Average Error: 14.3 → 0.3
Time: 39.0s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{\pi}{a} - \frac{\pi}{b}}{b - a}}{a + b} \cdot \frac{1}{2}\]
\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}{a} - \frac{\pi}{b}}{b - a}}{a + b} \cdot \frac{1}{2}
double f(double a, double b) {
        double r1057984 = atan2(1.0, 0.0);
        double r1057985 = 2.0;
        double r1057986 = r1057984 / r1057985;
        double r1057987 = 1.0;
        double r1057988 = b;
        double r1057989 = r1057988 * r1057988;
        double r1057990 = a;
        double r1057991 = r1057990 * r1057990;
        double r1057992 = r1057989 - r1057991;
        double r1057993 = r1057987 / r1057992;
        double r1057994 = r1057986 * r1057993;
        double r1057995 = r1057987 / r1057990;
        double r1057996 = r1057987 / r1057988;
        double r1057997 = r1057995 - r1057996;
        double r1057998 = r1057994 * r1057997;
        return r1057998;
}
↓
double f(double a, double b) {
        double r1057999 = atan2(1.0, 0.0);
        double r1058000 = a;
        double r1058001 = r1057999 / r1058000;
        double r1058002 = b;
        double r1058003 = r1057999 / r1058002;
        double r1058004 = r1058001 - r1058003;
        double r1058005 = r1058002 - r1058000;
        double r1058006 = r1058004 / r1058005;
        double r1058007 = r1058000 + r1058002;
        double r1058008 = r1058006 / r1058007;
        double r1058009 = 0.5;
        double r1058010 = r1058008 * r1058009;
        return r1058010;
}

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. Simplified0.3

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

  3. Taylor expanded around inf 0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019130 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))