Average Error: 14.3 → 0.3
Time: 4.4m
Precision: 64
Internal Precision: 384
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi}{a} - \frac{\pi}{b}}{\left(b - a\right) \cdot 2} \cdot \frac{1}{b + a}\]

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 simplify9.0

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

  4. Applied div-inv9.1

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

  5. Applied associate-*r*0.3

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

  6. Applied simplify0.3

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

Runtime

Time bar (total: 4.4m)Debug logProfile

herbie shell --seed '#(1070386091 2509006183 1430610344 1025408621 36622005 1425925650)' 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))