Average Error: 14.7 → 0.3
Time: 23.8s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{1 \cdot \left(\frac{\pi}{a} - \frac{\pi}{b}\right)}{b + a}}{2 \cdot \left(b - a\right)}\]
\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 r84269 = atan2(1.0, 0.0);
        double r84270 = 2.0;
        double r84271 = r84269 / r84270;
        double r84272 = 1.0;
        double r84273 = b;
        double r84274 = r84273 * r84273;
        double r84275 = a;
        double r84276 = r84275 * r84275;
        double r84277 = r84274 - r84276;
        double r84278 = r84272 / r84277;
        double r84279 = r84271 * r84278;
        double r84280 = r84272 / r84275;
        double r84281 = r84272 / r84273;
        double r84282 = r84280 - r84281;
        double r84283 = r84279 * r84282;
        return r84283;
}
↓
double f(double a, double b) {
        double r84284 = 1.0;
        double r84285 = atan2(1.0, 0.0);
        double r84286 = a;
        double r84287 = r84285 / r84286;
        double r84288 = b;
        double r84289 = r84285 / r84288;
        double r84290 = r84287 - r84289;
        double r84291 = r84284 * r84290;
        double r84292 = r84288 + r84286;
        double r84293 = r84291 / r84292;
        double r84294 = 2.0;
        double r84295 = r84288 - r84286;
        double r84296 = r84294 * r84295;
        double r84297 = r84293 / r84296;
        return r84297;
}

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 14.7

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

  3. Applied difference-of-squares9.8

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

  4. Applied associate-/r*9.3

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

  6. Applied frac-times9.2

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

  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 2019347 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))