Average Error: 0.1 → 0.2
Time: 17.2s
Precision: 64
Internal Precision: 576
\[\frac{e \cdot \sin v}{1 + e \cdot \cos v}\]
↓
\[\frac{\frac{e \cdot \sin v}{\sqrt{\cos v \cdot e + 1}}}{\sqrt{\cos v \cdot e + 1}}\]

Error

Bits error versus e

Bits error versus v

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Results

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Derivation

  1. Initial program 0.1

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

  2. Initial simplification0.1

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

  3. Taylor expanded around inf 0.1

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

  5. Applied add-sqr-sqrt0.1

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

  6. Applied associate-/r*0.2

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

  7. Final simplification0.2

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

Runtime

Time bar (total: 17.2s)Debug logProfile

herbie shell --seed 2018227 
(FPCore (e v)
  :name "Trigonometry A"
  :pre (<= 0 e 1)
  (/ (* e (sin v)) (+ 1 (* e (cos v)))))