Average Error: 0.7 → 0.5
Time: 17.2s
Precision: 64
Internal precision: 128
\[\cos^{-1} \left(\frac{1 - 5 \cdot {v}^2}{{v}^2 - 1}\right)\]
⬇
\[\cos^{-1} \left(4 \cdot \left({v}^{4} + {v}^2\right) - 1\right)\]

Error

Bits error versus v

Derivation

  1. Initial program 0.7

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

  2. Applied taylor 0.5

    \[\leadsto \cos^{-1} \left(\left(4 \cdot {v}^{4} + 4 \cdot {v}^2\right) - 1\right)\]

  3. Taylor expanded around 0 0.5

    \[\leadsto \cos^{-1} \color{blue}{\left(\left(4 \cdot {v}^{4} + 4 \cdot {v}^2\right) - 1\right)}\]

  4. Applied simplify 0.5

    \[\leadsto \color{blue}{\cos^{-1} \left(4 \cdot \left({v}^{4} + {v}^2\right) - 1\right)}\]
  5. Removed slow pow expressions

Runtime

Time bar (total: 17.2s) Debug log

Please include this information when filing a bug report:

herbie --seed '#(2619339579 2848021760 3869905955 3892467057 3534931764 3873819062)'
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (sqr v))) (- (sqr v) 1))))