Average Error: 0.1 → 0.2
Time: 36.7s
Precision: 64
Internal Precision: 576
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[e \cdot (e^{\log_* (1 + \frac{\sin v}{(e \cdot \left(\cos v\right) + 1)_*})} - 1)^*\]

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 *-un-lft-identity0.1

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

  4. Applied times-frac0.1

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

  5. Applied simplify0.1

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

  6. Applied simplify0.1

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

  8. Applied expm1-log1p-u0.2

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

Runtime

Time bar (total: 36.7s)Debug logProfile

herbie shell --seed '#(1071821486 549052472 3784827256 1559736200 3548510075 881134285)' +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))