Jmat.Real.dawson

Average Error: 29.8 → 0.0

Time: 24.2s

Precision: 64

Math

\[\frac{\left(\left(\left(\left(1 + 0.1049934946999999951788851149103720672429 \cdot \left(x \cdot x\right)\right) + 0.04240606040000000076517494562722276896238 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.007264418199999999985194687468492702464573 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 5.064034000000000243502107366566633572802 \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) + 1.789971000000000009994005623070734145585 \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 + 0.7715471018999999763821051601553335785866 \cdot \left(x \cdot x\right)\right) + 0.2909738639000000182122107617033179849386 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.06945557609999999937322456844412954524159 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.01400054419999999938406531896362139377743 \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) + 8.327945000000000442749725770852364803432 \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 1.789971000000000009994005623070734145585 \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\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1795823920150617.75 \lor \neg \left(x \le 59525946.964277647435665130615234375\right):\\ \;\;\;\;\frac{0.5}{x} + \left(\frac{0.1529819634592932686700805788859724998474}{{x}^{5}} + \frac{0.2514179000665375252054900556686334311962}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{4} \cdot \left(\left(1.789971000000000009994005623070734145585 \cdot 10^{-4} \cdot {x}^{6} + {x}^{4} \cdot 5.064034000000000243502107366566633572802 \cdot 10^{-4}\right) + \left(0.007264418199999999985194687468492702464573 \cdot \left(x \cdot x\right) + 0.04240606040000000076517494562722276896238\right)\right) + \left(1 + 0.1049934946999999951788851149103720672429 \cdot \left(x \cdot x\right)\right)}{\left(1 + {x}^{4} \cdot 0.2909738639000000182122107617033179849386\right) + \left(x \cdot x\right) \cdot \left(0.7715471018999999763821051601553335785866 + {x}^{4} \cdot \left(\left(0.06945557609999999937322456844412954524159 + 0.01400054419999999938406531896362139377743 \cdot {x}^{2}\right) + \left(8.327945000000000442749725770852364803432 \cdot 10^{-4} \cdot {x}^{4} + {x}^{6} \cdot \left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right)\right)\right)\right)} \cdot x\\ \end{array}\]

C

double f(double x) {
        double r142889 = 1.0;
        double r142890 = 0.1049934947;
        double r142891 = x;
        double r142892 = r142891 * r142891;
        double r142893 = r142890 * r142892;
        double r142894 = r142889 + r142893;
        double r142895 = 0.0424060604;
        double r142896 = r142892 * r142892;
        double r142897 = r142895 * r142896;
        double r142898 = r142894 + r142897;
        double r142899 = 0.0072644182;
        double r142900 = r142896 * r142892;
        double r142901 = r142899 * r142900;
        double r142902 = r142898 + r142901;
        double r142903 = 0.0005064034;
        double r142904 = r142900 * r142892;
        double r142905 = r142903 * r142904;
        double r142906 = r142902 + r142905;
        double r142907 = 0.0001789971;
        double r142908 = r142904 * r142892;
        double r142909 = r142907 * r142908;
        double r142910 = r142906 + r142909;
        double r142911 = 0.7715471019;
        double r142912 = r142911 * r142892;
        double r142913 = r142889 + r142912;
        double r142914 = 0.2909738639;
        double r142915 = r142914 * r142896;
        double r142916 = r142913 + r142915;
        double r142917 = 0.0694555761;
        double r142918 = r142917 * r142900;
        double r142919 = r142916 + r142918;
        double r142920 = 0.0140005442;
        double r142921 = r142920 * r142904;
        double r142922 = r142919 + r142921;
        double r142923 = 0.0008327945;
        double r142924 = r142923 * r142908;
        double r142925 = r142922 + r142924;
        double r142926 = 2.0;
        double r142927 = r142926 * r142907;
        double r142928 = r142908 * r142892;
        double r142929 = r142927 * r142928;
        double r142930 = r142925 + r142929;
        double r142931 = r142910 / r142930;
        double r142932 = r142931 * r142891;
        return r142932;
}

