Average Error: 16.9 → 8.1
Time: 26.3s
Precision: 64
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
↓
\[\sqrt{\sqrt{\mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)}} \cdot \left(\sqrt{\sqrt{\mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)}} \cdot \left(\sqrt{\mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right)\right)\right)\]
\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}}
↓
\sqrt{\sqrt{\mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)}} \cdot \left(\sqrt{\sqrt{\mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)}} \cdot \left(\sqrt{\mathsf{hypot}\left(1, \frac{U}{J \cdot \left(2 \cdot \cos \left(\frac{K}{2}\right)\right)}\right)} \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right)\right)\right)
double f(double J, double K, double U) {
        double r5007203 = -2.0;
        double r5007204 = J;
        double r5007205 = r5007203 * r5007204;
        double r5007206 = K;
        double r5007207 = 2.0;
        double r5007208 = r5007206 / r5007207;
        double r5007209 = cos(r5007208);
        double r5007210 = r5007205 * r5007209;
        double r5007211 = 1.0;
        double r5007212 = U;
        double r5007213 = r5007207 * r5007204;
        double r5007214 = r5007213 * r5007209;
        double r5007215 = r5007212 / r5007214;
        double r5007216 = pow(r5007215, r5007207);
        double r5007217 = r5007211 + r5007216;
        double r5007218 = sqrt(r5007217);
        double r5007219 = r5007210 * r5007218;
        return r5007219;
}
↓
double f(double J, double K, double U) {
        double r5007220 = 1.0;
        double r5007221 = U;
        double r5007222 = J;
        double r5007223 = 2.0;
        double r5007224 = K;
        double r5007225 = r5007224 / r5007223;
        double r5007226 = cos(r5007225);
        double r5007227 = r5007223 * r5007226;
        double r5007228 = r5007222 * r5007227;
        double r5007229 = r5007221 / r5007228;
        double r5007230 = hypot(r5007220, r5007229);
        double r5007231 = sqrt(r5007230);
        double r5007232 = sqrt(r5007231);
        double r5007233 = -2.0;
        double r5007234 = r5007222 * r5007233;
        double r5007235 = r5007234 * r5007226;
        double r5007236 = r5007231 * r5007235;
        double r5007237 = r5007232 * r5007236;
        double r5007238 = r5007232 * r5007237;
        return r5007238;
}

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 16.9

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

  2. Simplified7.9

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

  4. Applied add-sqr-sqrt8.0

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

  5. Applied associate-*l*8.0

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

  7. Applied add-sqr-sqrt8.0

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

  8. Applied sqrt-prod8.1

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

  9. Applied associate-*l*8.1

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

  10. Final simplification8.1

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (J K U)
  :name "Maksimov and Kolovsky, Equation (3)"
  (* (* (* -2 J) (cos (/ K 2))) (sqrt (+ 1 (pow (/ U (* (* 2 J) (cos (/ K 2)))) 2)))))