Average Error: 14.5 → 0.3
Time: 1.9m
Precision: 64
Internal Precision: 576
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi}{a} - \frac{\pi}{b}}{b + a} \cdot \frac{\frac{1}{2}}{b - a}\]

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

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

  4. Applied div-inv9.6

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

  5. Applied times-frac9.0

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

  6. Applied associate-*r*0.3

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

  7. Applied simplify0.3

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

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed '#(1071725047 233389029 2036512464 3988615230 2972226563 1111574017)' 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))