Average Error: 0.6 → 0.8
Time: 19.5s
Precision: 64
Internal Precision: 128
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\frac{\pi}{2} - \sin^{-1} \left(-1 + 4 \cdot \left(v \cdot v + {v}^{4}\right)\right)\]

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.6

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

  2. Taylor expanded around 0 0.8

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

  3. Simplified0.8

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

  5. Applied acos-asin0.8

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

  6. Final simplification0.8

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

Reproduce

herbie shell --seed 2019021 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))