Average Error: 14.4 → 0.2
Time: 9.8s
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 r76729 = atan2(1.0, 0.0);
        double r76730 = 2.0;
        double r76731 = r76729 / r76730;
        double r76732 = 1.0;
        double r76733 = b;
        double r76734 = r76733 * r76733;
        double r76735 = a;
        double r76736 = r76735 * r76735;
        double r76737 = r76734 - r76736;
        double r76738 = r76732 / r76737;
        double r76739 = r76731 * r76738;
        double r76740 = r76732 / r76735;
        double r76741 = r76732 / r76733;
        double r76742 = r76740 - r76741;
        double r76743 = r76739 * r76742;
        return r76743;
}
↓
double f(double a, double b) {
        double r76744 = 0.5;
        double r76745 = atan2(1.0, 0.0);
        double r76746 = a;
        double r76747 = r76745 / r76746;
        double r76748 = b;
        double r76749 = r76745 / r76748;
        double r76750 = r76747 - r76749;
        double r76751 = r76744 * r76750;
        double r76752 = r76748 + r76746;
        double r76753 = r76751 / r76752;
        double r76754 = r76748 - r76746;
        double r76755 = r76753 / r76754;
        return r76755;
}

Error

Bits error versus a

Bits error versus b

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.4

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

  3. Applied difference-of-squares9.8

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

  4. Applied associate-/r*9.3

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

  6. Applied associate-*r/9.2

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

  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. Applied associate-*l/0.3

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

  11. Taylor expanded around 0 0.2

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

  12. Simplified0.2

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

  13. 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 2019362 +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))))