Average Error: 0.1 → 0.2
Time: 20.6s
Precision: 64
Internal Precision: 576
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[e \cdot \log_* (1 + (e^{\frac{\sin v}{(\left(\cos v\right) \cdot e + 1)_*}} - 1)^*)\]

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. Initial simplification0.1

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

  4. Applied *-un-lft-identity0.1

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

  5. Applied times-frac0.1

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

  6. Simplified0.1

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

  8. Applied log1p-expm1-u0.2

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

  9. Final simplification0.2

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

Runtime

Time bar (total: 20.6s)Debug logProfile

herbie shell --seed 2018250 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))