}{y \cdot z}\right)\right)}\right)\right)\right)}^{-1} \]

Alternative 3: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return pow((3.0 / acos((x * (0.05555555555555555 * (sqrt(t) / (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((x * (0.05555555555555555d0 * (sqrt(t) / (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((x * (0.05555555555555555 * (Math.sqrt(t) / (y * z)))))), -1.0);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

   1. metadata-eval 97.3%

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

   2. *-commutative 97.3%

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

   3. times-frac 97.3%

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

   4. associate-*l* 97.3%

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

   5. associate-/l/ 98.4%

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

   6. *-commutative 98.4%

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

   7. associate-*r* 98.4%

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

   8. *-commutative 98.4%

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

   9. associate-/l/ 98.1%

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

   10. associate-*l* 98.1%

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

   11. times-frac 98.1%

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

   12. *-commutative 98.1%

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

3. Simplified 98.1%

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

   1. add-sqr-sqrt_binary64 98.1%

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

5. Applied rewrite-once 98.1%

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

7. Applied egg-rr 98.4%

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

   1. neg-mul-1 98.4%

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

   2. associate-/r* 98.4%

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

   3. metadata-eval 98.4%

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

   4. associate-*r/ 98.4%

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

   5. metadata-eval 98.4%

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

   6. *-commutative 98.4%

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

   7. associate-*l/ 98.4%

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

   8. associate-*r/ 98.4%

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

   9. *-commutative 98.4%

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

9. Simplified 98.4%

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

    1. clear-num 98.4%

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

    2. inv-pow 99.9%

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

    3. associate-/l/ 99.9%

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

    4. associate-/r* 99.9%

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

    5. metadata-eval 99.9%

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

11. Applied egg-rr 99.9%

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

12. Final simplification 99.9%

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

Alternative 4: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return -1.0 / (-3.0 / acos((x * (0.05555555555555555 * (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 = (-1.0d0) / ((-3.0d0) / acos((x * (0.05555555555555555d0 * (sqrt(t) / (y * z))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

   1. metadata-eval 97.3%

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

   2. *-commutative 97.3%

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

   3. times-frac 97.3%

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

   4. associate-*l* 97.3%

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

   5. associate-/l/ 98.4%

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

   6. *-commutative 98.4%

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

   7. associate-*r* 98.4%

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

   8. *-commutative 98.4%

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

   9. associate-/l/ 98.1%

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

   10. associate-*l* 98.1%

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

   11. times-frac 98.1%

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

   12. *-commutative 98.1%

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

3. Simplified 98.1%

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

   1. add-sqr-sqrt_binary64 98.1%

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

5. Applied rewrite-once 98.1%

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

7. Applied egg-rr 98.4%

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

   1. neg-mul-1 98.4%

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

   2. associate-/r* 98.4%

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

   3. metadata-eval 98.4%

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

   4. associate-*r/ 98.4%

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

   5. metadata-eval 98.4%

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

   6. *-commutative 98.4%

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

   7. associate-*l/ 98.4%

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

   8. associate-*r/ 98.4%

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

   9. *-commutative 98.4%

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

9. Simplified 98.4%

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

10. Final simplification 98.4%

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

Alternative 5: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

   1. metadata-eval 97.3%

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

   2. *-commutative 97.3%

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

   3. times-frac 97.3%

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

   4. associate-*l* 97.3%

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

   5. associate-/l/ 98.4%

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

   6. *-commutative 98.4%

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

   7. associate-*r* 98.4%

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

   8. *-commutative 98.4%

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

   9. associate-/l/ 98.1%

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

   10. associate-*l* 98.1%

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

   11. times-frac 98.1%

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

   12. *-commutative 98.1%

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

3. Simplified 97.3%

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

4. Final simplification 97.3%

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

Developer target: 98.0% 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 2023297 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, D"
  :precision binary64

  :herbie-target
  (/ (acos (* (/ (/ x 27.0) (* y z)) (/ (sqrt t) (/ 2.0 3.0)))) 3.0)

  (* (/ 1.0 3.0) (acos (* (/ (* 3.0 (/ x (* y 27.0))) (* z 2.0)) (sqrt t)))))