Error

Derivation

1. Initial program 14.7

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

2. Simplified14.6

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

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

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

5. Applied add-sqr-sqrt14.6

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

6. Applied times-frac14.6

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

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

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

8. Applied add-sqr-sqrt14.6

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

9. Applied times-frac14.6

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

10. Applied distribute-lft-out--14.6

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

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

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

12. Applied times-frac14.6

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

13. Applied difference-of-squares9.9

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

14. Applied *-un-lft-identity9.9

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

15. Applied times-frac9.5

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

16. Applied times-frac0.3

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

17. Simplified0.3

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

18. Simplified0.3

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

20. Applied associate-*l/0.3

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

21. Simplified0.3

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

22. Final simplification0.3

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

Reproduce

herbie shell --seed 2019091 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))