Average Error: 14.6 → 0.3
Time: 39.4s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi}{2} \cdot 1}{b + a} \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b - a}\]
\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}{2} \cdot 1}{b + a} \cdot \frac{\frac{1}{a} - \frac{1}{b}}{b - a}
double f(double a, double b) {
        double r4612413 = atan2(1.0, 0.0);
        double r4612414 = 2.0;
        double r4612415 = r4612413 / r4612414;
        double r4612416 = 1.0;
        double r4612417 = b;
        double r4612418 = r4612417 * r4612417;
        double r4612419 = a;
        double r4612420 = r4612419 * r4612419;
        double r4612421 = r4612418 - r4612420;
        double r4612422 = r4612416 / r4612421;
        double r4612423 = r4612415 * r4612422;
        double r4612424 = r4612416 / r4612419;
        double r4612425 = r4612416 / r4612417;
        double r4612426 = r4612424 - r4612425;
        double r4612427 = r4612423 * r4612426;
        return r4612427;
}
↓
double f(double a, double b) {
        double r4612428 = atan2(1.0, 0.0);
        double r4612429 = 2.0;
        double r4612430 = r4612428 / r4612429;
        double r4612431 = 1.0;
        double r4612432 = r4612430 * r4612431;
        double r4612433 = b;
        double r4612434 = a;
        double r4612435 = r4612433 + r4612434;
        double r4612436 = r4612432 / r4612435;
        double r4612437 = r4612431 / r4612434;
        double r4612438 = r4612431 / r4612433;
        double r4612439 = r4612437 - r4612438;
        double r4612440 = r4612433 - r4612434;
        double r4612441 = r4612439 / r4612440;
        double r4612442 = r4612436 * r4612441;
        return r4612442;
}

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 14.6

    \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
  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 *-un-lft-identity9.5

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

  5. Applied times-frac8.9

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

  6. Applied associate-*r*8.8

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

  7. Simplified8.8

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

  9. Applied associate-*r/8.8

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

  10. Applied associate-*l/0.3

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

  12. Applied *-un-lft-identity0.3

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

  13. Applied times-frac0.3

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

  14. Simplified0.3

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

  15. Final simplification0.3

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))