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. Initial simplification9.5

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

6. Applied *-un-lft-identity0.3

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

7. Applied *-un-lft-identity0.3

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

8. Applied times-frac0.3

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

9. Applied associate-/l*0.8

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

10. Simplified0.8

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

11. Final simplification0.8

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

Reproduce

herbie shell --seed 2018362 +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: 30.7s)Debug log