Maksimov and Kolovsky, Equation (3)

Average Error: 17.4 → 9.9

Time: 29.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(J \cdot 2\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) = -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(J \cdot 2\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \le 9.362287759345471 \cdot 10^{+294}:\\ \;\;\;\;\sqrt{{\left(\frac{U}{\left(J \cdot 2\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array}\]

C

double f(double J, double K, double U) {
        double r5446615 = -2.0;
        double r5446616 = J;
        double r5446617 = r5446615 * r5446616;
        double r5446618 = K;
        double r5446619 = 2.0;
        double r5446620 = r5446618 / r5446619;
        double r5446621 = cos(r5446620);
        double r5446622 = r5446617 * r5446621;
        double r5446623 = 1.0;
        double r5446624 = U;
        double r5446625 = r5446619 * r5446616;
        double r5446626 = r5446625 * r5446621;
        double r5446627 = r5446624 / r5446626;
        double r5446628 = pow(r5446627, r5446619);
        double r5446629 = r5446623 + r5446628;
        double r5446630 = sqrt(r5446629);
        double r5446631 = r5446622 * r5446630;
        return r5446631;
}

↓

double f(double J, double K, double U) {
        double r5446632 = U;
        double r5446633 = J;
        double r5446634 = 2.0;
        double r5446635 = r5446633 * r5446634;
        double r5446636 = K;
        double r5446637 = r5446636 / r5446634;
        double r5446638 = cos(r5446637);
        double r5446639 = r5446635 * r5446638;
        double r5446640 = r5446632 / r5446639;
        double r5446641 = pow(r5446640, r5446634);
        double r5446642 = 1.0;
        double r5446643 = r5446641 + r5446642;
        double r5446644 = sqrt(r5446643);
        double r5446645 = -2.0;
        double r5446646 = r5446645 * r5446633;
        double r5446647 = r5446638 * r5446646;
        double r5446648 = r5446644 * r5446647;
        double r5446649 = -inf.0;
        bool r5446650 = r5446648 <= r5446649;
        double r5446651 = -r5446632;
        double r5446652 = 9.362287759345471e+294;
        bool r5446653 = r5446648 <= r5446652;
        double r5446654 = r5446653 ? r5446648 : r5446651;
        double r5446655 = r5446650 ? r5446651 : r5446654;
        return r5446655;
}

Error

Bits error versus J

Bits error versus K

Bits error versus U

Derivation

1. Split input into 2 regimes

2. if (* (* (* -2 J) (cos (/ K 2))) (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))) < -inf.0 or 9.362287759345471e+294 < (* (* (* -2 J) (cos (/ K 2))) (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2))))

   1. Initial program 57.9

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

   3. Applied associate-*l* 57.9

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

   4. Simplified 57.9

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

   5. Taylor expanded around 0 32.6

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

   6. Simplified 32.6

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

   if -inf.0 < (* (* (* -2 J) (cos (/ K 2))) (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))) < 9.362287759345471e+294

   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 9.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(J \cdot 2\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) = -\infty:\\ \;\;\;\;-U\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(J \cdot 2\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \le 9.362287759345471 \cdot 10^{+294}:\\ \;\;\;\;\sqrt{{\left(\frac{U}{\left(J \cdot 2\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-U\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 
(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)))))