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

Percentage Accurate: 98.0% → 98.9%

Time: 32.3s

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: 98.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return exp(-log((3.0 / acos((0.05555555555555555 * (((sqrt(t) / z) / y) * x))))));
}

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(-log((3.0d0 / acos((0.05555555555555555d0 * (((sqrt(t) / z) / y) * x))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   3. lower-*.f64 97.7

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

4. Applied rewrites 97.7%

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

6. Applied rewrites 98.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 98.8

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-pow.f64 N/A

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

   8. unpow-1 N/A

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

   9. un-div-inv N/A

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

   10. lower-/.f64 98.8

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 98.8

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 98.8

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

8. Applied rewrites 98.8%

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

9. Final simplification 98.8%

   \[\leadsto e^{-\log \left(\frac{3}{\cos^{-1} \left(0.05555555555555555 \cdot \left(\frac{\frac{\sqrt{t}}{z}}{y} \cdot x\right)\right)}\right)} \]
10. Add Preprocessing

Alternative 2: 98.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   3. lower-*.f64 97.7

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

4. Applied rewrites 97.7%

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

6. Applied rewrites 98.8%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-log.f64 N/A

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

   4. pow-to-exp N/A

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

   5. lift-pow.f64 98.8

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

8. Applied rewrites 98.8%

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

9. Final simplification 98.8%

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

Alternative 3: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   3. lower-*.f64 97.7

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

4. Applied rewrites 97.7%

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

6. Applied rewrites 98.8%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-log.f64 N/A

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

   4. pow-to-exp N/A

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

   5. lift-pow.f64 98.8

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

8. Applied rewrites 98.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-/l/ N/A

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

   5. associate-*r/ N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 98.4

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

10. Applied rewrites 98.4%

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

11. Final simplification 98.4%

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

Alternative 4: 98.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

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

   3. lower-*.f64 97.7

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

4. Applied rewrites 97.7%

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

5. Final simplification 97.7%

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

Alternative 5: 97.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

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((0.05555555555555555d0 * (((sqrt(t) / z) / y) * x))) * 0.3333333333333333d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in t 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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 97.3%

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

6. Final simplification 97.3%

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

Alternative 6: 98.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

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((((0.05555555555555555d0 * x) / (y * z)) * sqrt(t))) * 0.3333333333333333d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

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

   3. lower-*.f64 97.7

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

4. Applied rewrites 97.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-/l/ N/A

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

   6. associate-*l/ N/A

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

   7. lower-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 96.9

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

6. Applied rewrites 96.9%

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

7. Final simplification 96.9%

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

Developer Target 1: 98.3% 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 2024270 
(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)))))