Average Error: 14.1 → 0.3
Time: 38.7s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{1 \cdot \frac{\frac{\pi}{2}}{a + b}}{\frac{b - a}{\frac{1}{a} - \frac{1}{b}}}\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{1 \cdot \frac{\frac{\pi}{2}}{a + b}}{\frac{b - a}{\frac{1}{a} - \frac{1}{b}}}
double f(double a, double b) {
        double r2845281 = atan2(1.0, 0.0);
        double r2845282 = 2.0;
        double r2845283 = r2845281 / r2845282;
        double r2845284 = 1.0;
        double r2845285 = b;
        double r2845286 = r2845285 * r2845285;
        double r2845287 = a;
        double r2845288 = r2845287 * r2845287;
        double r2845289 = r2845286 - r2845288;
        double r2845290 = r2845284 / r2845289;
        double r2845291 = r2845283 * r2845290;
        double r2845292 = r2845284 / r2845287;
        double r2845293 = r2845284 / r2845285;
        double r2845294 = r2845292 - r2845293;
        double r2845295 = r2845291 * r2845294;
        return r2845295;
}
↓
double f(double a, double b) {
        double r2845296 = 1.0;
        double r2845297 = atan2(1.0, 0.0);
        double r2845298 = 2.0;
        double r2845299 = r2845297 / r2845298;
        double r2845300 = a;
        double r2845301 = b;
        double r2845302 = r2845300 + r2845301;
        double r2845303 = r2845299 / r2845302;
        double r2845304 = r2845296 * r2845303;
        double r2845305 = r2845301 - r2845300;
        double r2845306 = r2845296 / r2845300;
        double r2845307 = r2845296 / r2845301;
        double r2845308 = r2845306 - r2845307;
        double r2845309 = r2845305 / r2845308;
        double r2845310 = r2845304 / r2845309;
        return r2845310;
}

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 14.1

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

  3. Applied difference-of-squares9.2

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

  4. Applied *-un-lft-identity9.2

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

  5. Applied times-frac8.8

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

  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/8.7

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

  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 associate-/l*0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2019172 +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))))