Average Error: 17.2 → 7.2
Time: 46.6s
Precision: 64
\[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}\]
↓
\[\mathsf{hypot}\left(1, \frac{U}{J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 2\right)}\right) \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right)\]
\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}}
↓
\mathsf{hypot}\left(1, \frac{U}{J \cdot \left(\cos \left(\frac{K}{2}\right) \cdot 2\right)}\right) \cdot \left(\left(J \cdot -2\right) \cdot \cos \left(\frac{K}{2}\right)\right)
double f(double J, double K, double U) {
        double r2769485 = -2.0;
        double r2769486 = J;
        double r2769487 = r2769485 * r2769486;
        double r2769488 = K;
        double r2769489 = 2.0;
        double r2769490 = r2769488 / r2769489;
        double r2769491 = cos(r2769490);
        double r2769492 = r2769487 * r2769491;
        double r2769493 = 1.0;
        double r2769494 = U;
        double r2769495 = r2769489 * r2769486;
        double r2769496 = r2769495 * r2769491;
        double r2769497 = r2769494 / r2769496;
        double r2769498 = pow(r2769497, r2769489);
        double r2769499 = r2769493 + r2769498;
        double r2769500 = sqrt(r2769499);
        double r2769501 = r2769492 * r2769500;
        return r2769501;
}
↓
double f(double J, double K, double U) {
        double r2769502 = 1.0;
        double r2769503 = U;
        double r2769504 = J;
        double r2769505 = K;
        double r2769506 = 2.0;
        double r2769507 = r2769505 / r2769506;
        double r2769508 = cos(r2769507);
        double r2769509 = r2769508 * r2769506;
        double r2769510 = r2769504 * r2769509;
        double r2769511 = r2769503 / r2769510;
        double r2769512 = hypot(r2769502, r2769511);
        double r2769513 = -2.0;
        double r2769514 = r2769504 * r2769513;
        double r2769515 = r2769514 * r2769508;
        double r2769516 = r2769512 * r2769515;
        return r2769516;
}

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 17.2

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

  2. Simplified7.2

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

  4. Applied *-un-lft-identity7.2

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

  5. Applied associate-/l*7.3

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

  7. Applied *-un-lft-identity7.3

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

  8. Applied associate-*l*7.3

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

  9. Simplified7.2

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

  10. Final simplification7.2

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

Reproduce

herbie shell --seed 2019144 +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)))))