Average Error: 0.6 → 0.6
Time: 51.2s
Precision: 64
Internal Precision: 128
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\sqrt{(-5 \cdot \left(v \cdot v\right) + 1)_*} \cdot \frac{\sqrt{(\left(-5 \cdot v\right) \cdot v + 1)_*}}{(v \cdot v + -1)_*}\right)\]

Error

Bits error versus v

Derivation

  1. Initial program 0.6

    \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]

  2. Initial simplification0.6

    \[\leadsto \cos^{-1} \left(\frac{(\left(-5 \cdot v\right) \cdot v + 1)_*}{(v \cdot v + -1)_*}\right)\]
  3. Using strategy rm

  4. Applied *-un-lft-identity0.6

    \[\leadsto \cos^{-1} \left(\frac{(\left(-5 \cdot v\right) \cdot v + 1)_*}{\color{blue}{1 \cdot (v \cdot v + -1)_*}}\right)\]

  5. Applied add-sqr-sqrt0.6

    \[\leadsto \cos^{-1} \left(\frac{\color{blue}{\sqrt{(\left(-5 \cdot v\right) \cdot v + 1)_*} \cdot \sqrt{(\left(-5 \cdot v\right) \cdot v + 1)_*}}}{1 \cdot (v \cdot v + -1)_*}\right)\]

  6. Applied times-frac0.6

    \[\leadsto \cos^{-1} \color{blue}{\left(\frac{\sqrt{(\left(-5 \cdot v\right) \cdot v + 1)_*}}{1} \cdot \frac{\sqrt{(\left(-5 \cdot v\right) \cdot v + 1)_*}}{(v \cdot v + -1)_*}\right)}\]

  7. Simplified0.6

    \[\leadsto \cos^{-1} \left(\color{blue}{\sqrt{(-5 \cdot \left(v \cdot v\right) + 1)_*}} \cdot \frac{\sqrt{(\left(-5 \cdot v\right) \cdot v + 1)_*}}{(v \cdot v + -1)_*}\right)\]

  8. Final simplification0.6

    \[\leadsto \cos^{-1} \left(\sqrt{(-5 \cdot \left(v \cdot v\right) + 1)_*} \cdot \frac{\sqrt{(\left(-5 \cdot v\right) \cdot v + 1)_*}}{(v \cdot v + -1)_*}\right)\]

Runtime

Time bar (total: 51.2s)Debug logProfile

herbie shell --seed 2018349 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))