Maksimov and Kolovsky, Equation (3)

Average Error: 18.4 → 8.2

Time: 8.7s

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 r178810 = -2.0;
        double r178811 = J;
        double r178812 = r178810 * r178811;
        double r178813 = K;
        double r178814 = 2.0;
        double r178815 = r178813 / r178814;
        double r178816 = cos(r178815);
        double r178817 = r178812 * r178816;
        double r178818 = 1.0;
        double r178819 = U;
        double r178820 = r178814 * r178811;
        double r178821 = r178820 * r178816;
        double r178822 = r178819 / r178821;
        double r178823 = pow(r178822, r178814);
        double r178824 = r178818 + r178823;
        double r178825 = sqrt(r178824);
        double r178826 = r178817 * r178825;
        return r178826;
}

↓

double f(double J, double K, double U) {
        double r178827 = -2.0;
        double r178828 = J;
        double r178829 = r178827 * r178828;
        double r178830 = K;
        double r178831 = 2.0;
        double r178832 = r178830 / r178831;
        double r178833 = cos(r178832);
        double r178834 = r178829 * r178833;
        double r178835 = 1.0;
        double r178836 = sqrt(r178835);
        double r178837 = U;
        double r178838 = r178831 * r178828;
        double r178839 = r178838 * r178833;
        double r178840 = r178837 / r178839;
        double r178841 = 2.0;
        double r178842 = r178831 / r178841;
        double r178843 = pow(r178840, r178842);
        double r178844 = hypot(r178836, r178843);
        double r178845 = r178834 * r178844;
        return r178845;
}

Error

Bits error versus J

Bits error versus K

Bits error versus U

Derivation

1. Initial program 18.4

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

3. Applied sqr-pow 18.4

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

4. Applied add-sqr-sqrt 18.4

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

5. Applied hypot-def 8.2

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

6. Final simplification 8.2

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

Reproduce

herbie shell --seed 2020057 +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)))))