Average Error: 29.6 → 0.0
Time: 30.0s
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 -656.5908303529681688814889639616012573242 \lor \neg \left(x \le 735244.049645020510070025920867919921875\right):\\ \;\;\;\;\mathsf{fma}\left(0.2514179000665375252054900556686334311962, \frac{1}{{x}^{3}}, \mathsf{fma}\left(0.1529819634592932686700805788859724998474, \frac{1}{{x}^{5}}, 0.5 \cdot \frac{1}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\left(\left(x \cdot \left({\left(x \cdot x\right)}^{3} \cdot {x}^{3}\right)\right) \cdot \left(8.327945000000000442749725770852364803432 \cdot 10^{-4} + \left(x \cdot x\right) \cdot \left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right)\right) + \mathsf{fma}\left(0.2909738639000000182122107617033179849386 \cdot x, {x}^{3}, \mathsf{fma}\left(0.7715471018999999763821051601553335785866 \cdot x, x, 1\right)\right)\right) + {x}^{6} \cdot \left(0.06945557609999999937322456844412954524159 + \left(x \cdot x\right) \cdot 0.01400054419999999938406531896362139377743\right)}} \cdot \frac{x}{\frac{-\sqrt{\left(\left(x \cdot \left({\left(x \cdot x\right)}^{3} \cdot {x}^{3}\right)\right) \cdot \left(8.327945000000000442749725770852364803432 \cdot 10^{-4} + \left(x \cdot x\right) \cdot \left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right)\right) + \mathsf{fma}\left(0.2909738639000000182122107617033179849386 \cdot x, {x}^{3}, \mathsf{fma}\left(0.7715471018999999763821051601553335785866 \cdot x, x, 1\right)\right)\right) + {x}^{6} \cdot \left(0.06945557609999999937322456844412954524159 + \left(x \cdot x\right) \cdot 0.01400054419999999938406531896362139377743\right)}}{\left(-{x}^{4}\right) \cdot \mathsf{fma}\left(x, x \cdot 0.007264418199999999985194687468492702464573, 0.04240606040000000076517494562722276896238\right) + \left(-\mathsf{fma}\left({\left(x \cdot x\right)}^{4}, 5.064034000000000243502107366566633572802 \cdot 10^{-4} + \left(x \cdot x\right) \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}, \mathsf{fma}\left(0.1049934946999999951788851149103720672429 \cdot x, x, 1\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 -656.5908303529681688814889639616012573242 \lor \neg \left(x \le 735244.049645020510070025920867919921875\right):\\
\;\;\;\;\mathsf{fma}\left(0.2514179000665375252054900556686334311962, \frac{1}{{x}^{3}}, \mathsf{fma}\left(0.1529819634592932686700805788859724998474, \frac{1}{{x}^{5}}, 0.5 \cdot \frac{1}{x}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\sqrt{\left(\left(x \cdot \left({\left(x \cdot x\right)}^{3} \cdot {x}^{3}\right)\right) \cdot \left(8.327945000000000442749725770852364803432 \cdot 10^{-4} + \left(x \cdot x\right) \cdot \left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right)\right) + \mathsf{fma}\left(0.2909738639000000182122107617033179849386 \cdot x, {x}^{3}, \mathsf{fma}\left(0.7715471018999999763821051601553335785866 \cdot x, x, 1\right)\right)\right) + {x}^{6} \cdot \left(0.06945557609999999937322456844412954524159 + \left(x \cdot x\right) \cdot 0.01400054419999999938406531896362139377743\right)}} \cdot \frac{x}{\frac{-\sqrt{\left(\left(x \cdot \left({\left(x \cdot x\right)}^{3} \cdot {x}^{3}\right)\right) \cdot \left(8.327945000000000442749725770852364803432 \cdot 10^{-4} + \left(x \cdot x\right) \cdot \left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right)\right) + \mathsf{fma}\left(0.2909738639000000182122107617033179849386 \cdot x, {x}^{3}, \mathsf{fma}\left(0.7715471018999999763821051601553335785866 \cdot x, x, 1\right)\right)\right) + {x}^{6} \cdot \left(0.06945557609999999937322456844412954524159 + \left(x \cdot x\right) \cdot 0.01400054419999999938406531896362139377743\right)}}{\left(-{x}^{4}\right) \cdot \mathsf{fma}\left(x, x \cdot 0.007264418199999999985194687468492702464573, 0.04240606040000000076517494562722276896238\right) + \left(-\mathsf{fma}\left({\left(x \cdot x\right)}^{4}, 5.064034000000000243502107366566633572802 \cdot 10^{-4} + \left(x \cdot x\right) \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}, \mathsf{fma}\left(0.1049934946999999951788851149103720672429 \cdot x, x, 1\right)\right)\right)}}\\

