Jmat.Real.dawson

Average Error: 28.7 → 0.0

Time: 2.4m

Precision: 64

Math

\[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -942486508.7001753:\\ \;\;\;\;\frac{0.5 + \frac{0.2514179000665375}{x \cdot x}}{x} + \frac{0.15298196345929327}{{x}^{5}}\\ \mathbf{elif}\;x \le 9551.679161702108:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left(\frac{x \cdot \mathsf{fma}\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.0001789971\right), \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), \left(\mathsf{fma}\left(0.0005064034, \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), \left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), 0.0072644182, \left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), 0.0424060604, \left(\mathsf{fma}\left(\left(x \cdot x\right), 0.1049934947, 1\right)\right)\right)\right)\right)\right)\right)\right)\right)}{\mathsf{fma}\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.0001789971\right) \cdot 2\right), \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), \left(\left(0.0140005442 + \left(x \cdot x\right) \cdot 0.0008327945\right) \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) + \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), 0.2909738639, \left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), 0.0694555761, \left(\mathsf{fma}\left(\left(x \cdot x\right), 0.7715471019, 1\right)\right)\right)\right)\right)\right)\right)}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{0.2514179000665375}{x \cdot x}}{x} + \frac{0.15298196345929327}{{x}^{5}}\\ \end{array}\]

C

double f(double x) {
        double r50024084 = 1.0;
        double r50024085 = 0.1049934947;
        double r50024086 = x;
        double r50024087 = r50024086 * r50024086;
        double r50024088 = r50024085 * r50024087;
        double r50024089 = r50024084 + r50024088;
        double r50024090 = 0.0424060604;
        double r50024091 = r50024087 * r50024087;
        double r50024092 = r50024090 * r50024091;
        double r50024093 = r50024089 + r50024092;
        double r50024094 = 0.0072644182;
        double r50024095 = r50024091 * r50024087;
        double r50024096 = r50024094 * r50024095;
        double r50024097 = r50024093 + r50024096;
        double r50024098 = 0.0005064034;
        double r50024099 = r50024095 * r50024087;
        double r50024100 = r50024098 * r50024099;
        double r50024101 = r50024097 + r50024100;
        double r50024102 = 0.0001789971;
        double r50024103 = r50024099 * r50024087;
        double r50024104 = r50024102 * r50024103;
        double r50024105 = r50024101 + r50024104;
        double r50024106 = 0.7715471019;
        double r50024107 = r50024106 * r50024087;
        double r50024108 = r50024084 + r50024107;
        double r50024109 = 0.2909738639;
        double r50024110 = r50024109 * r50024091;
        double r50024111 = r50024108 + r50024110;
        double r50024112 = 0.0694555761;
        double r50024113 = r50024112 * r50024095;
        double r50024114 = r50024111 + r50024113;
        double r50024115 = 0.0140005442;
        double r50024116 = r50024115 * r50024099;
        double r50024117 = r50024114 + r50024116;
        double r50024118 = 0.0008327945;
        double r50024119 = r50024118 * r50024103;
        double r50024120 = r50024117 + r50024119;
        double r50024121 = 2.0;
        double r50024122 = r50024121 * r50024102;
        double r50024123 = r50024103 * r50024087;
        double r50024124 = r50024122 * r50024123;
        double r50024125 = r50024120 + r50024124;
        double r50024126 = r50024105 / r50024125;
        double r50024127 = r50024126 * r50024086;
        return r50024127;
}

↓

double f(double x) {
        double r50024128 = x;
        double r50024129 = -942486508.7001753;
        bool r50024130 = r50024128 <= r50024129;
        double r50024131 = 0.5;
        double r50024132 = 0.2514179000665375;
        double r50024133 = r50024128 * r50024128;
        double r50024134 = r50024132 / r50024133;
        double r50024135 = r50024131 + r50024134;
        double r50024136 = r50024135 / r50024128;
        double r50024137 = 0.15298196345929327;
        double r50024138 = 5.0;
        double r50024139 = pow(r50024128, r50024138);
        double r50024140 = r50024137 / r50024139;
        double r50024141 = r50024136 + r50024140;
        double r50024142 = 9551.679161702108;
        bool r50024143 = r50024128 <= r50024142;
        double r50024144 = r50024133 * r50024133;
        double r50024145 = 0.0001789971;
        double r50024146 = r50024144 * r50024145;
        double r50024147 = r50024133 * r50024144;
        double r50024148 = 0.0005064034;
        double r50024149 = r50024144 * r50024144;
        double r50024150 = 0.0072644182;
        double r50024151 = 0.0424060604;
        double r50024152 = 0.1049934947;
        double r50024153 = 1.0;
        double r50024154 = fma(r50024133, r50024152, r50024153);
        double r50024155 = fma(r50024144, r50024151, r50024154);
        double r50024156 = fma(r50024147, r50024150, r50024155);
        double r50024157 = fma(r50024148, r50024149, r50024156);
        double r50024158 = fma(r50024146, r50024147, r50024157);
        double r50024159 = r50024128 * r50024158;
        double r50024160 = 2.0;
        double r50024161 = r50024146 * r50024160;
        double r50024162 = 0.0140005442;
        double r50024163 = 0.0008327945;
        double r50024164 = r50024133 * r50024163;
        double r50024165 = r50024162 + r50024164;
        double r50024166 = r50024165 * r50024149;
        double r50024167 = 0.2909738639;
        double r50024168 = 0.0694555761;
        double r50024169 = 0.7715471019;
        double r50024170 = fma(r50024133, r50024169, r50024153);
        double r50024171 = fma(r50024147, r50024168, r50024170);
        double r50024172 = fma(r50024144, r50024167, r50024171);
        double r50024173 = r50024166 + r50024172;
        double r50024174 = fma(r50024161, r50024149, r50024173);
        double r50024175 = r50024159 / r50024174;
        double r50024176 = expm1(r50024175);
        double r50024177 = log1p(r50024176);
        double r50024178 = r50024143 ? r50024177 : r50024141;
        double r50024179 = r50024130 ? r50024141 : r50024178;
        return r50024179;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -942486508.7001753 or 9551.679161702108 < x

