Average Error: 0.6 → 0.7
Time: 1.2m
Precision: 64
Internal Precision: 384
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)} \cdot \sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{1} \cdot \frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right)\]

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 *-un-lft-identity0.6

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

  4. Applied add-cube-cbrt0.7

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

  5. Applied times-frac0.7

    \[\leadsto \cos^{-1} \color{blue}{\left(\frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)} \cdot \sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{1} \cdot \frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right)}\]
  6. Removed slow pow expressions.

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed '#(1062930989 876886121 3990119081 3032829768 3060892583 1929069376)' 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))