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

Percentage Accurate: 98.0% → 99.6%

Time: 35.4s

Alternatives: 3

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: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\pi, 0.16666666666666666, \mathsf{fma}\left(\sqrt{\pi}, \sqrt{\pi} \cdot 0.5, -\cos^{-1} \left(\frac{x}{y \cdot \left(z \cdot 18\right)} \cdot \sqrt{t}\right)\right) \cdot -0.3333333333333333\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma
  PI
  0.16666666666666666
  (*
   (fma
    (sqrt PI)
    (* (sqrt PI) 0.5)
    (- (acos (* (/ x (* y (* z 18.0))) (sqrt t)))))
   -0.3333333333333333)))

C

double code(double x, double y, double z, double t) {
	return fma(((double) M_PI), 0.16666666666666666, (fma(sqrt(((double) M_PI)), (sqrt(((double) M_PI)) * 0.5), -acos(((x / (y * (z * 18.0))) * sqrt(t)))) * -0.3333333333333333));
}

Julia

function code(x, y, z, t)
	return fma(pi, 0.16666666666666666, Float64(fma(sqrt(pi), Float64(sqrt(pi) * 0.5), Float64(-acos(Float64(Float64(x / Float64(y * Float64(z * 18.0))) * sqrt(t))))) * -0.3333333333333333))
end

Wolfram

code[x_, y_, z_, t_] := N[(Pi * 0.16666666666666666 + N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[Sqrt[Pi], $MachinePrecision] * 0.5), $MachinePrecision] + (-N[ArcCos[N[(N[(x / N[(y * N[(z * 18.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $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. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 N/A

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

   2. /-lowering-/.f64 N/A

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

   3. *-lowering-*.f64 98.1

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

5. Simplified 98.1%

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

7. Applied egg-rr 96.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\pi, 0.16666666666666666, \sin^{-1} \left(\frac{0.05555555555555555 \cdot \left(x \cdot \sqrt{t}\right)}{y \cdot z}\right) \cdot -0.3333333333333333\right)} \]
8. Step-by-step derivation

   1. asin-acos N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{6}, \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} - \cos^{-1} \left(\frac{\frac{1}{18} \cdot \left(x \cdot \sqrt{t}\right)}{y \cdot z}\right)\right)} \cdot \frac{-1}{3}\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{6}, \left(\frac{\mathsf{PI}\left(\right)}{2} - \cos^{-1} \left(\frac{\color{blue}{\left(\frac{1}{18} \cdot x\right) \cdot \sqrt{t}}}{y \cdot z}\right)\right) \cdot \frac{-1}{3}\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{6}, \left(\frac{\mathsf{PI}\left(\right)}{2} - \cos^{-1} \color{blue}{\left(\frac{\frac{1}{18} \cdot x}{y \cdot z} \cdot \sqrt{t}\right)}\right) \cdot \frac{-1}{3}\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{6}, \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \left(\mathsf{neg}\left(\cos^{-1} \left(\frac{\frac{1}{18} \cdot x}{y \cdot z} \cdot \sqrt{t}\right)\right)\right)\right)} \cdot \frac{-1}{3}\right) \]

   5. add-sqr-sqrt N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{6}, \left(\frac{\color{blue}{\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}}}{2} + \left(\mathsf{neg}\left(\cos^{-1} \left(\frac{\frac{1}{18} \cdot x}{y \cdot z} \cdot \sqrt{t}\right)\right)\right)\right) \cdot \frac{-1}{3}\right) \]

   6. associate-/l* N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{6}, \left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{\sqrt{\mathsf{PI}\left(\right)}}{2}} + \left(\mathsf{neg}\left(\cos^{-1} \left(\frac{\frac{1}{18} \cdot x}{y \cdot z} \cdot \sqrt{t}\right)\right)\right)\right) \cdot \frac{-1}{3}\right) \]

   7. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{6}, \color{blue}{\mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \frac{\sqrt{\mathsf{PI}\left(\right)}}{2}, \mathsf{neg}\left(\cos^{-1} \left(\frac{\frac{1}{18} \cdot x}{y \cdot z} \cdot \sqrt{t}\right)\right)\right)} \cdot \frac{-1}{3}\right) \]

   8. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{6}, \mathsf{fma}\left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}, \frac{\sqrt{\mathsf{PI}\left(\right)}}{2}, \mathsf{neg}\left(\cos^{-1} \left(\frac{\frac{1}{18} \cdot x}{y \cdot z} \cdot \sqrt{t}\right)\right)\right) \cdot \frac{-1}{3}\right) \]

