Average Error: 17.4 → 0.4
Time: 31.3s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[U + \cos \left(\frac{K}{2}\right) \cdot \left(\mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, \ell \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{3} + 2\right)\right) \cdot J\right)\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
U + \cos \left(\frac{K}{2}\right) \cdot \left(\mathsf{fma}\left(\frac{1}{60}, {\ell}^{5}, \ell \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{3} + 2\right)\right) \cdot J\right)
double f(double J, double l, double K, double U) {
        double r2587196 = J;
        double r2587197 = l;
        double r2587198 = exp(r2587197);
        double r2587199 = -r2587197;
        double r2587200 = exp(r2587199);
        double r2587201 = r2587198 - r2587200;
        double r2587202 = r2587196 * r2587201;
        double r2587203 = K;
        double r2587204 = 2.0;
        double r2587205 = r2587203 / r2587204;
        double r2587206 = cos(r2587205);
        double r2587207 = r2587202 * r2587206;
        double r2587208 = U;
        double r2587209 = r2587207 + r2587208;
        return r2587209;
}
↓
double f(double J, double l, double K, double U) {
        double r2587210 = U;
        double r2587211 = K;
        double r2587212 = 2.0;
        double r2587213 = r2587211 / r2587212;
        double r2587214 = cos(r2587213);
        double r2587215 = 0.016666666666666666;
        double r2587216 = l;
        double r2587217 = 5.0;
        double r2587218 = pow(r2587216, r2587217);
        double r2587219 = r2587216 * r2587216;
        double r2587220 = 0.3333333333333333;
        double r2587221 = r2587219 * r2587220;
        double r2587222 = r2587221 + r2587212;
        double r2587223 = r2587216 * r2587222;
        double r2587224 = fma(r2587215, r2587218, r2587223);
        double r2587225 = J;
        double r2587226 = r2587224 * r2587225;
        double r2587227 = r2587214 * r2587226;
        double r2587228 = r2587210 + r2587227;
        return r2587228;
}

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. Taylor expanded around 0 0.4

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

  3. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))