Average Error: 17.2 → 0.4
Time: 25.4s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[U + \cos \left(\frac{K}{2}\right) \cdot \left(\mathsf{fma}\left(2, \ell, \left(\mathsf{fma}\left(\frac{1}{3}, \left(\left(\ell \cdot \ell\right) \cdot \ell\right), \left({\ell}^{5} \cdot \frac{1}{60}\right)\right)\right)\right) \cdot J\right)\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
U + \cos \left(\frac{K}{2}\right) \cdot \left(\mathsf{fma}\left(2, \ell, \left(\mathsf{fma}\left(\frac{1}{3}, \left(\left(\ell \cdot \ell\right) \cdot \ell\right), \left({\ell}^{5} \cdot \frac{1}{60}\right)\right)\right)\right) \cdot J\right)
double f(double J, double l, double K, double U) {
        double r1646107 = J;
        double r1646108 = l;
        double r1646109 = exp(r1646108);
        double r1646110 = -r1646108;
        double r1646111 = exp(r1646110);
        double r1646112 = r1646109 - r1646111;
        double r1646113 = r1646107 * r1646112;
        double r1646114 = K;
        double r1646115 = 2.0;
        double r1646116 = r1646114 / r1646115;
        double r1646117 = cos(r1646116);
        double r1646118 = r1646113 * r1646117;
        double r1646119 = U;
        double r1646120 = r1646118 + r1646119;
        return r1646120;
}
↓
double f(double J, double l, double K, double U) {
        double r1646121 = U;
        double r1646122 = K;
        double r1646123 = 2.0;
        double r1646124 = r1646122 / r1646123;
        double r1646125 = cos(r1646124);
        double r1646126 = l;
        double r1646127 = 0.3333333333333333;
        double r1646128 = r1646126 * r1646126;
        double r1646129 = r1646128 * r1646126;
        double r1646130 = 5.0;
        double r1646131 = pow(r1646126, r1646130);
        double r1646132 = 0.016666666666666666;
        double r1646133 = r1646131 * r1646132;
        double r1646134 = fma(r1646127, r1646129, r1646133);
        double r1646135 = fma(r1646123, r1646126, r1646134);
        double r1646136 = J;
        double r1646137 = r1646135 * r1646136;
        double r1646138 = r1646125 * r1646137;
        double r1646139 = r1646121 + r1646138;
        return r1646139;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.2

    \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]

  2. Taylor expanded around 0 0.4

    \[\leadsto \left(J \cdot \color{blue}{\left(2 \cdot \ell + \left(\frac{1}{3} \cdot {\ell}^{3} + \frac{1}{60} \cdot {\ell}^{5}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U\]

  3. Simplified0.4

    \[\leadsto \left(J \cdot \color{blue}{\mathsf{fma}\left(2, \ell, \left(\mathsf{fma}\left(\frac{1}{3}, \left(\left(\ell \cdot \ell\right) \cdot \ell\right), \left({\ell}^{5} \cdot \frac{1}{60}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U\]

  4. Final simplification0.4

    \[\leadsto U + \cos \left(\frac{K}{2}\right) \cdot \left(\mathsf{fma}\left(2, \ell, \left(\mathsf{fma}\left(\frac{1}{3}, \left(\left(\ell \cdot \ell\right) \cdot \ell\right), \left({\ell}^{5} \cdot \frac{1}{60}\right)\right)\right)\right) \cdot J\right)\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))