Average Error: 0.1 → 0.1
Time: 16.2s
Precision: 64
Internal Precision: 128
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e \cdot \sin v}{(\left(\cos v\right) \cdot e + 1)_*}\]

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. Simplified0.1

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

  3. Taylor expanded around -inf 0.1

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

  4. Final simplification0.1

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

Reproduce

herbie shell --seed 2019026 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))