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

Percentage Accurate: 98.0% → 97.8%

Time: 15.0s

Alternatives: 3

Speedup: N/A×

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: 97.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{t} \cdot -0.05555555555555555\\ t_2 := t\_1 \cdot \frac{\frac{x}{y}}{z}\\ t_3 := \frac{\mathsf{PI}\left(\right)}{2}\\ t_4 := \sin^{-1} \left(\frac{\left(-0.05555555555555555 \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right)\\ \frac{\left({t\_3}^{3} - {t\_4}^{3}\right) \cdot 0.3333333333333333}{{t\_3}^{4} - {\left(t\_4 \cdot \cos^{-1} t\_2\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} t\_2, \cos^{-1} \left(t\_1 \cdot \frac{x}{y \cdot z}\right), {t\_3}^{2}\right) \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (sqrt t) -0.05555555555555555))
        (t_2 (* t_1 (/ (/ x y) z)))
        (t_3 (/ (PI) 2.0))
        (t_4 (asin (/ (* (* -0.05555555555555555 (sqrt t)) x) (* z y)))))
   (*
    (/
     (* (- (pow t_3 3.0) (pow t_4 3.0)) 0.3333333333333333)
     (- (pow t_3 4.0) (pow (* t_4 (acos t_2)) 2.0)))
    (fma (asin t_2) (acos (* t_1 (/ x (* y z)))) (pow t_3 2.0)))))

Derivation

1. Initial program 98.5%

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

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right)}^{3}\right) \cdot 0.3333333333333333}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \color{blue}{\left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)}}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \color{blue}{\frac{\frac{x}{y}}{z}}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\color{blue}{\frac{x}{y}}}{z}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   4. associate-/l/ N/A

      \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \color{blue}{\frac{x}{y \cdot z}}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   5. associate-*r/ N/A

      \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \color{blue}{\left(\frac{\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot x}{y \cdot z}\right)}}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \color{blue}{\left(\frac{\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot x}{y \cdot z}\right)}}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\color{blue}{\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot x}}{y \cdot z}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\color{blue}{\left(\sqrt{t} \cdot \frac{-1}{18}\right)} \cdot x}{y \cdot z}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   9. *-commutative N/A

      \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\color{blue}{\left(\frac{-1}{18} \cdot \sqrt{t}\right)} \cdot x}{y \cdot z}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\color{blue}{\left(\frac{-1}{18} \cdot \sqrt{t}\right)} \cdot x}{y \cdot z}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   11. *-commutative N/A

       \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\left(\frac{-1}{18} \cdot \sqrt{t}\right) \cdot x}{\color{blue}{z \cdot y}}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   12. lower-*.f64 99.5

       \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\left(-0.05555555555555555 \cdot \sqrt{t}\right) \cdot x}{\color{blue}{z \cdot y}}\right)}^{3}\right) \cdot 0.3333333333333333}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

6. Applied rewrites 99.5%

   \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \color{blue}{\left(\frac{\left(-0.05555555555555555 \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right)}}^{3}\right) \cdot 0.3333333333333333}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\left(\frac{-1}{18} \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \color{blue}{\left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)} \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\left(\frac{-1}{18} \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\color{blue}{\left(\sqrt{t} \cdot \frac{-1}{18}\right)} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   3. *-commutative N/A

      \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\left(\frac{-1}{18} \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\color{blue}{\left(\frac{-1}{18} \cdot \sqrt{t}\right)} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\left(\frac{-1}{18} \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\color{blue}{\left(\frac{-1}{18} \cdot \sqrt{t}\right)} \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\left(\frac{-1}{18} \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\frac{-1}{18} \cdot \sqrt{t}\right) \cdot \color{blue}{\frac{\frac{x}{y}}{z}}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   6. lift-/.f64 N/A

      \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\left(\frac{-1}{18} \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\frac{-1}{18} \cdot \sqrt{t}\right) \cdot \frac{\color{blue}{\frac{x}{y}}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   7. associate-/l/ N/A

      \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\left(\frac{-1}{18} \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\frac{-1}{18} \cdot \sqrt{t}\right) \cdot \color{blue}{\frac{x}{y \cdot z}}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   8. *-commutative N/A

      \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\left(\frac{-1}{18} \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\frac{-1}{18} \cdot \sqrt{t}\right) \cdot \frac{x}{\color{blue}{z \cdot y}}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\left(\frac{-1}{18} \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\frac{-1}{18} \cdot \sqrt{t}\right) \cdot \frac{x}{\color{blue}{z \cdot y}}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   10. associate-/l* N/A

       \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\left(\frac{-1}{18} \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \color{blue}{\left(\frac{\left(\frac{-1}{18} \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right)} \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\left(\frac{-1}{18} \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\frac{\color{blue}{\left(\frac{-1}{18} \cdot \sqrt{t}\right) \cdot x}}{z \cdot y}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   12. lift-/.f64 99.5