\end{array}
double f(double x) {
        double r264982 = 1.0;
        double r264983 = 0.1049934947;
        double r264984 = x;
        double r264985 = r264984 * r264984;
        double r264986 = r264983 * r264985;
        double r264987 = r264982 + r264986;
        double r264988 = 0.0424060604;
        double r264989 = r264985 * r264985;
        double r264990 = r264988 * r264989;
        double r264991 = r264987 + r264990;
        double r264992 = 0.0072644182;
        double r264993 = r264989 * r264985;
        double r264994 = r264992 * r264993;
        double r264995 = r264991 + r264994;
        double r264996 = 0.0005064034;
        double r264997 = r264993 * r264985;
        double r264998 = r264996 * r264997;
        double r264999 = r264995 + r264998;
        double r265000 = 0.0001789971;
        double r265001 = r264997 * r264985;
        double r265002 = r265000 * r265001;
        double r265003 = r264999 + r265002;
        double r265004 = 0.7715471019;
        double r265005 = r265004 * r264985;
        double r265006 = r264982 + r265005;
        double r265007 = 0.2909738639;
        double r265008 = r265007 * r264989;
        double r265009 = r265006 + r265008;
        double r265010 = 0.0694555761;
        double r265011 = r265010 * r264993;
        double r265012 = r265009 + r265011;
        double r265013 = 0.0140005442;
        double r265014 = r265013 * r264997;
        double r265015 = r265012 + r265014;
        double r265016 = 0.0008327945;
        double r265017 = r265016 * r265001;
        double r265018 = r265015 + r265017;
        double r265019 = 2.0;
        double r265020 = r265019 * r265000;
        double r265021 = r265001 * r264985;
        double r265022 = r265020 * r265021;
        double r265023 = r265018 + r265022;
        double r265024 = r265003 / r265023;
        double r265025 = r265024 * r264984;
        return r265025;
}
↓
double f(double x) {
        double r265026 = x;
        double r265027 = -656.5908303529682;
        bool r265028 = r265026 <= r265027;
        double r265029 = 735244.0496450205;
        bool r265030 = r265026 <= r265029;
        double r265031 = !r265030;
        bool r265032 = r265028 || r265031;
        double r265033 = 0.2514179000665375;
        double r265034 = 1.0;
        double r265035 = 3.0;
        double r265036 = pow(r265026, r265035);
        double r265037 = r265034 / r265036;
        double r265038 = 0.15298196345929327;
        double r265039 = 5.0;
        double r265040 = pow(r265026, r265039);
        double r265041 = r265034 / r265040;
        double r265042 = 0.5;
        double r265043 = r265034 / r265026;
        double r265044 = r265042 * r265043;
        double r265045 = fma(r265038, r265041, r265044);
        double r265046 = fma(r265033, r265037, r265045);
        double r265047 = r265026 * r265026;
        double r265048 = pow(r265047, r265035);
        double r265049 = r265048 * r265036;
        double r265050 = r265026 * r265049;
        double r265051 = 0.0008327945;
        double r265052 = 2.0;
        double r265053 = 0.0001789971;
        double r265054 = r265052 * r265053;
        double r265055 = r265047 * r265054;
        double r265056 = r265051 + r265055;
        double r265057 = r265050 * r265056;
        double r265058 = 0.2909738639;
        double r265059 = r265058 * r265026;
        double r265060 = 0.7715471019;
        double r265061 = r265060 * r265026;
        double r265062 = 1.0;
        double r265063 = fma(r265061, r265026, r265062);
        double r265064 = fma(r265059, r265036, r265063);
        double r265065 = r265057 + r265064;
        double r265066 = 6.0;
        double r265067 = pow(r265026, r265066);
        double r265068 = 0.0694555761;
        double r265069 = 0.0140005442;
        double r265070 = r265047 * r265069;
        double r265071 = r265068 + r265070;
        double r265072 = r265067 * r265071;
        double r265073 = r265065 + r265072;
        double r265074 = sqrt(r265073);
        double r265075 = r265034 / r265074;
        double r265076 = -r265074;
        double r265077 = 4.0;
        double r265078 = pow(r265026, r265077);
        double r265079 = -r265078;
        double r265080 = 0.0072644182;
        double r265081 = r265026 * r265080;
        double r265082 = 0.0424060604;
        double r265083 = fma(r265026, r265081, r265082);
        double r265084 = r265079 * r265083;
        double r265085 = pow(r265047, r265077);
        double r265086 = 0.0005064034;
        double r265087 = r265047 * r265053;
        double r265088 = r265086 + r265087;
        double r265089 = 0.1049934947;
        double r265090 = r265089 * r265026;
        double r265091 = fma(r265090, r265026, r265062);
        double r265092 = fma(r265085, r265088, r265091);
        double r265093 = -r265092;
        double r265094 = r265084 + r265093;
        double r265095 = r265076 / r265094;
        double r265096 = r265026 / r265095;
        double r265097 = r265075 * r265096;
        double r265098 = r265032 ? r265046 : r265097;
        return r265098;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -656.5908303529682 or 735244.0496450205 < x

