Average Error: 14.2 → 0.7
Time: 3.8m
Precision: 64
Internal Precision: 384
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi}{2}}{\left(a \cdot b\right) \cdot \left(b + a\right)}\]

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.2

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

  2. Applied simplify9.1

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

  4. Applied frac-sub9.1

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

  5. Applied frac-times0.8

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

  6. Applied simplify0.7

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

Runtime

Time bar (total: 3.8m)Debug logProfile

herbie shell --seed '#(1070706311 3771791028 4128836681 4194990999 2341756049 504035650)' 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))