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Derivation

1. Initial program 14.4

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

2. Initial simplification9.7

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

4. Applied associate-/r*0.3

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

5. Taylor expanded around inf 0.3

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

6. Simplified0.3

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

8. Applied associate-/l*0.3

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

10. Applied div-inv0.3

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

11. Applied div-inv0.3

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

12. Applied distribute-lft-out--0.3

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

13. Applied associate-/l*0.3

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

14. Final simplification0.3

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

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018256 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))