    1. Initial program 59.9

      \[\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. Simplified59.8

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

    3. Taylor expanded around inf 0.0

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

    4. Simplified0.0

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

    if -656.5908303529682 < x < 735244.0496450205

    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

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

    4. Applied *-un-lft-identity0.0

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

    5. Applied add-sqr-sqrt0.0

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

    6. Applied times-frac0.0

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

    7. Applied *-un-lft-identity0.0

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

    8. Applied times-frac0.0

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

    9. Simplified0.0

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

    11. Applied frac-2neg0.0

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

    12. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -656.5908303529681688814889639616012573242 \lor \neg \left(x \le 735244.049645020510070025920867919921875\right):\\ \;\;\;\;\mathsf{fma}\left(0.2514179000665375252054900556686334311962, \frac{1}{{x}^{3}}, \mathsf{fma}\left(0.1529819634592932686700805788859724998474, \frac{1}{{x}^{5}}, 0.5 \cdot \frac{1}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\left(\left(x \cdot \left({\left(x \cdot x\right)}^{3} \cdot {x}^{3}\right)\right) \cdot \left(8.327945000000000442749725770852364803432 \cdot 10^{-4} + \left(x \cdot x\right) \cdot \left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right)\right) + \mathsf{fma}\left(0.2909738639000000182122107617033179849386 \cdot x, {x}^{3}, \mathsf{fma}\left(0.7715471018999999763821051601553335785866 \cdot x, x, 1\right)\right)\right) + {x}^{6} \cdot \left(0.06945557609999999937322456844412954524159 + \left(x \cdot x\right) \cdot 0.01400054419999999938406531896362139377743\right)}} \cdot \frac{x}{\frac{-\sqrt{\left(\left(x \cdot \left({\left(x \cdot x\right)}^{3} \cdot {x}^{3}\right)\right) \cdot \left(8.327945000000000442749725770852364803432 \cdot 10^{-4} + \left(x \cdot x\right) \cdot \left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right)\right) + \mathsf{fma}\left(0.2909738639000000182122107617033179849386 \cdot x, {x}^{3}, \mathsf{fma}\left(0.7715471018999999763821051601553335785866 \cdot x, x, 1\right)\right)\right) + {x}^{6} \cdot \left(0.06945557609999999937322456844412954524159 + \left(x \cdot x\right) \cdot 0.01400054419999999938406531896362139377743\right)}}{\left(-{x}^{4}\right) \cdot \mathsf{fma}\left(x, x \cdot 0.007264418199999999985194687468492702464573, 0.04240606040000000076517494562722276896238\right) + \left(-\mathsf{fma}\left({\left(x \cdot x\right)}^{4}, 5.064034000000000243502107366566633572802 \cdot 10^{-4} + \left(x \cdot x\right) \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}, \mathsf{fma}\left(0.1049934946999999951788851149103720672429 \cdot x, x, 1\right)\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.dawson"
  :precision binary64
  (* (/ (+ (+ (+ (+ (+ 1 (* 0.1049934947 (* x x))) (* 0.0424060604 (* (* x x) (* x x)))) (* 0.0072644182 (* (* (* x x) (* x x)) (* x x)))) (* 0.0005064034 (* (* (* (* x x) (* x x)) (* x x)) (* x x)))) (* 0.0001789971 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)))) (+ (+ (+ (+ (+ (+ 1 (* 0.7715471019 (* x x))) (* 0.2909738639 (* (* x x) (* x x)))) (* 0.0694555761 (* (* (* x x) (* x x)) (* x x)))) (* 0.0140005442 (* (* (* (* x x) (* x x)) (* x x)) (* x x)))) (* 0.0008327945 (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)))) (* (* 2 0.0001789971) (* (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)) (* x x))))) x))