Error

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

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

7. Simplified0.6

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

8. Final simplification0.6

   \[\leadsto \cos^{-1} \left(\frac{\sqrt{(\left(v \cdot v\right) \cdot -5 + 1)_*}}{(v \cdot v + -1)_*} \cdot \sqrt{(\left(v \cdot v\right) \cdot -5 + 1)_*}\right)\]

Reproduce

herbie shell --seed 2019004 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))

Details

Time bar (total: 1.3m)Debug log