Average Error: 16.9 → 0.3
Time: 27.3s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[U + \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{fma}\left({\ell}^{5}, \frac{1}{60}, \mathsf{fma}\left(\ell, \frac{1}{3} \cdot \ell, 2\right) \cdot \ell\right)\right) \cdot J\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
U + \left(\cos \left(\frac{K}{2}\right) \cdot \mathsf{fma}\left({\ell}^{5}, \frac{1}{60}, \mathsf{fma}\left(\ell, \frac{1}{3} \cdot \ell, 2\right) \cdot \ell\right)\right) \cdot J
double f(double J, double l, double K, double U) {
        double r4207761 = J;
        double r4207762 = l;
        double r4207763 = exp(r4207762);
        double r4207764 = -r4207762;
        double r4207765 = exp(r4207764);
        double r4207766 = r4207763 - r4207765;
        double r4207767 = r4207761 * r4207766;
        double r4207768 = K;
        double r4207769 = 2.0;
        double r4207770 = r4207768 / r4207769;
        double r4207771 = cos(r4207770);
        double r4207772 = r4207767 * r4207771;
        double r4207773 = U;
        double r4207774 = r4207772 + r4207773;
        return r4207774;
}
↓
double f(double J, double l, double K, double U) {
        double r4207775 = U;
        double r4207776 = K;
        double r4207777 = 2.0;
        double r4207778 = r4207776 / r4207777;
        double r4207779 = cos(r4207778);
        double r4207780 = l;
        double r4207781 = 5.0;
        double r4207782 = pow(r4207780, r4207781);
        double r4207783 = 0.016666666666666666;
        double r4207784 = 0.3333333333333333;
        double r4207785 = r4207784 * r4207780;
        double r4207786 = 2.0;
        double r4207787 = fma(r4207780, r4207785, r4207786);
        double r4207788 = r4207787 * r4207780;
        double r4207789 = fma(r4207782, r4207783, r4207788);
        double r4207790 = r4207779 * r4207789;
        double r4207791 = J;
        double r4207792 = r4207790 * r4207791;
        double r4207793 = r4207775 + r4207792;
        return r4207793;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 16.9

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

  2. Taylor expanded around 0 0.3

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

  3. Simplified0.3

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

  5. Applied associate-*l*0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))