Average Error: 0.5 → 0.5
Time: 46.6s
Precision: 64
Internal Precision: 576
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\frac{\pi}{2} - \sin^{-1} \left(\left(1 - \left(v \cdot v\right) \cdot 5\right) \cdot \frac{1}{v \cdot v - 1}\right)\]

Error

Bits error versus v

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Your Program's Arguments

Results

Enter valid numbers for all inputs

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 div-inv0.5

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

  5. Applied acos-asin0.5

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

  6. Final simplification0.5

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

Runtime

Time bar (total: 46.6s)Debug logProfile

herbie shell --seed 2018230 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))