Average Error: 0.0 → 0.0
Time: 1.8m
Precision: 64
Internal Precision: 128
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\sqrt{2} \cdot \sqrt{(\left(v \cdot v\right) \cdot -3 + 1)_*}}{e^{\log 4 - \log \left(1 - v \cdot v\right)}}\]

Error

Bits error versus v

Derivation

  1. Initial program 0.0

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

  2. Simplified0.0

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

  4. Applied associate-*l/0.0

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

  6. Applied add-exp-log0.0

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

  7. Applied add-exp-log0.0

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

  8. Applied div-exp0.0

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

  9. Final simplification0.0

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

Reproduce

herbie shell --seed 2019090 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))