Average Error: 14.2 → 0.3
Time: 24.3s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(-1, \frac{\pi}{b}, \frac{\pi}{a}\right)}{b + a}}{b - a}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\frac{1}{2} \cdot \frac{\mathsf{fma}\left(-1, \frac{\pi}{b}, \frac{\pi}{a}\right)}{b + a}}{b - a}
double f(double a, double b) {
        double r1394523 = atan2(1.0, 0.0);
        double r1394524 = 2.0;
        double r1394525 = r1394523 / r1394524;
        double r1394526 = 1.0;
        double r1394527 = b;
        double r1394528 = r1394527 * r1394527;
        double r1394529 = a;
        double r1394530 = r1394529 * r1394529;
        double r1394531 = r1394528 - r1394530;
        double r1394532 = r1394526 / r1394531;
        double r1394533 = r1394525 * r1394532;
        double r1394534 = r1394526 / r1394529;
        double r1394535 = r1394526 / r1394527;
        double r1394536 = r1394534 - r1394535;
        double r1394537 = r1394533 * r1394536;
        return r1394537;
}
↓
double f(double a, double b) {
        double r1394538 = 0.5;
        double r1394539 = -1.0;
        double r1394540 = atan2(1.0, 0.0);
        double r1394541 = b;
        double r1394542 = r1394540 / r1394541;
        double r1394543 = a;
        double r1394544 = r1394540 / r1394543;
        double r1394545 = fma(r1394539, r1394542, r1394544);
        double r1394546 = r1394541 + r1394543;
        double r1394547 = r1394545 / r1394546;
        double r1394548 = r1394538 * r1394547;
        double r1394549 = r1394541 - r1394543;
        double r1394550 = r1394548 / r1394549;
        return r1394550;
}

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.2

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

  2. Simplified0.3

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

  4. Applied div-inv0.3

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

  5. Applied times-frac0.3

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

  6. Simplified0.3

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

  7. Simplified0.3

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

  9. Applied associate-*r/0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2019141 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))