Average Error: 29.0 → 0.0
Time: 21.6s
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\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3158623.0682681673206388950347900390625 \lor \neg \left(x \le 9344017.82665434293448925018310546875\right):\\ \;\;\;\;\frac{0.5}{x} + \left(\frac{0.1529819634592932686700805788859724998474}{{x}^{5}} + \frac{0.2514179000665375252054900556686334311962}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left({\left(x \cdot x\right)}^{4}, \mathsf{fma}\left(1.789971000000000009994005623070734145585 \cdot 10^{-4}, x \cdot x, 5.064034000000000243502107366566633572802 \cdot 10^{-4}\right), {x}^{4} \cdot 0.04240606040000000076517494562722276896238\right) + \mathsf{fma}\left(0.1049934946999999951788851149103720672429, {x}^{2}, \mathsf{fma}\left(0.007264418199999999985194687468492702464573, {x}^{6}, 1\right)\right)\right) \cdot \frac{x}{\mathsf{fma}\left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{4}, \mathsf{fma}\left({x}^{2}, 8.327945000000000442749725770852364803432 \cdot 10^{-4}, 0.01400054419999999938406531896362139377743\right), \mathsf{fma}\left(0.06945557609999999937322456844412954524159, {x}^{6}, \mathsf{fma}\left(0.2909738639000000182122107617033179849386, {x}^{4}, \mathsf{fma}\left(x \cdot x, 0.7715471018999999763821051601553335785866, 1\right)\right)\right)\right)\right)}\\ \end{array}\]
\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 -3158623.0682681673206388950347900390625 \lor \neg \left(x \le 9344017.82665434293448925018310546875\right):\\
\;\;\;\;\frac{0.5}{x} + \left(\frac{0.1529819634592932686700805788859724998474}{{x}^{5}} + \frac{0.2514179000665375252054900556686334311962}{{x}^{3}}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\mathsf{fma}\left({\left(x \cdot x\right)}^{4}, \mathsf{fma}\left(1.789971000000000009994005623070734145585 \cdot 10^{-4}, x \cdot x, 5.064034000000000243502107366566633572802 \cdot 10^{-4}\right), {x}^{4} \cdot 0.04240606040000000076517494562722276896238\right) + \mathsf{fma}\left(0.1049934946999999951788851149103720672429, {x}^{2}, \mathsf{fma}\left(0.007264418199999999985194687468492702464573, {x}^{6}, 1\right)\right)\right) \cdot \frac{x}{\mathsf{fma}\left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{4}, \mathsf{fma}\left({x}^{2}, 8.327945000000000442749725770852364803432 \cdot 10^{-4}, 0.01400054419999999938406531896362139377743\right), \mathsf{fma}\left(0.06945557609999999937322456844412954524159, {x}^{6}, \mathsf{fma}\left(0.2909738639000000182122107617033179849386, {x}^{4}, \mathsf{fma}\left(x \cdot x, 0.7715471018999999763821051601553335785866, 1\right)\right)\right)\right)\right)}\\