   9. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{6}, \mathsf{fma}\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right)}}, \frac{\sqrt{\mathsf{PI}\left(\right)}}{2}, \mathsf{neg}\left(\cos^{-1} \left(\frac{\frac{1}{18} \cdot x}{y \cdot z} \cdot \sqrt{t}\right)\right)\right) \cdot \frac{-1}{3}\right) \]

   10. div-inv N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{6}, \mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \color{blue}{\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{1}{2}}, \mathsf{neg}\left(\cos^{-1} \left(\frac{\frac{1}{18} \cdot x}{y \cdot z} \cdot \sqrt{t}\right)\right)\right) \cdot \frac{-1}{3}\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{6}, \mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\frac{1}{2}}, \mathsf{neg}\left(\cos^{-1} \left(\frac{\frac{1}{18} \cdot x}{y \cdot z} \cdot \sqrt{t}\right)\right)\right) \cdot \frac{-1}{3}\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{6}, \mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \color{blue}{\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{1}{2}}, \mathsf{neg}\left(\cos^{-1} \left(\frac{\frac{1}{18} \cdot x}{y \cdot z} \cdot \sqrt{t}\right)\right)\right) \cdot \frac{-1}{3}\right) \]

   13. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{6}, \mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{1}{2}, \mathsf{neg}\left(\cos^{-1} \left(\frac{\frac{1}{18} \cdot x}{y \cdot z} \cdot \sqrt{t}\right)\right)\right) \cdot \frac{-1}{3}\right) \]

   14. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{6}, \mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \sqrt{\color{blue}{\mathsf{PI}\left(\right)}} \cdot \frac{1}{2}, \mathsf{neg}\left(\cos^{-1} \left(\frac{\frac{1}{18} \cdot x}{y \cdot z} \cdot \sqrt{t}\right)\right)\right) \cdot \frac{-1}{3}\right) \]

   15. neg-lowering-neg.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{6}, \mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \sqrt{\mathsf{PI}\left(\right)} \cdot \frac{1}{2}, \color{blue}{\mathsf{neg}\left(\cos^{-1} \left(\frac{\frac{1}{18} \cdot x}{y \cdot z} \cdot \sqrt{t}\right)\right)}\right) \cdot \frac{-1}{3}\right) \]

   16. acos-lowering-acos.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{6}, \mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \sqrt{\mathsf{PI}\left(\right)} \cdot \frac{1}{2}, \mathsf{neg}\left(\color{blue}{\cos^{-1} \left(\frac{\frac{1}{18} \cdot x}{y \cdot z} \cdot \sqrt{t}\right)}\right)\right) \cdot \frac{-1}{3}\right) \]

9. Applied egg-rr 98.4%

   \[\leadsto \mathsf{fma}\left(\pi, 0.16666666666666666, \color{blue}{\mathsf{fma}\left(\sqrt{\pi}, \sqrt{\pi} \cdot 0.5, -\cos^{-1} \left(\frac{0.05555555555555555 \cdot \left(x \cdot \sqrt{t}\right)}{y \cdot z}\right)\right)} \cdot -0.3333333333333333\right) \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{6}, \mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \sqrt{\mathsf{PI}\left(\right)} \cdot \frac{1}{2}, \mathsf{neg}\left(\cos^{-1} \left(\frac{\color{blue}{\left(x \cdot \sqrt{t}\right) \cdot \frac{1}{18}}}{y \cdot z}\right)\right)\right) \cdot \frac{-1}{3}\right) \]

    2. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{6}, \mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \sqrt{\mathsf{PI}\left(\right)} \cdot \frac{1}{2}, \mathsf{neg}\left(\cos^{-1} \left(\frac{\color{blue}{x \cdot \left(\sqrt{t} \cdot \frac{1}{18}\right)}}{y \cdot z}\right)\right)\right) \cdot \frac{-1}{3}\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{6}, \mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \sqrt{\mathsf{PI}\left(\right)} \cdot \frac{1}{2}, \mathsf{neg}\left(\cos^{-1} \left(\frac{x \cdot \color{blue}{\left(\frac{1}{18} \cdot \sqrt{t}\right)}}{y \cdot z}\right)\right)\right) \cdot \frac{-1}{3}\right) \]

