Average Error: 14.3 → 0.2
Time: 19.9s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{0.5 \cdot \frac{\frac{\pi}{a}}{b}}{b + a}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{0.5 \cdot \frac{\frac{\pi}{a}}{b}}{b + a}
double f(double a, double b) {
        double r72583 = atan2(1.0, 0.0);
        double r72584 = 2.0;
        double r72585 = r72583 / r72584;
        double r72586 = 1.0;
        double r72587 = b;
        double r72588 = r72587 * r72587;
        double r72589 = a;
        double r72590 = r72589 * r72589;
        double r72591 = r72588 - r72590;
        double r72592 = r72586 / r72591;
        double r72593 = r72585 * r72592;
        double r72594 = r72586 / r72589;
        double r72595 = r72586 / r72587;
        double r72596 = r72594 - r72595;
        double r72597 = r72593 * r72596;
        return r72597;
}
↓
double f(double a, double b) {
        double r72598 = 0.5;
        double r72599 = atan2(1.0, 0.0);
        double r72600 = a;
        double r72601 = r72599 / r72600;
        double r72602 = b;
        double r72603 = r72601 / r72602;
        double r72604 = r72598 * r72603;
        double r72605 = r72602 + r72600;
        double r72606 = r72604 / r72605;
        return r72606;
}

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 14.3

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

  3. Applied difference-of-squares9.4

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

  4. Applied *-un-lft-identity9.4

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

  5. Applied times-frac9.0

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

  6. Applied associate-*r*9.0

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

  7. Simplified8.9

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

  9. Applied associate-*l/8.9

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

  10. Applied associate-*l/0.3

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

  11. Taylor expanded around 0 0.2

    \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{\pi}{a \cdot b}}}{b + a}\]
  12. Using strategy rm

  13. Applied associate-/r*0.2

    \[\leadsto \frac{0.5 \cdot \color{blue}{\frac{\frac{\pi}{a}}{b}}}{b + a}\]

  14. Final simplification0.2

    \[\leadsto \frac{0.5 \cdot \frac{\frac{\pi}{a}}{b}}{b + a}\]

Reproduce

herbie shell --seed 2019326 +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))))