Maksimov and Kolovsky, Equation (3)

Average Error: 18.1 → 8.1

Time: 7.8s

Precision: 64

Math

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

↓

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

C

double f(double J, double K, double U) {
        double r149127 = -2.0;
        double r149128 = J;
        double r149129 = r149127 * r149128;
        double r149130 = K;
        double r149131 = 2.0;
        double r149132 = r149130 / r149131;
        double r149133 = cos(r149132);
        double r149134 = r149129 * r149133;
        double r149135 = 1.0;
        double r149136 = U;
        double r149137 = r149131 * r149128;
        double r149138 = r149137 * r149133;
        double r149139 = r149136 / r149138;
        double r149140 = pow(r149139, r149131);
        double r149141 = r149135 + r149140;
        double r149142 = sqrt(r149141);
        double r149143 = r149134 * r149142;
        return r149143;
}

↓

double f(double J, double K, double U) {
        double r149144 = -2.0;
        double r149145 = J;
        double r149146 = r149144 * r149145;
        double r149147 = K;
        double r149148 = 2.0;
        double r149149 = r149147 / r149148;
        double r149150 = cos(r149149);
        double r149151 = r149146 * r149150;
        double r149152 = 1.0;
        double r149153 = sqrt(r149152);
        double r149154 = U;
        double r149155 = r149148 * r149145;
        double r149156 = r149155 * r149150;
        double r149157 = r149154 / r149156;
        double r149158 = 2.0;
        double r149159 = r149148 / r149158;
        double r149160 = pow(r149157, r149159);
        double r149161 = hypot(r149153, r149160);
        double r149162 = r149151 * r149161;
        return r149162;
}

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 2020002 +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)))))