↓

double f(double x) {
        double r142933 = x;
        double r142934 = -1795823920150617.8;
        bool r142935 = r142933 <= r142934;
        double r142936 = 59525946.96427765;
        bool r142937 = r142933 <= r142936;
        double r142938 = !r142937;
        bool r142939 = r142935 || r142938;
        double r142940 = 0.5;
        double r142941 = r142940 / r142933;
        double r142942 = 0.15298196345929327;
        double r142943 = 5.0;
        double r142944 = pow(r142933, r142943);
        double r142945 = r142942 / r142944;
        double r142946 = 0.2514179000665375;
        double r142947 = 3.0;
        double r142948 = pow(r142933, r142947);
        double r142949 = r142946 / r142948;
        double r142950 = r142945 + r142949;
        double r142951 = r142941 + r142950;
        double r142952 = 4.0;
        double r142953 = pow(r142933, r142952);
        double r142954 = 0.0001789971;
        double r142955 = 6.0;
        double r142956 = pow(r142933, r142955);
        double r142957 = r142954 * r142956;
        double r142958 = 0.0005064034;
        double r142959 = r142953 * r142958;
        double r142960 = r142957 + r142959;
        double r142961 = 0.0072644182;
        double r142962 = r142933 * r142933;
        double r142963 = r142961 * r142962;
        double r142964 = 0.0424060604;
        double r142965 = r142963 + r142964;
        double r142966 = r142960 + r142965;
        double r142967 = r142953 * r142966;
        double r142968 = 1.0;
        double r142969 = 0.1049934947;
        double r142970 = r142969 * r142962;
        double r142971 = r142968 + r142970;
        double r142972 = r142967 + r142971;
        double r142973 = 0.2909738639;
        double r142974 = r142953 * r142973;
        double r142975 = r142968 + r142974;
        double r142976 = 0.7715471019;
        double r142977 = 0.0694555761;
        double r142978 = 0.0140005442;
        double r142979 = 2.0;
        double r142980 = pow(r142933, r142979);
        double r142981 = r142978 * r142980;
        double r142982 = r142977 + r142981;
        double r142983 = 0.0008327945;
        double r142984 = r142983 * r142953;
        double r142985 = 2.0;
        double r142986 = r142985 * r142954;
        double r142987 = r142956 * r142986;
        double r142988 = r142984 + r142987;
        double r142989 = r142982 + r142988;
        double r142990 = r142953 * r142989;
        double r142991 = r142976 + r142990;
        double r142992 = r142962 * r142991;
        double r142993 = r142975 + r142992;
        double r142994 = r142972 / r142993;
        double r142995 = r142994 * r142933;
        double r142996 = r142939 ? r142951 : r142995;
        return r142996;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -1795823920150617.8 or 59525946.96427765 < x

   1. Initial program 61.3

      \[\frac{\left(\left(\left(\left(1 + 0.1049934946999999951788851149103720672429 \cdot \left(x \cdot x\right)\right) + 0.04240606040000000076517494562722276896238 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.007264418199999999985194687468492702464573 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 5.064034000000000243502107366566633572802 \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) + 1.789971000000000009994005623070734145585 \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 + 0.7715471018999999763821051601553335785866 \cdot \left(x \cdot x\right)\right) + 0.2909738639000000182122107617033179849386 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.06945557609999999937322456844412954524159 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.01400054419999999938406531896362139377743 \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) + 8.327945000000000442749725770852364803432 \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 1.789971000000000009994005623070734145585 \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 61.2

      \[\leadsto \color{blue}{\frac{{x}^{4} \cdot \left(\left(1.789971000000000009994005623070734145585 \cdot 10^{-4} \cdot {x}^{6} + {x}^{4} \cdot 5.064034000000000243502107366566633572802 \cdot 10^{-4}\right) + \left(0.007264418199999999985194687468492702464573 \cdot \left(x \cdot x\right) + 0.04240606040000000076517494562722276896238\right)\right) + \left(1 + 0.1049934946999999951788851149103720672429 \cdot \left(x \cdot x\right)\right)}{\left(1 + {x}^{4} \cdot 0.2909738639000000182122107617033179849386\right) + \left(x \cdot x\right) \cdot \left(0.7715471018999999763821051601553335785866 + {x}^{4} \cdot \left(\left(0.06945557609999999937322456844412954524159 + \left(x \cdot x\right) \cdot 0.01400054419999999938406531896362139377743\right) + \left(8.327945000000000442749725770852364803432 \cdot 10^{-4} \cdot {x}^{4} + {x}^{6} \cdot \left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right)\right)\right)\right)} \cdot x}\]

   3. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{0.2514179000665375252054900556686334311962 \cdot \frac{1}{{x}^{3}} + \left(0.1529819634592932686700805788859724998474 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{x}\right)}\]

