Average Error: 29.5 → 0.0
Time: 22.5s
Precision: 64
\[\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 -276678.054250435321591794490814208984375:\\ \;\;\;\;\frac{0.2514179000665375252054900556686334311962}{x \cdot \left(x \cdot x\right)} + \left(\frac{0.5}{x} + \frac{0.1529819634592932686700805788859724998474}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)}\right)\\ \mathbf{elif}\;x \le 848.6300680576136983290780335664749145508:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \mathsf{fma}\left(1.789971000000000009994005623070734145585 \cdot 10^{-4}, x \cdot x, 5.064034000000000243502107366566633572802 \cdot 10^{-4}\right), \mathsf{fma}\left(\mathsf{fma}\left(0.04240606040000000076517494562722276896238, x \cdot x, \mathsf{fma}\left(0.007264418199999999985194687468492702464573, \left(x \cdot x\right) \cdot \left(x \cdot x\right), 0.1049934946999999951788851149103720672429\right)\right), x \cdot x, 1\right)\right) \cdot x}{\mathsf{fma}\left(\left(2 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), 1.789971000000000009994005623070734145585 \cdot 10^{-4}, \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \mathsf{fma}\left(8.327945000000000442749725770852364803432 \cdot 10^{-4}, x \cdot x, 0.01400054419999999938406531896362139377743\right), \mathsf{fma}\left(0.7715471018999999763821051601553335785866, x \cdot x, \mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right), 0.2909738639000000182122107617033179849386 + x \cdot \left(0.06945557609999999937322456844412954524159 \cdot x\right), 1\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.2514179000665375252054900556686334311962}{x \cdot \left(x \cdot x\right)} + \left(\frac{0.5}{x} + \frac{0.1529819634592932686700805788859724998474}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)}\right)\\ \end{array}\]
\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 -276678.054250435321591794490814208984375:\\
\;\;\;\;\frac{0.2514179000665375252054900556686334311962}{x \cdot \left(x \cdot x\right)} + \left(\frac{0.5}{x} + \frac{0.1529819634592932686700805788859724998474}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)}\right)\\

\mathbf{elif}\;x \le 848.6300680576136983290780335664749145508:\\
\;\;\;\;\frac{\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \mathsf{fma}\left(1.789971000000000009994005623070734145585 \cdot 10^{-4}, x \cdot x, 5.064034000000000243502107366566633572802 \cdot 10^{-4}\right), \mathsf{fma}\left(\mathsf{fma}\left(0.04240606040000000076517494562722276896238, x \cdot x, \mathsf{fma}\left(0.007264418199999999985194687468492702464573, \left(x \cdot x\right) \cdot \left(x \cdot x\right), 0.1049934946999999951788851149103720672429\right)\right), x \cdot x, 1\right)\right) \cdot x}{\mathsf{fma}\left(\left(2 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), 1.789971000000000009994005623070734145585 \cdot 10^{-4}, \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \mathsf{fma}\left(8.327945000000000442749725770852364803432 \cdot 10^{-4}, x \cdot x, 0.01400054419999999938406531896362139377743\right), \mathsf{fma}\left(0.7715471018999999763821051601553335785866, x \cdot x, \mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right), 0.2909738639000000182122107617033179849386 + x \cdot \left(0.06945557609999999937322456844412954524159 \cdot x\right), 1\right)\right)\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{0.2514179000665375252054900556686334311962}{x \cdot \left(x \cdot x\right)} + \left(\frac{0.5}{x} + \frac{0.1529819634592932686700805788859724998474}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)}\right)\\

