Average Error: 15.2 → 1.4
Time: 21.4s
Precision: 64
\[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\]
↓
\[e^{\left|m - n\right| - \mathsf{fma}\left(\frac{n + m}{2} - M, \frac{n + m}{2} - M, \ell\right)}\]
\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{n + m}{2} - M, \frac{n + m}{2} - M, \ell\right)}
double f(double K, double m, double n, double M, double l) {
        double r4218620 = K;
        double r4218621 = m;
        double r4218622 = n;
        double r4218623 = r4218621 + r4218622;
        double r4218624 = r4218620 * r4218623;
        double r4218625 = 2.0;
        double r4218626 = r4218624 / r4218625;
        double r4218627 = M;
        double r4218628 = r4218626 - r4218627;
        double r4218629 = cos(r4218628);
        double r4218630 = r4218623 / r4218625;
        double r4218631 = r4218630 - r4218627;
        double r4218632 = pow(r4218631, r4218625);
        double r4218633 = -r4218632;
        double r4218634 = l;
        double r4218635 = r4218621 - r4218622;
        double r4218636 = fabs(r4218635);
        double r4218637 = r4218634 - r4218636;
        double r4218638 = r4218633 - r4218637;
        double r4218639 = exp(r4218638);
        double r4218640 = r4218629 * r4218639;
        return r4218640;
}
↓
double f(double __attribute__((unused)) K, double m, double n, double M, double l) {
        double r4218641 = m;
        double r4218642 = n;
        double r4218643 = r4218641 - r4218642;
        double r4218644 = fabs(r4218643);
        double r4218645 = r4218642 + r4218641;
        double r4218646 = 2.0;
        double r4218647 = r4218645 / r4218646;
        double r4218648 = M;
        double r4218649 = r4218647 - r4218648;
        double r4218650 = l;
        double r4218651 = fma(r4218649, r4218649, r4218650);
        double r4218652 = r4218644 - r4218651;
        double r4218653 = exp(r4218652);
        return r4218653;
}

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 15.2

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

  2. Simplified15.1

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

  3. Taylor expanded around 0 1.4

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

  4. Final simplification1.4

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

Reproduce

herbie shell --seed 2019135 +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)))))))