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

Average Error: 1.3 → 0.2

Time: 8.3s

Precision: binary64

Math

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

↓

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

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
    (fma
     0.3333333333333333
     (acos (* (/ x (* y z)) (* (sqrt t) 0.05555555555555555)))
     1.0)))
  -1.0))

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(fma(0.3333333333333333, acos(((x / (y * z)) * (sqrt(t) * 0.05555555555555555))), 1.0))) + -1.0;
}

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(exp(log(fma(0.3333333333333333, acos(Float64(Float64(x / Float64(y * z)) * Float64(sqrt(t) * 0.05555555555555555))), 1.0))) + -1.0)
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[Exp[N[Log[N[(0.3333333333333333 * N[ArcCos[N[(N[(x / N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[t], $MachinePrecision] * 0.05555555555555555), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]

Target

Original	1.3
Target	1.2
Herbie	0.2

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

Derivation

1. Initial program 1.3

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

2. Simplified 1.3

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \cos^{-1} \left(\frac{\sqrt{t}}{y \cdot \frac{18}{\frac{x}{z}}}\right)} \]

3. Taylor expanded in t around 0 1.2

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

4. Applied egg-rr 0.3

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

5. Applied egg-rr 0.2

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

6. Final simplification 0.2

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

Reproduce

herbie shell --seed 2022210 
(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)))))