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

↓

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

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Derivation

Initial program 14.0
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
Initial simplification0.3
\[\leadsto \frac{\frac{1}{a} - \frac{1}{b}}{\left(a + b\right) \cdot 2} \cdot \frac{\pi}{b - a}\]
Using strategy rm
Applied clear-num0.3
\[\leadsto \frac{\frac{1}{a} - \frac{1}{b}}{\left(a + b\right) \cdot 2} \cdot \color{blue}{\frac{1}{\frac{b - a}{\pi}}}\]
Final simplification0.3
\[\leadsto \frac{\frac{1}{a} - \frac{1}{b}}{2 \cdot \left(a + b\right)} \cdot \frac{1}{\frac{b - a}{\pi}}\]

herbie shell --seed 2018219 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))