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

Percentage Accurate: 98.0% → 99.1%

Time: 14.7s

Alternatives: 6

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: 99.1% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (PI) 2.0))
        (t_2 (* (* (sqrt t) -0.05555555555555555) (/ (/ x y) z)))
        (t_3 (asin t_2))
        (t_4 (acos t_2)))
   (*
    (/
     (* (- (pow t_1 3.0) (pow t_3 3.0)) 0.3333333333333333)
     (- (pow t_1 4.0) (pow (* t_3 t_4) 2.0)))
    (fma t_3 t_4 (pow t_1 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 100.0%

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

Alternative 2: 98.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sqrt{t}}{z}\\ t_2 := \sin^{-1} \left(\left(\frac{x}{y} \cdot 0.05555555555555555\right) \cdot t\_1\right)\\ t_3 := {\mathsf{PI}\left(\right)}^{1.5}\\ \frac{t\_3 \cdot \frac{t\_3}{8} - {\sin^{-1} \left(0.05555555555555555 \cdot \left(\frac{t\_1}{y} \cdot x\right)\right)}^{3}}{\mathsf{fma}\left(0.25 \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right), \mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), t\_2\right) \cdot t\_2\right)} \cdot 0.3333333333333333 \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (sqrt t) z))
        (t_2 (asin (* (* (/ x y) 0.05555555555555555) t_1)))
        (t_3 (pow (PI) 1.5)))
   (*
    (/
     (-
      (* t_3 (/ t_3 8.0))
      (pow (asin (* 0.05555555555555555 (* (/ t_1 y) x))) 3.0))
     (fma (* 0.25 (PI)) (PI) (* (fma 0.5 (PI) t_2) t_2)))
    0.3333333333333333)))

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 98.1%

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

7. Applied rewrites 98.1%

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

9. Applied rewrites 99.6%

   \[\leadsto \frac{{\mathsf{PI}\left(\right)}^{1.5} \cdot \frac{{\mathsf{PI}\left(\right)}^{1.5}}{8} - {\sin^{-1} \left(0.05555555555555555 \cdot \left(\frac{\frac{\sqrt{t}}{z}}{y} \cdot x\right)\right)}^{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2} + \left({\sin^{-1} \left(0.05555555555555555 \cdot \left(\frac{\frac{\sqrt{t}}{z}}{y} \cdot x\right)\right)}^{2} + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} \left(0.05555555555555555 \cdot \left(\frac{\frac{\sqrt{t}}{z}}{y} \cdot x\right)\right)\right)} \cdot 0.3333333333333333 \]

10. Taylor expanded in x around 0

    \[\leadsto \frac{{\mathsf{PI}\left(\right)}^{\frac{3}{2}} \cdot \frac{{\mathsf{PI}\left(\right)}^{\frac{3}{2}}}{8} - {\sin^{-1} \left(\frac{1}{18} \cdot \left(\frac{\frac{\sqrt{t}}{z}}{y} \cdot x\right)\right)}^{3}}{\frac{1}{4} \cdot {\mathsf{PI}\left(\right)}^{2} + \left(\frac{1}{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \sin^{-1} \left(\frac{1}{18} \cdot \left(\sqrt{t} \cdot \frac{x}{y \cdot z}\right)\right)\right) + {\sin^{-1} \left(\frac{1}{18} \cdot \left(\sqrt{t} \cdot \frac{x}{y \cdot z}\right)\right)}^{2}\right)} \cdot \frac{1}{3} \]
11. Step-by-step derivation

12. Applied rewrites 99.6%

    \[\leadsto \frac{{\mathsf{PI}\left(\right)}^{1.5} \cdot \frac{{\mathsf{PI}\left(\right)}^{1.5}}{8} - {\sin^{-1} \left(0.05555555555555555 \cdot \left(\frac{\frac{\sqrt{t}}{z}}{y} \cdot x\right)\right)}^{3}}{\mathsf{fma}\left(0.25 \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right), \mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \sin^{-1} \left(\left(\frac{x}{y} \cdot 0.05555555555555555\right) \cdot \frac{\sqrt{t}}{z}\right)\right) \cdot \sin^{-1} \left(\left(\frac{x}{y} \cdot 0.05555555555555555\right) \cdot \frac{\sqrt{t}}{z}\right)\right)} \cdot 0.3333333333333333 \]
13. Add Preprocessing

