Maksimov and Kolovsky, Equation (3)

Average Error: 18.0 → 13.2

Time: 30.5s

Precision: 64

Math

\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]

↓

\[\begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(J \cdot 2\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) = -\infty:\\ \;\;\;\;\frac{\sqrt{0.25} \cdot U}{J \cdot \cos \left(K \cdot 0.5\right)} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(J \cdot 2\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \le 3.754265451791662071026116349996612187938 \cdot 10^{305}:\\ \;\;\;\;\sqrt{{\left(\frac{U}{\left(J \cdot 2\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{0.25} \cdot U}{J \cdot \cos \left(K \cdot 0.5\right)} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)\\ \end{array}\]

C

double f(double J, double K, double U) {
        double r5332168 = -2.0;
        double r5332169 = J;
        double r5332170 = r5332168 * r5332169;
        double r5332171 = K;
        double r5332172 = 2.0;
        double r5332173 = r5332171 / r5332172;
        double r5332174 = cos(r5332173);
        double r5332175 = r5332170 * r5332174;
        double r5332176 = 1.0;
        double r5332177 = U;
        double r5332178 = r5332172 * r5332169;
        double r5332179 = r5332178 * r5332174;
        double r5332180 = r5332177 / r5332179;
        double r5332181 = pow(r5332180, r5332172);
        double r5332182 = r5332176 + r5332181;
        double r5332183 = sqrt(r5332182);
        double r5332184 = r5332175 * r5332183;
        return r5332184;
}

↓

double f(double J, double K, double U) {
        double r5332185 = U;
        double r5332186 = J;
        double r5332187 = 2.0;
        double r5332188 = r5332186 * r5332187;
        double r5332189 = K;
        double r5332190 = r5332189 / r5332187;
        double r5332191 = cos(r5332190);
        double r5332192 = r5332188 * r5332191;
        double r5332193 = r5332185 / r5332192;
        double r5332194 = pow(r5332193, r5332187);
        double r5332195 = 1.0;
        double r5332196 = r5332194 + r5332195;
        double r5332197 = sqrt(r5332196);
        double r5332198 = -2.0;
        double r5332199 = r5332198 * r5332186;
        double r5332200 = r5332191 * r5332199;
        double r5332201 = r5332197 * r5332200;
        double r5332202 = -inf.0;
        bool r5332203 = r5332201 <= r5332202;
        double r5332204 = 0.25;
        double r5332205 = sqrt(r5332204);
        double r5332206 = r5332205 * r5332185;
        double r5332207 = 0.5;
        double r5332208 = r5332189 * r5332207;
        double r5332209 = cos(r5332208);
        double r5332210 = r5332186 * r5332209;
        double r5332211 = r5332206 / r5332210;
        double r5332212 = r5332211 * r5332200;
        double r5332213 = 3.754265451791662e+305;
        bool r5332214 = r5332201 <= r5332213;
        double r5332215 = r5332214 ? r5332201 : r5332212;
        double r5332216 = r5332203 ? r5332212 : r5332215;
        return r5332216;
}

Error

Bits error versus J

Bits error versus K

Bits error versus U

Derivation

1. Split input into 2 regimes

2. if (* (* (* -2.0 J) (cos (/ K 2.0))) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) (cos (/ K 2.0)))) 2.0)))) < -inf.0 or 3.754265451791662e+305 < (* (* (* -2.0 J) (cos (/ K 2.0))) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) (cos (/ K 2.0)))) 2.0))))

   1. Initial program 63.2

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]

   2. Taylor expanded around inf 46.1

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\frac{\sqrt{0.25} \cdot U}{\cos \left(0.5 \cdot K\right) \cdot J}}\]

   if -inf.0 < (* (* (* -2.0 J) (cos (/ K 2.0))) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) (cos (/ K 2.0)))) 2.0)))) < 3.754265451791662e+305

   1. Initial program 0.1

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 13.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(J \cdot 2\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) = -\infty:\\ \;\;\;\;\frac{\sqrt{0.25} \cdot U}{J \cdot \cos \left(K \cdot 0.5\right)} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(J \cdot 2\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \le 3.754265451791662071026116349996612187938 \cdot 10^{305}:\\ \;\;\;\;\sqrt{{\left(\frac{U}{\left(J \cdot 2\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{0.25} \cdot U}{J \cdot \cos \left(K \cdot 0.5\right)} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (J K U)
  :name "Maksimov and Kolovsky, Equation (3)"
  (* (* (* -2.0 J) (cos (/ K 2.0))) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) (cos (/ K 2.0)))) 2.0)))))