Average Error: 14.6 → 0.3
Time: 9.3s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi}{2}}{\frac{b - a}{1 \cdot b - a \cdot 1}} \cdot \frac{\frac{1}{b + a}}{a \cdot b}\]
\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{\pi}{2}}{\frac{b - a}{1 \cdot b - a \cdot 1}} \cdot \frac{\frac{1}{b + a}}{a \cdot b}
double f(double a, double b) {
        double r57576 = atan2(1.0, 0.0);
        double r57577 = 2.0;
        double r57578 = r57576 / r57577;
        double r57579 = 1.0;
        double r57580 = b;
        double r57581 = r57580 * r57580;
        double r57582 = a;
        double r57583 = r57582 * r57582;
        double r57584 = r57581 - r57583;
        double r57585 = r57579 / r57584;
        double r57586 = r57578 * r57585;
        double r57587 = r57579 / r57582;
        double r57588 = r57579 / r57580;
        double r57589 = r57587 - r57588;
        double r57590 = r57586 * r57589;
        return r57590;
}
↓
double f(double a, double b) {
        double r57591 = atan2(1.0, 0.0);
        double r57592 = 2.0;
        double r57593 = r57591 / r57592;
        double r57594 = b;
        double r57595 = a;
        double r57596 = r57594 - r57595;
        double r57597 = 1.0;
        double r57598 = r57597 * r57594;
        double r57599 = r57595 * r57597;
        double r57600 = r57598 - r57599;
        double r57601 = r57596 / r57600;
        double r57602 = r57593 / r57601;
        double r57603 = r57594 + r57595;
        double r57604 = r57597 / r57603;
        double r57605 = r57595 * r57594;
        double r57606 = r57604 / r57605;
        double r57607 = r57602 * r57606;
        return r57607;
}

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 14.6

    \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
  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*9.1

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

  6. Applied associate-*r/9.1

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

  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-/l*0.3

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

  11. Applied frac-sub0.3

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

  12. Applied associate-/r/0.3

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

  13. Applied times-frac0.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2020001 +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))))