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

Percentage Accurate: 97.9% → 99.6%

Time: 13.6s

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: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ e^{0.5 \cdot \log \left(0.1111111111111111 \cdot {\cos^{-1} \left(\left(0.05555555555555555 \cdot \frac{x}{y \cdot z}\right) \cdot \sqrt{t}\right)}^{2}\right)} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (exp
  (*
   0.5
   (log
    (*
     0.1111111111111111
     (pow (acos (* (* 0.05555555555555555 (/ x (* y z))) (sqrt t))) 2.0))))))

C

double code(double x, double y, double z, double t) {
	return exp((0.5 * log((0.1111111111111111 * pow(acos(((0.05555555555555555 * (x / (y * z))) * sqrt(t))), 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 = exp((0.5d0 * log((0.1111111111111111d0 * (acos(((0.05555555555555555d0 * (x / (y * z))) * sqrt(t))) ** 2.0d0)))))
end function

Java

public static double code(double x, double y, double z, double t) {
	return Math.exp((0.5 * Math.log((0.1111111111111111 * Math.pow(Math.acos(((0.05555555555555555 * (x / (y * z))) * Math.sqrt(t))), 2.0)))));
}

Python

def code(x, y, z, t):
	return math.exp((0.5 * math.log((0.1111111111111111 * math.pow(math.acos(((0.05555555555555555 * (x / (y * z))) * math.sqrt(t))), 2.0)))))

Julia

function code(x, y, z, t)
	return exp(Float64(0.5 * log(Float64(0.1111111111111111 * (acos(Float64(Float64(0.05555555555555555 * Float64(x / Float64(y * z))) * sqrt(t))) ^ 2.0)))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = exp((0.5 * log((0.1111111111111111 * (acos(((0.05555555555555555 * (x / (y * z))) * sqrt(t))) ^ 2.0)))));
end

Wolfram

code[x_, y_, z_, t_] := N[Exp[N[(0.5 * N[Log[N[(0.1111111111111111 * N[Power[N[ArcCos[N[(N[(0.05555555555555555 * N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 97.7%

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

   1. metadata-eval 97.7%

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

   2. associate-*l/ 97.7%

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

   3. *-commutative 97.7%

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

   4. associate-*l* 97.7%

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

   5. times-frac 97.7%

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

   6. *-commutative 97.7%

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

   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/ 98.1%

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

   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. add-cbrt-cube 98.4%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{\sqrt{t}}{y \cdot \left(18 \cdot z\right)} \cdot x\right)\right) \cdot \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{\sqrt{t}}{y \cdot \left(18 \cdot z\right)} \cdot x\right)\right)\right) \cdot \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{\sqrt{t}}{y \cdot \left(18 \cdot z\right)} \cdot x\right)\right)}} \]

   2. pow1/3 98.5%

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

   3. add-exp-log 98.5%

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

   4. pow-exp 98.5%

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

5. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{e^{\left(3 \cdot \log \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{\sqrt{t}}{y} \cdot \frac{x}{18 \cdot z}\right)\right)\right) \cdot 0.3333333333333333}} \]
6. Step-by-step derivation

   1. add-sqr-sqrt 98.8%

      \[\leadsto e^{\left(3 \cdot \log \color{blue}{\left(\sqrt{0.3333333333333333 \cdot \cos^{-1} \left(\frac{\sqrt{t}}{y} \cdot \frac{x}{18 \cdot z}\right)} \cdot \sqrt{0.3333333333333333 \cdot \cos^{-1} \left(\frac{\sqrt{t}}{y} \cdot \frac{x}{18 \cdot z}\right)}\right)}\right) \cdot 0.3333333333333333} \]

   2. sqrt-unprod 98.8%

      \[\leadsto e^{\left(3 \cdot \log \color{blue}{\left(\sqrt{\left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{\sqrt{t}}{y} \cdot \frac{x}{18 \cdot z}\right)\right) \cdot \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{\sqrt{t}}{y} \cdot \frac{x}{18 \cdot z}\right)\right)}\right)}\right) \cdot 0.3333333333333333} \]

   3. *-commutative 98.8%

      \[\leadsto e^{\left(3 \cdot \log \left(\sqrt{\color{blue}{\left(\cos^{-1} \left(\frac{\sqrt{t}}{y} \cdot \frac{x}{18 \cdot z}\right) \cdot 0.3333333333333333\right)} \cdot \left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{\sqrt{t}}{y} \cdot \frac{x}{18 \cdot z}\right)\right)}\right)\right) \cdot 0.3333333333333333} \]

   4. *-commutative 98.8%

      \[\leadsto e^{\left(3 \cdot \log \left(\sqrt{\left(\cos^{-1} \left(\frac{\sqrt{t}}{y} \cdot \frac{x}{18 \cdot z}\right) \cdot 0.3333333333333333\right) \cdot \color{blue}{\left(\cos^{-1} \left(\frac{\sqrt{t}}{y} \cdot \frac{x}{18 \cdot z}\right) \cdot 0.3333333333333333\right)}}\right)\right) \cdot 0.3333333333333333} \]

