Average Error: 0.5 → 0.5
Time: 1.9m
Precision: 64
Internal Precision: 576
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[(e^{\sqrt{\log_* (1 + \left(\sqrt[3]{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)} \cdot \sqrt[3]{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}\right) \cdot \sqrt[3]{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)})} \cdot \sqrt{\log_* (1 + \cos^{-1} \left(\frac{1 - \log \left(e^{5 \cdot \left(v \cdot v\right)}\right)}{v \cdot v - 1}\right))}} - 1)^*\]

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-sqr-sqrt0.5

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

  7. Applied add-log-exp0.5

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

  9. Applied add-cube-cbrt0.5

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

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed '#(1072840222 1305617769 1692503039 1353360431 4178980589 1488672652)' +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))