Average Error: 0.0 → 0.0
Time: 1.3m
Precision: 64
Internal Precision: 320
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(\sqrt{\sqrt{(\left(-3 \cdot v\right) \cdot v + 1)_*}} \cdot \sqrt{\sqrt{(\left(-3 \cdot v\right) \cdot v + 1)_*}}\right) \cdot \left(\left(1 - {v}^{4}\right) \cdot \frac{\sqrt{2}}{(\left(v \cdot v\right) \cdot 4 + 4)_*}\right)\]

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. 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 flip--0.0

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

  5. Applied associate-*l/0.0

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

  7. Applied add-sqr-sqrt0.0

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

  8. Applied sqrt-prod0.0

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

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

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

  11. Applied times-frac0.0

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

  12. Simplified0.0

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

  13. Simplified0.0

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

  14. Final simplification0.0

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

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018250 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))