Average Error: 0.1 → 0.2
Time: 26.2s
Precision: 64
Internal Precision: 576
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\log_* (1 + (e^{\frac{\sin v}{\sqrt{(\left(\cos v\right) \cdot e + 1)_*}}} - 1)^*) \cdot \frac{e}{\sqrt{(\left(\cos v\right) \cdot e + 1)_*}}\]

Error

Bits error versus e

Bits error versus v

Derivation

  1. Initial program 0.1

    \[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]

  2. Initial simplification0.1

    \[\leadsto \frac{e \cdot \sin v}{(\left(\cos v\right) \cdot e + 1)_*}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.1

    \[\leadsto \frac{e \cdot \sin v}{\color{blue}{\sqrt{(\left(\cos v\right) \cdot e + 1)_*} \cdot \sqrt{(\left(\cos v\right) \cdot e + 1)_*}}}\]

  5. Applied times-frac0.2

    \[\leadsto \color{blue}{\frac{e}{\sqrt{(\left(\cos v\right) \cdot e + 1)_*}} \cdot \frac{\sin v}{\sqrt{(\left(\cos v\right) \cdot e + 1)_*}}}\]
  6. Using strategy rm

  7. Applied log1p-expm1-u0.2

    \[\leadsto \frac{e}{\sqrt{(\left(\cos v\right) \cdot e + 1)_*}} \cdot \color{blue}{\log_* (1 + (e^{\frac{\sin v}{\sqrt{(\left(\cos v\right) \cdot e + 1)_*}}} - 1)^*)}\]

  8. Final simplification0.2

    \[\leadsto \log_* (1 + (e^{\frac{\sin v}{\sqrt{(\left(\cos v\right) \cdot e + 1)_*}}} - 1)^*) \cdot \frac{e}{\sqrt{(\left(\cos v\right) \cdot e + 1)_*}}\]

Runtime

Time bar (total: 26.2s)Debug logProfile

herbie shell --seed 2018255 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))