Average Error: 0.5 → 0.4
Time: 12.4s
Precision: binary64
\[\frac{2 + \left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left(\cos x - \cos y\right)}{3 \cdot \left(\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{3 - \sqrt{5}}{2} \cdot \cos y\right)}\]
↓
\[\frac{\frac{\frac{\left(\left(\sqrt{2} \cdot \left(\sin x - \frac{\sin y}{16}\right)\right) \cdot \left(\sin y - \frac{\sin x}{16}\right)\right) \cdot \left({\left(\cos x\right)}^{3} - {\left(\cos y\right)}^{3}\right)}{\cos y \cdot \left(\cos y + \cos x\right) + \cos x \cdot \cos x} + 2}{\left(1 + \frac{\sqrt{5} - 1}{2} \cdot \cos x\right) + \frac{\frac{3 \cdot 3 + \left(-5\right)}{3 + \sqrt{5}}}{2} \cdot \cos y}}{3}\]

Error

Bits error versus x

Bits error versus y

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 flip3--0.5

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

  6. Simplified0.5

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

  7. Simplified0.5

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

  9. Applied flip--0.5

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2020171 
(FPCore (x y)
  :name "Diagrams.TwoD.Path.Metafont.Internal:hobbyF from diagrams-contrib-1.3.0.5"
  :precision binary64
  (/ (+ 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))))))