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

Average Error: 1.4 → 1.2

Time: 8.1s

Precision: binary64

Cost: 13568

Math

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

↓

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

FPCore

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

↓

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

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)));
}

↓

double code(double x, double y, double z, double t) {
	return -(acos(((x / (y * z)) * (0.05555555555555555 * sqrt(t)))) * -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 = (1.0d0 / 3.0d0) * acos((((3.0d0 * (x / (y * 27.0d0))) / (z * 2.0d0)) * sqrt(t)))
end function

↓

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 / (y * z)) * (0.05555555555555555d0 * sqrt(t)))) * (-0.3333333333333333d0))
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)));
}

↓

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

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)))

↓

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

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

↓

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

↓

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

↓

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

Target

Original	1.4
Target	1.2
Herbie	1.2

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

Derivation

1. Initial program 1.4

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

2. Simplified 1.4

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z + z} \cdot \sqrt{t}\right)} \]
   Proof

   [Start] 1.4	\[ \frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \]
   metadata-eval [=>] 1.4	\[ \color{blue}{0.3333333333333333} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \]
   metadata-eval [<=] 1.4	\[ 0.3333333333333333 \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot \color{blue}{\left(1 + 1\right)}} \cdot \sqrt{t}\right) \]
   rational_best_oopsla_all_46_json_45_simplify-23 [<=] 1.4	\[ 0.3333333333333333 \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{\color{blue}{1 \cdot z + z \cdot 1}} \cdot \sqrt{t}\right) \]
   rational_best_oopsla_all_46_json_45_simplify-74 [<=] 1.4	\[ 0.3333333333333333 \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{\color{blue}{z \cdot 1} + z \cdot 1} \cdot \sqrt{t}\right) \]
   rational_best_oopsla_all_46_json_45_simplify-52 [=>] 1.4	\[ 0.3333333333333333 \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{\color{blue}{z} + z \cdot 1} \cdot \sqrt{t}\right) \]
   rational_best_oopsla_all_46_json_45_simplify-52 [=>] 1.4	\[ 0.3333333333333333 \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z + \color{blue}{z}} \cdot \sqrt{t}\right) \]

3. Taylor expanded in x around 0 1.2

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

4. Applied egg-rr 1.2

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

5. Final simplification 1.2

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

Alternatives

Alternative 1
Error	1.2
Cost	13504

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

Reproduce

herbie shell --seed 2023090 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, D"
  :precision binary64

  :herbie-target
  (/ (acos (* (/ (/ x 27.0) (* y z)) (/ (sqrt t) (/ 2.0 3.0)))) 3.0)

  (* (/ 1.0 3.0) (acos (* (/ (* 3.0 (/ x (* y 27.0))) (* z 2.0)) (sqrt t)))))