Average Error: 0.5 → 0.6
Time: 34.9s
Precision: 64
Internal Precision: 576
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\log \left(1 + (e^{\frac{(-5 \cdot \left(v \cdot v\right) + 1)_*}{(v \cdot v + -1)_*}} - 1)^*\right)\right)\]

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. Initial simplification0.5

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

  4. Applied log1p-expm1-u0.6

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

  6. Applied log1p-udef0.6

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

  7. Final simplification0.6

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

Runtime

Time bar (total: 34.9s)Debug logProfile

herbie shell --seed 2018234 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))