Average Error: 14.5 → 0.3
Time: 5.0m
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{1}{\frac{b + a}{\pi}}}{2 \cdot \left(a \cdot b\right)}\]

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. Using strategy rm

  3. Applied frac-sub14.5

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

  4. Applied associate-*l/14.5

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

  5. Applied frac-times14.5

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

  6. Simplified0.2

    \[\leadsto \frac{\color{blue}{\frac{\pi}{b + a}}}{2 \cdot \left(a \cdot b\right)}\]
  7. Using strategy rm

  8. Applied clear-num0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2019100 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))