Average Error: 13.8 → 0.3
Time: 5.6s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi}{2} \cdot 1}{b + a} \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b - a}\]
\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{\pi}{2} \cdot 1}{b + a} \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b - a}
double f(double a, double b) {
        double r43855 = atan2(1.0, 0.0);
        double r43856 = 2.0;
        double r43857 = r43855 / r43856;
        double r43858 = 1.0;
        double r43859 = b;
        double r43860 = r43859 * r43859;
        double r43861 = a;
        double r43862 = r43861 * r43861;
        double r43863 = r43860 - r43862;
        double r43864 = r43858 / r43863;
        double r43865 = r43857 * r43864;
        double r43866 = r43858 / r43861;
        double r43867 = r43858 / r43859;
        double r43868 = r43866 - r43867;
        double r43869 = r43865 * r43868;
        return r43869;
}
↓
double f(double a, double b) {
        double r43870 = atan2(1.0, 0.0);
        double r43871 = 2.0;
        double r43872 = r43870 / r43871;
        double r43873 = 1.0;
        double r43874 = r43872 * r43873;
        double r43875 = b;
        double r43876 = a;
        double r43877 = r43875 + r43876;
        double r43878 = r43874 / r43877;
        double r43879 = r43873 / r43876;
        double r43880 = r43873 / r43875;
        double r43881 = r43879 - r43880;
        double r43882 = r43875 - r43876;
        double r43883 = r43881 / r43882;
        double r43884 = r43878 * r43883;
        return r43884;
}

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 13.8

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

  3. Applied difference-of-squares9.4

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

  4. Applied associate-/r*8.9

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

  6. Applied associate-*r/8.9

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

  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 *-un-lft-identity0.3

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

  10. Applied times-frac0.3

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

  11. Simplified0.3

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

  13. Applied associate-*r/0.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2020003 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))