Maksimov and Kolovsky, Equation (3)

Average Error: 18.5 → 7.9

Time: 24.5s

Precision: 64

Math

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

↓

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

C

double f(double J, double K, double U) {
        double r129419 = -2.0;
        double r129420 = J;
        double r129421 = r129419 * r129420;
        double r129422 = K;
        double r129423 = 2.0;
        double r129424 = r129422 / r129423;
        double r129425 = cos(r129424);
        double r129426 = r129421 * r129425;
        double r129427 = 1.0;
        double r129428 = U;
        double r129429 = r129423 * r129420;
        double r129430 = r129429 * r129425;
        double r129431 = r129428 / r129430;
        double r129432 = pow(r129431, r129423);
        double r129433 = r129427 + r129432;
        double r129434 = sqrt(r129433);
        double r129435 = r129426 * r129434;
        return r129435;
}

↓

double f(double J, double K, double U) {
        double r129436 = -2.0;
        double r129437 = J;
        double r129438 = r129436 * r129437;
        double r129439 = K;
        double r129440 = 2.0;
        double r129441 = r129439 / r129440;
        double r129442 = cos(r129441);
        double r129443 = r129438 * r129442;
        double r129444 = 1.0;
        double r129445 = sqrt(r129444);
        double r129446 = U;
        double r129447 = r129440 * r129437;
        double r129448 = r129447 * r129442;
        double r129449 = r129446 / r129448;
        double r129450 = 2.0;
        double r129451 = r129440 / r129450;
        double r129452 = pow(r129449, r129451);
        double r129453 = hypot(r129445, r129452);
        double r129454 = r129443 * r129453;
        return r129454;
}

Error

Bits error versus J

Bits error versus K

Bits error versus U

Derivation

1. Initial program 18.5

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

3. Applied sqr-pow 18.5

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

4. Applied add-sqr-sqrt 18.5

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

5. Applied hypot-def 7.9

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

6. Final simplification 7.9

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(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)))))