Average Error: 0.5 → 0.6
Time: 38.8s
Precision: 64
Internal Precision: 576
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[{\left(e^{\sqrt{\log \left(\cos^{-1} \left(\frac{(\left(-5\right) \cdot \left(v \cdot v\right) + 1)_*}{v \cdot v - 1}\right)\right)}}\right)}^{\left(\sqrt{\log \left((e^{\log_* (1 + \cos^{-1} \left(\frac{1 - \log \left(e^{\left(v \cdot v\right) \cdot 5}\right)}{v \cdot v - 1}\right))} - 1)^*\right)}\right)}\]

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-exp-log0.5

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

  7. Applied add-sqr-sqrt0.6

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

  8. Applied exp-prod0.6

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

  9. Simplified0.6

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

  11. Applied add-log-exp0.6

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

  12. Final simplification0.6

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

Runtime

Time bar (total: 38.8s)Debug logProfile

herbie shell --seed 2018221 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))