Average Error: 0.5 → 0.5
Time: 1.6m
Precision: 64
Internal Precision: 576
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\frac{\pi}{2} - \sqrt[3]{{\left(\sin^{-1} \left(\frac{(\left(-5 \cdot v\right) \cdot v + 1)_*}{(v \cdot v + -1)_*}\right)\right)}^{3}}\]

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 acos-asin0.5

    \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}\]

  4. Applied simplify0.5

    \[\leadsto \frac{\pi}{2} - \color{blue}{\sin^{-1} \left(\frac{(\left(v \cdot -5\right) \cdot v + 1)_*}{(v \cdot v + -1)_*}\right)}\]
  5. Using strategy rm

  6. Applied add-cbrt-cube0.5

    \[\leadsto \frac{\pi}{2} - \color{blue}{\sqrt[3]{\left(\sin^{-1} \left(\frac{(\left(v \cdot -5\right) \cdot v + 1)_*}{(v \cdot v + -1)_*}\right) \cdot \sin^{-1} \left(\frac{(\left(v \cdot -5\right) \cdot v + 1)_*}{(v \cdot v + -1)_*}\right)\right) \cdot \sin^{-1} \left(\frac{(\left(v \cdot -5\right) \cdot v + 1)_*}{(v \cdot v + -1)_*}\right)}}\]

  7. Applied simplify0.5

    \[\leadsto \frac{\pi}{2} - \sqrt[3]{\color{blue}{{\left(\sin^{-1} \left(\frac{(\left(-5 \cdot v\right) \cdot v + 1)_*}{(v \cdot v + -1)_*}\right)\right)}^{3}}}\]

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))