Average Error: 14.8 → 0.3
Time: 6.5s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{\left(1 \cdot \left(b - a\right)\right) \cdot \left(\frac{\pi}{2} \cdot 1\right)}{b + a}}{a \cdot b}}{b - a}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\frac{\frac{\left(1 \cdot \left(b - a\right)\right) \cdot \left(\frac{\pi}{2} \cdot 1\right)}{b + a}}{a \cdot b}}{b - a}
double f(double a, double b) {
        double r46583 = atan2(1.0, 0.0);
        double r46584 = 2.0;
        double r46585 = r46583 / r46584;
        double r46586 = 1.0;
        double r46587 = b;
        double r46588 = r46587 * r46587;
        double r46589 = a;
        double r46590 = r46589 * r46589;
        double r46591 = r46588 - r46590;
        double r46592 = r46586 / r46591;
        double r46593 = r46585 * r46592;
        double r46594 = r46586 / r46589;
        double r46595 = r46586 / r46587;
        double r46596 = r46594 - r46595;
        double r46597 = r46593 * r46596;
        return r46597;
}
↓
double f(double a, double b) {
        double r46598 = 1.0;
        double r46599 = b;
        double r46600 = a;
        double r46601 = r46599 - r46600;
        double r46602 = r46598 * r46601;
        double r46603 = atan2(1.0, 0.0);
        double r46604 = 2.0;
        double r46605 = r46603 / r46604;
        double r46606 = r46605 * r46598;
        double r46607 = r46602 * r46606;
        double r46608 = r46599 + r46600;
        double r46609 = r46607 / r46608;
        double r46610 = r46600 * r46599;
        double r46611 = r46609 / r46610;
        double r46612 = r46611 / r46601;
        return r46612;
}

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 14.8

    \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
  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*9.0

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

  6. Applied associate-*r/9.0

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

  7. Applied associate-*l/0.3

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

  9. Applied associate-*r/0.3

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

  11. Applied frac-sub0.3

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

  12. Applied associate-*r/0.3

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

  13. Simplified0.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2019318 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))