Average Error: 14.3 → 0.3
Time: 18.6s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{1}{b + a} \cdot \left(\frac{\frac{\pi}{a}}{b} \cdot \frac{1}{2}\right)\]
\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{1}{b + a} \cdot \left(\frac{\frac{\pi}{a}}{b} \cdot \frac{1}{2}\right)
double f(double a, double b) {
        double r1338326 = atan2(1.0, 0.0);
        double r1338327 = 2.0;
        double r1338328 = r1338326 / r1338327;
        double r1338329 = 1.0;
        double r1338330 = b;
        double r1338331 = r1338330 * r1338330;
        double r1338332 = a;
        double r1338333 = r1338332 * r1338332;
        double r1338334 = r1338331 - r1338333;
        double r1338335 = r1338329 / r1338334;
        double r1338336 = r1338328 * r1338335;
        double r1338337 = r1338329 / r1338332;
        double r1338338 = r1338329 / r1338330;
        double r1338339 = r1338337 - r1338338;
        double r1338340 = r1338336 * r1338339;
        return r1338340;
}
↓
double f(double a, double b) {
        double r1338341 = 1.0;
        double r1338342 = b;
        double r1338343 = a;
        double r1338344 = r1338342 + r1338343;
        double r1338345 = r1338341 / r1338344;
        double r1338346 = atan2(1.0, 0.0);
        double r1338347 = r1338346 / r1338343;
        double r1338348 = r1338347 / r1338342;
        double r1338349 = 0.5;
        double r1338350 = r1338348 * r1338349;
        double r1338351 = r1338345 * r1338350;
        return r1338351;
}

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. Simplified14.3

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

  4. Applied difference-of-squares9.5

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

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

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

  6. Applied times-frac0.3

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

  7. Taylor expanded around 0 0.3

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

  9. Applied associate-/r*0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))