Average Error: 14.7 → 0.3
Time: 5.8s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi \cdot 1}{b + a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{2 \cdot \left(b - a\right)}\]
\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 1}{b + a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{2 \cdot \left(b - a\right)}
double f(double a, double b) {
        double r44608 = atan2(1.0, 0.0);
        double r44609 = 2.0;
        double r44610 = r44608 / r44609;
        double r44611 = 1.0;
        double r44612 = b;
        double r44613 = r44612 * r44612;
        double r44614 = a;
        double r44615 = r44614 * r44614;
        double r44616 = r44613 - r44615;
        double r44617 = r44611 / r44616;
        double r44618 = r44610 * r44617;
        double r44619 = r44611 / r44614;
        double r44620 = r44611 / r44612;
        double r44621 = r44619 - r44620;
        double r44622 = r44618 * r44621;
        return r44622;
}
↓
double f(double a, double b) {
        double r44623 = atan2(1.0, 0.0);
        double r44624 = 1.0;
        double r44625 = r44623 * r44624;
        double r44626 = b;
        double r44627 = a;
        double r44628 = r44626 + r44627;
        double r44629 = r44625 / r44628;
        double r44630 = r44624 / r44627;
        double r44631 = r44624 / r44626;
        double r44632 = r44630 - r44631;
        double r44633 = r44629 * r44632;
        double r44634 = 2.0;
        double r44635 = r44626 - r44627;
        double r44636 = r44634 * r44635;
        double r44637 = r44633 / r44636;
        return r44637;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.7

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

  3. Applied difference-of-squares9.6

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

  4. Applied associate-/r*9.1

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

  6. Applied frac-times9.1

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

  7. Applied associate-*l/0.3

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

  9. Applied associate-*r/0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2020081 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))