    4. frac-times N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{6}, \mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \sqrt{\mathsf{PI}\left(\right)} \cdot \frac{1}{2}, \mathsf{neg}\left(\cos^{-1} \color{blue}{\left(\frac{x}{y} \cdot \frac{\frac{1}{18} \cdot \sqrt{t}}{z}\right)}\right)\right) \cdot \frac{-1}{3}\right) \]

    5. associate-*l/ N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{6}, \mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \sqrt{\mathsf{PI}\left(\right)} \cdot \frac{1}{2}, \mathsf{neg}\left(\cos^{-1} \left(\frac{x}{y} \cdot \color{blue}{\left(\frac{\frac{1}{18}}{z} \cdot \sqrt{t}\right)}\right)\right)\right) \cdot \frac{-1}{3}\right) \]

    6. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{6}, \mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \sqrt{\mathsf{PI}\left(\right)} \cdot \frac{1}{2}, \mathsf{neg}\left(\cos^{-1} \color{blue}{\left(\left(\frac{x}{y} \cdot \frac{\frac{1}{18}}{z}\right) \cdot \sqrt{t}\right)}\right)\right) \cdot \frac{-1}{3}\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{6}, \mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \sqrt{\mathsf{PI}\left(\right)} \cdot \frac{1}{2}, \mathsf{neg}\left(\cos^{-1} \color{blue}{\left(\left(\frac{x}{y} \cdot \frac{\frac{1}{18}}{z}\right) \cdot \sqrt{t}\right)}\right)\right) \cdot \frac{-1}{3}\right) \]

    8. clear-num N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{6}, \mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \sqrt{\mathsf{PI}\left(\right)} \cdot \frac{1}{2}, \mathsf{neg}\left(\cos^{-1} \left(\left(\frac{x}{y} \cdot \color{blue}{\frac{1}{\frac{z}{\frac{1}{18}}}}\right) \cdot \sqrt{t}\right)\right)\right) \cdot \frac{-1}{3}\right) \]

    9. frac-times N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{6}, \mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \sqrt{\mathsf{PI}\left(\right)} \cdot \frac{1}{2}, \mathsf{neg}\left(\cos^{-1} \left(\color{blue}{\frac{x \cdot 1}{y \cdot \frac{z}{\frac{1}{18}}}} \cdot \sqrt{t}\right)\right)\right) \cdot \frac{-1}{3}\right) \]

    10. *-rgt-identity N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{6}, \mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \sqrt{\mathsf{PI}\left(\right)} \cdot \frac{1}{2}, \mathsf{neg}\left(\cos^{-1} \left(\frac{\color{blue}{x}}{y \cdot \frac{z}{\frac{1}{18}}} \cdot \sqrt{t}\right)\right)\right) \cdot \frac{-1}{3}\right) \]

    11. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{6}, \mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \sqrt{\mathsf{PI}\left(\right)} \cdot \frac{1}{2}, \mathsf{neg}\left(\cos^{-1} \left(\color{blue}{\frac{x}{y \cdot \frac{z}{\frac{1}{18}}}} \cdot \sqrt{t}\right)\right)\right) \cdot \frac{-1}{3}\right) \]

    12. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{6}, \mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \sqrt{\mathsf{PI}\left(\right)} \cdot \frac{1}{2}, \mathsf{neg}\left(\cos^{-1} \left(\frac{x}{\color{blue}{y \cdot \frac{z}{\frac{1}{18}}}} \cdot \sqrt{t}\right)\right)\right) \cdot \frac{-1}{3}\right) \]

    13. div-inv N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{6}, \mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \sqrt{\mathsf{PI}\left(\right)} \cdot \frac{1}{2}, \mathsf{neg}\left(\cos^{-1} \left(\frac{x}{y \cdot \color{blue}{\left(z \cdot \frac{1}{\frac{1}{18}}\right)}} \cdot \sqrt{t}\right)\right)\right) \cdot \frac{-1}{3}\right) \]

    14. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{6}, \mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \sqrt{\mathsf{PI}\left(\right)} \cdot \frac{1}{2}, \mathsf{neg}\left(\cos^{-1} \left(\frac{x}{y \cdot \color{blue}{\left(z \cdot \frac{1}{\frac{1}{18}}\right)}} \cdot \sqrt{t}\right)\right)\right) \cdot \frac{-1}{3}\right) \]

