Average Error: 14.6 → 0.2
Time: 1.8m
Precision: 64
Internal Precision: 384
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi \cdot \frac{1}{2}}{a + b}}{a \cdot b}\]

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 simplify14.6

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

  4. Applied frac-sub14.6

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

  5. Applied associate-*l/14.6

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

  6. Applied simplify0.2

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

Runtime

Time bar (total: 1.8m)Debug logProfile

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