Average Error: 14.5 → 0.3
Time: 21.1s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{1}{2} \cdot \frac{\frac{\pi}{b \cdot a}}{a + b}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{1}{2} \cdot \frac{\frac{\pi}{b \cdot a}}{a + b}
double f(double a, double b) {
        double r747372 = atan2(1.0, 0.0);
        double r747373 = 2.0;
        double r747374 = r747372 / r747373;
        double r747375 = 1.0;
        double r747376 = b;
        double r747377 = r747376 * r747376;
        double r747378 = a;
        double r747379 = r747378 * r747378;
        double r747380 = r747377 - r747379;
        double r747381 = r747375 / r747380;
        double r747382 = r747374 * r747381;
        double r747383 = r747375 / r747378;
        double r747384 = r747375 / r747376;
        double r747385 = r747383 - r747384;
        double r747386 = r747382 * r747385;
        return r747386;
}
↓
double f(double a, double b) {
        double r747387 = 0.5;
        double r747388 = atan2(1.0, 0.0);
        double r747389 = b;
        double r747390 = a;
        double r747391 = r747389 * r747390;
        double r747392 = r747388 / r747391;
        double r747393 = r747390 + r747389;
        double r747394 = r747392 / r747393;
        double r747395 = r747387 * r747394;
        return r747395;
}

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 14.5

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

  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

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019133 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))