Average Error: 0.6 → 1.0
Time: 2.5m
Precision: 64
Internal Precision: 384
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[e^{\log \left(\cos^{-1} \left(\frac{\frac{1 - \log \left(e^{5 \cdot \left(v \cdot v\right)}\right)}{v + 1}}{v - 1}\right)\right)}\]

Error

Bits error versus v

Derivation

  1. Initial program 0.6

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

  3. Applied difference-of-sqr-10.9

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

  4. Applied associate-/r*0.9

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

  6. Applied add-exp-log0.9

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

  8. Applied add-log-exp1.0

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

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed '#(1070833653 108281690 3330367898 3632331308 3494323072 43156186)' +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))