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Derivation

1. Initial program 14.5

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

2. Simplified0.3

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

4. Applied associate-*l/0.3

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

5. Simplified0.3

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

7. Applied div-inv0.3

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

8. Applied div-inv0.3

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

9. Applied distribute-lft-out--0.3

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

10. Applied times-frac0.3

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

11. Simplified0.3

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

12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019007 
(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: 29.7s)Debug log