Average Error: 15.6 → 1.3
Time: 36.6s
Precision: 64
\[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\]
↓
\[\log \left({\left(e^{\cos \left(\mathsf{fma}\left(\frac{K}{2}, n + m, -M\right)\right)}\right)}^{\left(e^{\left(-{\left(\frac{n + m}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\right)}\right)\]
\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)}
↓
\log \left({\left(e^{\cos \left(\mathsf{fma}\left(\frac{K}{2}, n + m, -M\right)\right)}\right)}^{\left(e^{\left(-{\left(\frac{n + m}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)}\right)}\right)
double f(double K, double m, double n, double M, double l) {
        double r127709 = K;
        double r127710 = m;
        double r127711 = n;
        double r127712 = r127710 + r127711;
        double r127713 = r127709 * r127712;
        double r127714 = 2.0;
        double r127715 = r127713 / r127714;
        double r127716 = M;
        double r127717 = r127715 - r127716;
        double r127718 = cos(r127717);
        double r127719 = r127712 / r127714;
        double r127720 = r127719 - r127716;
        double r127721 = pow(r127720, r127714);
        double r127722 = -r127721;
        double r127723 = l;
        double r127724 = r127710 - r127711;
        double r127725 = fabs(r127724);
        double r127726 = r127723 - r127725;
        double r127727 = r127722 - r127726;
        double r127728 = exp(r127727);
        double r127729 = r127718 * r127728;
        return r127729;
}
↓
double f(double K, double m, double n, double M, double l) {
        double r127730 = K;
        double r127731 = 2.0;
        double r127732 = r127730 / r127731;
        double r127733 = n;
        double r127734 = m;
        double r127735 = r127733 + r127734;
        double r127736 = M;
        double r127737 = -r127736;
        double r127738 = fma(r127732, r127735, r127737);
        double r127739 = cos(r127738);
        double r127740 = exp(r127739);
        double r127741 = r127735 / r127731;
        double r127742 = r127741 - r127736;
        double r127743 = pow(r127742, r127731);
        double r127744 = -r127743;
        double r127745 = l;
        double r127746 = r127734 - r127733;
        double r127747 = fabs(r127746);
        double r127748 = r127745 - r127747;
        double r127749 = r127744 - r127748;
        double r127750 = exp(r127749);
        double r127751 = pow(r127740, r127750);
        double r127752 = log(r127751);
        return r127752;
}

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 15.6

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

  3. Applied add-log-exp15.6

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

  4. Simplified1.3

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

  5. Final simplification1.3

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (K m n M l)
  :name "Maksimov and Kolovsky, Equation (32)"
  (* (cos (- (/ (* K (+ m n)) 2.0) M)) (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l (fabs (- m n)))))))