Average Error: 1.0 → 0.0
Time: 40.2s
Precision: 64
Internal Precision: 128
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{4}{3}}{\pi} \cdot \frac{\frac{1}{1 - v \cdot v}}{\sqrt{-6 \cdot \left(v \cdot v\right) + 2}}\]

Error

Bits error versus v

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Your Program's Arguments

Results

Enter valid numbers for all inputs

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

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

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

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

  5. Applied sqrt-prod0.0

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

  6. Applied div-inv0.0

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

  7. Applied times-frac0.0

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

  8. Simplified0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019021 
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))