Average Error: 14.5 → 0.3
Time: 3.4m
Precision: 64
Internal Precision: 384
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{b + a}}{\left(b - a\right) \cdot 2}\]

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. Applied simplify9.6

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

  4. Applied associate-/r*0.3

    \[\leadsto \color{blue}{\frac{\frac{\frac{1}{a} - \frac{1}{b}}{b + a}}{b - a}} \cdot \frac{\pi}{2}\]
  5. Using strategy rm

  6. Applied frac-times0.3

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

  7. Applied simplify0.3

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

Runtime

Time bar (total: 3.4m)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))