Average Error: 0.5 → 0.5
Time: 11.6s
Precision: 64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{2 + \log \left(e^{\left(\left(\sqrt{2} \cdot \left(\sqrt[3]{\sin x - \frac{\sin y}{16}} \cdot \sqrt[3]{\sin x - \frac{\sin y}{16}}\right)\right) \cdot \frac{\sqrt[3]{\sin x \cdot \sin x - \frac{\sin y}{16} \cdot \frac{\sin y}{16}}}{\sqrt[3]{\sin x + \frac{\sin y}{16}}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}
↓
\frac{2 + \log \left(e^{\left(\left(\sqrt{2} \cdot \left(\sqrt[3]{\sin x - \frac{\sin y}{16}} \cdot \sqrt[3]{\sin x - \frac{\sin y}{16}}\right)\right) \cdot \frac{\sqrt[3]{\sin x \cdot \sin x - \frac{\sin y}{16} \cdot \frac{\sin y}{16}}}{\sqrt[3]{\sin x + \frac{\sin y}{16}}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}
double code(double x, double y) {
	return ((2.0 + (((sqrt(2.0) * (sin(x) - (sin(y) / 16.0))) * (sin(y) - (sin(x) / 16.0))) * (cos(x) - cos(y)))) / (3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + (((3.0 - sqrt(5.0)) / 2.0) * cos(y)))));
}
↓
double code(double x, double y) {
	return ((2.0 + (log(exp((((sqrt(2.0) * (cbrt((sin(x) - (sin(y) / 16.0))) * cbrt((sin(x) - (sin(y) / 16.0))))) * (cbrt(((sin(x) * sin(x)) - ((sin(y) / 16.0) * (sin(y) / 16.0)))) / cbrt((sin(x) + (sin(y) / 16.0))))) * (sin(y) - (sin(x) / 16.0))))) * (cos(x) - cos(y)))) / (3.0 * ((1.0 + (((sqrt(5.0) - 1.0) / 2.0) * cos(x))) + (((3.0 - sqrt(5.0)) / 2.0) * cos(y)))));
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.5

    \[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
  2. Using strategy rm

  3. Applied add-log-exp0.5

    \[\leadsto \frac{2 + \color{blue}{\log \left(e^{\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)}\right)} \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
  4. Using strategy rm

  5. Applied add-cube-cbrt0.5

    \[\leadsto \frac{2 + \log \left(e^{\left(\sqrt{2} \cdot \color{blue}{\left(\left(\sqrt[3]{\sin x - \frac{\sin y}{16}} \cdot \sqrt[3]{\sin x - \frac{\sin y}{16}}\right) \cdot \sqrt[3]{\sin x - \frac{\sin y}{16}}\right)}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]

  6. Applied associate-*r*0.5

    \[\leadsto \frac{2 + \log \left(e^{\color{blue}{\left(\left(\sqrt{2} \cdot \left(\sqrt[3]{\sin x - \frac{\sin y}{16}} \cdot \sqrt[3]{\sin x - \frac{\sin y}{16}}\right)\right) \cdot \sqrt[3]{\sin x - \frac{\sin y}{16}}\right)} \cdot \left(\sin y - \frac{\sin x}{16}\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
  7. Using strategy rm

  8. Applied flip--0.5

    \[\leadsto \frac{2 + \log \left(e^{\left(\left(\sqrt{2} \cdot \left(\sqrt[3]{\sin x - \frac{\sin y}{16}} \cdot \sqrt[3]{\sin x - \frac{\sin y}{16}}\right)\right) \cdot \sqrt[3]{\color{blue}{\frac{\sin x \cdot \sin x - \frac{\sin y}{16} \cdot \frac{\sin y}{16}}{\sin x + \frac{\sin y}{16}}}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]

  9. Applied cbrt-div0.5

    \[\leadsto \frac{2 + \log \left(e^{\left(\left(\sqrt{2} \cdot \left(\sqrt[3]{\sin x - \frac{\sin y}{16}} \cdot \sqrt[3]{\sin x - \frac{\sin y}{16}}\right)\right) \cdot \color{blue}{\frac{\sqrt[3]{\sin x \cdot \sin x - \frac{\sin y}{16} \cdot \frac{\sin y}{16}}}{\sqrt[3]{\sin x + \frac{\sin y}{16}}}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]

  10. Final simplification0.5

    \[\leadsto \frac{2 + \log \left(e^{\left(\left(\sqrt{2} \cdot \left(\sqrt[3]{\sin x - \frac{\sin y}{16}} \cdot \sqrt[3]{\sin x - \frac{\sin y}{16}}\right)\right) \cdot \frac{\sqrt[3]{\sin x \cdot \sin x - \frac{\sin y}{16} \cdot \frac{\sin y}{16}}}{\sqrt[3]{\sin x + \frac{\sin y}{16}}}\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)}\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]

Reproduce

herbie shell --seed 2020102 
(FPCore (x y)
  :name "Diagrams.TwoD.Path.Metafont.Internal:hobbyF from diagrams-contrib-1.3.0.5"
  :precision binary64
  (/ (+ 2 (* (* (* (sqrt 2) (- (sin x) (/ (sin y) 16))) (- (sin y) (/ (sin x) 16))) (- (cos x) (cos y)))) (* 3 (+ (+ 1 (* (/ (- (sqrt 5) 1) 2) (cos x))) (* (/ (- 3 (sqrt 5)) 2) (cos y))))))