Average Error: 0.1 → 0.1
Time: 49.5s
Precision: 64
Internal Precision: 128
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[e \cdot \frac{\sin v}{e \cdot \cos v + 1}\]

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 *-commutative0.1

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

  4. Applied associate-/l*0.3

    \[\leadsto \color{blue}{\frac{\sin v}{\frac{1 + e \cdot \cos v}{e}}}\]
  5. Using strategy rm

  6. Applied associate-/r/0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019072 +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))