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

Percentage Accurate: 98.0% → 99.0%

Time: 15.0s

Alternatives: 2

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: 99.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{x}{y} \cdot -0.05555555555555555}{z} \cdot \sqrt{t}\\ t_2 := \sin^{-1} t\_1\\ t_3 := \cos^{-1} t\_1\\ \mathsf{fma}\left(t\_3, t\_2, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25\right) \cdot \frac{\left({\mathsf{PI}\left(\right)}^{3} \cdot 0.125 - {t\_2}^{3}\right) \cdot 0.3333333333333333}{{\mathsf{PI}\left(\right)}^{4} \cdot 0.0625 - {\left(t\_3 \cdot t\_2\right)}^{2}} \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ (* (/ x y) -0.05555555555555555) z) (sqrt t)))
        (t_2 (asin t_1))
        (t_3 (acos t_1)))
   (*
    (fma t_3 t_2 (* (* (PI) (PI)) 0.25))
    (/
     (* (- (* (pow (PI) 3.0) 0.125) (pow t_2 3.0)) 0.3333333333333333)
     (- (* (pow (PI) 4.0) 0.0625) (pow (* t_3 t_2) 2.0))))))

Derivation

1. Initial program 97.7%

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

4. Applied rewrites 98.6%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{3} \cdot \frac{\left(\frac{1}{4} \cdot {\mathsf{PI}\left(\right)}^{2} + \cos^{-1} \left(\frac{-1}{18} \cdot \left(\sqrt{t} \cdot \frac{x}{y \cdot z}\right)\right) \cdot \sin^{-1} \left(\frac{-1}{18} \cdot \left(\sqrt{t} \cdot \frac{x}{y \cdot z}\right)\right)\right) \cdot \left(\frac{1}{8} \cdot {\mathsf{PI}\left(\right)}^{3} - {\sin^{-1} \left(\frac{-1}{18} \cdot \left(\sqrt{t} \cdot \frac{x}{y \cdot z}\right)\right)}^{3}\right)}{\frac{1}{16} \cdot {\mathsf{PI}\left(\right)}^{4} - {\cos^{-1} \left(\frac{-1}{18} \cdot \left(\sqrt{t} \cdot \frac{x}{y \cdot z}\right)\right)}^{2} \cdot {\sin^{-1} \left(\frac{-1}{18} \cdot \left(\sqrt{t} \cdot \frac{x}{y \cdot z}\right)\right)}^{2}}} \]

7. Applied rewrites 95.9%

   \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(\sin^{-1} \left(\frac{-0.05555555555555555 \cdot x}{y} \cdot \frac{\sqrt{t}}{z}\right), \cos^{-1} \left(\frac{-0.05555555555555555 \cdot x}{y} \cdot \frac{\sqrt{t}}{z}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25\right) \cdot 0.3333333333333333\right) \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{-0.05555555555555555 \cdot x}{y} \cdot \frac{\sqrt{t}}{z}\right)}^{3}\right)}{\mathsf{fma}\left(-{\sin^{-1} \left(\frac{-0.05555555555555555 \cdot x}{y} \cdot \frac{\sqrt{t}}{z}\right)}^{2}, {\cos^{-1} \left(\frac{-0.05555555555555555 \cdot x}{y} \cdot \frac{\sqrt{t}}{z}\right)}^{2}, {\mathsf{PI}\left(\right)}^{4} \cdot 0.0625\right)}} \]

9. Applied rewrites 97.4%

   \[\leadsto \color{blue}{\frac{\left(\left(0.125 \cdot {\mathsf{PI}\left(\right)}^{3} - {\sin^{-1} \left(\left(\frac{\sqrt{t}}{z} \cdot -0.05555555555555555\right) \cdot \frac{x}{y}\right)}^{3}\right) \cdot \mathsf{fma}\left(\cos^{-1} \left(\left(\frac{\sqrt{t}}{z} \cdot -0.05555555555555555\right) \cdot \frac{x}{y}\right), \sin^{-1} \left(\left(\frac{\sqrt{t}}{z} \cdot -0.05555555555555555\right) \cdot \frac{x}{y}\right), 0.25 \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot 0.3333333333333333}{0.0625 \cdot {\mathsf{PI}\left(\right)}^{4} - {\left(\cos^{-1} \left(\left(\frac{\sqrt{t}}{z} \cdot -0.05555555555555555\right) \cdot \frac{x}{y}\right) \cdot \sin^{-1} \left(\left(\frac{\sqrt{t}}{z} \cdot -0.05555555555555555\right) \cdot \frac{x}{y}\right)\right)}^{2}}} \]

11. Applied rewrites 98.6%

    \[\leadsto \mathsf{fma}\left(\cos^{-1} \left(\frac{\frac{x}{y} \cdot -0.05555555555555555}{z} \cdot \sqrt{t}\right), \sin^{-1} \left(\frac{\frac{x}{y} \cdot -0.05555555555555555}{z} \cdot \sqrt{t}\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot 0.25\right) \cdot \color{blue}{\frac{\left({\mathsf{PI}\left(\right)}^{3} \cdot 0.125 - {\sin^{-1} \left(\frac{\frac{x}{y} \cdot -0.05555555555555555}{z} \cdot \sqrt{t}\right)}^{3}\right) \cdot 0.3333333333333333}{{\mathsf{PI}\left(\right)}^{4} \cdot 0.0625 - {\left(\cos^{-1} \left(\frac{\frac{x}{y} \cdot -0.05555555555555555}{z} \cdot \sqrt{t}\right) \cdot \sin^{-1} \left(\frac{\frac{x}{y} \cdot -0.05555555555555555}{z} \cdot \sqrt{t}\right)\right)}^{2}}} \]
12. Add Preprocessing

Alternative 2: 98.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(0.3333333333333333 * N[ArcCos[N[(N[(N[(x / N[(z * y), $MachinePrecision]), $MachinePrecision] * 0.05555555555555555), $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 98.4

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

5. Applied rewrites 98.4%

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

   1. lift-/.f64 N/A

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

   2. metadata-eval 98.4

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

7. Applied rewrites 98.4%

   \[\leadsto \color{blue}{0.3333333333333333} \cdot \cos^{-1} \left(\left(\frac{x}{z \cdot y} \cdot 0.05555555555555555\right) \cdot \sqrt{t}\right) \]
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 2024338 
(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)))))