Average Error: 14.1 → 0.3
Time: 1.1m
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{\pi}{2}}{a + b}}{\frac{a}{\frac{1}{b - a}}} - \frac{\frac{\frac{\pi}{2}}{a + b}}{b \cdot \left(b - a\right)}\]
\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{\frac{\pi}{2}}{a + b}}{\frac{a}{\frac{1}{b - a}}} - \frac{\frac{\frac{\pi}{2}}{a + b}}{b \cdot \left(b - a\right)}
double f(double a, double b) {
        double r2469880 = atan2(1.0, 0.0);
        double r2469881 = 2.0;
        double r2469882 = r2469880 / r2469881;
        double r2469883 = 1.0;
        double r2469884 = b;
        double r2469885 = r2469884 * r2469884;
        double r2469886 = a;
        double r2469887 = r2469886 * r2469886;
        double r2469888 = r2469885 - r2469887;
        double r2469889 = r2469883 / r2469888;
        double r2469890 = r2469882 * r2469889;
        double r2469891 = r2469883 / r2469886;
        double r2469892 = r2469883 / r2469884;
        double r2469893 = r2469891 - r2469892;
        double r2469894 = r2469890 * r2469893;
        return r2469894;
}
↓
double f(double a, double b) {
        double r2469895 = atan2(1.0, 0.0);
        double r2469896 = 2.0;
        double r2469897 = r2469895 / r2469896;
        double r2469898 = a;
        double r2469899 = b;
        double r2469900 = r2469898 + r2469899;
        double r2469901 = r2469897 / r2469900;
        double r2469902 = 1.0;
        double r2469903 = r2469899 - r2469898;
        double r2469904 = r2469902 / r2469903;
        double r2469905 = r2469898 / r2469904;
        double r2469906 = r2469901 / r2469905;
        double r2469907 = r2469899 * r2469903;
        double r2469908 = r2469901 / r2469907;
        double r2469909 = r2469906 - r2469908;
        return r2469909;
}

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 14.1

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

  2. Simplified9.0

    \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{\pi}{2}}{a + b}}{b - a}}{a} - \frac{\frac{\frac{\frac{\pi}{2}}{a + b}}{b - a}}{b}}\]
  3. Using strategy rm

  4. Applied associate-/l/4.5

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

  6. Applied div-inv4.6

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

  7. Applied associate-/l*0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019154 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))