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

Percentage Accurate: 98.0% → 98.6%

Time: 33.9s

Alternatives: 6

Speedup: 1.2×

Specification

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (/ 1.0 3.0) (acos (* (/ (* 3.0 (/ x (* y 27.0))) (* z 2.0)) (sqrt t)))))

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (1.0d0 / 3.0d0) * acos((((3.0d0 * (x / (y * 27.0d0))) / (z * 2.0d0)) * sqrt(t)))
end function

Java

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

Python

def code(x, y, z, t):
	return (1.0 / 3.0) * math.acos((((3.0 * (x / (y * 27.0))) / (z * 2.0)) * math.sqrt(t)))

Julia

function code(x, y, z, t)
	return Float64(Float64(1.0 / 3.0) * acos(Float64(Float64(Float64(3.0 * Float64(x / Float64(y * 27.0))) / Float64(z * 2.0)) * sqrt(t))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (1.0 / 3.0) * acos((((3.0 * (x / (y * 27.0))) / (z * 2.0)) * sqrt(t)));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(1.0 / 3.0), $MachinePrecision] * N[ArcCos[N[(N[(N[(3.0 * N[(x / N[(y * 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * 2.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 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: 98.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y, z, t)
	t_1 = asin(Float64(Float64(x * Float64(sqrt(t) * 0.05555555555555555)) / Float64(y * z)))
	return Float64(0.3333333333333333 * Float64(1.0 / Float64(fma(Float64(sqrt(pi) * sqrt(pi)), 0.5, t_1) / Float64(Float64(Float64(pi * pi) * 0.25) - (t_1 ^ 2.0)))))
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[(0.3333333333333333 * N[(1.0 / N[(N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * 0.5 + t$95$1), $MachinePrecision] / N[(N[(N[(Pi * Pi), $MachinePrecision] * 0.25), $MachinePrecision] - N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 98.3%

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

   1. lift-/.f64 N/A

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

   2. metadata-eval 98.3

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

4. Applied rewrites 98.3%

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

6. Applied rewrites 97.3%

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

   1. lift-PI.f64 N/A

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

   2. add-sqr-sqrt N/A

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

   3. lower-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lower-sqrt.f64 98.8

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

8. Applied rewrites 98.8%

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

Alternative 2: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot {\left(\sqrt{\sqrt{\cos^{-1} \left(\frac{x \cdot \left(\sqrt{t} \cdot 0.05555555555555555\right)}{y \cdot z}\right)}}\right)}^{4} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (*
  0.3333333333333333
  (pow
   (sqrt (sqrt (acos (/ (* x (* (sqrt t) 0.05555555555555555)) (* y z)))))
   4.0)))

C

double code(double x, double y, double z, double t) {
	return 0.3333333333333333 * pow(sqrt(sqrt(acos(((x * (sqrt(t) * 0.05555555555555555)) / (y * z))))), 4.0);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 0.3333333333333333d0 * (sqrt(sqrt(acos(((x * (sqrt(t) * 0.05555555555555555d0)) / (y * z))))) ** 4.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return 0.3333333333333333 * Math.pow(Math.sqrt(Math.sqrt(Math.acos(((x * (Math.sqrt(t) * 0.05555555555555555)) / (y * z))))), 4.0);
}

Python

def code(x, y, z, t):
	return 0.3333333333333333 * math.pow(math.sqrt(math.sqrt(math.acos(((x * (math.sqrt(t) * 0.05555555555555555)) / (y * z))))), 4.0)

Julia

function code(x, y, z, t)
	return Float64(0.3333333333333333 * (sqrt(sqrt(acos(Float64(Float64(x * Float64(sqrt(t) * 0.05555555555555555)) / Float64(y * z))))) ^ 4.0))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 0.3333333333333333 * (sqrt(sqrt(acos(((x * (sqrt(t) * 0.05555555555555555)) / (y * z))))) ^ 4.0);
end

