Average Error: 13.8 → 0.3
Time: 2.3m
Precision: 64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{1}{a + b}}{\frac{a}{\pi} \cdot \left(b - a\right)} + \frac{\frac{-\pi}{b - a}}{\left(a + b\right) \cdot b}}{2}\]
\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{\frac{1}{a + b}}{\frac{a}{\pi} \cdot \left(b - a\right)} + \frac{\frac{-\pi}{b - a}}{\left(a + b\right) \cdot b}}{2}
double f(double a, double b) {
        double r6884412 = atan2(1.0, 0.0);
        double r6884413 = 2.0;
        double r6884414 = r6884412 / r6884413;
        double r6884415 = 1.0;
        double r6884416 = b;
        double r6884417 = r6884416 * r6884416;
        double r6884418 = a;
        double r6884419 = r6884418 * r6884418;
        double r6884420 = r6884417 - r6884419;
        double r6884421 = r6884415 / r6884420;
        double r6884422 = r6884414 * r6884421;
        double r6884423 = r6884415 / r6884418;
        double r6884424 = r6884415 / r6884416;
        double r6884425 = r6884423 - r6884424;
        double r6884426 = r6884422 * r6884425;
        return r6884426;
}
↓
double f(double a, double b) {
        double r6884427 = 1.0;
        double r6884428 = a;
        double r6884429 = b;
        double r6884430 = r6884428 + r6884429;
        double r6884431 = r6884427 / r6884430;
        double r6884432 = atan2(1.0, 0.0);
        double r6884433 = r6884428 / r6884432;
        double r6884434 = r6884429 - r6884428;
        double r6884435 = r6884433 * r6884434;
        double r6884436 = r6884431 / r6884435;
        double r6884437 = -r6884432;
        double r6884438 = r6884437 / r6884434;
        double r6884439 = r6884430 * r6884429;
        double r6884440 = r6884438 / r6884439;
        double r6884441 = r6884436 + r6884440;
        double r6884442 = 2.0;
        double r6884443 = r6884441 / r6884442;
        return r6884443;
}

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 13.8

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

  2. Simplified13.8

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

  4. Applied difference-of-squares13.8

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

  5. Applied *-un-lft-identity13.8

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

  6. Applied times-frac13.6

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

  7. Applied associate-/l*9.5

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

  9. Applied difference-of-squares5.1

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

  10. Applied associate-/r*4.8

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

  12. Applied fma-udef4.8

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

  13. Simplified0.3

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

  15. Applied associate-/r/0.3

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

  16. Final simplification0.3

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

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))