Average Error: 0.6 → 1.0
Time: 1.8m
Precision: 64
Internal Precision: 576
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[(e^{\log_* (1 + \left(\frac{\pi}{2} - \sin^{-1} \left(\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{1 + v}}{v - 1}\right)\right))} - 1)^*\]

Error

Bits error versus v

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.6

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

  3. Applied acos-asin0.6

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

  5. Applied difference-of-sqr-11.0

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

  6. Applied associate-/r*1.0

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

  8. Applied expm1-log1p-u1.0

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

  9. Final simplification1.0

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

Runtime

Time bar (total: 1.8m)Debug logProfile

herbie shell --seed 2018242 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))