Average Error: 17.4 → 0.3
Time: 34.1s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[U + \cos \left(\frac{K}{2}\right) \cdot \left(\mathsf{fma}\left(2, \ell, \mathsf{fma}\left(\frac{1}{60}, \ell \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\ell \cdot \ell\right)\right), \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} \cdot \ell\right)\right)\right) \cdot J\right)\]
\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, \mathsf{fma}\left(\frac{1}{60}, \ell \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\ell \cdot \ell\right)\right), \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} \cdot \ell\right)\right)\right) \cdot J\right)
double f(double J, double l, double K, double U) {
        double r1965764 = J;
        double r1965765 = l;
        double r1965766 = exp(r1965765);
        double r1965767 = -r1965765;
        double r1965768 = exp(r1965767);
        double r1965769 = r1965766 - r1965768;
        double r1965770 = r1965764 * r1965769;
        double r1965771 = K;
        double r1965772 = 2.0;
        double r1965773 = r1965771 / r1965772;
        double r1965774 = cos(r1965773);
        double r1965775 = r1965770 * r1965774;
        double r1965776 = U;
        double r1965777 = r1965775 + r1965776;
        return r1965777;
}
↓
double f(double J, double l, double K, double U) {
        double r1965778 = U;
        double r1965779 = K;
        double r1965780 = 2.0;
        double r1965781 = r1965779 / r1965780;
        double r1965782 = cos(r1965781);
        double r1965783 = l;
        double r1965784 = 0.016666666666666666;
        double r1965785 = r1965783 * r1965783;
        double r1965786 = r1965785 * r1965785;
        double r1965787 = r1965783 * r1965786;
        double r1965788 = 0.3333333333333333;
        double r1965789 = r1965788 * r1965783;
        double r1965790 = r1965785 * r1965789;
        double r1965791 = fma(r1965784, r1965787, r1965790);
        double r1965792 = fma(r1965780, r1965783, r1965791);
        double r1965793 = J;
        double r1965794 = r1965792 * r1965793;
        double r1965795 = r1965782 * r1965794;
        double r1965796 = r1965778 + r1965795;
        return r1965796;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.4

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

  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

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

  4. Taylor expanded around inf 0.3

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

  5. Simplified0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2019142 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))