Maksimov and Kolovsky, Equation (3)

Average Error: 18.0 → 8.3

Time: 34.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;U \le 2.350079613172404302600140982159304734472 \cdot 10^{257}:\\ \;\;\;\;J \cdot \left(\mathsf{hypot}\left({\left(\frac{\frac{\frac{U}{J}}{\cos \left(\frac{K}{2}\right)}}{2}\right)}^{\left(\frac{2}{2}\right)}, \sqrt{1}\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(U \cdot \sqrt{0.25}\right) \cdot -2\\ \end{array}\]

C

double f(double J, double K, double U) {
        double r5865301 = -2.0;
        double r5865302 = J;
        double r5865303 = r5865301 * r5865302;
        double r5865304 = K;
        double r5865305 = 2.0;
        double r5865306 = r5865304 / r5865305;
        double r5865307 = cos(r5865306);
        double r5865308 = r5865303 * r5865307;
        double r5865309 = 1.0;
        double r5865310 = U;
        double r5865311 = r5865305 * r5865302;
        double r5865312 = r5865311 * r5865307;
        double r5865313 = r5865310 / r5865312;
        double r5865314 = pow(r5865313, r5865305);
        double r5865315 = r5865309 + r5865314;
        double r5865316 = sqrt(r5865315);
        double r5865317 = r5865308 * r5865316;
        return r5865317;
}

↓

double f(double J, double K, double U) {
        double r5865318 = U;
        double r5865319 = 2.3500796131724043e+257;
        bool r5865320 = r5865318 <= r5865319;
        double r5865321 = J;
        double r5865322 = r5865318 / r5865321;
        double r5865323 = K;
        double r5865324 = 2.0;
        double r5865325 = r5865323 / r5865324;
        double r5865326 = cos(r5865325);
        double r5865327 = r5865322 / r5865326;
        double r5865328 = r5865327 / r5865324;
        double r5865329 = 2.0;
        double r5865330 = r5865324 / r5865329;
        double r5865331 = pow(r5865328, r5865330);
        double r5865332 = 1.0;
        double r5865333 = sqrt(r5865332);
        double r5865334 = hypot(r5865331, r5865333);
        double r5865335 = -2.0;
        double r5865336 = r5865326 * r5865335;
        double r5865337 = r5865334 * r5865336;
        double r5865338 = r5865321 * r5865337;
        double r5865339 = 0.25;
        double r5865340 = sqrt(r5865339);
        double r5865341 = r5865318 * r5865340;
        double r5865342 = r5865341 * r5865335;
        double r5865343 = r5865320 ? r5865338 : r5865342;
        return r5865343;
}

Error

Bits error versus J

Bits error versus K

Bits error versus U

Derivation

1. Split input into 2 regimes

2. if U < 2.3500796131724043e+257

   1. Initial program 16.7

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

   2. Simplified 16.7

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

   4. Applied add-sqr-sqrt 16.7

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

   5. Applied sqr-pow 16.7

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

   6. Applied hypot-def 7.1

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

   8. Applied add-sqr-sqrt 7.2

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

   9. Applied associate-*l* 7.2

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

   11. Applied *-un-lft-identity 7.2

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

   12. Applied associate-*l* 7.2

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

   13. Simplified 7.1

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

   if 2.3500796131724043e+257 < U

   1. Initial program 46.0

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

   2. Simplified 46.0

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

   3. Taylor expanded around inf 33.5

      \[\leadsto \color{blue}{-2 \cdot \left(\sqrt{0.25} \cdot U\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 8.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \le 2.350079613172404302600140982159304734472 \cdot 10^{257}:\\ \;\;\;\;J \cdot \left(\mathsf{hypot}\left({\left(\frac{\frac{\frac{U}{J}}{\cos \left(\frac{K}{2}\right)}}{2}\right)}^{\left(\frac{2}{2}\right)}, \sqrt{1}\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot -2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(U \cdot \sqrt{0.25}\right) \cdot -2\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +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)))))