Average Error: 14.3 → 0.3
Time: 13.6s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\left(\pi \cdot 1\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b + a}}{2 \cdot \left(b - a\right)}\]
\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{\left(\pi \cdot 1\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b + a}}{2 \cdot \left(b - a\right)}
double f(double a, double b) {
        double r39727 = atan2(1.0, 0.0);
        double r39728 = 2.0;
        double r39729 = r39727 / r39728;
        double r39730 = 1.0;
        double r39731 = b;
        double r39732 = r39731 * r39731;
        double r39733 = a;
        double r39734 = r39733 * r39733;
        double r39735 = r39732 - r39734;
        double r39736 = r39730 / r39735;
        double r39737 = r39729 * r39736;
        double r39738 = r39730 / r39733;
        double r39739 = r39730 / r39731;
        double r39740 = r39738 - r39739;
        double r39741 = r39737 * r39740;
        return r39741;
}
↓
double f(double a, double b) {
        double r39742 = atan2(1.0, 0.0);
        double r39743 = 1.0;
        double r39744 = r39742 * r39743;
        double r39745 = a;
        double r39746 = r39743 / r39745;
        double r39747 = b;
        double r39748 = r39743 / r39747;
        double r39749 = r39746 - r39748;
        double r39750 = r39744 * r39749;
        double r39751 = r39747 + r39745;
        double r39752 = r39750 / r39751;
        double r39753 = 2.0;
        double r39754 = r39747 - r39745;
        double r39755 = r39753 * r39754;
        double r39756 = r39752 / r39755;
        return r39756;
}

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-squares9.5

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

  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 frac-times9.1

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

  7. Applied associate-*l/0.3

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

  9. Applied associate-*r/0.3

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

  10. Applied associate-*l/0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019291 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))