Error

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Derivation

1. Initial program 0.5

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
2. Using strategy rm

3. Applied *-un-lft-identity0.5

   \[\leadsto \frac{\color{blue}{1 \cdot \left(1 - 5 \cdot \left(v \cdot v\right)\right)}}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]

4. Applied times-frac0.5

   \[\leadsto \color{blue}{\frac{1}{\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v}}\]

5. Simplified0.4

   \[\leadsto \color{blue}{\frac{\frac{\frac{1}{t}}{\pi}}{\sqrt{2 + \left(v \cdot -6\right) \cdot v}}} \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v}\]
6. Using strategy rm

7. Applied *-un-lft-identity0.4

   \[\leadsto \frac{\frac{\frac{1}{t}}{\pi}}{\color{blue}{1 \cdot \sqrt{2 + \left(v \cdot -6\right) \cdot v}}} \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v}\]

8. Applied div-inv0.4

   \[\leadsto \frac{\color{blue}{\frac{1}{t} \cdot \frac{1}{\pi}}}{1 \cdot \sqrt{2 + \left(v \cdot -6\right) \cdot v}} \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v}\]

9. Applied times-frac0.3

   \[\leadsto \color{blue}{\left(\frac{\frac{1}{t}}{1} \cdot \frac{\frac{1}{\pi}}{\sqrt{2 + \left(v \cdot -6\right) \cdot v}}\right)} \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v}\]

10. Simplified0.3

    \[\leadsto \left(\color{blue}{\frac{1}{t}} \cdot \frac{\frac{1}{\pi}}{\sqrt{2 + \left(v \cdot -6\right) \cdot v}}\right) \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v}\]
11. Using strategy rm

12. Applied associate-*l/0.1

    \[\leadsto \color{blue}{\frac{1 \cdot \frac{\frac{1}{\pi}}{\sqrt{2 + \left(v \cdot -6\right) \cdot v}}}{t}} \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v}\]

13. Simplified0.1

    \[\leadsto \frac{\color{blue}{\frac{\frac{1}{\pi}}{\sqrt{2 + \left(v \cdot -6\right) \cdot v}}}}{t} \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v}\]

14. Final simplification0.1

    \[\leadsto \frac{1 - \left(v \cdot v\right) \cdot 5}{1 - v \cdot v} \cdot \frac{\frac{\frac{1}{\pi}}{\sqrt{2 + \left(v \cdot -6\right) \cdot v}}}{t}\]

Reproduce

herbie shell --seed 2019005 
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  (/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v)))))

Details

Time bar (total: 30.8s)Debug log