Average Error: 17.9 → 8.5
Time: 1.0m
Precision: 64
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
↓
\[\left(\sqrt{\mathsf{hypot}\left({\left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2}\right)}^{\left(\frac{2}{2}\right)}, \sqrt{1}\right)} \cdot \left(\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot -2\right)\right) \cdot \sqrt{\mathsf{hypot}\left({\left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2}\right)}^{\left(\frac{2}{2}\right)}, \sqrt{1}\right)}\]
\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(\sqrt{\mathsf{hypot}\left({\left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2}\right)}^{\left(\frac{2}{2}\right)}, \sqrt{1}\right)} \cdot \left(\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot -2\right)\right) \cdot \sqrt{\mathsf{hypot}\left({\left(\frac{U}{\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot 2}\right)}^{\left(\frac{2}{2}\right)}, \sqrt{1}\right)}
double f(double J, double K, double U) {
        double r7725132 = -2.0;
        double r7725133 = J;
        double r7725134 = r7725132 * r7725133;
        double r7725135 = K;
        double r7725136 = 2.0;
        double r7725137 = r7725135 / r7725136;
        double r7725138 = cos(r7725137);
        double r7725139 = r7725134 * r7725138;
        double r7725140 = 1.0;
        double r7725141 = U;
        double r7725142 = r7725136 * r7725133;
        double r7725143 = r7725142 * r7725138;
        double r7725144 = r7725141 / r7725143;
        double r7725145 = pow(r7725144, r7725136);
        double r7725146 = r7725140 + r7725145;
        double r7725147 = sqrt(r7725146);
        double r7725148 = r7725139 * r7725147;
        return r7725148;
}
↓
double f(double J, double K, double U) {
        double r7725149 = U;
        double r7725150 = K;
        double r7725151 = 2.0;
        double r7725152 = r7725150 / r7725151;
        double r7725153 = cos(r7725152);
        double r7725154 = J;
        double r7725155 = r7725153 * r7725154;
        double r7725156 = r7725155 * r7725151;
        double r7725157 = r7725149 / r7725156;
        double r7725158 = 2.0;
        double r7725159 = r7725151 / r7725158;
        double r7725160 = pow(r7725157, r7725159);
        double r7725161 = 1.0;
        double r7725162 = sqrt(r7725161);
        double r7725163 = hypot(r7725160, r7725162);
        double r7725164 = sqrt(r7725163);
        double r7725165 = -2.0;
        double r7725166 = r7725155 * r7725165;
        double r7725167 = r7725164 * r7725166;
        double r7725168 = r7725167 * r7725164;
        return r7725168;
}

Error

Bits error versus J

Bits error versus K

Bits error versus U

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 17.9

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

  2. Simplified17.9

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

  4. Applied add-sqr-sqrt17.9

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

  5. Applied sqr-pow17.9

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

  6. Applied hypot-def8.4

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

  8. Applied add-sqr-sqrt8.5

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

  9. Applied associate-*l*8.5

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

  10. Final simplification8.5

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(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)))))