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

Percentage Accurate: 98.0% → 99.5%

Time: 29.2s

Alternatives: 5

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: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[Power[N[(N[(1.0 / N[ArcCos[N[(N[Sqrt[t], $MachinePrecision] * N[(x * N[(0.05555555555555555 / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision], -1.0], $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. *-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 98.3

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

4. Applied rewrites 98.2%

   \[\leadsto \color{blue}{\cos^{-1} \left(\sqrt{t} \cdot \left(x \cdot \frac{3}{\left(54 \cdot y\right) \cdot z}\right)\right) \cdot 0.3333333333333333} \]

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 98.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (3.0d0 / acos(((sqrt(t) * x) * (0.05555555555555555d0 / (y * z))))) ** (-1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[Power[N[(3.0 / N[ArcCos[N[(N[(N[Sqrt[t], $MachinePrecision] * x), $MachinePrecision] * N[(0.05555555555555555 / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $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. *-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 98.3

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

4. Applied rewrites 98.2%

   \[\leadsto \color{blue}{\cos^{-1} \left(\sqrt{t} \cdot \left(x \cdot \frac{3}{\left(54 \cdot y\right) \cdot z}\right)\right) \cdot 0.3333333333333333} \]

6. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. un-div-inv N/A

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

   4. lower-/.f64 99.7

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 99.6

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 99.6

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

8. Applied rewrites 99.6%

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

Alternative 3: 98.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return acos(((((x / (27.0 * y)) * 3.0) / (2.0 * 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(((((x / (27.0d0 * y)) * 3.0d0) / (2.0d0 * z)) * sqrt(t))) * 0.3333333333333333d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 4: 98.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[ArcCos[N[(N[(N[Sqrt[t], $MachinePrecision] * N[(0.05555555555555555 / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $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. *-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 98.3

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

4. Applied rewrites 98.2%

   \[\leadsto \color{blue}{\cos^{-1} \left(\sqrt{t} \cdot \left(x \cdot \frac{3}{\left(54 \cdot y\right) \cdot z}\right)\right) \cdot 0.3333333333333333} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\sqrt{t} \cdot \left(x \cdot \frac{3}{\left(54 \cdot y\right) \cdot z}\right)\right)} \cdot \frac{1}{3} \]

   2. *-commutative N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\left(x \cdot \frac{3}{\left(54 \cdot y\right) \cdot z}\right) \cdot \sqrt{t}\right)} \cdot \frac{1}{3} \]

   3. lift-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{\left(x \cdot \frac{3}{\left(54 \cdot y\right) \cdot z}\right)} \cdot \sqrt{t}\right) \cdot \frac{1}{3} \]

   4. *-commutative N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{\left(\frac{3}{\left(54 \cdot y\right) \cdot z} \cdot x\right)} \cdot \sqrt{t}\right) \cdot \frac{1}{3} \]

   5. associate-*l* N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\frac{3}{\left(54 \cdot y\right) \cdot z} \cdot \left(x \cdot \sqrt{t}\right)\right)} \cdot \frac{1}{3} \]

   6. lift-/.f64 N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{\frac{3}{\left(54 \cdot y\right) \cdot z}} \cdot \left(x \cdot \sqrt{t}\right)\right) \cdot \frac{1}{3} \]

   7. lift-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{3}{\color{blue}{\left(54 \cdot y\right) \cdot z}} \cdot \left(x \cdot \sqrt{t}\right)\right) \cdot \frac{1}{3} \]

   8. lift-*.f64 N/A

      \[\leadsto \cos^{-1} \left(\frac{3}{\color{blue}{\left(54 \cdot y\right)} \cdot z} \cdot \left(x \cdot \sqrt{t}\right)\right) \cdot \frac{1}{3} \]

   9. associate-*l* N/A

      \[\leadsto \cos^{-1} \left(\frac{3}{\color{blue}{54 \cdot \left(y \cdot z\right)}} \cdot \left(x \cdot \sqrt{t}\right)\right) \cdot \frac{1}{3} \]

   10. *-commutative N/A

       \[\leadsto \cos^{-1} \left(\frac{3}{54 \cdot \color{blue}{\left(z \cdot y\right)}} \cdot \left(x \cdot \sqrt{t}\right)\right) \cdot \frac{1}{3} \]

   11. lift-*.f64 N/A

       \[\leadsto \cos^{-1} \left(\frac{3}{54 \cdot \color{blue}{\left(z \cdot y\right)}} \cdot \left(x \cdot \sqrt{t}\right)\right) \cdot \frac{1}{3} \]

   12. associate-/r* N/A

       \[\leadsto \cos^{-1} \left(\color{blue}{\frac{\frac{3}{54}}{z \cdot y}} \cdot \left(x \cdot \sqrt{t}\right)\right) \cdot \frac{1}{3} \]

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. *-commutative N/A

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

   16. associate-*r* N/A

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

6. Applied rewrites 98.2%

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

7. Final simplification 98.2%

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

Alternative 5: 96.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   3. lower-acos.f64 N/A

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

   4. associate-*r/ N/A

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

   5. associate-*r/ N/A

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

   6. associate-*l/ N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sqrt.f64 98.1

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

5. Applied rewrites 98.1%

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

6. Final simplification 98.1%

   \[\leadsto \cos^{-1} \left(\left(\sqrt{t} \cdot x\right) \cdot \frac{0.05555555555555555}{y \cdot z}\right) \cdot 0.3333333333333333 \]
7. 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 2024240 
(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)))))