Average Error: 0.1 → 0.2
Time: 22.0s
Precision: 64
Internal Precision: 384
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{\frac{e \cdot \sin v}{\sqrt{(e \cdot \left(\cos v\right) + 1)_*}}}{\sqrt{1 + e \cdot \cos v}}\]

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 associate-/r*0.2

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

  5. Applied simplify0.2

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

Runtime

Time bar (total: 22.0s)Debug logProfile

herbie shell --seed '#(1070100504 930361288 1279167582 284574201 1450237281 2578255382)' +o rules:numerics
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))