Average Error: 0.5 → 0.5
Time: 2.2m
Precision: 64
Internal Precision: 576
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[(e^{(e^{\log_* (1 + \log_* (1 + \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)))} - 1)^*} - 1)^*\]

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 expm1-log1p-u0.5

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

Runtime

Time bar (total: 2.2m)Debug logProfile

herbie shell --seed '#(1071979731 1496239409 439705970 2863295848 982327776 189749553)' +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))