Average Error: 0.6 → 0.6
Time: 4.4m
Precision: 64
Internal Precision: 576
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\sqrt[3]{{\left(\frac{(\left(-5\right) \cdot \left(v \cdot v\right) + 1)_*}{v \cdot v - 1}\right)}^{3}}\right)\]

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 add-cbrt-cube0.6

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

  4. Applied add-cbrt-cube0.6

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

  5. Applied cbrt-undiv0.6

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

  6. Applied simplify0.6

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

Runtime

Time bar (total: 4.4m)Debug logProfile

herbie shell --seed 2018170 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))