Average Error: 14.8 → 1.3
Time: 28.5s
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 r3313176 = K;
        double r3313177 = m;
        double r3313178 = n;
        double r3313179 = r3313177 + r3313178;
        double r3313180 = r3313176 * r3313179;
        double r3313181 = 2.0;
        double r3313182 = r3313180 / r3313181;
        double r3313183 = M;
        double r3313184 = r3313182 - r3313183;
        double r3313185 = cos(r3313184);
        double r3313186 = r3313179 / r3313181;
        double r3313187 = r3313186 - r3313183;
        double r3313188 = pow(r3313187, r3313181);
        double r3313189 = -r3313188;
        double r3313190 = l;
        double r3313191 = r3313177 - r3313178;
        double r3313192 = fabs(r3313191);
        double r3313193 = r3313190 - r3313192;
        double r3313194 = r3313189 - r3313193;
        double r3313195 = exp(r3313194);
        double r3313196 = r3313185 * r3313195;
        return r3313196;
}
↓
double f(double __attribute__((unused)) K, double m, double n, double M, double l) {
        double r3313197 = m;
        double r3313198 = n;
        double r3313199 = r3313197 - r3313198;
        double r3313200 = fabs(r3313199);
        double r3313201 = r3313197 + r3313198;
        double r3313202 = 2.0;
        double r3313203 = r3313201 / r3313202;
        double r3313204 = M;
        double r3313205 = r3313203 - r3313204;
        double r3313206 = l;
        double r3313207 = fma(r3313205, r3313205, r3313206);
        double r3313208 = r3313200 - r3313207;
        double r3313209 = exp(r3313208);
        return r3313209;
}

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 14.8

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

  2. Simplified14.8

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

  3. Taylor expanded around 0 1.3

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

  4. Final simplification1.3

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

Reproduce

herbie shell --seed 2019142 +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)))))))