Average Error: 14.4 → 0.3
Time: 14.2s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{1}{b + a}}{a \cdot b} \cdot \frac{\pi \cdot 1}{2}\]
\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{1}{b + a}}{a \cdot b} \cdot \frac{\pi \cdot 1}{2}
double f(double a, double b) {
        double r43351 = atan2(1.0, 0.0);
        double r43352 = 2.0;
        double r43353 = r43351 / r43352;
        double r43354 = 1.0;
        double r43355 = b;
        double r43356 = r43355 * r43355;
        double r43357 = a;
        double r43358 = r43357 * r43357;
        double r43359 = r43356 - r43358;
        double r43360 = r43354 / r43359;
        double r43361 = r43353 * r43360;
        double r43362 = r43354 / r43357;
        double r43363 = r43354 / r43355;
        double r43364 = r43362 - r43363;
        double r43365 = r43361 * r43364;
        return r43365;
}
↓
double f(double a, double b) {
        double r43366 = 1.0;
        double r43367 = b;
        double r43368 = a;
        double r43369 = r43367 + r43368;
        double r43370 = r43366 / r43369;
        double r43371 = r43368 * r43367;
        double r43372 = r43370 / r43371;
        double r43373 = atan2(1.0, 0.0);
        double r43374 = r43373 * r43366;
        double r43375 = 2.0;
        double r43376 = r43374 / r43375;
        double r43377 = r43372 * r43376;
        return r43377;
}

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 14.4

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

  3. Applied difference-of-squares9.7

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

  4. Applied associate-/r*9.2

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

  6. Applied associate-*r/9.2

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

  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-/l*0.3

    \[\leadsto \color{blue}{\frac{\frac{\pi}{2} \cdot \frac{1}{b + a}}{\frac{b - a}{\frac{1}{a} - \frac{1}{b}}}}\]
  10. Using strategy rm

  11. Applied frac-sub0.3

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

  12. Applied associate-/r/0.3

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

  13. Applied times-frac0.3

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

  14. Simplified0.4

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

  15. Final simplification0.3

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

Reproduce

herbie shell --seed 2019297 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))