Average Error: 1.0 → 0.0
Time: 1.5m
Precision: 64
Internal Precision: 128
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\sqrt{\frac{\frac{\frac{4}{3}}{\pi}}{1 - v \cdot v}}}{\frac{\sqrt{(-6 \cdot \left(v \cdot v\right) + 2)_*}}{\sqrt{\frac{\frac{\frac{4}{3}}{\pi}}{1 - v \cdot v}}}}\]

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. Initial simplification0.0

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

  4. Applied add-sqr-sqrt0.0

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

  5. Applied associate-/l*0.0

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

  6. Final simplification0.0

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

Runtime

Time bar (total: 1.5m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes0.00.00.00.00%
herbie shell --seed 2018353 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))