Average Error: 14.3 → 0.2
Time: 8.5s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{0.5 \cdot \left(\frac{\pi}{a} - \frac{\pi}{b}\right)}{b + a}}{b - a}\]
\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{0.5 \cdot \left(\frac{\pi}{a} - \frac{\pi}{b}\right)}{b + a}}{b - a}
double f(double a, double b) {
        double r55582 = atan2(1.0, 0.0);
        double r55583 = 2.0;
        double r55584 = r55582 / r55583;
        double r55585 = 1.0;
        double r55586 = b;
        double r55587 = r55586 * r55586;
        double r55588 = a;
        double r55589 = r55588 * r55588;
        double r55590 = r55587 - r55589;
        double r55591 = r55585 / r55590;
        double r55592 = r55584 * r55591;
        double r55593 = r55585 / r55588;
        double r55594 = r55585 / r55586;
        double r55595 = r55593 - r55594;
        double r55596 = r55592 * r55595;
        return r55596;
}
↓
double f(double a, double b) {
        double r55597 = 0.5;
        double r55598 = atan2(1.0, 0.0);
        double r55599 = a;
        double r55600 = r55598 / r55599;
        double r55601 = b;
        double r55602 = r55598 / r55601;
        double r55603 = r55600 - r55602;
        double r55604 = r55597 * r55603;
        double r55605 = r55601 + r55599;
        double r55606 = r55604 / r55605;
        double r55607 = r55601 - r55599;
        double r55608 = r55606 / r55607;
        return r55608;
}

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-squares9.3

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

  4. Applied *-un-lft-identity9.3

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

  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-*r/8.9

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

  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. Taylor expanded around 0 0.2

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

  15. Simplified0.2

    \[\leadsto \frac{\frac{\color{blue}{0.5 \cdot \left(\frac{\pi}{a} - \frac{\pi}{b}\right)}}{b + a}}{b - a}\]

  16. Final simplification0.2

    \[\leadsto \frac{\frac{0.5 \cdot \left(\frac{\pi}{a} - \frac{\pi}{b}\right)}{b + a}}{b - a}\]

Reproduce

herbie shell --seed 2020024 +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))))