Average Error: 14.8 → 0.2
Time: 14.6s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b + a}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b + a}
double f(double a, double b) {
        double r41392 = atan2(1.0, 0.0);
        double r41393 = 2.0;
        double r41394 = r41392 / r41393;
        double r41395 = 1.0;
        double r41396 = b;
        double r41397 = r41396 * r41396;
        double r41398 = a;
        double r41399 = r41398 * r41398;
        double r41400 = r41397 - r41399;
        double r41401 = r41395 / r41400;
        double r41402 = r41394 * r41401;
        double r41403 = r41395 / r41398;
        double r41404 = r41395 / r41396;
        double r41405 = r41403 - r41404;
        double r41406 = r41402 * r41405;
        return r41406;
}
↓
double f(double a, double b) {
        double r41407 = 0.5;
        double r41408 = atan2(1.0, 0.0);
        double r41409 = a;
        double r41410 = b;
        double r41411 = r41409 * r41410;
        double r41412 = r41408 / r41411;
        double r41413 = r41407 * r41412;
        double r41414 = r41410 + r41409;
        double r41415 = r41413 / r41414;
        return r41415;
}

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 14.8

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

  3. Applied difference-of-squares9.9

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

  4. Applied add-cube-cbrt9.9

    \[\leadsto \left(\frac{\pi}{2} \cdot \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\left(b + a\right) \cdot \left(b - a\right)}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]

  5. Applied times-frac9.5

    \[\leadsto \left(\frac{\pi}{2} \cdot \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{b + a} \cdot \frac{\sqrt[3]{1}}{b - a}\right)}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
  6. Using strategy rm

  7. Applied associate-*l/9.5

    \[\leadsto \left(\frac{\pi}{2} \cdot \color{blue}{\frac{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \frac{\sqrt[3]{1}}{b - a}}{b + a}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]

  8. Applied associate-*r/9.4

    \[\leadsto \color{blue}{\frac{\frac{\pi}{2} \cdot \left(\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \frac{\sqrt[3]{1}}{b - a}\right)}{b + a}} \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]

  9. Applied associate-*l/0.3

    \[\leadsto \color{blue}{\frac{\left(\frac{\pi}{2} \cdot \left(\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \frac{\sqrt[3]{1}}{b - a}\right)\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)}{b + a}}\]

  10. Taylor expanded around 0 0.2

    \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{\pi}{a \cdot b}}}{b + a}\]

  11. Final simplification0.2

    \[\leadsto \frac{0.5 \cdot \frac{\pi}{a \cdot b}}{b + a}\]

Reproduce

herbie shell --seed 2019212 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))