Average Error: 14.1 → 4.7
Time: 56.2s
Precision: 64
Internal Precision: 128
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{(\left(\frac{\pi}{b - a} \cdot \frac{1}{b + a}\right) \cdot \left(\frac{-1}{b}\right) + \left(\frac{\frac{1}{b + a}}{a \cdot \left(\frac{1}{\pi} \cdot \left(b - a\right)\right)}\right))_*}{2}\]

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.1

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

  2. Simplified14.1

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

  4. Applied difference-of-squares14.1

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

  5. Applied *-un-lft-identity14.1

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

  6. Applied times-frac13.9

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

  7. Applied associate-/l*9.9

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

  9. Applied difference-of-squares5.0

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

  10. Applied *-un-lft-identity5.0

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

  11. Applied times-frac4.7

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

  13. Applied associate-/r/4.7

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

  15. Applied div-inv4.7

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

  16. Applied associate-*l*4.7

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

  17. Final simplification4.7

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

Reproduce

herbie shell --seed 2019090 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))