Average Error: 14.2 → 0.3
Time: 11.8s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{1 \cdot \left(\frac{\pi}{a} - \frac{\pi}{b}\right)}{b - a}}{2 \cdot \left(b + a\right)}\]
\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 \cdot \left(\frac{\pi}{a} - \frac{\pi}{b}\right)}{b - a}}{2 \cdot \left(b + a\right)}
double f(double a, double b) {
        double r50776 = atan2(1.0, 0.0);
        double r50777 = 2.0;
        double r50778 = r50776 / r50777;
        double r50779 = 1.0;
        double r50780 = b;
        double r50781 = r50780 * r50780;
        double r50782 = a;
        double r50783 = r50782 * r50782;
        double r50784 = r50781 - r50783;
        double r50785 = r50779 / r50784;
        double r50786 = r50778 * r50785;
        double r50787 = r50779 / r50782;
        double r50788 = r50779 / r50780;
        double r50789 = r50787 - r50788;
        double r50790 = r50786 * r50789;
        return r50790;
}
↓
double f(double a, double b) {
        double r50791 = 1.0;
        double r50792 = atan2(1.0, 0.0);
        double r50793 = a;
        double r50794 = r50792 / r50793;
        double r50795 = b;
        double r50796 = r50792 / r50795;
        double r50797 = r50794 - r50796;
        double r50798 = r50791 * r50797;
        double r50799 = r50795 - r50793;
        double r50800 = r50798 / r50799;
        double r50801 = 2.0;
        double r50802 = r50795 + r50793;
        double r50803 = r50801 * r50802;
        double r50804 = r50800 / r50803;
        return r50804;
}

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 14.2

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

  3. Applied associate-*l*14.2

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

  4. Simplified0.3

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

  6. Applied associate-*r/0.3

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

  7. Applied frac-times0.3

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

  9. Applied associate-*l/0.3

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

  10. Applied associate-*r/0.3

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

  11. Taylor expanded around 0 0.3

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

  12. Simplified0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2019351 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))