Average Error: 14.5 → 4.7
Time: 19.4s
Precision: 64
Internal Precision: 128
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{(\left(\frac{\frac{\pi}{b + a}}{b - a}\right) \cdot \left(\frac{-1}{b}\right) + \left(\frac{\frac{1}{b + a}}{\left(b - a\right) \cdot \frac{a}{\pi}}\right))_*}{2}\]

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.5

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

  2. Simplified14.5

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

  4. Applied *-un-lft-identity14.5

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

  5. Applied associate-/l*14.6

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

  6. Taylor expanded around inf 20.4

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

  7. Simplified10.0

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

  9. Applied difference-of-squares5.2

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

  10. Applied associate-/r*5.0

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

  12. Applied associate-/r*4.7

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

  13. Final simplification4.7

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

Reproduce

herbie shell --seed 2019053 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))