Average Error: 14.2 → 0.3
Time: 12.1s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi \cdot \left(\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1\right)}{b - a}}{2 \cdot \left(b + a\right)}\]
\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{\pi \cdot \left(\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1\right)}{b - a}}{2 \cdot \left(b + a\right)}
double f(double a, double b) {
        double r49900 = atan2(1.0, 0.0);
        double r49901 = 2.0;
        double r49902 = r49900 / r49901;
        double r49903 = 1.0;
        double r49904 = b;
        double r49905 = r49904 * r49904;
        double r49906 = a;
        double r49907 = r49906 * r49906;
        double r49908 = r49905 - r49907;
        double r49909 = r49903 / r49908;
        double r49910 = r49902 * r49909;
        double r49911 = r49903 / r49906;
        double r49912 = r49903 / r49904;
        double r49913 = r49911 - r49912;
        double r49914 = r49910 * r49913;
        return r49914;
}
↓
double f(double a, double b) {
        double r49915 = atan2(1.0, 0.0);
        double r49916 = 1.0;
        double r49917 = a;
        double r49918 = r49916 / r49917;
        double r49919 = b;
        double r49920 = r49916 / r49919;
        double r49921 = r49918 - r49920;
        double r49922 = r49921 * r49916;
        double r49923 = r49915 * r49922;
        double r49924 = r49919 - r49917;
        double r49925 = r49923 / r49924;
        double r49926 = 2.0;
        double r49927 = r49919 + r49917;
        double r49928 = r49926 * r49927;
        double r49929 = r49925 / r49928;
        return r49929;
}

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. Final simplification0.3

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

Reproduce

herbie shell --seed 2019350 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))