Average Error: 14.2 → 4.5
Time: 19.5s
Precision: 64
Internal Precision: 128
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{(\left(\frac{\frac{\pi}{b + a}}{b - a}\right) \cdot \left(\frac{-1}{b}\right) + \left(\frac{\frac{\pi}{a \cdot \left(b + a\right)}}{b - a}\right))_*}{2}\]

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.2

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

  2. Simplified14.2

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

  4. Applied associate-/l/14.2

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

  6. Applied difference-of-squares14.2

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

  7. Applied associate-*r*9.8

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

  9. Applied difference-of-squares5.1

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

  10. Applied associate-/r*4.8

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

  12. Applied associate-/r*4.5

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

  13. Final simplification4.5

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

Reproduce

herbie shell --seed 2019089 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))