Average Error: 14.5 → 0.3
Time: 19.7s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\pi}{2} \cdot \frac{\frac{\frac{1}{a} - \frac{1}{b}}{b - a} \cdot 1}{b + a}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\pi}{2} \cdot \frac{\frac{\frac{1}{a} - \frac{1}{b}}{b - a} \cdot 1}{b + a}
double f(double a, double b) {
        double r41501 = atan2(1.0, 0.0);
        double r41502 = 2.0;
        double r41503 = r41501 / r41502;
        double r41504 = 1.0;
        double r41505 = b;
        double r41506 = r41505 * r41505;
        double r41507 = a;
        double r41508 = r41507 * r41507;
        double r41509 = r41506 - r41508;
        double r41510 = r41504 / r41509;
        double r41511 = r41503 * r41510;
        double r41512 = r41504 / r41507;
        double r41513 = r41504 / r41505;
        double r41514 = r41512 - r41513;
        double r41515 = r41511 * r41514;
        return r41515;
}
↓
double f(double a, double b) {
        double r41516 = atan2(1.0, 0.0);
        double r41517 = 2.0;
        double r41518 = r41516 / r41517;
        double r41519 = 1.0;
        double r41520 = a;
        double r41521 = r41519 / r41520;
        double r41522 = b;
        double r41523 = r41519 / r41522;
        double r41524 = r41521 - r41523;
        double r41525 = r41522 - r41520;
        double r41526 = r41524 / r41525;
        double r41527 = r41526 * r41519;
        double r41528 = r41522 + r41520;
        double r41529 = r41527 / r41528;
        double r41530 = r41518 * r41529;
        return r41530;
}

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 14.5

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

  3. Applied associate-*l*14.5

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

  4. Simplified0.3

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

  6. Applied associate-*r/0.3

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

  7. Applied associate-*r/0.3

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

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

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

  10. Applied times-frac0.3

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

  11. Simplified0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019325 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))