Average Error: 14.2 → 0.2
Time: 32.3s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi}{a \cdot b} \cdot \frac{1}{2}}{b + a}\]
\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}{a \cdot b} \cdot \frac{1}{2}}{b + a}
double f(double a, double b) {
        double r1688345 = atan2(1.0, 0.0);
        double r1688346 = 2.0;
        double r1688347 = r1688345 / r1688346;
        double r1688348 = 1.0;
        double r1688349 = b;
        double r1688350 = r1688349 * r1688349;
        double r1688351 = a;
        double r1688352 = r1688351 * r1688351;
        double r1688353 = r1688350 - r1688352;
        double r1688354 = r1688348 / r1688353;
        double r1688355 = r1688347 * r1688354;
        double r1688356 = r1688348 / r1688351;
        double r1688357 = r1688348 / r1688349;
        double r1688358 = r1688356 - r1688357;
        double r1688359 = r1688355 * r1688358;
        return r1688359;
}
↓
double f(double a, double b) {
        double r1688360 = atan2(1.0, 0.0);
        double r1688361 = a;
        double r1688362 = b;
        double r1688363 = r1688361 * r1688362;
        double r1688364 = r1688360 / r1688363;
        double r1688365 = 0.5;
        double r1688366 = r1688364 * r1688365;
        double r1688367 = r1688362 + r1688361;
        double r1688368 = r1688366 / r1688367;
        return r1688368;
}

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. Simplified0.3

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

  3. Taylor expanded around 0 0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019135 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))