Error

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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. Simplified9.6

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

4. Applied *-un-lft-identity9.6

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

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 associate-*l/0.3

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

9. Simplified0.3

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

10. Taylor expanded around inf 0.2

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

11. Final simplification0.2

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

Reproduce

herbie shell --seed 2019007 +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: 40.5s)Debug log