Average Error: 0.1 → 0.2
Time: 32.0s
Precision: 64
Internal Precision: 576
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{e}{\sqrt{(\left(\cos v\right) \cdot e + 1)_*}} \cdot \log_* (1 + (e^{\frac{\sin v}{\sqrt{(e \cdot \left(\cos v\right) + 1)_*}}} - 1)^*)\]

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 log1p-expm1-u0.2

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

Runtime

Time bar (total: 32.0s)Debug logProfile

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))