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Derivation

1. Initial program 14.6

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]

2. Simplified9.8

   \[\leadsto \color{blue}{\frac{(\left(\frac{\pi}{2}\right) \cdot \left(\frac{-1}{b}\right) + \left(\frac{\frac{\pi}{2}}{a}\right))_*}{\left(a + b\right) \cdot \left(b - a\right)}}\]
3. Using strategy rm

4. Applied *-un-lft-identity9.8

   \[\leadsto \frac{\color{blue}{1 \cdot (\left(\frac{\pi}{2}\right) \cdot \left(\frac{-1}{b}\right) + \left(\frac{\frac{\pi}{2}}{a}\right))_*}}{\left(a + b\right) \cdot \left(b - a\right)}\]

5. Applied times-frac0.3

   \[\leadsto \color{blue}{\frac{1}{a + b} \cdot \frac{(\left(\frac{\pi}{2}\right) \cdot \left(\frac{-1}{b}\right) + \left(\frac{\frac{\pi}{2}}{a}\right))_*}{b - a}}\]

6. Simplified0.3

   \[\leadsto \frac{1}{a + b} \cdot \color{blue}{\frac{\frac{\pi}{a \cdot 2} - \frac{\pi}{2 \cdot b}}{b - a}}\]
7. Using strategy rm

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

   \[\leadsto \frac{1}{a + b} \cdot \frac{\frac{\pi}{a \cdot 2} - \frac{\pi}{2 \cdot b}}{\color{blue}{1 \cdot \left(b - a\right)}}\]

9. Applied div-inv0.3

   \[\leadsto \frac{1}{a + b} \cdot \frac{\frac{\pi}{a \cdot 2} - \color{blue}{\pi \cdot \frac{1}{2 \cdot b}}}{1 \cdot \left(b - a\right)}\]

10. Applied div-inv0.3

    \[\leadsto \frac{1}{a + b} \cdot \frac{\color{blue}{\pi \cdot \frac{1}{a \cdot 2}} - \pi \cdot \frac{1}{2 \cdot b}}{1 \cdot \left(b - a\right)}\]

11. Applied distribute-lft-out--0.3

    \[\leadsto \frac{1}{a + b} \cdot \frac{\color{blue}{\pi \cdot \left(\frac{1}{a \cdot 2} - \frac{1}{2 \cdot b}\right)}}{1 \cdot \left(b - a\right)}\]

12. Applied times-frac0.3

    \[\leadsto \frac{1}{a + b} \cdot \color{blue}{\left(\frac{\pi}{1} \cdot \frac{\frac{1}{a \cdot 2} - \frac{1}{2 \cdot b}}{b - a}\right)}\]

13. Simplified0.3

    \[\leadsto \frac{1}{a + b} \cdot \left(\color{blue}{\pi} \cdot \frac{\frac{1}{a \cdot 2} - \frac{1}{2 \cdot b}}{b - a}\right)\]

14. Simplified0.3

    \[\leadsto \frac{1}{a + b} \cdot \left(\pi \cdot \color{blue}{\frac{\frac{1}{2 \cdot a} - \frac{\frac{1}{2}}{b}}{b - a}}\right)\]
15. Using strategy rm

16. Applied associate-*l/0.3

    \[\leadsto \color{blue}{\frac{1 \cdot \left(\pi \cdot \frac{\frac{1}{2 \cdot a} - \frac{\frac{1}{2}}{b}}{b - a}\right)}{a + b}}\]

17. Simplified0.3

    \[\leadsto \frac{\color{blue}{\frac{\pi}{b - a} \cdot \left(\frac{\frac{1}{2}}{a} - \frac{\frac{1}{2}}{b}\right)}}{a + b}\]

18. Final simplification0.3

    \[\leadsto \frac{\frac{\pi}{b - a} \cdot \left(\frac{\frac{1}{2}}{a} - \frac{\frac{1}{2}}{b}\right)}{b + a}\]

Reproduce

herbie shell --seed 2019008 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))

Details

Time bar (total: 30.9s)Debug log