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

Percentage Accurate: 97.9% → 98.7%

Time: 32.4s

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: 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: 98.7% accurate, 0.7× speedup

Math

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

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (pow (/ 3.0 (acos (* (/ 0.05555555555555555 z) (/ (* x (sqrt t)) y)))) -1.0))

C

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

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
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 / z) * ((x * sqrt(t)) / y)))) ** (-1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = (3.0 / acos(((0.05555555555555555 / z) * ((x * sqrt(t)) / y)))) ^ -1.0;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[Power[N[(3.0 / N[ArcCos[N[(N[(0.05555555555555555 / z), $MachinePrecision] * N[(N[(x * N[Sqrt[t], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

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. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. frac-2neg N/A

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

   9. frac-times N/A

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

   10. *-commutative N/A

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

   11. lower-/.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. distribute-lft-neg-in N/A

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

   14. lower-*.f64 N/A

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

   15. metadata-eval N/A

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

   16. lift-*.f64 N/A

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

   17. associate-*l* N/A

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

   18. lower-*.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. distribute-lft-neg-in N/A

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

4. Applied rewrites 98.1%

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

6. Applied rewrites 99.2%

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

8. Applied rewrites 100.0%

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

9. Final simplification 100.0%

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

Alternative 2: 98.7% accurate, 0.9× speedup

Math

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

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (PI))))
   (*
    0.3333333333333333
    (fma
     (PI)
     0.5
     (fma
      (* t_1 -0.5)
      t_1
      (acos (* (/ 0.05555555555555555 z) (/ (* x (sqrt t)) y))))))))

Derivation

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. 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 98.1

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

4. Applied rewrites 98.1%

   \[\leadsto \color{blue}{\cos^{-1} \left(\sqrt{t} \cdot \left(0.05555555555555555 \cdot \frac{\frac{x}{y}}{z}\right)\right) \cdot 0.3333333333333333} \]
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. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-/l/ N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r/ N/A

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

   7. *-commutative N/A

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

   8. lower-/.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 98.1

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 98.1

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

6. Applied rewrites 98.1%

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

   1. lift-acos.f64 N/A

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

   2. acos-asin N/A

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

   3. lift-PI.f64 N/A

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

   4. div-inv N/A

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

   5. metadata-eval N/A

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

   6. sub-neg N/A

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

   7. lift-*.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. associate-*r/ N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. frac-times N/A

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

   16. lift-/.f64 N/A

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

   17. lift-/.f64 N/A

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

   18. lift-*.f64 N/A

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

8. Applied rewrites 98.4%

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

   1. lift-neg.f64 N/A

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

   2. neg-sub0 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \color{blue}{0 - \sin^{-1} \left(\frac{\sqrt{t} \cdot x}{y} \cdot \frac{\frac{1}{18}}{z}\right)}\right) \cdot \frac{1}{3} \]

   3. lift-asin.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, 0 - \color{blue}{\sin^{-1} \left(\frac{\sqrt{t} \cdot x}{y} \cdot \frac{\frac{1}{18}}{z}\right)}\right) \cdot \frac{1}{3} \]

   4. asin-acos N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, 0 - \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} - \cos^{-1} \left(\frac{\sqrt{t} \cdot x}{y} \cdot \frac{\frac{1}{18}}{z}\right)\right)}\right) \cdot \frac{1}{3} \]

   5. lift-acos.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, 0 - \left(\frac{\mathsf{PI}\left(\right)}{2} - \color{blue}{\cos^{-1} \left(\frac{\sqrt{t} \cdot x}{y} \cdot \frac{\frac{1}{18}}{z}\right)}\right)\right) \cdot \frac{1}{3} \]

   6. lift-PI.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, 0 - \left(\frac{\color{blue}{\mathsf{PI}\left(\right)}}{2} - \cos^{-1} \left(\frac{\sqrt{t} \cdot x}{y} \cdot \frac{\frac{1}{18}}{z}\right)\right)\right) \cdot \frac{1}{3} \]

   7. div-inv N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, 0 - \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} - \cos^{-1} \left(\frac{\sqrt{t} \cdot x}{y} \cdot \frac{\frac{1}{18}}{z}\right)\right)\right) \cdot \frac{1}{3} \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, 0 - \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} - \cos^{-1} \left(\frac{\sqrt{t} \cdot x}{y} \cdot \frac{\frac{1}{18}}{z}\right)\right)\right) \cdot \frac{1}{3} \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, 0 - \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} - \cos^{-1} \left(\frac{\sqrt{t} \cdot x}{y} \cdot \frac{\frac{1}{18}}{z}\right)\right)\right) \cdot \frac{1}{3} \]

   10. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, 0 - \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} - \cos^{-1} \left(\frac{\sqrt{t} \cdot x}{y} \cdot \frac{\frac{1}{18}}{z}\right)\right)\right) \cdot \frac{1}{3} \]

   11. associate--r- N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{2}, \color{blue}{\left(0 - \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) + \cos^{-1} \left(\frac{\sqrt{t} \cdot x}{y} \cdot \frac{\frac{1}{18}}{z}\right)}\right) \cdot \frac{1}{3} \]

