Average Error: 14.3 → 0.3
Time: 1.5m
Precision: 64
Internal Precision: 384
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{1}{a} - \frac{1}{b}}{b + a}}{b - a} \cdot \frac{\pi}{2}\]

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.3

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

  2. Applied simplify9.5

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

  4. Applied associate-/r*0.3

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

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed '#(1064397287 3527694221 3797617954 1138343853 2854031332 1153838279)' +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))