Error

Derivation

1. Initial program 14.2

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

2. Simplified14.2

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

4. Applied *-un-lft-identity14.2

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

5. Applied difference-of-squares14.2

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

6. Applied *-un-lft-identity14.2

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

7. Applied times-frac13.9

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

8. Applied times-frac9.6

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

9. Simplified9.6

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

11. Applied difference-of-squares4.8

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

12. Applied associate-/r*4.5

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

14. Applied pow14.5

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

15. Applied pow14.5

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

16. Applied pow-prod-down4.5

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

17. Simplified4.5

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

19. Applied div-inv4.5

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

20. Final simplification4.5

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

Reproduce

herbie shell --seed 2019089 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))