Maksimov and Kolovsky, Equation (3)

Average Error: 16.8 → 16.3

Time: 27.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;J \le -1.1866131181783549 \cdot 10^{-211}:\\ \;\;\;\;\left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{\frac{\frac{U}{J + J}}{\left(J + J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\cos \left(\frac{K}{2}\right)} + 1}\right) \cdot \left(J \cdot -2\right)\\ \mathbf{elif}\;J \le 7.984518687285712 \cdot 10^{-287}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\frac{U}{\left(J + J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(J + J\right) \cdot \cos \left(\frac{K}{2}\right)} + 1}} \cdot \left(\left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\sqrt{\frac{U}{\left(J + J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(J + J\right) \cdot \cos \left(\frac{K}{2}\right)} + 1}}\right)\\ \end{array}\]

C

double f(double J, double K, double U) {
        double r3679688 = -2.0;
        double r3679689 = J;
        double r3679690 = r3679688 * r3679689;
        double r3679691 = K;
        double r3679692 = 2.0;
        double r3679693 = r3679691 / r3679692;
        double r3679694 = cos(r3679693);
        double r3679695 = r3679690 * r3679694;
        double r3679696 = 1.0;
        double r3679697 = U;
        double r3679698 = r3679692 * r3679689;
        double r3679699 = r3679698 * r3679694;
        double r3679700 = r3679697 / r3679699;
        double r3679701 = pow(r3679700, r3679692);
        double r3679702 = r3679696 + r3679701;
        double r3679703 = sqrt(r3679702);
        double r3679704 = r3679695 * r3679703;
        return r3679704;
}

↓

double f(double J, double K, double U) {
        double r3679705 = J;
        double r3679706 = -1.1866131181783549e-211;
        bool r3679707 = r3679705 <= r3679706;
        double r3679708 = K;
        double r3679709 = 2.0;
        double r3679710 = r3679708 / r3679709;
        double r3679711 = cos(r3679710);
        double r3679712 = U;
        double r3679713 = r3679705 + r3679705;
        double r3679714 = r3679712 / r3679713;
        double r3679715 = r3679713 * r3679711;
        double r3679716 = r3679714 / r3679715;
        double r3679717 = r3679712 / r3679711;
        double r3679718 = r3679716 * r3679717;
        double r3679719 = 1.0;
        double r3679720 = r3679718 + r3679719;
        double r3679721 = sqrt(r3679720);
        double r3679722 = r3679711 * r3679721;
        double r3679723 = -2.0;
        double r3679724 = r3679705 * r3679723;
        double r3679725 = r3679722 * r3679724;
        double r3679726 = 7.984518687285712e-287;
        bool r3679727 = r3679705 <= r3679726;
        double r3679728 = -r3679712;
        double r3679729 = r3679712 / r3679715;
        double r3679730 = r3679729 * r3679729;
        double r3679731 = r3679730 + r3679719;
        double r3679732 = sqrt(r3679731);
        double r3679733 = sqrt(r3679732);
        double r3679734 = r3679724 * r3679711;
        double r3679735 = r3679734 * r3679733;
        double r3679736 = r3679733 * r3679735;
        double r3679737 = r3679727 ? r3679728 : r3679736;
        double r3679738 = r3679707 ? r3679725 : r3679737;
        return r3679738;
}

Error

Bits error versus J

Bits error versus K

Bits error versus U

Derivation

1. Split input into 3 regimes

2. if J < -1.1866131181783549e-211

   1. Initial program 12.2

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

   2. Simplified 12.2

      \[\leadsto \color{blue}{\sqrt{\frac{U}{\left(J + J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(J + J\right) \cdot \cos \left(\frac{K}{2}\right)} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)}\]
   3. Using strategy rm

   4. Applied associate-*r* 12.2

      \[\leadsto \color{blue}{\left(\sqrt{\frac{U}{\left(J + J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(J + J\right) \cdot \cos \left(\frac{K}{2}\right)} + 1} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(-2 \cdot J\right)}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 12.2

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

   7. Applied times-frac 12.3

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

   8. Applied associate-*r* 12.6

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

   9. Simplified 12.6

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

   if -1.1866131181783549e-211 < J < 7.984518687285712e-287

   1. Initial program 40.8

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

   2. Simplified 40.8

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

   3. Taylor expanded around -inf 33.0

      \[\leadsto \color{blue}{-1 \cdot U}\]

   4. Simplified 33.0

      \[\leadsto \color{blue}{-U}\]

   if 7.984518687285712e-287 < J

   1. Initial program 16.3

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

   2. Simplified 16.3

      \[\leadsto \color{blue}{\sqrt{\frac{U}{\left(J + J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(J + J\right) \cdot \cos \left(\frac{K}{2}\right)} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt 16.3

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

   5. Applied sqrt-prod 16.4

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

   6. Applied associate-*l* 16.4

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

4. Final simplification 16.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \le -1.1866131181783549 \cdot 10^{-211}:\\ \;\;\;\;\left(\cos \left(\frac{K}{2}\right) \cdot \sqrt{\frac{\frac{U}{J + J}}{\left(J + J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\cos \left(\frac{K}{2}\right)} + 1}\right) \cdot \left(J \cdot -2\right)\\ \mathbf{elif}\;J \le 7.984518687285712 \cdot 10^{-287}:\\ \;\;\;\;-U\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{\frac{U}{\left(J + J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(J + J\right) \cdot \cos \left(\frac{K}{2}\right)} + 1}} \cdot \left(\left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\sqrt{\frac{U}{\left(J + J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(J + J\right) \cdot \cos \left(\frac{K}{2}\right)} + 1}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019132 
(FPCore (J K U)
  :name "Maksimov and Kolovsky, Equation (3)"
  (* (* (* -2 J) (cos (/ K 2))) (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))))