Average Error: 15.1 → 4.8
Time: 5.0m
Precision: 64
Internal Precision: 128
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{(\left(\frac{\frac{1}{\frac{b + a}{\pi}}}{b - a}\right) \cdot \left(\frac{-1}{b}\right) + \left(\frac{\frac{\pi}{b + a}}{\left(b - a\right) \cdot a}\right))_*}{2}\]

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.1

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

  2. Simplified15.0

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

  4. Applied *-commutative15.0

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

  5. Applied difference-of-squares15.0

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

  6. Applied associate-/r*14.7

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

  7. Applied associate-/l/10.2

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

  9. Applied *-commutative10.2

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

  10. Applied difference-of-squares5.1

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

  11. Applied associate-/r*4.8

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

  13. Applied *-un-lft-identity4.8

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

  14. Applied associate-/l*4.8

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

  15. Final simplification4.8

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

Reproduce

herbie shell --seed 2019088 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))