Average Error: 14.4 → 0.3
Time: 1.5m
Precision: 64
Internal Precision: 384
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi}{2}}{b + a} \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b - a}\]

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.4

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

  2. Applied simplify9.4

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

  4. Applied times-frac0.3

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

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))