Average Error: 0.1 → 0.2
Time: 21.4s
Precision: 64
Internal Precision: 128
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e}{\sqrt{(\left(\cos v\right) \cdot e + 1)_*}} \cdot \frac{\sin v}{\sqrt{(\left(\cos v\right) \cdot e + 1)_*}}\]

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. Simplified0.1

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

  4. Applied add-sqr-sqrt0.2

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

  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. Final simplification0.2

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

Reproduce

herbie shell --seed 2019030 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))