Average Error: 14.6 → 0.3
Time: 1.9m
Precision: 64
Internal Precision: 384
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{\pi}{a \cdot 2} - \frac{1}{\frac{2 \cdot b}{\pi}}}{b + a}}{b - a}\]

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. Applied simplify9.9

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

  4. Applied associate-/r*0.3

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

  5. Applied simplify0.3

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

  7. Applied clear-num0.3

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

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed '#(1070991898 1055468627 4280279443 640792587 928206309 3646738750)' +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))