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

Average Error: 1.4 → 1.0

Time: 10.0s

Precision: binary64

Cost: 32896

Math

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

↓

\[e^{\log \left({\cos^{-1} \left(\frac{0.05555555555555555}{y \cdot z} \cdot \left(x \cdot \sqrt{t}\right)\right)}^{2} \cdot 0.1111111111111111\right) \cdot 0.5} \]

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
 (exp
  (*
   (log
    (*
     (pow (acos (* (/ 0.05555555555555555 (* y z)) (* x (sqrt t)))) 2.0)
     0.1111111111111111))
   0.5)))

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 exp((log((pow(acos(((0.05555555555555555 / (y * z)) * (x * sqrt(t)))), 2.0) * 0.1111111111111111)) * 0.5));
}

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 = exp((log(((acos(((0.05555555555555555d0 / (y * z)) * (x * sqrt(t)))) ** 2.0d0) * 0.1111111111111111d0)) * 0.5d0))
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.exp((Math.log((Math.pow(Math.acos(((0.05555555555555555 / (y * z)) * (x * Math.sqrt(t)))), 2.0) * 0.1111111111111111)) * 0.5));
}

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.exp((math.log((math.pow(math.acos(((0.05555555555555555 / (y * z)) * (x * math.sqrt(t)))), 2.0) * 0.1111111111111111)) * 0.5))

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 exp(Float64(log(Float64((acos(Float64(Float64(0.05555555555555555 / Float64(y * z)) * Float64(x * sqrt(t)))) ^ 2.0) * 0.1111111111111111)) * 0.5))
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 = exp((log(((acos(((0.05555555555555555 / (y * z)) * (x * sqrt(t)))) ^ 2.0) * 0.1111111111111111)) * 0.5));
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[Exp[N[(N[Log[N[(N[Power[N[ArcCos[N[(N[(0.05555555555555555 / N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(x * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * 0.1111111111111111), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]

Target

Original	1.4
Target	1.2
Herbie	1.0

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

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 1.2

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \cos^{-1} \left(\left(\frac{x}{y \cdot z} \cdot 0.05555555555555555\right) \cdot \sqrt{t}\right)} \]
   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	\[ \color{blue}{0.3333333333333333} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right) \]
   *-commutative [=>] 1.4	\[ 0.3333333333333333 \cdot \cos^{-1} \left(\frac{\color{blue}{\frac{x}{y \cdot 27} \cdot 3}}{z \cdot 2} \cdot \sqrt{t}\right) \]
   associate-*l/ [=>] 1.4	\[ 0.3333333333333333 \cdot \cos^{-1} \left(\frac{\color{blue}{\frac{x \cdot 3}{y \cdot 27}}}{z \cdot 2} \cdot \sqrt{t}\right) \]
   times-frac [=>] 1.4	\[ 0.3333333333333333 \cdot \cos^{-1} \left(\frac{\color{blue}{\frac{x}{y} \cdot \frac{3}{27}}}{z \cdot 2} \cdot \sqrt{t}\right) \]
   times-frac [=>] 1.4	\[ 0.3333333333333333 \cdot \cos^{-1} \left(\color{blue}{\left(\frac{\frac{x}{y}}{z} \cdot \frac{\frac{3}{27}}{2}\right)} \cdot \sqrt{t}\right) \]
   associate-/l/ [=>] 1.2	\[ 0.3333333333333333 \cdot \cos^{-1} \left(\left(\color{blue}{\frac{x}{z \cdot y}} \cdot \frac{\frac{3}{27}}{2}\right) \cdot \sqrt{t}\right) \]
   *-commutative [=>] 1.2	\[ 0.3333333333333333 \cdot \cos^{-1} \left(\left(\frac{x}{\color{blue}{y \cdot z}} \cdot \frac{\frac{3}{27}}{2}\right) \cdot \sqrt{t}\right) \]
   metadata-eval [=>] 1.2	\[ 0.3333333333333333 \cdot \cos^{-1} \left(\left(\frac{x}{y \cdot z} \cdot \frac{\color{blue}{0.1111111111111111}}{2}\right) \cdot \sqrt{t}\right) \]
   metadata-eval [=>] 1.2	\[ 0.3333333333333333 \cdot \cos^{-1} \left(\left(\frac{x}{y \cdot z} \cdot \color{blue}{0.05555555555555555}\right) \cdot \sqrt{t}\right) \]

3. Applied egg-rr 1.2

   \[\leadsto \color{blue}{\sqrt{{\cos^{-1} \left(\frac{0.05555555555555555}{\frac{\frac{z}{x} \cdot y}{\sqrt{t}}}\right)}^{2} \cdot 0.1111111111111111}} \]

4. Simplified 1.2

   \[\leadsto \color{blue}{\sqrt{{\cos^{-1} \left(\frac{0.05555555555555555}{\frac{y \cdot z}{x}} \cdot \sqrt{t}\right)}^{2} \cdot 0.1111111111111111}} \]
   Proof

   [Start] 1.2	\[ \sqrt{{\cos^{-1} \left(\frac{0.05555555555555555}{\frac{\frac{z}{x} \cdot y}{\sqrt{t}}}\right)}^{2} \cdot 0.1111111111111111} \]
   associate-/r/ [=>] 1.2	\[ \sqrt{{\cos^{-1} \color{blue}{\left(\frac{0.05555555555555555}{\frac{z}{x} \cdot y} \cdot \sqrt{t}\right)}}^{2} \cdot 0.1111111111111111} \]
   associate-*l/ [=>] 1.2	\[ \sqrt{{\cos^{-1} \left(\frac{0.05555555555555555}{\color{blue}{\frac{z \cdot y}{x}}} \cdot \sqrt{t}\right)}^{2} \cdot 0.1111111111111111} \]
   *-commutative [<=] 1.2	\[ \sqrt{{\cos^{-1} \left(\frac{0.05555555555555555}{\frac{\color{blue}{y \cdot z}}{x}} \cdot \sqrt{t}\right)}^{2} \cdot 0.1111111111111111} \]

5. Applied egg-rr 1.0

   \[\leadsto \color{blue}{e^{\log \left({\cos^{-1} \left(\frac{0.05555555555555555}{y \cdot z} \cdot \left(x \cdot \sqrt{t}\right)\right)}^{2} \cdot 0.1111111111111111\right) \cdot 0.5}} \]

6. Final simplification 1.0

   \[\leadsto e^{\log \left({\cos^{-1} \left(\frac{0.05555555555555555}{y \cdot z} \cdot \left(x \cdot \sqrt{t}\right)\right)}^{2} \cdot 0.1111111111111111\right) \cdot 0.5} \]

Alternatives

Alternative 1
Error	0.2
Cost	26432

\[e^{\mathsf{log1p}\left(0.3333333333333333 \cdot \cos^{-1} \left(\frac{0.05555555555555555}{\frac{y \cdot \frac{z}{x}}{\sqrt{t}}}\right)\right)} + -1 \]

Alternative 2
Error	1.2
Cost	26368

\[\sqrt{0.1111111111111111 \cdot {\cos^{-1} \left(\sqrt{t} \cdot \frac{0.05555555555555555}{\frac{y \cdot z}{x}}\right)}^{2}} \]

Alternative 3
Error	1.4
Cost	13504

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

Alternative 4
Error	1.2
Cost	13504

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

Reproduce

herbie shell --seed 2023054 
(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)))))