    15. metadata-eval N/A

        \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right), \frac{1}{6}, \mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \sqrt{\mathsf{PI}\left(\right)} \cdot \frac{1}{2}, \mathsf{neg}\left(\cos^{-1} \left(\frac{x}{y \cdot \left(z \cdot \color{blue}{18}\right)} \cdot \sqrt{t}\right)\right)\right) \cdot \frac{-1}{3}\right) \]

    16. sqrt-lowering-sqrt.f64 99.6

        \[\leadsto \mathsf{fma}\left(\pi, 0.16666666666666666, \mathsf{fma}\left(\sqrt{\pi}, \sqrt{\pi} \cdot 0.5, -\cos^{-1} \left(\frac{x}{y \cdot \left(z \cdot 18\right)} \cdot \color{blue}{\sqrt{t}}\right)\right) \cdot -0.3333333333333333\right) \]

11. Applied egg-rr 99.6%

    \[\leadsto \mathsf{fma}\left(\pi, 0.16666666666666666, \mathsf{fma}\left(\sqrt{\pi}, \sqrt{\pi} \cdot 0.5, -\cos^{-1} \color{blue}{\left(\frac{x}{y \cdot \left(z \cdot 18\right)} \cdot \sqrt{t}\right)}\right) \cdot -0.3333333333333333\right) \]
12. Add Preprocessing

Alternative 2: 97.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 0.3333333333333333d0 * acos(((x * (0.05555555555555555d0 * (sqrt(t) / y))) / z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(0.3333333333333333 * N[ArcCos[N[(N[(x * N[(0.05555555555555555 * N[(N[Sqrt[t], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]], $MachinePrecision]), $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. Taylor expanded in x 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. *-lowering-*.f64 N/A

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

   2. acos-lowering-acos.f64 N/A

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

   3. associate-*r* N/A

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

   4. associate-*r/ N/A

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

   5. /-lowering-/.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-lowering-*.f64 N/A

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

   8. *-commutative N/A

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

   9. *-lowering-*.f64 N/A

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

   10. sqrt-lowering-sqrt.f64 N/A

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

   11. *-lowering-*.f64 96.9

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

5. Simplified 96.9%

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

   1. associate-/l* N/A

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

   2. metadata-eval N/A

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

   3. times-frac N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. associate-*l* N/A

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

   7. times-frac N/A

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

   8. *-commutative N/A

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

   9. associate-*l/ N/A

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

   10. /-lowering-/.f64 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. times-frac N/A

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

   13. metadata-eval N/A

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

   14. *-lowering-*.f64 N/A

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

   15. /-lowering-/.f64 N/A

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

   16. sqrt-lowering-sqrt.f64 97.7

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

7. Applied egg-rr 97.7%

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

Alternative 3: 98.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 0.3333333333333333d0 * acos((sqrt(t) * ((x * 0.05555555555555555d0) / (y * z))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(0.3333333333333333 * N[ArcCos[N[(N[Sqrt[t], $MachinePrecision] * N[(N[(x * 0.05555555555555555), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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. Taylor expanded in x 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. *-lowering-*.f64 N/A

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

   2. acos-lowering-acos.f64 N/A

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

   3. associate-*r* N/A

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

   4. associate-*r/ N/A

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

   5. /-lowering-/.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-lowering-*.f64 N/A

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

   8. *-commutative N/A

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

   9. *-lowering-*.f64 N/A

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

   10. sqrt-lowering-sqrt.f64 N/A

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

   11. *-lowering-*.f64 96.9

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

5. Simplified 96.9%

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

   1. associate-*r* N/A

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

   2. associate-*l/ N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

      \[\leadsto \frac{1}{3} \cdot \cos^{-1} \left(\left(\color{blue}{\frac{\frac{1}{9}}{2}} \cdot \frac{x}{y \cdot z}\right) \cdot \sqrt{t}\right) \]

   5. metadata-eval N/A

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

   6. associate-/r* N/A

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

   7. times-frac N/A

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

   8. times-frac N/A

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

   9. *-commutative N/A

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

   10. associate-*r/ N/A

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

   11. *-commutative N/A

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

   12. *-lowering-*.f64 N/A

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

7. Applied egg-rr 98.1%

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

8. Final simplification 98.1%

   \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \left(\sqrt{t} \cdot \frac{x \cdot 0.05555555555555555}{y \cdot z}\right) \]
9. 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 2024204 
(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)))))