Average Error: 14.1 → 0.3
Time: 17.7s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{1 \cdot \left(\frac{\pi}{a} - \frac{\pi}{b}\right)}{b + a}}{2 \cdot \left(b - a\right)}\]
\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 \cdot \left(\frac{\pi}{a} - \frac{\pi}{b}\right)}{b + a}}{2 \cdot \left(b - a\right)}
double f(double a, double b) {
        double r41400 = atan2(1.0, 0.0);
        double r41401 = 2.0;
        double r41402 = r41400 / r41401;
        double r41403 = 1.0;
        double r41404 = b;
        double r41405 = r41404 * r41404;
        double r41406 = a;
        double r41407 = r41406 * r41406;
        double r41408 = r41405 - r41407;
        double r41409 = r41403 / r41408;
        double r41410 = r41402 * r41409;
        double r41411 = r41403 / r41406;
        double r41412 = r41403 / r41404;
        double r41413 = r41411 - r41412;
        double r41414 = r41410 * r41413;
        return r41414;
}
↓
double f(double a, double b) {
        double r41415 = 1.0;
        double r41416 = atan2(1.0, 0.0);
        double r41417 = a;
        double r41418 = r41416 / r41417;
        double r41419 = b;
        double r41420 = r41416 / r41419;
        double r41421 = r41418 - r41420;
        double r41422 = r41415 * r41421;
        double r41423 = r41419 + r41417;
        double r41424 = r41422 / r41423;
        double r41425 = 2.0;
        double r41426 = r41419 - r41417;
        double r41427 = r41425 * r41426;
        double r41428 = r41424 / r41427;
        return r41428;
}

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 14.1

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

  3. Applied difference-of-squares9.2

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

  4. Applied associate-/r*8.7

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

  6. Applied frac-times8.7

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

  7. Applied associate-*l/0.3

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

  9. Applied associate-*r/0.3

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

  10. Applied associate-*l/0.3

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

  11. Taylor expanded around 0 0.3

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

  12. Simplified0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2019303 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))