       \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\left(-0.05555555555555555 \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right)}^{3}\right) \cdot 0.3333333333333333}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \color{blue}{\left(\frac{\left(-0.05555555555555555 \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right)} \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

8. Applied rewrites 99.5%

   \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\left(-0.05555555555555555 \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right)}^{3}\right) \cdot 0.3333333333333333}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \color{blue}{\left(\frac{\left(-0.05555555555555555 \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right)} \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]
9. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\left(\frac{-1}{18} \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\frac{\left(\frac{-1}{18} \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \color{blue}{\frac{\frac{x}{y}}{z}}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\left(\frac{-1}{18} \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\frac{\left(\frac{-1}{18} \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\color{blue}{\frac{x}{y}}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   3. associate-/l/ N/A

      \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\left(\frac{-1}{18} \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\frac{\left(\frac{-1}{18} \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \color{blue}{\frac{x}{y \cdot z}}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   4. *-commutative N/A

      \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\left(\frac{-1}{18} \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\frac{\left(\frac{-1}{18} \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{x}{\color{blue}{z \cdot y}}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\left(\frac{-1}{18} \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\frac{\left(\frac{-1}{18} \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{x}{\color{blue}{z \cdot y}}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   6. lower-/.f64 99.5

      \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\left(-0.05555555555555555 \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right)}^{3}\right) \cdot 0.3333333333333333}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\frac{\left(-0.05555555555555555 \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \color{blue}{\frac{x}{z \cdot y}}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\left(\frac{-1}{18} \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\frac{\left(\frac{-1}{18} \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{x}{\color{blue}{z \cdot y}}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   8. *-commutative N/A

      \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\left(\frac{-1}{18} \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\frac{\left(\frac{-1}{18} \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{x}{\color{blue}{y \cdot z}}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   9. lower-*.f64 99.5

      \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\left(-0.05555555555555555 \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right)}^{3}\right) \cdot 0.3333333333333333}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\frac{\left(-0.05555555555555555 \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{x}{\color{blue}{y \cdot z}}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

10. Applied rewrites 99.5%

    \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\left(-0.05555555555555555 \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right)}^{3}\right) \cdot 0.3333333333333333}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\frac{\left(-0.05555555555555555 \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \color{blue}{\frac{x}{y \cdot z}}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]
11. Add Preprocessing

Alternative 2: 97.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{t} \cdot -0.05555555555555555\\ t_2 := t\_1 \cdot \frac{\frac{x}{y}}{z}\\ t_3 := \sin^{-1} t\_2\\ t_4 := \frac{\mathsf{PI}\left(\right)}{2}\\ \frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{\mathsf{PI}\left(\right)}{8}, {\sin^{-1} \left(\frac{x \cdot -0.05555555555555555}{z} \cdot \frac{\sqrt{t}}{y}\right)}^{3}\right) \cdot 0.3333333333333333}{{t\_4}^{4} - {\left(t\_3 \cdot \cos^{-1} t\_2\right)}^{2}} \cdot \mathsf{fma}\left(t\_3, \cos^{-1} \left(t\_1 \cdot \frac{x}{y \cdot z}\right), {t\_4}^{2}\right) \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (sqrt t) -0.05555555555555555))
        (t_2 (* t_1 (/ (/ x y) z)))
        (t_3 (asin t_2))
        (t_4 (/ (PI) 2.0)))
   (*
    (/
     (*
      (fma
       (* (PI) (PI))
       (/ (PI) 8.0)
       (pow (asin (* (/ (* x -0.05555555555555555) z) (/ (sqrt t) y))) 3.0))
      0.3333333333333333)
     (- (pow t_4 4.0) (pow (* t_3 (acos t_2)) 2.0)))
    (fma t_3 (acos (* t_1 (/ x (* y z)))) (pow t_4 2.0)))))

Derivation

1. Initial program 98.5%

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

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right)}^{3}\right) \cdot 0.3333333333333333}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \color{blue}{\left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)}}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \color{blue}{\frac{\frac{x}{y}}{z}}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\color{blue}{\frac{x}{y}}}{z}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   4. associate-/l/ N/A

      \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \color{blue}{\frac{x}{y \cdot z}}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   5. associate-*r/ N/A

      \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \color{blue}{\left(\frac{\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot x}{y \cdot z}\right)}}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \color{blue}{\left(\frac{\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot x}{y \cdot z}\right)}}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\color{blue}{\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot x}}{y \cdot z}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\color{blue}{\left(\sqrt{t} \cdot \frac{-1}{18}\right)} \cdot x}{y \cdot z}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   9. *-commutative N/A

      \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\color{blue}{\left(\frac{-1}{18} \cdot \sqrt{t}\right)} \cdot x}{y \cdot z}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\color{blue}{\left(\frac{-1}{18} \cdot \sqrt{t}\right)} \cdot x}{y \cdot z}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   11. *-commutative N/A

