Average Error: 1.0 → 0.0
Time: 2.0m
Precision: 64
Internal Precision: 320
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{4}{\pi \cdot 3}}{(\left(-\sqrt{1}\right) \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) + \left(\sqrt{1}\right))_*} \cdot \frac{(v \cdot v + 1)_*}{\sqrt{(\left(-6\right) \cdot \left(v \cdot v\right) + 2)_*}}\]

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. Initial simplification0.0

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

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

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

  5. Applied sqrt-prod0.0

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

  6. Applied flip--0.0

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

  7. Applied associate-/r/0.0

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

  8. Applied times-frac0.0

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

  9. Simplified0.0

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

  10. Simplified0.0

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

  11. Final simplification0.0

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

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed 2018230 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))