Average Error: 0.1 → 0.2
Time: 1.1m
Precision: 64
Internal Precision: 320
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e}{\sqrt{(\left(\cos v\right) \cdot e + 1)_*}} \cdot (e^{\log_* (1 + \frac{\sin v}{\sqrt{(e \cdot \left(\cos v\right) + 1)_*}})} - 1)^*\]

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. Using strategy rm

  3. Applied add-sqr-sqrt0.2

    \[\leadsto \frac{e \cdot \sin v}{\color{blue}{\sqrt{1 + e \cdot \cos v} \cdot \sqrt{1 + e \cdot \cos v}}}\]

  4. Applied times-frac0.2

    \[\leadsto \color{blue}{\frac{e}{\sqrt{1 + e \cdot \cos v}} \cdot \frac{\sin v}{\sqrt{1 + e \cdot \cos v}}}\]

  5. Applied simplify0.2

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

  6. Applied simplify0.2

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

  8. Applied expm1-log1p-u0.2

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018214 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))