Jmat.Real.dawson

Average Error: 29.1 → 0.2

Time: 6.8s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.042406060400000001 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.00726441819999999999 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 5.0640340000000002 \cdot 10^{-4} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 1.789971 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.77154710189999998 \cdot \left(x \cdot x\right)\right) + 0.29097386390000002 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.069455576099999999 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.014000544199999999 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 8.32794500000000044 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 1.789971 \cdot 10^{-4}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.147765270182274 \lor \neg \left(x \le 1.1589205875478283\right):\\ \;\;\;\;\frac{0.1529819634592933}{{x}^{5}} + \left(\frac{0.5}{x} + \frac{0.25141790006653753}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, x, 0.265709700396150994 \cdot {x}^{5}\right) - 0.66655360720000001 \cdot {x}^{3}\\ \end{array}\]

C

double f(double x) {
        double r206796 = 1.0;
        double r206797 = 0.1049934947;
        double r206798 = x;
        double r206799 = r206798 * r206798;
        double r206800 = r206797 * r206799;
        double r206801 = r206796 + r206800;
        double r206802 = 0.0424060604;
        double r206803 = r206799 * r206799;
        double r206804 = r206802 * r206803;
        double r206805 = r206801 + r206804;
        double r206806 = 0.0072644182;
        double r206807 = r206803 * r206799;
        double r206808 = r206806 * r206807;
        double r206809 = r206805 + r206808;
        double r206810 = 0.0005064034;
        double r206811 = r206807 * r206799;
        double r206812 = r206810 * r206811;
        double r206813 = r206809 + r206812;
        double r206814 = 0.0001789971;
        double r206815 = r206811 * r206799;
        double r206816 = r206814 * r206815;
        double r206817 = r206813 + r206816;
        double r206818 = 0.7715471019;
        double r206819 = r206818 * r206799;
        double r206820 = r206796 + r206819;
        double r206821 = 0.2909738639;
        double r206822 = r206821 * r206803;
        double r206823 = r206820 + r206822;
        double r206824 = 0.0694555761;
        double r206825 = r206824 * r206807;
        double r206826 = r206823 + r206825;
        double r206827 = 0.0140005442;
        double r206828 = r206827 * r206811;
        double r206829 = r206826 + r206828;
        double r206830 = 0.0008327945;
        double r206831 = r206830 * r206815;
        double r206832 = r206829 + r206831;
        double r206833 = 2.0;
        double r206834 = r206833 * r206814;
        double r206835 = r206815 * r206799;
        double r206836 = r206834 * r206835;
        double r206837 = r206832 + r206836;
        double r206838 = r206817 / r206837;
        double r206839 = r206838 * r206798;
        return r206839;
}

↓

double f(double x) {
        double r206840 = x;
        double r206841 = -1.147765270182274;
        bool r206842 = r206840 <= r206841;
        double r206843 = 1.1589205875478283;
        bool r206844 = r206840 <= r206843;
        double r206845 = !r206844;
        bool r206846 = r206842 || r206845;
        double r206847 = 0.15298196345929327;
        double r206848 = 5.0;
        double r206849 = pow(r206840, r206848);
        double r206850 = r206847 / r206849;
        double r206851 = 0.5;
        double r206852 = r206851 / r206840;
        double r206853 = 0.2514179000665375;
        double r206854 = 3.0;
        double r206855 = pow(r206840, r206854);
        double r206856 = r206853 / r206855;
        double r206857 = r206852 + r206856;
        double r206858 = r206850 + r206857;
        double r206859 = 1.0;
        double r206860 = 0.265709700396151;
        double r206861 = r206860 * r206849;
        double r206862 = fma(r206859, r206840, r206861);
        double r206863 = 0.6665536072;
        double r206864 = r206863 * r206855;
        double r206865 = r206862 - r206864;
        double r206866 = r206846 ? r206858 : r206865;
        return r206866;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -1.147765270182274 or 1.1589205875478283 < x

   1. Initial program 58.5

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.042406060400000001 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.00726441819999999999 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 5.0640340000000002 \cdot 10^{-4} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 1.789971 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.77154710189999998 \cdot \left(x \cdot x\right)\right) + 0.29097386390000002 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.069455576099999999 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.014000544199999999 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 8.32794500000000044 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 1.789971 \cdot 10^{-4}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\]

