Average Error: 14.6 → 1.4
Time: 18.5s
Precision: 64
\[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\]
↓
\[e^{\left|m - n\right| - \mathsf{fma}\left(\frac{m + n}{2} - M, \frac{m + n}{2} - M, \ell\right)}\]
\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}
↓
e^{\left|m - n\right| - \mathsf{fma}\left(\frac{m + n}{2} - M, \frac{m + n}{2} - M, \ell\right)}
double f(double K, double m, double n, double M, double l) {
        double r2950563 = K;
        double r2950564 = m;
        double r2950565 = n;
        double r2950566 = r2950564 + r2950565;
        double r2950567 = r2950563 * r2950566;
        double r2950568 = 2.0;
        double r2950569 = r2950567 / r2950568;
        double r2950570 = M;
        double r2950571 = r2950569 - r2950570;
        double r2950572 = cos(r2950571);
        double r2950573 = r2950566 / r2950568;
        double r2950574 = r2950573 - r2950570;
        double r2950575 = pow(r2950574, r2950568);
        double r2950576 = -r2950575;
        double r2950577 = l;
        double r2950578 = r2950564 - r2950565;
        double r2950579 = fabs(r2950578);
        double r2950580 = r2950577 - r2950579;
        double r2950581 = r2950576 - r2950580;
        double r2950582 = exp(r2950581);
        double r2950583 = r2950572 * r2950582;
        return r2950583;
}
↓
double f(double __attribute__((unused)) K, double m, double n, double M, double l) {
        double r2950584 = m;
        double r2950585 = n;
        double r2950586 = r2950584 - r2950585;
        double r2950587 = fabs(r2950586);
        double r2950588 = r2950584 + r2950585;
        double r2950589 = 2.0;
        double r2950590 = r2950588 / r2950589;
        double r2950591 = M;
        double r2950592 = r2950590 - r2950591;
        double r2950593 = l;
        double r2950594 = fma(r2950592, r2950592, r2950593);
        double r2950595 = r2950587 - r2950594;
        double r2950596 = exp(r2950595);
        return r2950596;
}

Error

Bits error versus K

Bits error versus m

Bits error versus n

Bits error versus M

Bits error versus l

Derivation

  1. Initial program 14.6

    \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\]

  2. Simplified14.6

    \[\leadsto \color{blue}{\cos \left(K \cdot \frac{m + n}{2} - M\right) \cdot e^{\left|m - n\right| - \mathsf{fma}\left(\frac{m + n}{2} - M, \frac{m + n}{2} - M, \ell\right)}}\]

  3. Taylor expanded around 0 1.4

    \[\leadsto \color{blue}{1} \cdot e^{\left|m - n\right| - \mathsf{fma}\left(\frac{m + n}{2} - M, \frac{m + n}{2} - M, \ell\right)}\]

  4. Final simplification1.4

    \[\leadsto e^{\left|m - n\right| - \mathsf{fma}\left(\frac{m + n}{2} - M, \frac{m + n}{2} - M, \ell\right)}\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (K m n M l)
  :name "Maksimov and Kolovsky, Equation (32)"
  (* (cos (- (/ (* K (+ m n)) 2) M)) (exp (- (- (pow (- (/ (+ m n) 2) M) 2)) (- l (fabs (- m n)))))))