Alternative 3: 97.7% accurate, 0.2× speedup

Math

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

FPCore

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

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 98.1%

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

7. Applied rewrites 98.1%

   \[\leadsto \frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(0.05555555555555555 \cdot \left(\frac{\frac{\sqrt{t}}{z}}{y} \cdot x\right)\right)}^{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2} + \left({\sin^{-1} \left(0.05555555555555555 \cdot \left(\frac{\frac{\sqrt{t}}{z}}{y} \cdot x\right)\right)}^{2} + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} \left(0.05555555555555555 \cdot \left(\frac{\frac{\sqrt{t}}{z}}{y} \cdot x\right)\right)\right)} \cdot 0.3333333333333333 \]

9. Applied rewrites 99.2%

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

11. Applied rewrites 99.2%

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

12. Final simplification 99.2%

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

Alternative 4: 97.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin^{-1} \left(\left(x \cdot \frac{\frac{\sqrt{t}}{z}}{y}\right) \cdot 0.05555555555555555\right)\\ \frac{\left(0.125 \cdot {\mathsf{PI}\left(\right)}^{3} - {t\_1}^{3}\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), t\_1\right), t\_1, \left(0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (asin (* (* x (/ (/ (sqrt t) z) y)) 0.05555555555555555))))
   (/
    (* (- (* 0.125 (pow (PI) 3.0)) (pow t_1 3.0)) 0.3333333333333333)
    (fma (fma 0.5 (PI) t_1) t_1 (* (* 0.25 (PI)) (PI))))))

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 98.1%

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

7. Applied rewrites 98.1%

   \[\leadsto \frac{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{3} - {\sin^{-1} \left(0.05555555555555555 \cdot \left(\frac{\frac{\sqrt{t}}{z}}{y} \cdot x\right)\right)}^{3}}{{\left(\frac{\mathsf{PI}\left(\right)}{2}\right)}^{2} + \left({\sin^{-1} \left(0.05555555555555555 \cdot \left(\frac{\frac{\sqrt{t}}{z}}{y} \cdot x\right)\right)}^{2} + \frac{\mathsf{PI}\left(\right)}{2} \cdot \sin^{-1} \left(0.05555555555555555 \cdot \left(\frac{\frac{\sqrt{t}}{z}}{y} \cdot x\right)\right)\right)} \cdot 0.3333333333333333 \]

8. Taylor expanded in x around 0

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

10. Applied rewrites 97.7%

    \[\leadsto \left({\mathsf{PI}\left(\right)}^{3} \cdot 0.125 - {\sin^{-1} \left(\left(0.05555555555555555 \cdot \frac{x}{z}\right) \cdot \frac{\sqrt{t}}{y}\right)}^{3}\right) \cdot \color{blue}{\frac{0.3333333333333333}{\mathsf{fma}\left(0.25 \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right), \mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \sin^{-1} \left(\left(0.05555555555555555 \cdot \frac{x}{z}\right) \cdot \frac{\sqrt{t}}{y}\right)\right) \cdot \sin^{-1} \left(\left(0.05555555555555555 \cdot \frac{x}{z}\right) \cdot \frac{\sqrt{t}}{y}\right)\right)}} \]

12. Applied rewrites 98.1%

    \[\leadsto \frac{\left(0.125 \cdot {\mathsf{PI}\left(\right)}^{3} - {\sin^{-1} \left(\left(x \cdot \frac{\frac{\sqrt{t}}{z}}{y}\right) \cdot 0.05555555555555555\right)}^{3}\right) \cdot 0.3333333333333333}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \sin^{-1} \left(\left(x \cdot \frac{\frac{\sqrt{t}}{z}}{y}\right) \cdot 0.05555555555555555\right)\right), \color{blue}{\sin^{-1} \left(\left(x \cdot \frac{\frac{\sqrt{t}}{z}}{y}\right) \cdot 0.05555555555555555\right)}, \left(0.25 \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)} \]
13. Add Preprocessing

Alternative 5: 97.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[ArcCos[N[(N[(N[(N[(N[Sqrt[t], $MachinePrecision] / z), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision] * 0.05555555555555555), $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 98.1%

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

Alternative 6: 98.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[ArcCos[N[(N[(N[(N[Sqrt[t], $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * 0.05555555555555555), $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 98.1%

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

7. Applied rewrites 98.3%

   \[\leadsto \cos^{-1} \left(\left(\frac{\sqrt{t}}{z \cdot y} \cdot x\right) \cdot 0.05555555555555555\right) \cdot 0.3333333333333333 \]
8. 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 2024320 
(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)))))