Average Error: 14.5 → 0.3
Time: 4.9s
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 r44594 = atan2(1.0, 0.0);
        double r44595 = 2.0;
        double r44596 = r44594 / r44595;
        double r44597 = 1.0;
        double r44598 = b;
        double r44599 = r44598 * r44598;
        double r44600 = a;
        double r44601 = r44600 * r44600;
        double r44602 = r44599 - r44601;
        double r44603 = r44597 / r44602;
        double r44604 = r44596 * r44603;
        double r44605 = r44597 / r44600;
        double r44606 = r44597 / r44598;
        double r44607 = r44605 - r44606;
        double r44608 = r44604 * r44607;
        return r44608;
}
↓
double f(double a, double b) {
        double r44609 = atan2(1.0, 0.0);
        double r44610 = 2.0;
        double r44611 = r44609 / r44610;
        double r44612 = 1.0;
        double r44613 = r44611 * r44612;
        double r44614 = a;
        double r44615 = r44612 / r44614;
        double r44616 = b;
        double r44617 = r44612 / r44616;
        double r44618 = r44615 - r44617;
        double r44619 = r44616 + r44614;
        double r44620 = r44618 / r44619;
        double r44621 = r44616 - r44614;
        double r44622 = r44620 / r44621;
        double r44623 = r44613 * r44622;
        return r44623;
}

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