   2. Simplified 58.5

      \[\leadsto \color{blue}{\frac{\left({\left(x \cdot x\right)}^{4} \cdot \left(5.0640340000000002 \cdot 10^{-4} + \left(x \cdot x\right) \cdot 1.789971 \cdot 10^{-4}\right) + \mathsf{fma}\left(0.00726441819999999999, {x}^{6}, \mathsf{fma}\left({x}^{4}, 0.042406060400000001, \mathsf{fma}\left(x \cdot x, 0.1049934947, 1\right)\right)\right)\right) \cdot x}{\mathsf{fma}\left(1.789971 \cdot 10^{-4}, 2 \cdot {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left(0.29097386390000002, {x}^{4}, \mathsf{fma}\left(0.77154710189999998 \cdot x, x, 1\right)\right) + \mathsf{fma}\left({x}^{6}, 0.069455576099999999, {\left(x \cdot x\right)}^{4} \cdot \left(\left(x \cdot x\right) \cdot 8.32794500000000044 \cdot 10^{-4} + 0.014000544199999999\right)\right)\right)}}\]

   3. Taylor expanded around inf 0.4

      \[\leadsto \color{blue}{0.25141790006653753 \cdot \frac{1}{{x}^{3}} + \left(0.1529819634592933 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{x}\right)}\]

   4. Simplified 0.4

      \[\leadsto \color{blue}{\frac{0.1529819634592933}{{x}^{5}} + \left(\frac{0.5}{x} + \frac{0.25141790006653753}{{x}^{3}}\right)}\]

   if -1.147765270182274 < x < 1.1589205875478283

   1. Initial program 0.0

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.042406060400000001 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.00726441819999999999 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 5.0640340000000002 \cdot 10^{-4} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 1.789971 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.77154710189999998 \cdot \left(x \cdot x\right)\right) + 0.29097386390000002 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.069455576099999999 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.014000544199999999 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 8.32794500000000044 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 1.789971 \cdot 10^{-4}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\]

   2. Simplified 0.0

      \[\leadsto \color{blue}{\frac{\left({\left(x \cdot x\right)}^{4} \cdot \left(5.0640340000000002 \cdot 10^{-4} + \left(x \cdot x\right) \cdot 1.789971 \cdot 10^{-4}\right) + \mathsf{fma}\left(0.00726441819999999999, {x}^{6}, \mathsf{fma}\left({x}^{4}, 0.042406060400000001, \mathsf{fma}\left(x \cdot x, 0.1049934947, 1\right)\right)\right)\right) \cdot x}{\mathsf{fma}\left(1.789971 \cdot 10^{-4}, 2 \cdot {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left(0.29097386390000002, {x}^{4}, \mathsf{fma}\left(0.77154710189999998 \cdot x, x, 1\right)\right) + \mathsf{fma}\left({x}^{6}, 0.069455576099999999, {\left(x \cdot x\right)}^{4} \cdot \left(\left(x \cdot x\right) \cdot 8.32794500000000044 \cdot 10^{-4} + 0.014000544199999999\right)\right)\right)}}\]

   3. Taylor expanded around 0 0.1

      \[\leadsto \color{blue}{\left(1 \cdot x + 0.265709700396150994 \cdot {x}^{5}\right) - 0.66655360720000001 \cdot {x}^{3}}\]

   4. Simplified 0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, x, 0.265709700396150994 \cdot {x}^{5}\right) - 0.66655360720000001 \cdot {x}^{3}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.147765270182274 \lor \neg \left(x \le 1.1589205875478283\right):\\ \;\;\;\;\frac{0.1529819634592933}{{x}^{5}} + \left(\frac{0.5}{x} + \frac{0.25141790006653753}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, x, 0.265709700396150994 \cdot {x}^{5}\right) - 0.66655360720000001 \cdot {x}^{3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.dawson"
  :precision binary64
  (* (/ (+ (+ (+ (+ (+ 1 (* 0.1049934947 (* x x))) (* 0.0424060604 (* (* x x) (* x x)))) (* 0.0072644182 (* (* (* x x) (* x x)) (* x x)))) (* 0.0005064034 (* (* (* (* x x) (* x x)) (* x x)) (* x x)))) (* 0.0001789971 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)))) (+ (+ (+ (+ (+ (+ 1 (* 0.7715471019 (* x x))) (* 0.2909738639 (* (* x x) (* x x)))) (* 0.0694555761 (* (* (* x x) (* x x)) (* x x)))) (* 0.0140005442 (* (* (* (* x x) (* x x)) (* x x)) (* x x)))) (* 0.0008327945 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)))) (* (* 2 0.0001789971) (* (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)) (* x x))))) x))