Average Error: 14.3 → 0.2
Time: 21.7s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\left(\frac{\frac{\pi}{2}}{b + a} \cdot 1\right) \cdot \left(1 \cdot b - a \cdot 1\right)}{b - a}}{a \cdot b}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\frac{\left(\frac{\frac{\pi}{2}}{b + a} \cdot 1\right) \cdot \left(1 \cdot b - a \cdot 1\right)}{b - a}}{a \cdot b}
double f(double a, double b) {
        double r86342 = atan2(1.0, 0.0);
        double r86343 = 2.0;
        double r86344 = r86342 / r86343;
        double r86345 = 1.0;
        double r86346 = b;
        double r86347 = r86346 * r86346;
        double r86348 = a;
        double r86349 = r86348 * r86348;
        double r86350 = r86347 - r86349;
        double r86351 = r86345 / r86350;
        double r86352 = r86344 * r86351;
        double r86353 = r86345 / r86348;
        double r86354 = r86345 / r86346;
        double r86355 = r86353 - r86354;
        double r86356 = r86352 * r86355;
        return r86356;
}
↓
double f(double a, double b) {
        double r86357 = atan2(1.0, 0.0);
        double r86358 = 2.0;
        double r86359 = r86357 / r86358;
        double r86360 = b;
        double r86361 = a;
        double r86362 = r86360 + r86361;
        double r86363 = r86359 / r86362;
        double r86364 = 1.0;
        double r86365 = r86363 * r86364;
        double r86366 = r86364 * r86360;
        double r86367 = r86361 * r86364;
        double r86368 = r86366 - r86367;
        double r86369 = r86365 * r86368;
        double r86370 = r86360 - r86361;
        double r86371 = r86369 / r86370;
        double r86372 = r86361 * r86360;
        double r86373 = r86371 / r86372;
        return r86373;
}

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. Using strategy rm

  3. Applied difference-of-squares9.4

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

  4. Applied *-un-lft-identity9.4

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

  5. Applied times-frac9.0

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

  6. Applied associate-*r*9.0

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

  7. Simplified8.9

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

  9. Applied frac-sub8.9

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

  10. Applied associate-*r/8.9

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

  11. Applied frac-times0.8

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

  13. Applied associate-/r*0.2

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

  14. Final simplification0.2

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))