Average Error: 14.4 → 0.3
Time: 19.9s
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\pi}{2}}{b + a} \cdot \frac{1}{a \cdot b}\]
\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{\pi}{2}}{b + a} \cdot \frac{1}{a \cdot b}
double f(double a, double b) {
        double r43448 = atan2(1.0, 0.0);
        double r43449 = 2.0;
        double r43450 = r43448 / r43449;
        double r43451 = 1.0;
        double r43452 = b;
        double r43453 = r43452 * r43452;
        double r43454 = a;
        double r43455 = r43454 * r43454;
        double r43456 = r43453 - r43455;
        double r43457 = r43451 / r43456;
        double r43458 = r43450 * r43457;
        double r43459 = r43451 / r43454;
        double r43460 = r43451 / r43452;
        double r43461 = r43459 - r43460;
        double r43462 = r43458 * r43461;
        return r43462;
}
↓
double f(double a, double b) {
        double r43463 = atan2(1.0, 0.0);
        double r43464 = 2.0;
        double r43465 = r43463 / r43464;
        double r43466 = b;
        double r43467 = a;
        double r43468 = r43466 + r43467;
        double r43469 = r43465 / r43468;
        double r43470 = 1.0;
        double r43471 = r43467 * r43466;
        double r43472 = r43470 / r43471;
        double r43473 = r43469 * r43472;
        return r43473;
}

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 14.4

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

  3. Applied difference-of-squares9.3

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

  4. Applied *-un-lft-identity9.3

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

  5. Applied times-frac9.0

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

  6. Applied associate-*r*9.0

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

  7. Simplified8.9

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

  9. Applied associate-*l*0.3

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

  10. Taylor expanded around 0 0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019199 
(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))))