Average Error: 15.3 → 1.3
Time: 11.7s
Precision: 64
\[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\]
↓
\[\left(\sqrt[3]{\frac{1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \cdot \sqrt[3]{\frac{1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}}\right) \cdot \sqrt[3]{\frac{1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}}\]
\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)}
↓
\left(\sqrt[3]{\frac{1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \cdot \sqrt[3]{\frac{1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}}\right) \cdot \sqrt[3]{\frac{1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}}
double f(double K, double m, double n, double M, double l) {
        double r192081 = K;
        double r192082 = m;
        double r192083 = n;
        double r192084 = r192082 + r192083;
        double r192085 = r192081 * r192084;
        double r192086 = 2.0;
        double r192087 = r192085 / r192086;
        double r192088 = M;
        double r192089 = r192087 - r192088;
        double r192090 = cos(r192089);
        double r192091 = r192084 / r192086;
        double r192092 = r192091 - r192088;
        double r192093 = pow(r192092, r192086);
        double r192094 = -r192093;
        double r192095 = l;
        double r192096 = r192082 - r192083;
        double r192097 = fabs(r192096);
        double r192098 = r192095 - r192097;
        double r192099 = r192094 - r192098;
        double r192100 = exp(r192099);
        double r192101 = r192090 * r192100;
        return r192101;
}
↓
double f(double __attribute__((unused)) K, double m, double n, double M, double l) {
        double r192102 = 1.0;
        double r192103 = m;
        double r192104 = n;
        double r192105 = r192103 + r192104;
        double r192106 = 2.0;
        double r192107 = r192105 / r192106;
        double r192108 = M;
        double r192109 = r192107 - r192108;
        double r192110 = pow(r192109, r192106);
        double r192111 = l;
        double r192112 = r192103 - r192104;
        double r192113 = fabs(r192112);
        double r192114 = r192111 - r192113;
        double r192115 = r192110 + r192114;
        double r192116 = exp(r192115);
        double r192117 = r192102 / r192116;
        double r192118 = cbrt(r192117);
        double r192119 = r192118 * r192118;
        double r192120 = r192119 * r192118;
        return r192120;
}

Error

Bits error versus K

Bits error versus m

Bits error versus n

Bits error versus M

Bits error versus l

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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

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

  3. Taylor expanded around 0 1.3

    \[\leadsto \frac{\color{blue}{1}}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}\]
  4. Using strategy rm

  5. Applied add-cube-cbrt1.3

    \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \cdot \sqrt[3]{\frac{1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}}\right) \cdot \sqrt[3]{\frac{1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}}}\]

  6. Final simplification1.3

    \[\leadsto \left(\sqrt[3]{\frac{1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}} \cdot \sqrt[3]{\frac{1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}}\right) \cdot \sqrt[3]{\frac{1}{e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}}}\]

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (K m n M l)
  :name "Maksimov and Kolovsky, Equation (32)"
  :precision binary64
  (* (cos (- (/ (* K (+ m n)) 2) M)) (exp (- (- (pow (- (/ (+ m n) 2) M) 2)) (- l (fabs (- m n)))))))