Jmat.Real.dawson

Average Error: 29.2 → 0.0

Time: 30.6s

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 -13083725713432352 \lor \neg \left(x \le 662.4918751995975298996199853718280792236\right):\\ \;\;\;\;\left(\frac{0.5}{x} + \frac{0.1529819634592932686700805788859724998474}{{x}^{5}}\right) + \frac{0.2514179000665375252054900556686334311962}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\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 \sqrt[3]{\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 \left(\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)}{\sqrt[3]{\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\right)\\ \end{array}\]

C

double f(double x) {
        double r145055 = 1.0;
        double r145056 = 0.1049934947;
        double r145057 = x;
        double r145058 = r145057 * r145057;
        double r145059 = r145056 * r145058;
        double r145060 = r145055 + r145059;
        double r145061 = 0.0424060604;
        double r145062 = r145058 * r145058;
        double r145063 = r145061 * r145062;
        double r145064 = r145060 + r145063;
        double r145065 = 0.0072644182;
        double r145066 = r145062 * r145058;
        double r145067 = r145065 * r145066;
        double r145068 = r145064 + r145067;
        double r145069 = 0.0005064034;
        double r145070 = r145066 * r145058;
        double r145071 = r145069 * r145070;
        double r145072 = r145068 + r145071;
        double r145073 = 0.0001789971;
        double r145074 = r145070 * r145058;
        double r145075 = r145073 * r145074;
        double r145076 = r145072 + r145075;
        double r145077 = 0.7715471019;
        double r145078 = r145077 * r145058;
        double r145079 = r145055 + r145078;
        double r145080 = 0.2909738639;
        double r145081 = r145080 * r145062;
        double r145082 = r145079 + r145081;
        double r145083 = 0.0694555761;
        double r145084 = r145083 * r145066;
        double r145085 = r145082 + r145084;
        double r145086 = 0.0140005442;
        double r145087 = r145086 * r145070;
        double r145088 = r145085 + r145087;
        double r145089 = 0.0008327945;
        double r145090 = r145089 * r145074;
        double r145091 = r145088 + r145090;
        double r145092 = 2.0;
        double r145093 = r145092 * r145073;
        double r145094 = r145074 * r145058;
        double r145095 = r145093 * r145094;
        double r145096 = r145091 + r145095;
        double r145097 = r145076 / r145096;
        double r145098 = r145097 * r145057;
        return r145098;
}

↓

double f(double x) {
        double r145099 = x;
        double r145100 = -13083725713432352.0;
        bool r145101 = r145099 <= r145100;
        double r145102 = 662.4918751995975;
        bool r145103 = r145099 <= r145102;
        double r145104 = !r145103;
        bool r145105 = r145101 || r145104;
        double r145106 = 0.5;
        double r145107 = r145106 / r145099;
        double r145108 = 0.15298196345929327;
        double r145109 = 5.0;
        double r145110 = pow(r145099, r145109);
        double r145111 = r145108 / r145110;
        double r145112 = r145107 + r145111;
        double r145113 = 0.2514179000665375;
        double r145114 = 3.0;
        double r145115 = pow(r145099, r145114);
        double r145116 = r145113 / r145115;
        double r145117 = r145112 + r145116;
        double r145118 = 1.0;
        double r145119 = 1.0;
        double r145120 = 4.0;
        double r145121 = pow(r145099, r145120);
        double r145122 = 0.2909738639;
        double r145123 = r145121 * r145122;
        double r145124 = r145119 + r145123;
        double r145125 = r145099 * r145099;
        double r145126 = 0.7715471019;
        double r145127 = 0.0694555761;
        double r145128 = 0.0140005442;
        double r145129 = r145125 * r145128;
        double r145130 = r145127 + r145129;
        double r145131 = 0.0008327945;
        double r145132 = r145131 * r145121;
        double r145133 = 6.0;
        double r145134 = pow(r145099, r145133);
        double r145135 = 2.0;
        double r145136 = 0.0001789971;
        double r145137 = r145135 * r145136;
        double r145138 = r145134 * r145137;
        double r145139 = r145132 + r145138;
        double r145140 = r145130 + r145139;
        double r145141 = r145121 * r145140;
        double r145142 = r145126 + r145141;
        double r145143 = r145125 * r145142;
        double r145144 = r145124 + r145143;
        double r145145 = cbrt(r145144);
        double r145146 = r145145 * r145145;
        double r145147 = r145118 / r145146;
        double r145148 = r145136 * r145134;
        double r145149 = 0.0005064034;
        double r145150 = r145121 * r145149;
        double r145151 = r145148 + r145150;
        double r145152 = 0.0072644182;
        double r145153 = r145152 * r145125;
        double r145154 = 0.0424060604;
        double r145155 = r145153 + r145154;
        double r145156 = r145151 + r145155;
        double r145157 = r145121 * r145156;
        double r145158 = 0.1049934947;
        double r145159 = r145158 * r145125;
        double r145160 = r145119 + r145159;
        double r145161 = r145157 + r145160;
        double r145162 = r145161 / r145145;
        double r145163 = r145162 * r145099;
        double r145164 = r145147 * r145163;
        double r145165 = r145105 ? r145117 : r145164;
        return r145165;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -13083725713432352.0 or 662.4918751995975 < x

