Average Error: 17.1 → 0.4
Time: 7.9s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\left(J \cdot \left(\frac{1}{3} \cdot {\ell}^{3}\right) + J \cdot \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
\left(J \cdot \left(\frac{1}{3} \cdot {\ell}^{3}\right) + J \cdot \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
double f(double J, double l, double K, double U) {
        double r170753 = J;
        double r170754 = l;
        double r170755 = exp(r170754);
        double r170756 = -r170754;
        double r170757 = exp(r170756);
        double r170758 = r170755 - r170757;
        double r170759 = r170753 * r170758;
        double r170760 = K;
        double r170761 = 2.0;
        double r170762 = r170760 / r170761;
        double r170763 = cos(r170762);
        double r170764 = r170759 * r170763;
        double r170765 = U;
        double r170766 = r170764 + r170765;
        return r170766;
}
↓
double f(double J, double l, double K, double U) {
        double r170767 = J;
        double r170768 = 0.3333333333333333;
        double r170769 = l;
        double r170770 = 3.0;
        double r170771 = pow(r170769, r170770);
        double r170772 = r170768 * r170771;
        double r170773 = r170767 * r170772;
        double r170774 = 0.016666666666666666;
        double r170775 = 5.0;
        double r170776 = pow(r170769, r170775);
        double r170777 = 2.0;
        double r170778 = r170777 * r170769;
        double r170779 = fma(r170774, r170776, r170778);
        double r170780 = r170767 * r170779;
        double r170781 = r170773 + r170780;
        double r170782 = K;
        double r170783 = 2.0;
        double r170784 = r170782 / r170783;
        double r170785 = cos(r170784);
        double r170786 = r170781 * r170785;
        double r170787 = U;
        double r170788 = r170786 + r170787;
        return r170788;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.1

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

  2. Taylor expanded around 0 0.4

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

  3. Simplified0.4

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

  5. Applied fma-udef0.4

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

  6. Applied distribute-lft-in0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2020060 +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))