   4. Simplified 0.0

      \[\leadsto \color{blue}{\frac{0.5}{x} + \left(\frac{0.1529819634592932686700805788859724998474}{{x}^{5}} + \frac{0.2514179000665375252054900556686334311962}{{x}^{3}}\right)}\]

   if -1795823920150617.8 < x < 59525946.96427765

   1. Initial program 0.0

      \[\frac{\left(\left(\left(\left(1 + 0.1049934946999999951788851149103720672429 \cdot \left(x \cdot x\right)\right) + 0.04240606040000000076517494562722276896238 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.007264418199999999985194687468492702464573 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 5.064034000000000243502107366566633572802 \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) + 1.789971000000000009994005623070734145585 \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 + 0.7715471018999999763821051601553335785866 \cdot \left(x \cdot x\right)\right) + 0.2909738639000000182122107617033179849386 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.06945557609999999937322456844412954524159 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.01400054419999999938406531896362139377743 \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) + 8.327945000000000442749725770852364803432 \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 1.789971000000000009994005623070734145585 \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

      \[\leadsto \color{blue}{\frac{{x}^{4} \cdot \left(\left(1.789971000000000009994005623070734145585 \cdot 10^{-4} \cdot {x}^{6} + {x}^{4} \cdot 5.064034000000000243502107366566633572802 \cdot 10^{-4}\right) + \left(0.007264418199999999985194687468492702464573 \cdot \left(x \cdot x\right) + 0.04240606040000000076517494562722276896238\right)\right) + \left(1 + 0.1049934946999999951788851149103720672429 \cdot \left(x \cdot x\right)\right)}{\left(1 + {x}^{4} \cdot 0.2909738639000000182122107617033179849386\right) + \left(x \cdot x\right) \cdot \left(0.7715471018999999763821051601553335785866 + {x}^{4} \cdot \left(\left(0.06945557609999999937322456844412954524159 + \left(x \cdot x\right) \cdot 0.01400054419999999938406531896362139377743\right) + \left(8.327945000000000442749725770852364803432 \cdot 10^{-4} \cdot {x}^{4} + {x}^{6} \cdot \left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right)\right)\right)\right)} \cdot x}\]

   3. Taylor expanded around 0 0.0

      \[\leadsto \frac{{x}^{4} \cdot \left(\left(1.789971000000000009994005623070734145585 \cdot 10^{-4} \cdot {x}^{6} + {x}^{4} \cdot 5.064034000000000243502107366566633572802 \cdot 10^{-4}\right) + \left(0.007264418199999999985194687468492702464573 \cdot \left(x \cdot x\right) + 0.04240606040000000076517494562722276896238\right)\right) + \left(1 + 0.1049934946999999951788851149103720672429 \cdot \left(x \cdot x\right)\right)}{\left(1 + {x}^{4} \cdot 0.2909738639000000182122107617033179849386\right) + \left(x \cdot x\right) \cdot \left(0.7715471018999999763821051601553335785866 + {x}^{4} \cdot \left(\left(0.06945557609999999937322456844412954524159 + \color{blue}{0.01400054419999999938406531896362139377743 \cdot {x}^{2}}\right) + \left(8.327945000000000442749725770852364803432 \cdot 10^{-4} \cdot {x}^{4} + {x}^{6} \cdot \left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right)\right)\right)\right)} \cdot x\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1795823920150617.75 \lor \neg \left(x \le 59525946.964277647435665130615234375\right):\\ \;\;\;\;\frac{0.5}{x} + \left(\frac{0.1529819634592932686700805788859724998474}{{x}^{5}} + \frac{0.2514179000665375252054900556686334311962}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{4} \cdot \left(\left(1.789971000000000009994005623070734145585 \cdot 10^{-4} \cdot {x}^{6} + {x}^{4} \cdot 5.064034000000000243502107366566633572802 \cdot 10^{-4}\right) + \left(0.007264418199999999985194687468492702464573 \cdot \left(x \cdot x\right) + 0.04240606040000000076517494562722276896238\right)\right) + \left(1 + 0.1049934946999999951788851149103720672429 \cdot \left(x \cdot x\right)\right)}{\left(1 + {x}^{4} \cdot 0.2909738639000000182122107617033179849386\right) + \left(x \cdot x\right) \cdot \left(0.7715471018999999763821051601553335785866 + {x}^{4} \cdot \left(\left(0.06945557609999999937322456844412954524159 + 0.01400054419999999938406531896362139377743 \cdot {x}^{2}\right) + \left(8.327945000000000442749725770852364803432 \cdot 10^{-4} \cdot {x}^{4} + {x}^{6} \cdot \left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right)\right)\right)\right)} \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(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))