   5. swap-sqr 98.8%

      \[\leadsto e^{\left(3 \cdot \log \left(\sqrt{\color{blue}{\left(\cos^{-1} \left(\frac{\sqrt{t}}{y} \cdot \frac{x}{18 \cdot z}\right) \cdot \cos^{-1} \left(\frac{\sqrt{t}}{y} \cdot \frac{x}{18 \cdot z}\right)\right) \cdot \left(0.3333333333333333 \cdot 0.3333333333333333\right)}}\right)\right) \cdot 0.3333333333333333} \]

7. Applied egg-rr 99.7%

   \[\leadsto e^{\left(3 \cdot \log \color{blue}{\left(\sqrt{{\cos^{-1} \left(\frac{\sqrt{t}}{\frac{y}{x \cdot \frac{0.05555555555555555}{z}}}\right)}^{2} \cdot 0.1111111111111111}\right)}\right) \cdot 0.3333333333333333} \]
8. Step-by-step derivation

   1. pow1/2 99.7%

      \[\leadsto e^{\left(3 \cdot \log \color{blue}{\left({\left({\cos^{-1} \left(\frac{\sqrt{t}}{\frac{y}{x \cdot \frac{0.05555555555555555}{z}}}\right)}^{2} \cdot 0.1111111111111111\right)}^{0.5}\right)}\right) \cdot 0.3333333333333333} \]

   2. log-pow 99.7%

      \[\leadsto e^{\left(3 \cdot \color{blue}{\left(0.5 \cdot \log \left({\cos^{-1} \left(\frac{\sqrt{t}}{\frac{y}{x \cdot \frac{0.05555555555555555}{z}}}\right)}^{2} \cdot 0.1111111111111111\right)\right)}\right) \cdot 0.3333333333333333} \]

   3. associate-/r/ 98.8%

      \[\leadsto e^{\left(3 \cdot \left(0.5 \cdot \log \left({\cos^{-1} \color{blue}{\left(\frac{\sqrt{t}}{y} \cdot \left(x \cdot \frac{0.05555555555555555}{z}\right)\right)}}^{2} \cdot 0.1111111111111111\right)\right)\right) \cdot 0.3333333333333333} \]

   4. *-commutative 98.8%

      \[\leadsto e^{\left(3 \cdot \left(0.5 \cdot \log \left({\cos^{-1} \color{blue}{\left(\left(x \cdot \frac{0.05555555555555555}{z}\right) \cdot \frac{\sqrt{t}}{y}\right)}}^{2} \cdot 0.1111111111111111\right)\right)\right) \cdot 0.3333333333333333} \]

9. Applied egg-rr 98.8%

   \[\leadsto e^{\left(3 \cdot \color{blue}{\left(0.5 \cdot \log \left({\cos^{-1} \left(\left(x \cdot \frac{0.05555555555555555}{z}\right) \cdot \frac{\sqrt{t}}{y}\right)}^{2} \cdot 0.1111111111111111\right)\right)}\right) \cdot 0.3333333333333333} \]
10. Step-by-step derivation

    1. associate-*r/ 99.7%

       \[\leadsto e^{\left(3 \cdot \left(0.5 \cdot \log \left({\cos^{-1} \color{blue}{\left(\frac{\left(x \cdot \frac{0.05555555555555555}{z}\right) \cdot \sqrt{t}}{y}\right)}}^{2} \cdot 0.1111111111111111\right)\right)\right) \cdot 0.3333333333333333} \]

    2. associate-*r* 98.8%

       \[\leadsto e^{\left(3 \cdot \left(0.5 \cdot \log \left({\cos^{-1} \left(\frac{\color{blue}{x \cdot \left(\frac{0.05555555555555555}{z} \cdot \sqrt{t}\right)}}{y}\right)}^{2} \cdot 0.1111111111111111\right)\right)\right) \cdot 0.3333333333333333} \]

    3. *-commutative 98.8%

       \[\leadsto e^{\left(3 \cdot \left(0.5 \cdot \log \left({\cos^{-1} \left(\frac{\color{blue}{\left(\frac{0.05555555555555555}{z} \cdot \sqrt{t}\right) \cdot x}}{y}\right)}^{2} \cdot 0.1111111111111111\right)\right)\right) \cdot 0.3333333333333333} \]

    4. associate-/l* 98.0%

       \[\leadsto e^{\left(3 \cdot \left(0.5 \cdot \log \left({\cos^{-1} \color{blue}{\left(\frac{\frac{0.05555555555555555}{z} \cdot \sqrt{t}}{\frac{y}{x}}\right)}}^{2} \cdot 0.1111111111111111\right)\right)\right) \cdot 0.3333333333333333} \]

    5. associate-/l* 99.2%

       \[\leadsto e^{\left(3 \cdot \left(0.5 \cdot \log \left({\cos^{-1} \color{blue}{\left(\frac{\frac{0.05555555555555555}{z}}{\frac{\frac{y}{x}}{\sqrt{t}}}\right)}}^{2} \cdot 0.1111111111111111\right)\right)\right) \cdot 0.3333333333333333} \]

