Average Error: 1.0 → 0.0
Time: 36.0s
Precision: 64
Internal Precision: 128
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{\frac{1}{\sqrt{\pi - v \cdot \left(\pi \cdot v\right)}}}{\sqrt{\pi - v \cdot \left(\pi \cdot v\right)}} \cdot \frac{4}{3}}{\sqrt{(\left(v \cdot -6\right) \cdot v + 2)_*}}\]

Error

Bits error versus v

Derivation

  1. Initial program 1.0

    \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\frac{\frac{\frac{4}{3}}{\pi - v \cdot \left(\pi \cdot v\right)}}{\sqrt{(\left(v \cdot -6\right) \cdot v + 2)_*}}}\]
  3. Using strategy rm

  4. Applied div-inv0.0

    \[\leadsto \frac{\color{blue}{\frac{4}{3} \cdot \frac{1}{\pi - v \cdot \left(\pi \cdot v\right)}}}{\sqrt{(\left(v \cdot -6\right) \cdot v + 2)_*}}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt1.0

    \[\leadsto \frac{\frac{4}{3} \cdot \frac{1}{\color{blue}{\sqrt{\pi - v \cdot \left(\pi \cdot v\right)} \cdot \sqrt{\pi - v \cdot \left(\pi \cdot v\right)}}}}{\sqrt{(\left(v \cdot -6\right) \cdot v + 2)_*}}\]

  7. Applied associate-/r*0.0

    \[\leadsto \frac{\frac{4}{3} \cdot \color{blue}{\frac{\frac{1}{\sqrt{\pi - v \cdot \left(\pi \cdot v\right)}}}{\sqrt{\pi - v \cdot \left(\pi \cdot v\right)}}}}{\sqrt{(\left(v \cdot -6\right) \cdot v + 2)_*}}\]

  8. Final simplification0.0

    \[\leadsto \frac{\frac{\frac{1}{\sqrt{\pi - v \cdot \left(\pi \cdot v\right)}}}{\sqrt{\pi - v \cdot \left(\pi \cdot v\right)}} \cdot \frac{4}{3}}{\sqrt{(\left(v \cdot -6\right) \cdot v + 2)_*}}\]

Reproduce

herbie shell --seed 2019093 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))