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

Average Error: 1.2 → 1.2

Time: 9.2s

Precision: binary64

Cost: 13504

Math

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

↓

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

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
 (*
  0.3333333333333333
  (acos (/ (sqrt t) (/ (/ z 0.05555555555555555) (/ x y))))))

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 0.3333333333333333 * acos((sqrt(t) / ((z / 0.05555555555555555) / (x / y))));
}

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 = 0.3333333333333333d0 * acos((sqrt(t) / ((z / 0.05555555555555555d0) / (x / y))))
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 0.3333333333333333 * Math.acos((Math.sqrt(t) / ((z / 0.05555555555555555) / (x / y))));
}

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 0.3333333333333333 * math.acos((math.sqrt(t) / ((z / 0.05555555555555555) / (x / y))))

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

Target

Original	1.2
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 1.2

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

2. Simplified 1.2

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \cos^{-1} \left(\frac{3}{z} \cdot \left(\frac{\frac{x}{y}}{54} \cdot \sqrt{t}\right)\right)} \]
   Proof

   [Start] 1.2	\[ \frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \]
   metadata-eval [=>] 1.2	\[ \color{blue}{0.3333333333333333} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \]
   rational.json-simplify-2 [=>] 1.2	\[ 0.3333333333333333 \cdot \cos^{-1} \color{blue}{\left(\sqrt{t} \cdot \frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2}\right)} \]
   rational.json-simplify-46 [=>] 1.2	\[ 0.3333333333333333 \cdot \cos^{-1} \left(\sqrt{t} \cdot \color{blue}{\frac{\frac{3 \cdot \frac{x}{y \cdot 27}}{z}}{2}}\right) \]
   rational.json-simplify-49 [=>] 1.2	\[ 0.3333333333333333 \cdot \cos^{-1} \left(\sqrt{t} \cdot \frac{\color{blue}{\frac{x}{y \cdot 27} \cdot \frac{3}{z}}}{2}\right) \]
   rational.json-simplify-49 [=>] 1.2	\[ 0.3333333333333333 \cdot \cos^{-1} \left(\sqrt{t} \cdot \color{blue}{\left(\frac{3}{z} \cdot \frac{\frac{x}{y \cdot 27}}{2}\right)}\right) \]
   rational.json-simplify-43 [=>] 1.2	\[ 0.3333333333333333 \cdot \cos^{-1} \color{blue}{\left(\frac{3}{z} \cdot \left(\frac{\frac{x}{y \cdot 27}}{2} \cdot \sqrt{t}\right)\right)} \]
   rational.json-simplify-46 [=>] 1.2	\[ 0.3333333333333333 \cdot \cos^{-1} \left(\frac{3}{z} \cdot \left(\frac{\color{blue}{\frac{\frac{x}{y}}{27}}}{2} \cdot \sqrt{t}\right)\right) \]
   rational.json-simplify-47 [=>] 1.2	\[ 0.3333333333333333 \cdot \cos^{-1} \left(\frac{3}{z} \cdot \left(\color{blue}{\frac{\frac{x}{y}}{27 \cdot 2}} \cdot \sqrt{t}\right)\right) \]
   metadata-eval [=>] 1.2	\[ 0.3333333333333333 \cdot \cos^{-1} \left(\frac{3}{z} \cdot \left(\frac{\frac{x}{y}}{\color{blue}{54}} \cdot \sqrt{t}\right)\right) \]

3. Applied egg-rr 1.2

   \[\leadsto 0.3333333333333333 \cdot \cos^{-1} \color{blue}{\left(\frac{\frac{3}{z}}{\frac{y \cdot \frac{54}{x}}{\sqrt{t}}}\right)} \]