\end{array}
double f(double x) {
        double r7550127 = 1.0;
        double r7550128 = 0.1049934947;
        double r7550129 = x;
        double r7550130 = r7550129 * r7550129;
        double r7550131 = r7550128 * r7550130;
        double r7550132 = r7550127 + r7550131;
        double r7550133 = 0.0424060604;
        double r7550134 = r7550130 * r7550130;
        double r7550135 = r7550133 * r7550134;
        double r7550136 = r7550132 + r7550135;
        double r7550137 = 0.0072644182;
        double r7550138 = r7550134 * r7550130;
        double r7550139 = r7550137 * r7550138;
        double r7550140 = r7550136 + r7550139;
        double r7550141 = 0.0005064034;
        double r7550142 = r7550138 * r7550130;
        double r7550143 = r7550141 * r7550142;
        double r7550144 = r7550140 + r7550143;
        double r7550145 = 0.0001789971;
        double r7550146 = r7550142 * r7550130;
        double r7550147 = r7550145 * r7550146;
        double r7550148 = r7550144 + r7550147;
        double r7550149 = 0.7715471019;
        double r7550150 = r7550149 * r7550130;
        double r7550151 = r7550127 + r7550150;
        double r7550152 = 0.2909738639;
        double r7550153 = r7550152 * r7550134;
        double r7550154 = r7550151 + r7550153;
        double r7550155 = 0.0694555761;
        double r7550156 = r7550155 * r7550138;
        double r7550157 = r7550154 + r7550156;
        double r7550158 = 0.0140005442;
        double r7550159 = r7550158 * r7550142;
        double r7550160 = r7550157 + r7550159;
        double r7550161 = 0.0008327945;
        double r7550162 = r7550161 * r7550146;
        double r7550163 = r7550160 + r7550162;
        double r7550164 = 2.0;
        double r7550165 = r7550164 * r7550145;
        double r7550166 = r7550146 * r7550130;
        double r7550167 = r7550165 * r7550166;
        double r7550168 = r7550163 + r7550167;
        double r7550169 = r7550148 / r7550168;
        double r7550170 = r7550169 * r7550129;
        return r7550170;
}
↓
double f(double x) {
        double r7550171 = x;
        double r7550172 = -276678.0542504353;
        bool r7550173 = r7550171 <= r7550172;
        double r7550174 = 0.2514179000665375;
        double r7550175 = r7550171 * r7550171;
        double r7550176 = r7550171 * r7550175;
        double r7550177 = r7550174 / r7550176;
        double r7550178 = 0.5;
        double r7550179 = r7550178 / r7550171;
        double r7550180 = 0.15298196345929327;
        double r7550181 = r7550176 * r7550175;
        double r7550182 = r7550180 / r7550181;
        double r7550183 = r7550179 + r7550182;
        double r7550184 = r7550177 + r7550183;
        double r7550185 = 848.6300680576137;
        bool r7550186 = r7550171 <= r7550185;
        double r7550187 = r7550175 * r7550175;
        double r7550188 = r7550187 * r7550187;
        double r7550189 = 0.0001789971;
        double r7550190 = 0.0005064034;
        double r7550191 = fma(r7550189, r7550175, r7550190);
        double r7550192 = 0.0424060604;
        double r7550193 = 0.0072644182;
        double r7550194 = 0.1049934947;
        double r7550195 = fma(r7550193, r7550187, r7550194);
        double r7550196 = fma(r7550192, r7550175, r7550195);
        double r7550197 = 1.0;
        double r7550198 = fma(r7550196, r7550175, r7550197);
        double r7550199 = fma(r7550188, r7550191, r7550198);
        double r7550200 = r7550199 * r7550171;
        double r7550201 = 2.0;
        double r7550202 = r7550201 * r7550188;
        double r7550203 = r7550202 * r7550187;
        double r7550204 = 0.0008327945;
        double r7550205 = 0.0140005442;
        double r7550206 = fma(r7550204, r7550175, r7550205);
        double r7550207 = 0.7715471019;
        double r7550208 = 0.2909738639;
        double r7550209 = 0.0694555761;
        double r7550210 = r7550209 * r7550171;
        double r7550211 = r7550171 * r7550210;
        double r7550212 = r7550208 + r7550211;
        double r7550213 = fma(r7550187, r7550212, r7550197);
        double r7550214 = fma(r7550207, r7550175, r7550213);
        double r7550215 = fma(r7550188, r7550206, r7550214);
        double r7550216 = fma(r7550203, r7550189, r7550215);
        double r7550217 = r7550200 / r7550216;
        double r7550218 = r7550186 ? r7550217 : r7550184;
        double r7550219 = r7550173 ? r7550184 : r7550218;
        return r7550219;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -276678.0542504353 or 848.6300680576137 < x

    1. Initial program 59.4

      \[\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. Simplified59.4

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

    3. Taylor expanded around inf 0.0

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

    4. Simplified0.0

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

    if -276678.0542504353 < x < 848.6300680576137

    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. Simplified0.0

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

    3. Taylor expanded around 0 0.0

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

    4. Simplified0.0

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

    6. Applied associate-*l/0.0

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

    7. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -276678.054250435321591794490814208984375:\\ \;\;\;\;\frac{0.2514179000665375252054900556686334311962}{x \cdot \left(x \cdot x\right)} + \left(\frac{0.5}{x} + \frac{0.1529819634592932686700805788859724998474}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)}\right)\\ \mathbf{elif}\;x \le 848.6300680576136983290780335664749145508:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \mathsf{fma}\left(1.789971000000000009994005623070734145585 \cdot 10^{-4}, x \cdot x, 5.064034000000000243502107366566633572802 \cdot 10^{-4}\right), \mathsf{fma}\left(\mathsf{fma}\left(0.04240606040000000076517494562722276896238, x \cdot x, \mathsf{fma}\left(0.007264418199999999985194687468492702464573, \left(x \cdot x\right) \cdot \left(x \cdot x\right), 0.1049934946999999951788851149103720672429\right)\right), x \cdot x, 1\right)\right) \cdot x}{\mathsf{fma}\left(\left(2 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), 1.789971000000000009994005623070734145585 \cdot 10^{-4}, \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \mathsf{fma}\left(8.327945000000000442749725770852364803432 \cdot 10^{-4}, x \cdot x, 0.01400054419999999938406531896362139377743\right), \mathsf{fma}\left(0.7715471018999999763821051601553335785866, x \cdot x, \mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right), 0.2909738639000000182122107617033179849386 + x \cdot \left(0.06945557609999999937322456844412954524159 \cdot x\right), 1\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.2514179000665375252054900556686334311962}{x \cdot \left(x \cdot x\right)} + \left(\frac{0.5}{x} + \frac{0.1529819634592932686700805788859724998474}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.dawson"
  (* (/ (+ (+ (+ (+ (+ 1.0 (* 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 (* 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 0.0001789971) (* (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)) (* x x))))) x))