Average Error: 17.4 → 0.4
Time: 7.9s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\mathsf{fma}\left(J \cdot \left(\frac{1}{3} \cdot {\ell}^{3}\right) + J \cdot \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right), \cos \left(\frac{K}{2}\right), U\right)\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
\mathsf{fma}\left(J \cdot \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)
double f(double J, double l, double K, double U) {
        double r135606 = J;
        double r135607 = l;
        double r135608 = exp(r135607);
        double r135609 = -r135607;
        double r135610 = exp(r135609);
        double r135611 = r135608 - r135610;
        double r135612 = r135606 * r135611;
        double r135613 = K;
        double r135614 = 2.0;
        double r135615 = r135613 / r135614;
        double r135616 = cos(r135615);
        double r135617 = r135612 * r135616;
        double r135618 = U;
        double r135619 = r135617 + r135618;
        return r135619;
}
↓
double f(double J, double l, double K, double U) {
        double r135620 = J;
        double r135621 = 0.3333333333333333;
        double r135622 = l;
        double r135623 = 3.0;
        double r135624 = pow(r135622, r135623);
        double r135625 = r135621 * r135624;
        double r135626 = r135620 * r135625;
        double r135627 = 0.016666666666666666;
        double r135628 = 5.0;
        double r135629 = pow(r135622, r135628);
        double r135630 = 2.0;
        double r135631 = r135630 * r135622;
        double r135632 = fma(r135627, r135629, r135631);
        double r135633 = r135620 * r135632;
        double r135634 = r135626 + r135633;
        double r135635 = K;
        double r135636 = 2.0;
        double r135637 = r135635 / r135636;
        double r135638 = cos(r135637);
        double r135639 = U;
        double r135640 = fma(r135634, r135638, r135639);
        return r135640;
}

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. Simplified17.4

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

  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. Final simplification0.4

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

Reproduce

herbie shell --seed 2020046 +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))