Maksimov and Kolovsky, Equation (3)

Average Error: 18.1 → 16.7

Time: 21.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;J \le -5.928674869168628889484109583876842263371 \cdot 10^{-224}:\\ \;\;\;\;\left(\left|\sqrt[3]{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right| \cdot \left(-2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\right)\right) \cdot \left|\sqrt[3]{\sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}}\right|\\ \mathbf{elif}\;J \le 1.006919644133390623029514297475878068522 \cdot 10^{-222}:\\ \;\;\;\;-2 \cdot \left(\sqrt{0.25} \cdot U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \left(\sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \cdot \cos \left(\frac{K}{2}\right)\right)\\ \end{array}\]

C

double f(double J, double K, double U) {
        double r130641 = -2.0;
        double r130642 = J;
        double r130643 = r130641 * r130642;
        double r130644 = K;
        double r130645 = 2.0;
        double r130646 = r130644 / r130645;
        double r130647 = cos(r130646);
        double r130648 = r130643 * r130647;
        double r130649 = 1.0;
        double r130650 = U;
        double r130651 = r130645 * r130642;
        double r130652 = r130651 * r130647;
        double r130653 = r130650 / r130652;
        double r130654 = pow(r130653, r130645);
        double r130655 = r130649 + r130654;
        double r130656 = sqrt(r130655);
        double r130657 = r130648 * r130656;
        return r130657;
}

↓

double f(double J, double K, double U) {
        double r130658 = J;
        double r130659 = -5.928674869168629e-224;
        bool r130660 = r130658 <= r130659;
        double r130661 = 1.0;
        double r130662 = U;
        double r130663 = 2.0;
        double r130664 = r130663 * r130658;
        double r130665 = K;
        double r130666 = r130665 / r130663;
        double r130667 = cos(r130666);
        double r130668 = r130664 * r130667;
        double r130669 = r130662 / r130668;
        double r130670 = pow(r130669, r130663);
        double r130671 = r130661 + r130670;
        double r130672 = cbrt(r130671);
        double r130673 = fabs(r130672);
        double r130674 = -2.0;
        double r130675 = r130658 * r130667;
        double r130676 = r130674 * r130675;
        double r130677 = r130673 * r130676;
        double r130678 = sqrt(r130671);
        double r130679 = cbrt(r130678);
        double r130680 = fabs(r130679);
        double r130681 = r130677 * r130680;
        double r130682 = 1.0069196441333906e-222;
        bool r130683 = r130658 <= r130682;
        double r130684 = 0.25;
        double r130685 = sqrt(r130684);
        double r130686 = r130685 * r130662;
        double r130687 = r130674 * r130686;
        double r130688 = r130674 * r130658;
        double r130689 = r130678 * r130667;
        double r130690 = r130688 * r130689;
        double r130691 = r130683 ? r130687 : r130690;
        double r130692 = r130660 ? r130681 : r130691;
        return r130692;
}

Error

Bits error versus J

Bits error versus K

Bits error versus U

Derivation

1. Split input into 3 regimes

2. if J < -5.928674869168629e-224

   1. Initial program 13.9

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

   3. Applied add-cube-cbrt 14.1

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

   4. Applied sqrt-prod 14.1

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

   5. Applied associate-*r* 14.1

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

   6. Simplified 14.1

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

   8. Applied add-sqr-sqrt 14.1

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

   9. Applied cbrt-prod 14.1

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

   10. Applied rem-sqrt-square 14.1

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

   if -5.928674869168629e-224 < J < 1.0069196441333906e-222

   1. Initial program 43.6

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

   3. Applied associate-*l* 43.7

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

   4. Simplified 43.7

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

   5. Taylor expanded around 0 33.1

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{0.25} \cdot U\right)}\]

   if 1.0069196441333906e-222 < J

   1. Initial program 13.9

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

   3. Applied associate-*l* 13.9

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

   4. Simplified 13.9

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

4. Final simplification 16.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \le -5.928674869168628889484109583876842263371 \cdot 10^{-224}:\\ \;\;\;\;\left(\left|\sqrt[3]{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\right| \cdot \left(-2 \cdot \left(J \cdot \cos \left(\frac{K}{2}\right)\right)\right)\right) \cdot \left|\sqrt[3]{\sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}}\right|\\ \mathbf{elif}\;J \le 1.006919644133390623029514297475878068522 \cdot 10^{-222}:\\ \;\;\;\;-2 \cdot \left(\sqrt{0.25} \cdot U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot J\right) \cdot \left(\sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \cdot \cos \left(\frac{K}{2}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 
(FPCore (J K U)
  :name "Maksimov and Kolovsky, Equation (3)"
  :precision binary64
  (* (* (* -2 J) (cos (/ K 2))) (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))))