Average Error: 16.9 → 0.5
Time: 46.7s
Precision: 64
\[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
↓
\[U + \sqrt[3]{\cos \left(\frac{K}{2}\right)} \cdot \left(\left(J \cdot (\frac{1}{60} \cdot \left({\ell}^{5}\right) + \left(\ell \cdot (\frac{1}{3} \cdot \left(\ell \cdot \ell\right) + 2)_*\right))_*\right) \cdot \left(\sqrt[3]{\cos \left(\frac{K}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{K}{2}\right)}\right)\right)\]
\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U
↓
U + \sqrt[3]{\cos \left(\frac{K}{2}\right)} \cdot \left(\left(J \cdot (\frac{1}{60} \cdot \left({\ell}^{5}\right) + \left(\ell \cdot (\frac{1}{3} \cdot \left(\ell \cdot \ell\right) + 2)_*\right))_*\right) \cdot \left(\sqrt[3]{\cos \left(\frac{K}{2}\right)} \cdot \sqrt[3]{\cos \left(\frac{K}{2}\right)}\right)\right)
double f(double J, double l, double K, double U) {
        double r35388241 = J;
        double r35388242 = l;
        double r35388243 = exp(r35388242);
        double r35388244 = -r35388242;
        double r35388245 = exp(r35388244);
        double r35388246 = r35388243 - r35388245;
        double r35388247 = r35388241 * r35388246;
        double r35388248 = K;
        double r35388249 = 2.0;
        double r35388250 = r35388248 / r35388249;
        double r35388251 = cos(r35388250);
        double r35388252 = r35388247 * r35388251;
        double r35388253 = U;
        double r35388254 = r35388252 + r35388253;
        return r35388254;
}
↓
double f(double J, double l, double K, double U) {
        double r35388255 = U;
        double r35388256 = K;
        double r35388257 = 2.0;
        double r35388258 = r35388256 / r35388257;
        double r35388259 = cos(r35388258);
        double r35388260 = cbrt(r35388259);
        double r35388261 = J;
        double r35388262 = 0.016666666666666666;
        double r35388263 = l;
        double r35388264 = 5.0;
        double r35388265 = pow(r35388263, r35388264);
        double r35388266 = 0.3333333333333333;
        double r35388267 = r35388263 * r35388263;
        double r35388268 = fma(r35388266, r35388267, r35388257);
        double r35388269 = r35388263 * r35388268;
        double r35388270 = fma(r35388262, r35388265, r35388269);
        double r35388271 = r35388261 * r35388270;
        double r35388272 = r35388260 * r35388260;
        double r35388273 = r35388271 * r35388272;
        double r35388274 = r35388260 * r35388273;
        double r35388275 = r35388255 + r35388274;
        return r35388275;
}

Error

Bits error versus J

Bits error versus l

Bits error versus K

Bits error versus U

Derivation

  1. Initial program 16.9

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

  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

    \[\leadsto \left(J \cdot \color{blue}{(\frac{1}{60} \cdot \left({\ell}^{5}\right) + \left(\ell \cdot (\frac{1}{3} \cdot \left(\ell \cdot \ell\right) + 2)_*\right))_*}\right) \cdot \cos \left(\frac{K}{2}\right) + U\]
  4. Using strategy rm

  5. Applied add-cube-cbrt0.5

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

  6. Applied associate-*r*0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2))) U))