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

Percentage Accurate: 97.9% → 99.7%

Time: 17.5s

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: 97.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (/ 1.0 3.0) (acos (* (/ (* 3.0 (/ x (* y 27.0))) (* z 2.0)) (sqrt t)))))

C

double code(double x, double y, double z, double t) {
	return (1.0 / 3.0) * acos((((3.0 * (x / (y * 27.0))) / (z * 2.0)) * sqrt(t)));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (1.0d0 / 3.0d0) * acos((((3.0d0 * (x / (y * 27.0d0))) / (z * 2.0d0)) * sqrt(t)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (1.0 / 3.0) * Math.acos((((3.0 * (x / (y * 27.0))) / (z * 2.0)) * Math.sqrt(t)));
}

Python

def code(x, y, z, t):
	return (1.0 / 3.0) * math.acos((((3.0 * (x / (y * 27.0))) / (z * 2.0)) * math.sqrt(t)))

Julia

function code(x, y, z, t)
	return Float64(Float64(1.0 / 3.0) * acos(Float64(Float64(Float64(3.0 * Float64(x / Float64(y * 27.0))) / Float64(z * 2.0)) * sqrt(t))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (1.0 / 3.0) * acos((((3.0 * (x / (y * 27.0))) / (z * 2.0)) * sqrt(t)));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(1.0 / 3.0), $MachinePrecision] * N[ArcCos[N[(N[(N[(3.0 * N[(x / N[(y * 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * 2.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.5 \cdot \sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}, 0.3333333333333333, \sin^{-1} \left(\sqrt{t} \cdot \left(0.05555555555555555 \cdot \frac{x}{z \cdot y}\right)\right) \cdot \left(-0.3333333333333333\right)\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma
  (* 0.5 (cbrt (* (* (PI) (PI)) (PI))))
  0.3333333333333333
  (*
   (asin (* (sqrt t) (* 0.05555555555555555 (/ x (* z y)))))
   (- 0.3333333333333333))))

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. lift-acos.f64 N/A

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

   3. acos-asin N/A

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

   4. sub-neg N/A

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

   5. distribute-rgt-in N/A

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

   6. lower-fma.f64 N/A

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

   7. clear-num N/A

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

   8. associate-/r/ N/A

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

   9. metadata-eval N/A

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

   10. lower-*.f64 N/A

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

   11. lower-PI.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

4. Applied rewrites 98.1%

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

   1. lift-PI.f64 N/A

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

   2. add-cbrt-cube N/A

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

   3. lower-cbrt.f64 N/A

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

   4. rem-cube-cbrt N/A

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

   5. add-cbrt-cube N/A

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

   6. lift-PI.f64 N/A

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

   7. lower-pow.f64 99.6

      \[\leadsto \mathsf{fma}\left(0.5 \cdot \sqrt[3]{\color{blue}{{\mathsf{PI}\left(\right)}^{3}}}, 0.3333333333333333, \left(-\sin^{-1} \left(\sqrt{t} \cdot \left(0.05555555555555555 \cdot \frac{\frac{x}{y}}{z}\right)\right)\right) \cdot 0.3333333333333333\right) \]

6. Applied rewrites 99.6%

   \[\leadsto \mathsf{fma}\left(0.5 \cdot \color{blue}{\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}}, 0.3333333333333333, \left(-\sin^{-1} \left(\sqrt{t} \cdot \left(0.05555555555555555 \cdot \frac{\frac{x}{y}}{z}\right)\right)\right) \cdot 0.3333333333333333\right) \]
7. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. unpow3 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 99.6

      \[\leadsto \mathsf{fma}\left(0.5 \cdot \sqrt[3]{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \mathsf{PI}\left(\right)}, 0.3333333333333333, \left(-\sin^{-1} \left(\sqrt{t} \cdot \left(0.05555555555555555 \cdot \frac{\frac{x}{y}}{z}\right)\right)\right) \cdot 0.3333333333333333\right) \]

8. Applied rewrites 99.6%

   \[\leadsto \mathsf{fma}\left(0.5 \cdot \sqrt[3]{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}}, 0.3333333333333333, \left(-\sin^{-1} \left(\sqrt{t} \cdot \left(0.05555555555555555 \cdot \frac{\frac{x}{y}}{z}\right)\right)\right) \cdot 0.3333333333333333\right) \]
9. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-*.f64 N/A

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

   5. lower-/.f64 99.4

      \[\leadsto \mathsf{fma}\left(0.5 \cdot \sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}, 0.3333333333333333, \left(-\sin^{-1} \left(\sqrt{t} \cdot \left(0.05555555555555555 \cdot \color{blue}{\frac{x}{y \cdot z}}\right)\right)\right) \cdot 0.3333333333333333\right) \]

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.4

      \[\leadsto \mathsf{fma}\left(0.5 \cdot \sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}, 0.3333333333333333, \left(-\sin^{-1} \left(\sqrt{t} \cdot \left(0.05555555555555555 \cdot \frac{x}{\color{blue}{z \cdot y}}\right)\right)\right) \cdot 0.3333333333333333\right) \]

10. Applied rewrites 99.4%

    \[\leadsto \mathsf{fma}\left(0.5 \cdot \sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}, 0.3333333333333333, \left(-\sin^{-1} \left(\sqrt{t} \cdot \left(0.05555555555555555 \cdot \color{blue}{\frac{x}{z \cdot y}}\right)\right)\right) \cdot 0.3333333333333333\right) \]

11. Final simplification 99.4%

    \[\leadsto \mathsf{fma}\left(0.5 \cdot \sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}, 0.3333333333333333, \sin^{-1} \left(\sqrt{t} \cdot \left(0.05555555555555555 \cdot \frac{x}{z \cdot y}\right)\right) \cdot \left(-0.3333333333333333\right)\right) \]
12. Add Preprocessing

Alternative 2: 98.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}, 0.16666666666666666, -0.3333333333333333 \cdot \sin^{-1} \left(0.05555555555555555 \cdot \left(\frac{\frac{\sqrt{t}}{z}}{y} \cdot x\right)\right)\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma
  (cbrt (* (* (PI) (PI)) (PI)))
  0.16666666666666666
  (*
   -0.3333333333333333
   (asin (* 0.05555555555555555 (* (/ (/ (sqrt t) z) y) x))))))

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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 98.1%

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

7. Applied rewrites 98.1%

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

9. Applied rewrites 99.6%

   \[\leadsto \mathsf{fma}\left(\sqrt[3]{{\mathsf{PI}\left(\right)}^{3}}, 0.16666666666666666, -0.3333333333333333 \cdot \sin^{-1} \left(0.05555555555555555 \cdot \left(\frac{\frac{\sqrt{t}}{z}}{y} \cdot x\right)\right)\right) \]
10. Step-by-step derivation

11. Applied rewrites 99.6%

    \[\leadsto \mathsf{fma}\left(\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}, 0.16666666666666666, -0.3333333333333333 \cdot \sin^{-1} \left(0.05555555555555555 \cdot \left(\frac{\frac{\sqrt{t}}{z}}{y} \cdot x\right)\right)\right) \]
12. Add Preprocessing

Alternative 3: 98.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. *-commutative N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 98.0

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

6. Applied rewrites 98.0%

   \[\leadsto \cos^{-1} \left(\sqrt{t} \cdot \left(0.05555555555555555 \cdot \color{blue}{\frac{x}{z \cdot y}}\right)\right) \cdot 0.3333333333333333 \]
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 2024324 
(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)))))