Average Error: 17.1 → 0.4
Time: 24.8s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\mathsf{fma}\left(\left(\frac{1}{3} \cdot J\right) \cdot {\ell}^{3} + J \cdot \mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, 2 \cdot \ell\right), \cos \left(\frac{K}{2}\right), U\right)\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
\mathsf{fma}\left(\left(\frac{1}{3} \cdot J\right) \cdot {\ell}^{3} + 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 r88190 = J;
        double r88191 = l;
        double r88192 = exp(r88191);
        double r88193 = -r88191;
        double r88194 = exp(r88193);
        double r88195 = r88192 - r88194;
        double r88196 = r88190 * r88195;
        double r88197 = K;
        double r88198 = 2.0;
        double r88199 = r88197 / r88198;
        double r88200 = cos(r88199);
        double r88201 = r88196 * r88200;
        double r88202 = U;
        double r88203 = r88201 + r88202;
        return r88203;
}
↓
double f(double J, double l, double K, double U) {
        double r88204 = 0.3333333333333333;
        double r88205 = J;
        double r88206 = r88204 * r88205;
        double r88207 = l;
        double r88208 = 3.0;
        double r88209 = pow(r88207, r88208);
        double r88210 = r88206 * r88209;
        double r88211 = 0.016666666666666666;
        double r88212 = 5.0;
        double r88213 = pow(r88207, r88212);
        double r88214 = 2.0;
        double r88215 = r88214 * r88207;
        double r88216 = fma(r88211, r88213, r88215);
        double r88217 = r88205 * r88216;
        double r88218 = r88210 + r88217;
        double r88219 = K;
        double r88220 = 2.0;
        double r88221 = r88219 / r88220;
        double r88222 = cos(r88221);
        double r88223 = U;
        double r88224 = fma(r88218, r88222, r88223);
        return r88224;
}

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. Simplified17.1

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

  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

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019209 +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))