Average Error: 17.2 → 0.4
Time: 29.8s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\mathsf{fma}\left(\left(\frac{1}{3} \cdot {\ell}^{3}\right) \cdot J + \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right) \cdot J, \cos \left(\frac{K}{2}\right), U\right)\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
\mathsf{fma}\left(\left(\frac{1}{3} \cdot {\ell}^{3}\right) \cdot J + \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right) \cdot J, \cos \left(\frac{K}{2}\right), U\right)
double f(double J, double l, double K, double U) {
        double r173779 = J;
        double r173780 = l;
        double r173781 = exp(r173780);
        double r173782 = -r173780;
        double r173783 = exp(r173782);
        double r173784 = r173781 - r173783;
        double r173785 = r173779 * r173784;
        double r173786 = K;
        double r173787 = 2.0;
        double r173788 = r173786 / r173787;
        double r173789 = cos(r173788);
        double r173790 = r173785 * r173789;
        double r173791 = U;
        double r173792 = r173790 + r173791;
        return r173792;
}
↓
double f(double J, double l, double K, double U) {
        double r173793 = 0.3333333333333333;
        double r173794 = l;
        double r173795 = 3.0;
        double r173796 = pow(r173794, r173795);
        double r173797 = r173793 * r173796;
        double r173798 = J;
        double r173799 = r173797 * r173798;
        double r173800 = 0.016666666666666666;
        double r173801 = 5.0;
        double r173802 = pow(r173794, r173801);
        double r173803 = 2.0;
        double r173804 = r173803 * r173794;
        double r173805 = fma(r173800, r173802, r173804);
        double r173806 = r173805 * r173798;
        double r173807 = r173799 + r173806;
        double r173808 = K;
        double r173809 = 2.0;
        double r173810 = r173808 / r173809;
        double r173811 = cos(r173810);
        double r173812 = U;
        double r173813 = fma(r173807, r173811, r173812);
        return r173813;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 17.2

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

  2. Simplified17.2

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

  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. Using strategy rm

  6. Applied fma-udef0.4

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

  7. Applied distribute-lft-in0.4

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

  8. Simplified0.4

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

  9. Simplified0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019351 +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))