Average Error: 14.6 → 0.3
Time: 18.9s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{1}{a} - \frac{1}{b}}{b - a} \cdot \frac{\frac{\pi}{2} \cdot 1}{a + b}\]
\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}{a} - \frac{1}{b}}{b - a} \cdot \frac{\frac{\pi}{2} \cdot 1}{a + b}
double f(double a, double b) {
        double r2691281 = atan2(1.0, 0.0);
        double r2691282 = 2.0;
        double r2691283 = r2691281 / r2691282;
        double r2691284 = 1.0;
        double r2691285 = b;
        double r2691286 = r2691285 * r2691285;
        double r2691287 = a;
        double r2691288 = r2691287 * r2691287;
        double r2691289 = r2691286 - r2691288;
        double r2691290 = r2691284 / r2691289;
        double r2691291 = r2691283 * r2691290;
        double r2691292 = r2691284 / r2691287;
        double r2691293 = r2691284 / r2691285;
        double r2691294 = r2691292 - r2691293;
        double r2691295 = r2691291 * r2691294;
        return r2691295;
}
↓
double f(double a, double b) {
        double r2691296 = 1.0;
        double r2691297 = a;
        double r2691298 = r2691296 / r2691297;
        double r2691299 = b;
        double r2691300 = r2691296 / r2691299;
        double r2691301 = r2691298 - r2691300;
        double r2691302 = r2691299 - r2691297;
        double r2691303 = r2691301 / r2691302;
        double r2691304 = atan2(1.0, 0.0);
        double r2691305 = 2.0;
        double r2691306 = r2691304 / r2691305;
        double r2691307 = r2691306 * r2691296;
        double r2691308 = r2691297 + r2691299;
        double r2691309 = r2691307 / r2691308;
        double r2691310 = r2691303 * r2691309;
        return r2691310;
}

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 14.6

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

  3. Applied difference-of-squares9.5

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

  4. Applied *-un-lft-identity9.5

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

  5. Applied times-frac8.9

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

  6. Applied associate-*r*8.8

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

  7. Simplified8.8

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

  9. Applied associate-*r/8.8

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

  10. Applied associate-*l/0.3

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

  12. Applied *-un-lft-identity0.3

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

  13. Applied times-frac0.3

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

  14. Simplified0.3

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

  15. Final simplification0.3

    \[\leadsto \frac{\frac{1}{a} - \frac{1}{b}}{b - a} \cdot \frac{\frac{\pi}{2} \cdot 1}{a + b}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))