Average Error: 14.6 → 0.3
Time: 2.5m
Precision: 64
Internal Precision: 128
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi}{\left(b + a\right) \cdot \frac{2}{b - a}}}{b - a} \cdot \frac{1}{a \cdot b}\]

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.6

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

  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 difference-of-squares9.6

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

  5. Applied associate-/r*9.1

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

  7. Applied frac-sub9.1

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

  8. Applied associate-/r/9.1

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

  9. Applied *-un-lft-identity9.1

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

  10. Applied flip-+14.6

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

  11. Applied associate-/r/14.6

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

  12. Applied times-frac14.6

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

  13. Applied times-frac14.6

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

  14. Simplified0.3

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

  15. Simplified0.3

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

  16. Taylor expanded around inf 0.3

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

  17. Final simplification0.3

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

Reproduce

herbie shell --seed 2019068 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))