Average Error: 1.0 → 0.0
Time: 3.6m
Precision: 64
Internal Precision: 128
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{\frac{4}{3}}{\sqrt{\pi - v \cdot \left(v \cdot \pi\right)}}}{\sqrt{(v \cdot \left(v \cdot -6\right) + 2)_*}} \cdot \frac{1}{\sqrt{\pi - v \cdot \left(v \cdot \pi\right)}}\]

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 *-un-lft-identity0.0

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

  5. Applied *-un-lft-identity0.0

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

  6. Applied times-frac0.0

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

  7. Applied associate-/l*0.0

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

  8. Simplified0.0

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

  10. Applied add-sqr-sqrt1.0

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

  11. Applied *-un-lft-identity1.0

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

  12. Applied times-frac1.0

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

  13. Applied *-un-lft-identity1.0

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

  14. Applied sqrt-prod1.0

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

  15. Applied times-frac0.0

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

  16. Applied *-un-lft-identity0.0

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

  17. Applied times-frac0.0

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

  18. Simplified0.0

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

  19. Simplified0.0

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

  20. Final simplification0.0

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

Reproduce

herbie shell --seed 2019089 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))