\end{array}
double f(double x) {
        double r123057 = 1.0;
        double r123058 = 0.1049934947;
        double r123059 = x;
        double r123060 = r123059 * r123059;
        double r123061 = r123058 * r123060;
        double r123062 = r123057 + r123061;
        double r123063 = 0.0424060604;
        double r123064 = r123060 * r123060;
        double r123065 = r123063 * r123064;
        double r123066 = r123062 + r123065;
        double r123067 = 0.0072644182;
        double r123068 = r123064 * r123060;
        double r123069 = r123067 * r123068;
        double r123070 = r123066 + r123069;
        double r123071 = 0.0005064034;
        double r123072 = r123068 * r123060;
        double r123073 = r123071 * r123072;
        double r123074 = r123070 + r123073;
        double r123075 = 0.0001789971;
        double r123076 = r123072 * r123060;
        double r123077 = r123075 * r123076;
        double r123078 = r123074 + r123077;
        double r123079 = 0.7715471019;
        double r123080 = r123079 * r123060;
        double r123081 = r123057 + r123080;
        double r123082 = 0.2909738639;
        double r123083 = r123082 * r123064;
        double r123084 = r123081 + r123083;
        double r123085 = 0.0694555761;
        double r123086 = r123085 * r123068;
        double r123087 = r123084 + r123086;
        double r123088 = 0.0140005442;
        double r123089 = r123088 * r123072;
        double r123090 = r123087 + r123089;
        double r123091 = 0.0008327945;
        double r123092 = r123091 * r123076;
        double r123093 = r123090 + r123092;
        double r123094 = 2.0;
        double r123095 = r123094 * r123075;
        double r123096 = r123076 * r123060;
        double r123097 = r123095 * r123096;
        double r123098 = r123093 + r123097;
        double r123099 = r123078 / r123098;
        double r123100 = r123099 * r123059;
        return r123100;
}
↓
double f(double x) {
        double r123101 = x;
        double r123102 = -3158623.0682681673;
        bool r123103 = r123101 <= r123102;
        double r123104 = 9344017.826654343;
        bool r123105 = r123101 <= r123104;
        double r123106 = !r123105;
        bool r123107 = r123103 || r123106;
        double r123108 = 0.5;
        double r123109 = r123108 / r123101;
        double r123110 = 0.15298196345929327;
        double r123111 = 5.0;
        double r123112 = pow(r123101, r123111);
        double r123113 = r123110 / r123112;
        double r123114 = 0.2514179000665375;
        double r123115 = 3.0;
        double r123116 = pow(r123101, r123115);
        double r123117 = r123114 / r123116;
        double r123118 = r123113 + r123117;
        double r123119 = r123109 + r123118;
        double r123120 = r123101 * r123101;
        double r123121 = 4.0;
        double r123122 = pow(r123120, r123121);
        double r123123 = 0.0001789971;
        double r123124 = 0.0005064034;
        double r123125 = fma(r123123, r123120, r123124);
        double r123126 = pow(r123101, r123121);
        double r123127 = 0.0424060604;
        double r123128 = r123126 * r123127;
        double r123129 = fma(r123122, r123125, r123128);
        double r123130 = 0.1049934947;
        double r123131 = 2.0;
        double r123132 = pow(r123101, r123131);
        double r123133 = 0.0072644182;
        double r123134 = 6.0;
        double r123135 = pow(r123101, r123134);
        double r123136 = 1.0;
        double r123137 = fma(r123133, r123135, r123136);
        double r123138 = fma(r123130, r123132, r123137);
        double r123139 = r123129 + r123138;
        double r123140 = 2.0;
        double r123141 = r123140 * r123123;
        double r123142 = pow(r123120, r123134);
        double r123143 = 0.0008327945;
        double r123144 = 0.0140005442;
        double r123145 = fma(r123132, r123143, r123144);
        double r123146 = 0.0694555761;
        double r123147 = 0.2909738639;
        double r123148 = 0.7715471019;
        double r123149 = fma(r123120, r123148, r123136);
        double r123150 = fma(r123147, r123126, r123149);
        double r123151 = fma(r123146, r123135, r123150);
        double r123152 = fma(r123122, r123145, r123151);
        double r123153 = fma(r123141, r123142, r123152);
        double r123154 = r123101 / r123153;
        double r123155 = r123139 * r123154;
        double r123156 = r123107 ? r123119 : r123155;
        return r123156;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -3158623.0682681673 or 9344017.826654343 < x

    1. Initial program 60.0

      \[\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. Simplified60.0

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

    3. Taylor expanded around 0 60.0

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

    4. Simplified60.0

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

    6. Applied div-inv60.0

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

    7. Applied associate-*l*60.0

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

    8. Simplified60.0

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

    9. Taylor expanded around inf 0.0

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

    10. Simplified0.0

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

    if -3158623.0682681673 < x < 9344017.826654343

    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

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

    3. Taylor expanded around 0 0.0

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

    4. Simplified0.0

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

    6. Applied div-inv0.0

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

    7. Applied associate-*l*0.0

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

    8. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3158623.0682681673206388950347900390625 \lor \neg \left(x \le 9344017.82665434293448925018310546875\right):\\ \;\;\;\;\frac{0.5}{x} + \left(\frac{0.1529819634592932686700805788859724998474}{{x}^{5}} + \frac{0.2514179000665375252054900556686334311962}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left({\left(x \cdot x\right)}^{4}, \mathsf{fma}\left(1.789971000000000009994005623070734145585 \cdot 10^{-4}, x \cdot x, 5.064034000000000243502107366566633572802 \cdot 10^{-4}\right), {x}^{4} \cdot 0.04240606040000000076517494562722276896238\right) + \mathsf{fma}\left(0.1049934946999999951788851149103720672429, {x}^{2}, \mathsf{fma}\left(0.007264418199999999985194687468492702464573, {x}^{6}, 1\right)\right)\right) \cdot \frac{x}{\mathsf{fma}\left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}, {\left(x \cdot x\right)}^{6}, \mathsf{fma}\left({\left(x \cdot x\right)}^{4}, \mathsf{fma}\left({x}^{2}, 8.327945000000000442749725770852364803432 \cdot 10^{-4}, 0.01400054419999999938406531896362139377743\right), \mathsf{fma}\left(0.06945557609999999937322456844412954524159, {x}^{6}, \mathsf{fma}\left(0.2909738639000000182122107617033179849386, {x}^{4}, \mathsf{fma}\left(x \cdot x, 0.7715471018999999763821051601553335785866, 1\right)\right)\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(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))