Average Error: 0.5 → 0.5
Time: 2.9m
Precision: 64
Internal Precision: 576
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[(e^{\log_* (1 + \sqrt[3]{{\left(\cos^{-1} \left(\frac{(\left(-5\right) \cdot \left(v \cdot v\right) + 1)_*}{v \cdot v - 1}\right)\right)}^{3}})} - 1)^*\]

Error

Bits error versus v

Derivation

  1. Initial program 0.5

    \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
  2. Using strategy rm

  3. Applied expm1-log1p-u0.5

    \[\leadsto \color{blue}{(e^{\log_* (1 + \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right))} - 1)^*}\]
  4. Using strategy rm

  5. Applied add-cbrt-cube0.5

    \[\leadsto (e^{\log_* (1 + \color{blue}{\sqrt[3]{\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \cdot \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right) \cdot \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}})} - 1)^*\]

  6. Applied simplify0.5

    \[\leadsto (e^{\log_* (1 + \sqrt[3]{\color{blue}{{\left(\cos^{-1} \left(\frac{(\left(-5\right) \cdot \left(v \cdot v\right) + 1)_*}{v \cdot v - 1}\right)\right)}^{3}}})} - 1)^*\]

Runtime

Time bar (total: 2.9m)Debug logProfile

herbie shell --seed 2018199 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))