   12. neg-sub0 N/A

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

   13. lift-*.f64 N/A

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

   14. distribute-lft-neg-in N/A

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

   15. metadata-eval N/A

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

   16. lift-PI.f64 N/A

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

   17. add-sqr-sqrt N/A

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

   18. associate-*r* N/A

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

10. Applied rewrites 99.9%

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

11. Final simplification 99.9%

    \[\leadsto 0.3333333333333333 \cdot \mathsf{fma}\left(\mathsf{PI}\left(\right), 0.5, \mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)} \cdot -0.5, \sqrt{\mathsf{PI}\left(\right)}, \cos^{-1} \left(\frac{0.05555555555555555}{z} \cdot \frac{x \cdot \sqrt{t}}{y}\right)\right)\right) \]
12. Add Preprocessing

Alternative 3: 97.3% accurate, 1.1× speedup

Math

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

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (/ (acos (* (/ 0.05555555555555555 z) (/ (* x (sqrt t)) y))) 3.0))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return acos(((0.05555555555555555 / z) * ((x * sqrt(t)) / y))) / 3.0;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
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 / z) * ((x * sqrt(t)) / y))) / 3.0d0
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return Math.acos(((0.05555555555555555 / z) * ((x * Math.sqrt(t)) / y))) / 3.0;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return math.acos(((0.05555555555555555 / z) * ((x * math.sqrt(t)) / y))) / 3.0

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(acos(Float64(Float64(0.05555555555555555 / z) * Float64(Float64(x * sqrt(t)) / y))) / 3.0)
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = acos(((0.05555555555555555 / z) * ((x * sqrt(t)) / y))) / 3.0;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[ArcCos[N[(N[(0.05555555555555555 / z), $MachinePrecision] * N[(N[(x * N[Sqrt[t], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 3.0), $MachinePrecision]

Derivation

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. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. frac-2neg N/A

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

   9. frac-times N/A

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

   10. *-commutative N/A

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

   11. lower-/.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. distribute-lft-neg-in N/A

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

   14. lower-*.f64 N/A

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

   15. metadata-eval N/A

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

   16. lift-*.f64 N/A

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

   17. associate-*l* N/A

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

   18. lower-*.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. distribute-lft-neg-in N/A

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

4. Applied rewrites 98.1%

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

6. Applied rewrites 99.2%

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

   1. lift-pow.f64 N/A

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

   2. inv-pow N/A

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

   3. lift-*.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow-1 N/A

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

   6. un-div-inv N/A

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

   7. clear-num N/A

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

   8. lower-/.f64 97.7

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

8. Applied rewrites 98.5%

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

9. Final simplification 98.5%

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

Alternative 4: 98.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \cos^{-1} \left(\frac{-3 \cdot x}{\left(-54 \cdot y\right) \cdot z} \cdot \sqrt{t}\right) \cdot 0.3333333333333333 \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (* (acos (* (/ (* -3.0 x) (* (* -54.0 y) z)) (sqrt t))) 0.3333333333333333))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	return acos((((-3.0 * x) / ((-54.0 * y) * z)) * sqrt(t))) * 0.3333333333333333;
}

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
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(((((-3.0d0) * x) / (((-54.0d0) * y) * z)) * sqrt(t))) * 0.3333333333333333d0
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	return Math.acos((((-3.0 * x) / ((-54.0 * y) * z)) * Math.sqrt(t))) * 0.3333333333333333;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	return math.acos((((-3.0 * x) / ((-54.0 * y) * z)) * math.sqrt(t))) * 0.3333333333333333

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(acos(Float64(Float64(Float64(-3.0 * x) / Float64(Float64(-54.0 * y) * z)) * sqrt(t))) * 0.3333333333333333)
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = acos((((-3.0 * x) / ((-54.0 * y) * z)) * sqrt(t))) * 0.3333333333333333;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[ArcCos[N[(N[(N[(-3.0 * x), $MachinePrecision] / N[(N[(-54.0 * y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

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. 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 98.1

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

4. Applied rewrites 98.1%

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

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-/l/ N/A

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

   6. lift-*.f64 N/A

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

   7. times-frac N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lower-/.f64 98.1

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 98.1

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. lower-*.f64 98.1

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

   22. lift-*.f64 N/A

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

   23. *-commutative N/A

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

   24. lower-*.f64 98.1

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

6. Applied rewrites 98.1%

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

7. Final simplification 98.1%

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

Alternative 5: 98.1% accurate, 1.2× speedup

Math

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

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (*
  (acos (* (/ (* 0.05555555555555555 x) (* z y)) (sqrt t)))
  0.3333333333333333))

C

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

Fortran

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
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) / (z * y)) * sqrt(t))) * 0.3333333333333333d0
end function

Java

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

Python

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

Julia

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

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = acos((((0.05555555555555555 * x) / (z * y)) * sqrt(t))) * 0.3333333333333333;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(N[ArcCos[N[(N[(N[(0.05555555555555555 * x), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

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. 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 98.1

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

4. Applied rewrites 98.1%

   \[\leadsto \color{blue}{\cos^{-1} \left(\sqrt{t} \cdot \left(0.05555555555555555 \cdot \frac{\frac{x}{y}}{z}\right)\right) \cdot 0.3333333333333333} \]
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. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-/l/ N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r/ N/A

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

   7. *-commutative N/A

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

   8. lower-/.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 98.1

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 98.1

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

6. Applied rewrites 98.1%

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

7. Final simplification 98.1%

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

Developer Target 1: 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 2024277 
(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)))))