11. Simplified 99.2%

    \[\leadsto e^{\left(3 \cdot \color{blue}{\left(0.5 \cdot \log \left({\cos^{-1} \left(\frac{\frac{0.05555555555555555}{z}}{\frac{\frac{y}{x}}{\sqrt{t}}}\right)}^{2} \cdot 0.1111111111111111\right)\right)}\right) \cdot 0.3333333333333333} \]

12. Taylor expanded in z around 0 100.0%

    \[\leadsto e^{\color{blue}{0.5 \cdot \log \left(0.1111111111111111 \cdot {\cos^{-1} \left(0.05555555555555555 \cdot \left(\frac{x}{y \cdot z} \cdot \sqrt{t}\right)\right)}^{2}\right)}} \]
13. Step-by-step derivation

    1. associate-*r* 100.0%

       \[\leadsto e^{0.5 \cdot \log \left(0.1111111111111111 \cdot {\cos^{-1} \color{blue}{\left(\left(0.05555555555555555 \cdot \frac{x}{y \cdot z}\right) \cdot \sqrt{t}\right)}}^{2}\right)} \]

14. Simplified 100.0%

    \[\leadsto e^{\color{blue}{0.5 \cdot \log \left(0.1111111111111111 \cdot {\cos^{-1} \left(\left(0.05555555555555555 \cdot \frac{x}{y \cdot z}\right) \cdot \sqrt{t}\right)}^{2}\right)}} \]

15. Final simplification 100.0%

    \[\leadsto e^{0.5 \cdot \log \left(0.1111111111111111 \cdot {\cos^{-1} \left(\left(0.05555555555555555 \cdot \frac{x}{y \cdot z}\right) \cdot \sqrt{t}\right)}^{2}\right)} \]

Alternative 2: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{\sqrt{t}}{y} \cdot \frac{x}{z \cdot 18}\right)\right)} + -1 \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+
  (exp
   (log1p (* 0.3333333333333333 (acos (* (/ (sqrt t) y) (/ x (* z 18.0)))))))
  -1.0))

C

double code(double x, double y, double z, double t) {
	return exp(log1p((0.3333333333333333 * acos(((sqrt(t) / y) * (x / (z * 18.0))))))) + -1.0;
}

Java

public static double code(double x, double y, double z, double t) {
	return Math.exp(Math.log1p((0.3333333333333333 * Math.acos(((Math.sqrt(t) / y) * (x / (z * 18.0))))))) + -1.0;
}

Python

def code(x, y, z, t):
	return math.exp(math.log1p((0.3333333333333333 * math.acos(((math.sqrt(t) / y) * (x / (z * 18.0))))))) + -1.0

Julia

function code(x, y, z, t)
	return Float64(exp(log1p(Float64(0.3333333333333333 * acos(Float64(Float64(sqrt(t) / y) * Float64(x / Float64(z * 18.0))))))) + -1.0)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[Exp[N[Log[1 + N[(0.3333333333333333 * N[ArcCos[N[(N[(N[Sqrt[t], $MachinePrecision] / y), $MachinePrecision] * N[(x / N[(z * 18.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 97.7%

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

   1. metadata-eval 97.7%

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

   2. associate-*l/ 97.7%

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

   3. *-commutative 97.7%

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

   4. associate-*l* 97.7%

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

   5. times-frac 97.7%

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

   6. *-commutative 97.7%

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

   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/ 98.1%

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

   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. expm1-log1p-u 98.5%

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

   2. expm1-udef 100.0%

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

   3. associate-*l/ 99.6%

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

   4. times-frac 98.8%

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

5. Applied egg-rr 98.8%

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

6. Final simplification 98.8%

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

Alternative 3: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

   1. metadata-eval 97.7%

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

   2. associate-*l/ 97.7%

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

   3. *-commutative 97.7%

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

   4. associate-*l* 97.7%

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

   5. times-frac 97.7%

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

   6. *-commutative 97.7%

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

   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/ 98.1%

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

   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. expm1-log1p-u 98.5%

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

   2. expm1-udef 100.0%

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

   3. associate-*l/ 99.6%

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

   4. times-frac 98.8%

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

5. Applied egg-rr 98.8%

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

   1. associate-*l/ 99.6%

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

   2. div-inv 99.6%

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

   3. associate-/r* 99.6%

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

   4. metadata-eval 99.6%

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

7. Applied egg-rr 99.6%

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

8. Final simplification 99.6%

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

Alternative 4: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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((x * (sqrt(t) / (y * (z * 18.0d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

   1. metadata-eval 97.7%

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

   2. associate-*l/ 97.7%

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

   3. *-commutative 97.7%

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

   4. associate-*l* 97.7%

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

   5. times-frac 97.7%

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

   6. *-commutative 97.7%

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

   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/ 98.1%

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

   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. Final simplification 98.5%

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

Developer target: 98.1% 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 2023223 
(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)))))