       \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\left(\frac{-1}{18} \cdot \sqrt{t}\right) \cdot x}{\color{blue}{z \cdot y}}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   12. lower-*.f64 99.5

       \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(\frac{\left(-0.05555555555555555 \cdot \sqrt{t}\right) \cdot x}{\color{blue}{z \cdot y}}\right)}^{3}\right) \cdot 0.3333333333333333}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

6. Applied rewrites 99.5%

   \[\leadsto \frac{\left({\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \color{blue}{\left(\frac{\left(-0.05555555555555555 \cdot \sqrt{t}\right) \cdot x}{z \cdot y}\right)}}^{3}\right) \cdot 0.3333333333333333}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

8. Applied rewrites 99.2%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{\mathsf{PI}\left(\right)}{8}, {\sin^{-1} \left(\frac{x \cdot -0.05555555555555555}{z} \cdot \frac{\sqrt{t}}{y}\right)}^{3}\right)} \cdot 0.3333333333333333}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]
9. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{\mathsf{PI}\left(\right)}{8}, {\sin^{-1} \left(\frac{x \cdot \frac{-1}{18}}{z} \cdot \frac{\sqrt{t}}{y}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \color{blue}{\frac{\frac{x}{y}}{z}}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{\mathsf{PI}\left(\right)}{8}, {\sin^{-1} \left(\frac{x \cdot \frac{-1}{18}}{z} \cdot \frac{\sqrt{t}}{y}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\color{blue}{\frac{x}{y}}}{z}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   3. associate-/l/ N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{\mathsf{PI}\left(\right)}{8}, {\sin^{-1} \left(\frac{x \cdot \frac{-1}{18}}{z} \cdot \frac{\sqrt{t}}{y}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \color{blue}{\frac{x}{y \cdot z}}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   4. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{\mathsf{PI}\left(\right)}{8}, {\sin^{-1} \left(\frac{x \cdot \frac{-1}{18}}{z} \cdot \frac{\sqrt{t}}{y}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{x}{\color{blue}{z \cdot y}}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{\mathsf{PI}\left(\right)}{8}, {\sin^{-1} \left(\frac{x \cdot \frac{-1}{18}}{z} \cdot \frac{\sqrt{t}}{y}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{x}{\color{blue}{z \cdot y}}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   6. lower-/.f64 99.2

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{\mathsf{PI}\left(\right)}{8}, {\sin^{-1} \left(\frac{x \cdot -0.05555555555555555}{z} \cdot \frac{\sqrt{t}}{y}\right)}^{3}\right) \cdot 0.3333333333333333}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \color{blue}{\frac{x}{z \cdot y}}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{\mathsf{PI}\left(\right)}{8}, {\sin^{-1} \left(\frac{x \cdot \frac{-1}{18}}{z} \cdot \frac{\sqrt{t}}{y}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{x}{\color{blue}{z \cdot y}}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   8. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{\mathsf{PI}\left(\right)}{8}, {\sin^{-1} \left(\frac{x \cdot \frac{-1}{18}}{z} \cdot \frac{\sqrt{t}}{y}\right)}^{3}\right) \cdot \frac{1}{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot \frac{-1}{18}\right) \cdot \frac{x}{\color{blue}{y \cdot z}}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

   9. lower-*.f64 99.2

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{\mathsf{PI}\left(\right)}{8}, {\sin^{-1} \left(\frac{x \cdot -0.05555555555555555}{z} \cdot \frac{\sqrt{t}}{y}\right)}^{3}\right) \cdot 0.3333333333333333}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{x}{\color{blue}{y \cdot z}}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]

10. Applied rewrites 99.2%

    \[\leadsto \frac{\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), \frac{\mathsf{PI}\left(\right)}{8}, {\sin^{-1} \left(\frac{x \cdot -0.05555555555555555}{z} \cdot \frac{\sqrt{t}}{y}\right)}^{3}\right) \cdot 0.3333333333333333}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{4} - {\left(\sin^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right) \cdot \cos^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right)\right)}^{2}} \cdot \mathsf{fma}\left(\sin^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \frac{\frac{x}{y}}{z}\right), \cos^{-1} \left(\left(\sqrt{t} \cdot -0.05555555555555555\right) \cdot \color{blue}{\frac{x}{y \cdot z}}\right), {\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2}\right) \]
11. Add Preprocessing

Alternative 3: 97.2% accurate, 1.2× speedup

Math

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

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(Float64(sqrt(t) * Float64(0.05555555555555555 * 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[(N[Sqrt[t], $MachinePrecision] * N[(0.05555555555555555 * x), $MachinePrecision]), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]

Derivation

1. Initial program 98.5%

   \[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \]
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 97.7%

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

7. Applied rewrites 98.5%

   \[\leadsto \cos^{-1} \left(\frac{\sqrt{t} \cdot \left(0.05555555555555555 \cdot x\right)}{z \cdot y}\right) \cdot 0.3333333333333333 \]
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 2024339 
(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)))))