Average Error: 0.5 → 0.6
Time: 53.6s
Precision: 64
Internal Precision: 576
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\frac{\pi}{2} - \sin^{-1} \left(\frac{\sqrt{1 - \left(v \cdot v\right) \cdot 5}}{v \cdot v - 1} \cdot \sqrt{1 - \left(v \cdot v\right) \cdot 5}\right)\]

Error

Bits error versus v

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Results

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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 *-un-lft-identity0.5

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

  4. Applied add-sqr-sqrt0.6

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

  5. Applied times-frac0.6

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

  6. Simplified0.6

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

  8. Applied acos-asin0.6

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

  9. Final simplification0.6

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

Runtime

Time bar (total: 53.6s)Debug logProfile

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