Average Error: 14.5 → 0.3
Time: 19.8s
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 r41402 = atan2(1.0, 0.0);
        double r41403 = 2.0;
        double r41404 = r41402 / r41403;
        double r41405 = 1.0;
        double r41406 = b;
        double r41407 = r41406 * r41406;
        double r41408 = a;
        double r41409 = r41408 * r41408;
        double r41410 = r41407 - r41409;
        double r41411 = r41405 / r41410;
        double r41412 = r41404 * r41411;
        double r41413 = r41405 / r41408;
        double r41414 = r41405 / r41406;
        double r41415 = r41413 - r41414;
        double r41416 = r41412 * r41415;
        return r41416;
}
↓
double f(double a, double b) {
        double r41417 = atan2(1.0, 0.0);
        double r41418 = 2.0;
        double r41419 = r41417 / r41418;
        double r41420 = 1.0;
        double r41421 = a;
        double r41422 = r41420 / r41421;
        double r41423 = b;
        double r41424 = r41420 / r41423;
        double r41425 = r41422 - r41424;
        double r41426 = r41423 - r41421;
        double r41427 = r41425 / r41426;
        double r41428 = r41427 * r41420;
        double r41429 = r41423 + r41421;
        double r41430 = r41428 / r41429;
        double r41431 = r41419 * r41430;
        return r41431;
}

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))))