Average Error: 14.3 → 0.3
Time: 20.0s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{\pi}{2} \cdot \left(\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1\right)}{b + a}}{b - a}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\frac{\frac{\pi}{2} \cdot \left(\left(\frac{1}{a} - \frac{1}{b}\right) \cdot 1\right)}{b + a}}{b - a}
double f(double a, double b) {
        double r49294 = atan2(1.0, 0.0);
        double r49295 = 2.0;
        double r49296 = r49294 / r49295;
        double r49297 = 1.0;
        double r49298 = b;
        double r49299 = r49298 * r49298;
        double r49300 = a;
        double r49301 = r49300 * r49300;
        double r49302 = r49299 - r49301;
        double r49303 = r49297 / r49302;
        double r49304 = r49296 * r49303;
        double r49305 = r49297 / r49300;
        double r49306 = r49297 / r49298;
        double r49307 = r49305 - r49306;
        double r49308 = r49304 * r49307;
        return r49308;
}
↓
double f(double a, double b) {
        double r49309 = atan2(1.0, 0.0);
        double r49310 = 2.0;
        double r49311 = r49309 / r49310;
        double r49312 = 1.0;
        double r49313 = a;
        double r49314 = r49312 / r49313;
        double r49315 = b;
        double r49316 = r49312 / r49315;
        double r49317 = r49314 - r49316;
        double r49318 = r49317 * r49312;
        double r49319 = r49311 * r49318;
        double r49320 = r49315 + r49313;
        double r49321 = r49319 / r49320;
        double r49322 = r49315 - r49313;
        double r49323 = r49321 / r49322;
        return r49323;
}

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 14.3

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

  3. Applied associate-*l*14.3

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

  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-*l/0.3

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

  7. Applied associate-*r/0.3

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

  9. Applied associate-*r/0.3

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

  10. Applied associate-*r/0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019326 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))