Average Error: 14.3 → 0.3
Time: 38.9s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \left(\pi \cdot 1\right)}{2 \cdot \left(a + b\right)}}{b - a}\]
\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{\left(\frac{1}{a} - \frac{1}{b}\right) \cdot \left(\pi \cdot 1\right)}{2 \cdot \left(a + b\right)}}{b - a}
double f(double a, double b) {
        double r2614834 = atan2(1.0, 0.0);
        double r2614835 = 2.0;
        double r2614836 = r2614834 / r2614835;
        double r2614837 = 1.0;
        double r2614838 = b;
        double r2614839 = r2614838 * r2614838;
        double r2614840 = a;
        double r2614841 = r2614840 * r2614840;
        double r2614842 = r2614839 - r2614841;
        double r2614843 = r2614837 / r2614842;
        double r2614844 = r2614836 * r2614843;
        double r2614845 = r2614837 / r2614840;
        double r2614846 = r2614837 / r2614838;
        double r2614847 = r2614845 - r2614846;
        double r2614848 = r2614844 * r2614847;
        return r2614848;
}
↓
double f(double a, double b) {
        double r2614849 = 1.0;
        double r2614850 = a;
        double r2614851 = r2614849 / r2614850;
        double r2614852 = b;
        double r2614853 = r2614849 / r2614852;
        double r2614854 = r2614851 - r2614853;
        double r2614855 = atan2(1.0, 0.0);
        double r2614856 = r2614855 * r2614849;
        double r2614857 = r2614854 * r2614856;
        double r2614858 = 2.0;
        double r2614859 = r2614850 + r2614852;
        double r2614860 = r2614858 * r2614859;
        double r2614861 = r2614857 / r2614860;
        double r2614862 = r2614852 - r2614850;
        double r2614863 = r2614861 / r2614862;
        return r2614863;
}

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 associate-/r*8.8

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

  6. Applied associate-*r/8.8

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

  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 frac-times0.3

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

  10. Applied associate-*l/0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))