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

Percentage Accurate: 97.9% → 99.3%

Time: 11.6s

Alternatives: 4

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{t} \leq 4.5 \cdot 10^{+45}:\\ \;\;\;\;e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(x \cdot \left(\frac{0.05555555555555555}{y} \cdot \frac{\sqrt{t}}{z}\right)\right)\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \cos^{-1} \left(x \cdot \left(\sqrt{t} \cdot \frac{\frac{0.05555555555555555}{y}}{z}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (sqrt t) 4.5e+45)
   (+
    (exp
     (log1p
      (*
       0.3333333333333333
       (acos (* x (* (/ 0.05555555555555555 y) (/ (sqrt t) z)))))))
    -1.0)
   (*
    0.3333333333333333
    (acos (* x (* (sqrt t) (/ (/ 0.05555555555555555 y) z)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (sqrt(t) <= 4.5e+45) {
		tmp = exp(log1p((0.3333333333333333 * acos((x * ((0.05555555555555555 / y) * (sqrt(t) / z))))))) + -1.0;
	} else {
		tmp = 0.3333333333333333 * acos((x * (sqrt(t) * ((0.05555555555555555 / y) / z))));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (Math.sqrt(t) <= 4.5e+45) {
		tmp = Math.exp(Math.log1p((0.3333333333333333 * Math.acos((x * ((0.05555555555555555 / y) * (Math.sqrt(t) / z))))))) + -1.0;
	} else {
		tmp = 0.3333333333333333 * Math.acos((x * (Math.sqrt(t) * ((0.05555555555555555 / y) / z))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if math.sqrt(t) <= 4.5e+45:
		tmp = math.exp(math.log1p((0.3333333333333333 * math.acos((x * ((0.05555555555555555 / y) * (math.sqrt(t) / z))))))) + -1.0
	else:
		tmp = 0.3333333333333333 * math.acos((x * (math.sqrt(t) * ((0.05555555555555555 / y) / z))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (sqrt(t) <= 4.5e+45)
		tmp = Float64(exp(log1p(Float64(0.3333333333333333 * acos(Float64(x * Float64(Float64(0.05555555555555555 / y) * Float64(sqrt(t) / z))))))) + -1.0);
	else
		tmp = Float64(0.3333333333333333 * acos(Float64(x * Float64(sqrt(t) * Float64(Float64(0.05555555555555555 / y) / z)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[Sqrt[t], $MachinePrecision], 4.5e+45], N[(N[Exp[N[Log[1 + N[(0.3333333333333333 * N[ArcCos[N[(x * N[(N[(0.05555555555555555 / y), $MachinePrecision] * N[(N[Sqrt[t], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision], N[(0.3333333333333333 * N[ArcCos[N[(x * N[(N[Sqrt[t], $MachinePrecision] * N[(N[(0.05555555555555555 / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 t) < 4.4999999999999998e45

   1. Initial program 98.0%

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

      1. metadata-eval 98.0%

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

      2. times-frac 98.0%

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

      3. associate-*r/ 98.0%

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

      4. times-frac 96.9%

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

      5. *-commutative 96.9%

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

      6. times-frac 98.0%

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

      7. associate-/l* 98.0%

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

      8. associate-*l/ 98.0%

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

      9. associate-*r/ 98.0%

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

      10. associate-/r/ 98.0%

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

      11. *-commutative 98.0%

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

      12. associate-*r* 98.0%

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

      13. metadata-eval 98.0%

          \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(\frac{x}{z} \cdot \frac{\sqrt{t}}{\left(\color{blue}{0.6666666666666666} \cdot 27\right) \cdot y}\right) \]

      14. metadata-eval 98.0%

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

   3. Simplified 98.0%

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

      1. expm1-log1p-u 98.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{x}{z} \cdot \frac{\sqrt{t}}{18 \cdot y}\right)\right)\right)} \]

      2. expm1-udef 99.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{x}{z} \cdot \frac{\sqrt{t}}{18 \cdot y}\right)\right)} - 1} \]

   5. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(x \cdot \left(\frac{0.05555555555555555}{y} \cdot \frac{\sqrt{t}}{z}\right)\right)\right)} - 1} \]

   if 4.4999999999999998e45 < (sqrt.f64 t)

   1. Initial program 98.1%

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

      1. metadata-eval 98.1%

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

      2. associate-*l/ 98.1%

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

      3. *-commutative 98.1%

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

      4. associate-*l* 98.1%

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

      5. times-frac 98.1%

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

      6. *-commutative 98.1%

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

      7. associate-/l/ 98.5%

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

      8. associate-*r/ 95.7%

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

      9. associate-*l/ 98.5%

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

   3. Simplified 98.5%

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

      1. clear-num 98.5%

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

      2. associate-/r/ 98.5%

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

      3. associate-*r* 98.5%

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

      4. *-commutative 98.5%

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

      5. associate-/r* 98.5%

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

      6. associate-/r* 98.5%

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

      7. metadata-eval 98.5%

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

   5. Applied egg-rr 98.5%

      \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(\color{blue}{\left(\frac{\frac{0.05555555555555555}{y}}{z} \cdot \sqrt{t}\right)} \cdot x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{t} \leq 4.5 \cdot 10^{+45}:\\ \;\;\;\;e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(x \cdot \left(\frac{0.05555555555555555}{y} \cdot \frac{\sqrt{t}}{z}\right)\right)\right)} + -1\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \cos^{-1} \left(x \cdot \left(\sqrt{t} \cdot \frac{\frac{0.05555555555555555}{y}}{z}\right)\right)\\ \end{array} \]

