Average Error: 15.3 → 1.3
Time: 20.8s
Precision: 64
\[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\]
↓
\[\frac{1}{e^{(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell)_* - \left|m - n\right|}}\]
double f(double K, double m, double n, double M, double l) {
        double r13184166 = K;
        double r13184167 = m;
        double r13184168 = n;
        double r13184169 = r13184167 + r13184168;
        double r13184170 = r13184166 * r13184169;
        double r13184171 = 2.0;
        double r13184172 = r13184170 / r13184171;
        double r13184173 = M;
        double r13184174 = r13184172 - r13184173;
        double r13184175 = cos(r13184174);
        double r13184176 = r13184169 / r13184171;
        double r13184177 = r13184176 - r13184173;
        double r13184178 = pow(r13184177, r13184171);
        double r13184179 = -r13184178;
        double r13184180 = l;
        double r13184181 = r13184167 - r13184168;
        double r13184182 = fabs(r13184181);
        double r13184183 = r13184180 - r13184182;
        double r13184184 = r13184179 - r13184183;
        double r13184185 = exp(r13184184);
        double r13184186 = r13184175 * r13184185;
        return r13184186;
}
↓
double f(double __attribute__((unused)) K, double m, double n, double M, double l) {
        double r13184187 = 1.0;
        double r13184188 = m;
        double r13184189 = n;
        double r13184190 = r13184188 + r13184189;
        double r13184191 = 2.0;
        double r13184192 = r13184190 / r13184191;
        double r13184193 = M;
        double r13184194 = r13184192 - r13184193;
        double r13184195 = l;
        double r13184196 = fma(r13184194, r13184194, r13184195);
        double r13184197 = r13184188 - r13184189;
        double r13184198 = fabs(r13184197);
        double r13184199 = r13184196 - r13184198;
        double r13184200 = exp(r13184199);
        double r13184201 = r13184187 / r13184200;
        return r13184201;
}


\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)}
↓
\frac{1}{e^{(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell)_* - \left|m - n\right|}}

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 15.3

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

  2. Simplified15.3

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

  3. Taylor expanded around 0 1.3

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

  4. Final simplification1.3

    \[\leadsto \frac{1}{e^{(\left(\frac{m + n}{2} - M\right) \cdot \left(\frac{m + n}{2} - M\right) + \ell)_* - \left|m - n\right|}}\]

Reproduce

herbie shell --seed 2019102 +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)))))))