Average Error: 14.2 → 0.3
Time: 11.5s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{\pi}{2} \cdot 1}{b + a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b - a}\]
\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{\frac{\pi}{2} \cdot 1}{b + a} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b - a}
double f(double a, double b) {
        double r43375 = atan2(1.0, 0.0);
        double r43376 = 2.0;
        double r43377 = r43375 / r43376;
        double r43378 = 1.0;
        double r43379 = b;
        double r43380 = r43379 * r43379;
        double r43381 = a;
        double r43382 = r43381 * r43381;
        double r43383 = r43380 - r43382;
        double r43384 = r43378 / r43383;
        double r43385 = r43377 * r43384;
        double r43386 = r43378 / r43381;
        double r43387 = r43378 / r43379;
        double r43388 = r43386 - r43387;
        double r43389 = r43385 * r43388;
        return r43389;
}
↓
double f(double a, double b) {
        double r43390 = atan2(1.0, 0.0);
        double r43391 = 2.0;
        double r43392 = r43390 / r43391;
        double r43393 = 1.0;
        double r43394 = r43392 * r43393;
        double r43395 = b;
        double r43396 = a;
        double r43397 = r43395 + r43396;
        double r43398 = r43394 / r43397;
        double r43399 = r43393 / r43396;
        double r43400 = r43393 / r43395;
        double r43401 = r43399 - r43400;
        double r43402 = r43398 * r43401;
        double r43403 = r43395 - r43396;
        double r43404 = r43402 / r43403;
        return r43404;
}

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 difference-of-squares9.5

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

  4. Applied associate-/r*9.0

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

  6. Applied associate-*r/9.0

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

  7. Applied associate-*l/0.3

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

  9. Applied associate-*r/0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2020047 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))