4. Simplified 1.2

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

   [Start] 1.2	\[ 0.3333333333333333 \cdot \cos^{-1} \left(\frac{\frac{3}{z}}{\frac{y \cdot \frac{54}{x}}{\sqrt{t}}}\right) \]
   rational.json-simplify-61 [<=] 1.2	\[ 0.3333333333333333 \cdot \cos^{-1} \color{blue}{\left(\frac{\sqrt{t}}{\frac{y \cdot \frac{54}{x}}{\frac{3}{z}}}\right)} \]
   rational.json-simplify-7 [<=] 1.2	\[ 0.3333333333333333 \cdot \cos^{-1} \left(\frac{\sqrt{t}}{\color{blue}{\frac{\frac{y \cdot \frac{54}{x}}{\frac{3}{z}}}{1}}}\right) \]
   rational.json-simplify-61 [<=] 1.2	\[ 0.3333333333333333 \cdot \cos^{-1} \color{blue}{\left(\frac{1}{\frac{\frac{y \cdot \frac{54}{x}}{\frac{3}{z}}}{\sqrt{t}}}\right)} \]
   rational.json-simplify-46 [<=] 1.9	\[ 0.3333333333333333 \cdot \cos^{-1} \left(\frac{1}{\color{blue}{\frac{y \cdot \frac{54}{x}}{\frac{3}{z} \cdot \sqrt{t}}}}\right) \]
   rational.json-simplify-61 [=>] 1.9	\[ 0.3333333333333333 \cdot \cos^{-1} \color{blue}{\left(\frac{\frac{3}{z} \cdot \sqrt{t}}{\frac{y \cdot \frac{54}{x}}{1}}\right)} \]
   rational.json-simplify-7 [=>] 1.9	\[ 0.3333333333333333 \cdot \cos^{-1} \left(\frac{\frac{3}{z} \cdot \sqrt{t}}{\color{blue}{y \cdot \frac{54}{x}}}\right) \]
   rational.json-simplify-2 [=>] 1.9	\[ 0.3333333333333333 \cdot \cos^{-1} \left(\frac{\frac{3}{z} \cdot \sqrt{t}}{\color{blue}{\frac{54}{x} \cdot y}}\right) \]
   rational.json-simplify-46 [=>] 1.7	\[ 0.3333333333333333 \cdot \cos^{-1} \color{blue}{\left(\frac{\frac{\frac{3}{z} \cdot \sqrt{t}}{\frac{54}{x}}}{y}\right)} \]
   rational.json-simplify-61 [<=] 1.7	\[ 0.3333333333333333 \cdot \cos^{-1} \left(\frac{\color{blue}{\frac{x}{\frac{54}{\frac{3}{z} \cdot \sqrt{t}}}}}{y}\right) \]
   rational.json-simplify-44 [<=] 1.9	\[ 0.3333333333333333 \cdot \cos^{-1} \color{blue}{\left(\frac{\frac{x}{y}}{\frac{54}{\frac{3}{z} \cdot \sqrt{t}}}\right)} \]
   rational.json-simplify-61 [=>] 1.9	\[ 0.3333333333333333 \cdot \cos^{-1} \color{blue}{\left(\frac{\frac{3}{z} \cdot \sqrt{t}}{\frac{54}{\frac{x}{y}}}\right)} \]
   metadata-eval [<=] 1.9	\[ 0.3333333333333333 \cdot \cos^{-1} \left(\frac{\frac{3}{z} \cdot \sqrt{t}}{\frac{\color{blue}{\frac{108}{2}}}{\frac{x}{y}}}\right) \]
   rational.json-simplify-44 [<=] 1.9	\[ 0.3333333333333333 \cdot \cos^{-1} \left(\frac{\frac{3}{z} \cdot \sqrt{t}}{\color{blue}{\frac{\frac{108}{\frac{x}{y}}}{2}}}\right) \]
   rational.json-simplify-61 [<=] 1.9	\[ 0.3333333333333333 \cdot \cos^{-1} \color{blue}{\left(\frac{2}{\frac{\frac{108}{\frac{x}{y}}}{\frac{3}{z} \cdot \sqrt{t}}}\right)} \]
   rational.json-simplify-46 [=>] 1.3	\[ 0.3333333333333333 \cdot \cos^{-1} \left(\frac{2}{\color{blue}{\frac{\frac{\frac{108}{\frac{x}{y}}}{\frac{3}{z}}}{\sqrt{t}}}}\right) \]
   rational.json-simplify-61 [<=] 1.3	\[ 0.3333333333333333 \cdot \cos^{-1} \color{blue}{\left(\frac{\sqrt{t}}{\frac{\frac{\frac{108}{\frac{x}{y}}}{\frac{3}{z}}}{2}}\right)} \]
   rational.json-simplify-44 [=>] 1.3	\[ 0.3333333333333333 \cdot \cos^{-1} \left(\frac{\sqrt{t}}{\color{blue}{\frac{\frac{\frac{108}{\frac{x}{y}}}{2}}{\frac{3}{z}}}}\right) \]
   rational.json-simplify-47 [=>] 1.3	\[ 0.3333333333333333 \cdot \cos^{-1} \left(\frac{\sqrt{t}}{\color{blue}{\frac{\frac{108}{\frac{x}{y}}}{2 \cdot \frac{3}{z}}}}\right) \]

5. Final simplification 1.2

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

Alternatives

Alternative 1
Error	1.2
Cost	13504

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

Reproduce

herbie shell --seed 2023074 
(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)))))