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{\left(x \cdot \frac{0.05555555555555555}{y}\right) \cdot \sqrt{t}}{z}\right)\right)} + -1 \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+
  (exp
   (log1p
    (*
     0.3333333333333333
     (acos (/ (* (* x (/ 0.05555555555555555 y)) (sqrt t)) z)))))
  -1.0))

C

double code(double x, double y, double z, double t) {
	return exp(log1p((0.3333333333333333 * acos((((x * (0.05555555555555555 / y)) * sqrt(t)) / z))))) + -1.0;
}

Java

public static double code(double x, double y, double z, double t) {
	return Math.exp(Math.log1p((0.3333333333333333 * Math.acos((((x * (0.05555555555555555 / y)) * Math.sqrt(t)) / z))))) + -1.0;
}

Python

def code(x, y, z, t):
	return math.exp(math.log1p((0.3333333333333333 * math.acos((((x * (0.05555555555555555 / y)) * math.sqrt(t)) / z))))) + -1.0

Julia

function code(x, y, z, t)
	return Float64(exp(log1p(Float64(0.3333333333333333 * acos(Float64(Float64(Float64(x * Float64(0.05555555555555555 / y)) * sqrt(t)) / z))))) + -1.0)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[Exp[N[Log[1 + N[(0.3333333333333333 * N[ArcCos[N[(N[(N[(x * N[(0.05555555555555555 / y), $MachinePrecision]), $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 98.0%

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

   1. metadata-eval 98.0%

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

   2. times-frac 98.0%

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

   3. associate-*r/ 98.0%

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

   4. times-frac 97.3%

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

   5. *-commutative 97.3%

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

   6. times-frac 97.9%

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

   7. associate-/l* 97.9%

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

   8. associate-*l/ 97.9%

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

   9. associate-*r/ 96.9%

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

   10. associate-/r/ 96.9%

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

   11. *-commutative 96.9%

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

   12. associate-*r* 96.9%

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

   13. metadata-eval 96.9%

       \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(\frac{x}{z} \cdot \frac{\sqrt{t}}{\left(\color{blue}{0.6666666666666666} \cdot 27\right) \cdot y}\right) \]

   14. metadata-eval 96.9%

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

3. Simplified 96.9%

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

   1. expm1-log1p-u 96.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{x}{z} \cdot \frac{\sqrt{t}}{18 \cdot y}\right)\right)\right)} \]

   2. expm1-udef 98.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{x}{z} \cdot \frac{\sqrt{t}}{18 \cdot y}\right)\right)} - 1} \]

5. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(x \cdot \left(\frac{0.05555555555555555}{y} \cdot \frac{\sqrt{t}}{z}\right)\right)\right)} - 1} \]
6. Step-by-step derivation

   1. associate-*r* 97.6%

      \[\leadsto e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \color{blue}{\left(\left(x \cdot \frac{0.05555555555555555}{y}\right) \cdot \frac{\sqrt{t}}{z}\right)}\right)} - 1 \]

   2. associate-*r/ 99.4%

      \[\leadsto e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \color{blue}{\left(\frac{\left(x \cdot \frac{0.05555555555555555}{y}\right) \cdot \sqrt{t}}{z}\right)}\right)} - 1 \]

7. Applied egg-rr 99.4%

   \[\leadsto e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \color{blue}{\left(\frac{\left(x \cdot \frac{0.05555555555555555}{y}\right) \cdot \sqrt{t}}{z}\right)}\right)} - 1 \]

8. Final simplification 99.4%

   \[\leadsto e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{\left(x \cdot \frac{0.05555555555555555}{y}\right) \cdot \sqrt{t}}{z}\right)\right)} + -1 \]

Alternative 3: 97.9% accurate, 1.0× 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 98.0%

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

2. Final simplification 98.0%

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

Alternative 4: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

   1. metadata-eval 98.0%

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

   2. associate-*l/ 98.0%

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

   3. *-commutative 98.0%

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

   4. associate-*l* 98.0%

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

   5. times-frac 98.0%

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

   6. *-commutative 98.0%

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

   7. associate-/l/ 97.3%

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

   8. associate-*r/ 96.6%

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

   9. associate-*l/ 97.3%

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

3. Simplified 97.3%

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

   1. clear-num 97.3%

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

   2. associate-/r/ 97.3%

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

   3. associate-*r* 97.3%

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

   4. *-commutative 97.3%

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

   5. associate-/r* 97.3%

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

   6. associate-/r* 97.3%

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

   7. metadata-eval 97.3%

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

5. Applied egg-rr 97.3%

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

6. Final simplification 97.3%

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

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 2023200 
(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)))))