Average Error: 0.0 → 0.0
Time: 1.5m
Precision: 64
Internal Precision: 320
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[e^{\log \left(1 - v \cdot v\right) + \log \left(\frac{\sqrt{2}}{4}\right)} \cdot \sqrt{(\left(v \cdot -3\right) \cdot v + 1)_*}\]

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. Initial simplification0.0

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

  4. Applied add-exp-log0.0

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

  5. Applied add-exp-log0.0

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

  6. Applied prod-exp0.0

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

  7. Final simplification0.0

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

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed 2018248 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))