Maksimov and Kolovsky, Equation (3)

Average Error: 18.1 → 8.1

Time: 27.6s

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 r160333 = -2.0;
        double r160334 = J;
        double r160335 = r160333 * r160334;
        double r160336 = K;
        double r160337 = 2.0;
        double r160338 = r160336 / r160337;
        double r160339 = cos(r160338);
        double r160340 = r160335 * r160339;
        double r160341 = 1.0;
        double r160342 = U;
        double r160343 = r160337 * r160334;
        double r160344 = r160343 * r160339;
        double r160345 = r160342 / r160344;
        double r160346 = pow(r160345, r160337);
        double r160347 = r160341 + r160346;
        double r160348 = sqrt(r160347);
        double r160349 = r160340 * r160348;
        return r160349;
}

↓

double f(double J, double K, double U) {
        double r160350 = -2.0;
        double r160351 = J;
        double r160352 = r160350 * r160351;
        double r160353 = K;
        double r160354 = 2.0;
        double r160355 = r160353 / r160354;
        double r160356 = cos(r160355);
        double r160357 = r160352 * r160356;
        double r160358 = 1.0;
        double r160359 = sqrt(r160358);
        double r160360 = U;
        double r160361 = r160354 * r160351;
        double r160362 = r160361 * r160356;
        double r160363 = r160360 / r160362;
        double r160364 = 2.0;
        double r160365 = r160354 / r160364;
        double r160366 = pow(r160363, r160365);
        double r160367 = hypot(r160359, r160366);
        double r160368 = r160357 * r160367;
        return r160368;
}

Error

Bits error versus J

Bits error versus K

Bits error versus U

Derivation

1. Initial program 18.1

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

3. Applied sqr-pow 18.1

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

4. Applied add-sqr-sqrt 18.1

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

5. Applied hypot-def 8.1

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

6. Final simplification 8.1

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

Reproduce

herbie shell --seed 2019305 +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)))))