Average Error: 17.3 → 0.6
Time: 21.6s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[\cos \left(\frac{K}{2}\right) \cdot \left(\left(\ell \cdot J\right) \cdot 2\right) + U\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
\cos \left(\frac{K}{2}\right) \cdot \left(\left(\ell \cdot J\right) \cdot 2\right) + U
double f(double J, double l, double K, double U) {
        double r1378169 = J;
        double r1378170 = l;
        double r1378171 = exp(r1378170);
        double r1378172 = -r1378170;
        double r1378173 = exp(r1378172);
        double r1378174 = r1378171 - r1378173;
        double r1378175 = r1378169 * r1378174;
        double r1378176 = K;
        double r1378177 = 2.0;
        double r1378178 = r1378176 / r1378177;
        double r1378179 = cos(r1378178);
        double r1378180 = r1378175 * r1378179;
        double r1378181 = U;
        double r1378182 = r1378180 + r1378181;
        return r1378182;
}
↓
double f(double J, double l, double K, double U) {
        double r1378183 = K;
        double r1378184 = 2.0;
        double r1378185 = r1378183 / r1378184;
        double r1378186 = cos(r1378185);
        double r1378187 = l;
        double r1378188 = J;
        double r1378189 = r1378187 * r1378188;
        double r1378190 = r1378189 * r1378184;
        double r1378191 = r1378186 * r1378190;
        double r1378192 = U;
        double r1378193 = r1378191 + r1378192;
        return r1378193;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 17.3

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

  2. Simplified17.3

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

  3. Taylor expanded around 0 0.6

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

  5. Applied fma-udef0.6

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

  6. Final simplification0.6

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

Reproduce

herbie shell --seed 2019154 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))