Average Error: 17.3 → 0.4
Time: 25.6s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\mathsf{fma}\left(J \cdot \mathsf{fma}\left(2, \ell, \mathsf{fma}\left(\frac{1}{3}, {\ell}^{3}, \frac{1}{60} \cdot {\ell}^{5}\right)\right), \cos \left(\frac{K}{2}\right), U\right)\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
\mathsf{fma}\left(J \cdot \mathsf{fma}\left(2, \ell, \mathsf{fma}\left(\frac{1}{3}, {\ell}^{3}, \frac{1}{60} \cdot {\ell}^{5}\right)\right), \cos \left(\frac{K}{2}\right), U\right)
double f(double J, double l, double K, double U) {
        double r85188 = J;
        double r85189 = l;
        double r85190 = exp(r85189);
        double r85191 = -r85189;
        double r85192 = exp(r85191);
        double r85193 = r85190 - r85192;
        double r85194 = r85188 * r85193;
        double r85195 = K;
        double r85196 = 2.0;
        double r85197 = r85195 / r85196;
        double r85198 = cos(r85197);
        double r85199 = r85194 * r85198;
        double r85200 = U;
        double r85201 = r85199 + r85200;
        return r85201;
}
↓
double f(double J, double l, double K, double U) {
        double r85202 = J;
        double r85203 = 2.0;
        double r85204 = l;
        double r85205 = 0.3333333333333333;
        double r85206 = 3.0;
        double r85207 = pow(r85204, r85206);
        double r85208 = 0.016666666666666666;
        double r85209 = 5.0;
        double r85210 = pow(r85204, r85209);
        double r85211 = r85208 * r85210;
        double r85212 = fma(r85205, r85207, r85211);
        double r85213 = fma(r85203, r85204, r85212);
        double r85214 = r85202 * r85213;
        double r85215 = K;
        double r85216 = 2.0;
        double r85217 = r85215 / r85216;
        double r85218 = cos(r85217);
        double r85219 = U;
        double r85220 = fma(r85214, r85218, r85219);
        return r85220;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.3

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

  2. Simplified17.3

    \[\leadsto \color{blue}{\mathsf{fma}\left(J \cdot \left(e^{\ell} - e^{-\ell}\right), \cos \left(\frac{K}{2}\right), U\right)}\]

  3. Taylor expanded around 0 0.4

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

  4. Simplified0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))