   1. Initial program 60.5

      \[\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 60.4

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

   4. Applied add-cube-cbrt 60.5

      \[\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)}{\color{blue}{\left(\sqrt[3]{\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 \sqrt[3]{\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)}\right) \cdot \sqrt[3]{\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\]

   5. Applied *-un-lft-identity 60.5

      \[\leadsto \frac{\color{blue}{1 \cdot \left({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)\right)}}{\left(\sqrt[3]{\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 \sqrt[3]{\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)}\right) \cdot \sqrt[3]{\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\]

   6. Applied times-frac 60.5

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt[3]{\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 \sqrt[3]{\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 \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)}{\sqrt[3]{\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)}}\right)} \cdot x\]

   7. Applied associate-*l* 60.5

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{\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 \sqrt[3]{\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 \left(\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)}{\sqrt[3]{\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\right)}\]

   8. Taylor expanded around inf 0.0

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

   9. Simplified 0.0

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

   if -13083725713432352.0 < x < 662.4918751995975

   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. Using strategy rm

   4. Applied add-cube-cbrt 0.1

      \[\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)}{\color{blue}{\left(\sqrt[3]{\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 \sqrt[3]{\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)}\right) \cdot \sqrt[3]{\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\]

   5. Applied *-un-lft-identity 0.1

      \[\leadsto \frac{\color{blue}{1 \cdot \left({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)\right)}}{\left(\sqrt[3]{\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 \sqrt[3]{\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)}\right) \cdot \sqrt[3]{\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\]

   6. Applied times-frac 0.1

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt[3]{\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 \sqrt[3]{\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 \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)}{\sqrt[3]{\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)}}\right)} \cdot x\]

   7. Applied associate-*l* 0.1

      \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{\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 \sqrt[3]{\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 \left(\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)}{\sqrt[3]{\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\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -13083725713432352 \lor \neg \left(x \le 662.4918751995975298996199853718280792236\right):\\ \;\;\;\;\left(\frac{0.5}{x} + \frac{0.1529819634592932686700805788859724998474}{{x}^{5}}\right) + \frac{0.2514179000665375252054900556686334311962}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{\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 \sqrt[3]{\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 \left(\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)}{\sqrt[3]{\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\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (x)
  :name "Jmat.Real.dawson"
  :precision binary64
  (* (/ (+ (+ (+ (+ (+ 1 (* 0.1049934947 (* x x))) (* 0.042406060400000001 (* (* x x) (* x x)))) (* 0.00726441819999999999 (* (* (* x x) (* x x)) (* x x)))) (* 5.0640340000000002e-4 (* (* (* (* x x) (* x x)) (* x x)) (* x x)))) (* 1.789971e-4 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)))) (+ (+ (+ (+ (+ (+ 1 (* 0.77154710189999998 (* x x))) (* 0.29097386390000002 (* (* x x) (* x x)))) (* 0.069455576099999999 (* (* (* x x) (* x x)) (* x x)))) (* 0.014000544199999999 (* (* (* (* x x) (* x x)) (* x x)) (* x x)))) (* 8.32794500000000044e-4 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)))) (* (* 2 1.789971e-4) (* (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)) (* x x))))) x))