Average Error: 17.5 → 0.4
Time: 9.3s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\mathsf{fma}\left(J \cdot \mathsf{fma}\left(\frac{1}{3}, {\ell}^{3}, \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right), \cos \left(\frac{K}{2}\right), U\right)\]
\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(\frac{1}{3}, {\ell}^{3}, \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right), \cos \left(\frac{K}{2}\right), U\right)
double f(double J, double l, double K, double U) {
        double r169800 = J;
        double r169801 = l;
        double r169802 = exp(r169801);
        double r169803 = -r169801;
        double r169804 = exp(r169803);
        double r169805 = r169802 - r169804;
        double r169806 = r169800 * r169805;
        double r169807 = K;
        double r169808 = 2.0;
        double r169809 = r169807 / r169808;
        double r169810 = cos(r169809);
        double r169811 = r169806 * r169810;
        double r169812 = U;
        double r169813 = r169811 + r169812;
        return r169813;
}
↓
double f(double J, double l, double K, double U) {
        double r169814 = J;
        double r169815 = 0.3333333333333333;
        double r169816 = l;
        double r169817 = 3.0;
        double r169818 = pow(r169816, r169817);
        double r169819 = 0.016666666666666666;
        double r169820 = 5.0;
        double r169821 = pow(r169816, r169820);
        double r169822 = 2.0;
        double r169823 = r169822 * r169816;
        double r169824 = fma(r169819, r169821, r169823);
        double r169825 = fma(r169815, r169818, r169824);
        double r169826 = r169814 * r169825;
        double r169827 = K;
        double r169828 = 2.0;
        double r169829 = r169827 / r169828;
        double r169830 = cos(r169829);
        double r169831 = U;
        double r169832 = fma(r169826, r169830, r169831);
        return r169832;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.5

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

  2. Simplified17.5

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

  3. Taylor expanded around 0 0.4

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

  4. Simplified0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))