Wolfram

code[x_, y_, z_, t_] := N[(0.3333333333333333 * N[Power[N[Sqrt[N[Sqrt[N[ArcCos[N[(N[(x * N[(N[Sqrt[t], $MachinePrecision] * 0.05555555555555555), $MachinePrecision]), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.3%

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

   1. lift-/.f64 N/A

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

   2. metadata-eval 98.3

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

4. Applied rewrites 98.3%

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

6. Applied rewrites 97.3%

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

8. Applied rewrites 97.4%

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

   1. lift-/.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-log.f64 N/A

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

   5. exp-to-pow N/A

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

   6. pow-flip N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. pow-pow N/A

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

   10. sqr-pow N/A

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

   11. unpow-prod-down N/A

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

10. Applied rewrites 98.8%

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

11. Final simplification 98.8%

    \[\leadsto 0.3333333333333333 \cdot {\left(\sqrt{\sqrt{\cos^{-1} \left(\frac{x \cdot \left(\sqrt{t} \cdot 0.05555555555555555\right)}{y \cdot z}\right)}}\right)}^{4} \]
12. Add Preprocessing

Alternative 3: 98.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \cdot \cos^{-1} \left(\sqrt{t} \cdot \frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (*
  0.3333333333333333
  (acos (* (sqrt t) (/ (* 3.0 (/ x (* y 27.0))) (* z 2.0))))))

C

double code(double x, double y, double z, double t) {
	return 0.3333333333333333 * acos((sqrt(t) * ((3.0 * (x / (y * 27.0))) / (z * 2.0))));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 0.3333333333333333d0 * acos((sqrt(t) * ((3.0d0 * (x / (y * 27.0d0))) / (z * 2.0d0))))
end function

Java

public static double code(double x, double y, double z, double t) {
	return 0.3333333333333333 * Math.acos((Math.sqrt(t) * ((3.0 * (x / (y * 27.0))) / (z * 2.0))));
}

Python

def code(x, y, z, t):
	return 0.3333333333333333 * math.acos((math.sqrt(t) * ((3.0 * (x / (y * 27.0))) / (z * 2.0))))

Julia

function code(x, y, z, t)
	return Float64(0.3333333333333333 * acos(Float64(sqrt(t) * Float64(Float64(3.0 * Float64(x / Float64(y * 27.0))) / Float64(z * 2.0)))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 0.3333333333333333 * acos((sqrt(t) * ((3.0 * (x / (y * 27.0))) / (z * 2.0))));
end

Wolfram

code[x_, y_, z_, t_] := N[(0.3333333333333333 * N[ArcCos[N[(N[Sqrt[t], $MachinePrecision] * N[(N[(3.0 * N[(x / N[(y * 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.3%

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

   1. lift-/.f64 N/A

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

   2. metadata-eval 98.3

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

4. Applied rewrites 98.3%

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

5. Final simplification 98.3%

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

Alternative 4: 98.0% accurate, 1.1× speedup

Math

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

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(Float64(sqrt(t) * Float64(x * Float64(0.05555555555555555 / 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[(N[Sqrt[t], $MachinePrecision] * N[(x * N[(0.05555555555555555 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.3%

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

   1. lift-/.f64 N/A

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

   2. metadata-eval 98.3

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

4. Applied rewrites 98.3%

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

6. Applied rewrites 98.4%

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

7. Final simplification 98.4%

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

Alternative 5: 98.1% accurate, 1.2× speedup

Math

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

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(x / Float64(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[(x / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.3%

   \[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \]
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 98.4

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

5. Applied rewrites 98.4%

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

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

   4. *-commutative N/A

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

   5. lower-*.f64 98.4

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 98.4

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

7. Applied rewrites 98.4%

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

8. Final simplification 98.4%

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

Alternative 6: 97.1% 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 98.3%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-acos.f64 N/A

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

   3. associate-*r/ N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-*.f64 97.3

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

5. Applied rewrites 97.3%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \cos^{-1} \left(\frac{0.05555555555555555 \cdot \left(x \cdot \sqrt{t}\right)}{y \cdot z}\right)} \]
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 2024221 
(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)))))