Maksimov and Kolovsky, Equation (3)

Average Error: 17.7 → 13.0

Time: 29.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(J \cdot 2\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) = -\infty:\\ \;\;\;\;\frac{\sqrt{0.25} \cdot U}{J \cdot \cos \left(K \cdot 0.5\right)} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(J \cdot 2\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \le 3.684639207742759661770964572575773256411 \cdot 10^{304}:\\ \;\;\;\;\sqrt{{\left(\frac{U}{\left(J \cdot 2\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{0.25} \cdot U}{J \cdot \cos \left(K \cdot 0.5\right)} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)\\ \end{array}\]

C

double f(double J, double K, double U) {
        double r4797373 = -2.0;
        double r4797374 = J;
        double r4797375 = r4797373 * r4797374;
        double r4797376 = K;
        double r4797377 = 2.0;
        double r4797378 = r4797376 / r4797377;
        double r4797379 = cos(r4797378);
        double r4797380 = r4797375 * r4797379;
        double r4797381 = 1.0;
        double r4797382 = U;
        double r4797383 = r4797377 * r4797374;
        double r4797384 = r4797383 * r4797379;
        double r4797385 = r4797382 / r4797384;
        double r4797386 = pow(r4797385, r4797377);
        double r4797387 = r4797381 + r4797386;
        double r4797388 = sqrt(r4797387);
        double r4797389 = r4797380 * r4797388;
        return r4797389;
}

↓

double f(double J, double K, double U) {
        double r4797390 = U;
        double r4797391 = J;
        double r4797392 = 2.0;
        double r4797393 = r4797391 * r4797392;
        double r4797394 = K;
        double r4797395 = r4797394 / r4797392;
        double r4797396 = cos(r4797395);
        double r4797397 = r4797393 * r4797396;
        double r4797398 = r4797390 / r4797397;
        double r4797399 = pow(r4797398, r4797392);
        double r4797400 = 1.0;
        double r4797401 = r4797399 + r4797400;
        double r4797402 = sqrt(r4797401);
        double r4797403 = -2.0;
        double r4797404 = r4797403 * r4797391;
        double r4797405 = r4797396 * r4797404;
        double r4797406 = r4797402 * r4797405;
        double r4797407 = -inf.0;
        bool r4797408 = r4797406 <= r4797407;
        double r4797409 = 0.25;
        double r4797410 = sqrt(r4797409);
        double r4797411 = r4797410 * r4797390;
        double r4797412 = 0.5;
        double r4797413 = r4797394 * r4797412;
        double r4797414 = cos(r4797413);
        double r4797415 = r4797391 * r4797414;
        double r4797416 = r4797411 / r4797415;
        double r4797417 = r4797416 * r4797405;
        double r4797418 = 3.6846392077427597e+304;
        bool r4797419 = r4797406 <= r4797418;
        double r4797420 = r4797419 ? r4797406 : r4797417;
        double r4797421 = r4797408 ? r4797417 : r4797420;
        return r4797421;
}

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 3.6846392077427597e+304 < (* (* (* -2.0 J) (cos (/ K 2.0))) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) (cos (/ K 2.0)))) 2.0))))

   1. Initial program 63.1

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

   2. Taylor expanded around inf 46.4

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

   if -inf.0 < (* (* (* -2.0 J) (cos (/ K 2.0))) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) (cos (/ K 2.0)))) 2.0)))) < 3.6846392077427597e+304

   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 13.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{{\left(\frac{U}{\left(J \cdot 2\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) = -\infty:\\ \;\;\;\;\frac{\sqrt{0.25} \cdot U}{J \cdot \cos \left(K \cdot 0.5\right)} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)\\ \mathbf{elif}\;\sqrt{{\left(\frac{U}{\left(J \cdot 2\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right) \le 3.684639207742759661770964572575773256411 \cdot 10^{304}:\\ \;\;\;\;\sqrt{{\left(\frac{U}{\left(J \cdot 2\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{0.25} \cdot U}{J \cdot \cos \left(K \cdot 0.5\right)} \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \left(-2 \cdot J\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(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)))))