Average Error: 14.6 → 0.3
Time: 37.6s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{1}{a} - \frac{1}{b}}{b - a} \cdot \frac{\frac{\pi}{2} \cdot 1}{a + b}\]
\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 r2712304 = atan2(1.0, 0.0);
        double r2712305 = 2.0;
        double r2712306 = r2712304 / r2712305;
        double r2712307 = 1.0;
        double r2712308 = b;
        double r2712309 = r2712308 * r2712308;
        double r2712310 = a;
        double r2712311 = r2712310 * r2712310;
        double r2712312 = r2712309 - r2712311;
        double r2712313 = r2712307 / r2712312;
        double r2712314 = r2712306 * r2712313;
        double r2712315 = r2712307 / r2712310;
        double r2712316 = r2712307 / r2712308;
        double r2712317 = r2712315 - r2712316;
        double r2712318 = r2712314 * r2712317;
        return r2712318;
}
↓
double f(double a, double b) {
        double r2712319 = 1.0;
        double r2712320 = a;
        double r2712321 = r2712319 / r2712320;
        double r2712322 = b;
        double r2712323 = r2712319 / r2712322;
        double r2712324 = r2712321 - r2712323;
        double r2712325 = r2712322 - r2712320;
        double r2712326 = r2712324 / r2712325;
        double r2712327 = atan2(1.0, 0.0);
        double r2712328 = 2.0;
        double r2712329 = r2712327 / r2712328;
        double r2712330 = r2712329 * r2712319;
        double r2712331 = r2712320 + r2712322;
        double r2712332 = r2712330 / r2712331;
        double r2712333 = r2712326 * r2712332;
        return r2712333;
}

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 +o rules:numerics
(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))))