Average Error: 1.0 → 0.0
Time: 30.6s
Precision: 64
Internal Precision: 128
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{\sqrt[3]{\frac{4}{3}} \cdot \sqrt[3]{\frac{4}{3}}}{\frac{\pi - \pi \cdot \left(v \cdot v\right)}{\sqrt[3]{\frac{4}{3}}}}}{\sqrt{v \cdot \left(-6 \cdot v\right) + 2}}\]

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

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

  4. Applied add-cube-cbrt0.0

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

  5. Applied associate-/l*0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019090 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))