Average Error: 14.5 → 0.3
Time: 8.2s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\left(\frac{\pi}{2} \cdot 1\right) \cdot \frac{\frac{\frac{1}{a} - \frac{1}{b}}{b + a}}{b - a}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\left(\frac{\pi}{2} \cdot 1\right) \cdot \frac{\frac{\frac{1}{a} - \frac{1}{b}}{b + a}}{b - a}
double f(double a, double b) {
        double r53593 = atan2(1.0, 0.0);
        double r53594 = 2.0;
        double r53595 = r53593 / r53594;
        double r53596 = 1.0;
        double r53597 = b;
        double r53598 = r53597 * r53597;
        double r53599 = a;
        double r53600 = r53599 * r53599;
        double r53601 = r53598 - r53600;
        double r53602 = r53596 / r53601;
        double r53603 = r53595 * r53602;
        double r53604 = r53596 / r53599;
        double r53605 = r53596 / r53597;
        double r53606 = r53604 - r53605;
        double r53607 = r53603 * r53606;
        return r53607;
}
↓
double f(double a, double b) {
        double r53608 = atan2(1.0, 0.0);
        double r53609 = 2.0;
        double r53610 = r53608 / r53609;
        double r53611 = 1.0;
        double r53612 = r53610 * r53611;
        double r53613 = a;
        double r53614 = r53611 / r53613;
        double r53615 = b;
        double r53616 = r53611 / r53615;
        double r53617 = r53614 - r53616;
        double r53618 = r53615 + r53613;
        double r53619 = r53617 / r53618;
        double r53620 = r53615 - r53613;
        double r53621 = r53619 / r53620;
        double r53622 = r53612 * r53621;
        return r53622;
}

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 14.5

    \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
  2. 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 *-un-lft-identity9.6

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

  5. Applied times-frac9.1

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

  6. Applied associate-*r*9.1

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

  7. Simplified9.1

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

  9. Applied associate-*r/9.0

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

  10. Applied associate-*l/0.3

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

  12. Applied associate-*l/0.3

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

  13. Applied associate-*l/0.3

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

  15. Applied *-un-lft-identity0.3

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

  16. Applied *-un-lft-identity0.3

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

  17. Applied times-frac0.3

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

  18. Applied times-frac0.3

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

  19. Simplified0.3

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

  20. Final simplification0.3

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

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))