   1. Initial program 58.9

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\]

   2. Simplified 58.9

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.0001789971, \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), \left(\mathsf{fma}\left(0.0005064034, \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), \left(\mathsf{fma}\left(0.0072644182, \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), \left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), 0.0424060604, \left(\mathsf{fma}\left(0.1049934947, \left(x \cdot x\right), 1\right)\right)\right)\right)\right)\right)\right)\right)\right)}{\mathsf{fma}\left(\left(2 \cdot 0.0001789971\right), \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right), \left(\mathsf{fma}\left(\left(0.0008327945 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right), \left(x \cdot x\right), \left(\mathsf{fma}\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), 0.0140005442, \left(\mathsf{fma}\left(0.2909738639, \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right) + \mathsf{fma}\left(\left(x \cdot x\right), 0.7715471019, 1\right)\right)\right)\right)\right)\right)\right)} \cdot x}\]

   3. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{0.15298196345929327 \cdot \frac{1}{{x}^{5}} + \left(0.2514179000665375 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}\right)}\]

   4. Simplified 0.0

      \[\leadsto \color{blue}{\frac{0.15298196345929327}{{x}^{5}} + \frac{\frac{0.2514179000665375}{x \cdot x} + 0.5}{x}}\]

   if -942486508.7001753 < x < 9551.679161702108

   1. Initial program 0.0

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\]

   2. Simplified 0.0

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.0001789971, \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), \left(\mathsf{fma}\left(0.0005064034, \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), \left(\mathsf{fma}\left(0.0072644182, \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), \left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), 0.0424060604, \left(\mathsf{fma}\left(0.1049934947, \left(x \cdot x\right), 1\right)\right)\right)\right)\right)\right)\right)\right)\right)}{\mathsf{fma}\left(\left(2 \cdot 0.0001789971\right), \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right), \left(\mathsf{fma}\left(\left(0.0008327945 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right), \left(x \cdot x\right), \left(\mathsf{fma}\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), 0.0140005442, \left(\mathsf{fma}\left(0.2909738639, \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right) + \mathsf{fma}\left(\left(x \cdot x\right), 0.7715471019, 1\right)\right)\right)\right)\right)\right)\right)} \cdot x}\]
   3. Using strategy rm

   4. Applied div-inv 0.0

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.0001789971, \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), \left(\mathsf{fma}\left(0.0005064034, \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), \left(\mathsf{fma}\left(0.0072644182, \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), \left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), 0.0424060604, \left(\mathsf{fma}\left(0.1049934947, \left(x \cdot x\right), 1\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot \frac{1}{\mathsf{fma}\left(\left(2 \cdot 0.0001789971\right), \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right), \left(\mathsf{fma}\left(\left(0.0008327945 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right), \left(x \cdot x\right), \left(\mathsf{fma}\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), 0.0140005442, \left(\mathsf{fma}\left(0.2909738639, \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right) + \mathsf{fma}\left(\left(x \cdot x\right), 0.7715471019, 1\right)\right)\right)\right)\right)\right)\right)}\right)} \cdot x\]
   5. Using strategy rm

   6. Applied log1p-expm1-u 0.0

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left(\left(\mathsf{fma}\left(0.0001789971, \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), \left(\mathsf{fma}\left(0.0005064034, \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), \left(\mathsf{fma}\left(0.0072644182, \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), \left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), 0.0424060604, \left(\mathsf{fma}\left(0.1049934947, \left(x \cdot x\right), 1\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot \frac{1}{\mathsf{fma}\left(\left(2 \cdot 0.0001789971\right), \left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right), \left(\mathsf{fma}\left(\left(0.0008327945 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right), \left(x \cdot x\right), \left(\mathsf{fma}\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), 0.0140005442, \left(\mathsf{fma}\left(0.2909738639, \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)\right) + \mathsf{fma}\left(\left(x \cdot x\right), 0.7715471019, 1\right)\right)\right)\right)\right)\right)\right)}\right) \cdot x\right)\right)\right)\right)}\]

   7. Simplified 0.0

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(\mathsf{expm1}\left(\left(\frac{\mathsf{fma}\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.0001789971\right), \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), \left(\mathsf{fma}\left(0.0005064034, \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), \left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), 0.0072644182, \left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), 0.0424060604, \left(\mathsf{fma}\left(\left(x \cdot x\right), 0.1049934947, 1\right)\right)\right)\right)\right)\right)\right)\right)\right) \cdot x}{\mathsf{fma}\left(\left(2 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.0001789971\right)\right), \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), \left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), 0.2909738639, \left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), 0.0694555761, \left(\mathsf{fma}\left(\left(x \cdot x\right), 0.7715471019, 1\right)\right)\right)\right)\right) + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot 0.0008327945 + 0.0140005442\right)\right)\right)}\right)\right)\right)}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -942486508.7001753:\\ \;\;\;\;\frac{0.5 + \frac{0.2514179000665375}{x \cdot x}}{x} + \frac{0.15298196345929327}{{x}^{5}}\\ \mathbf{elif}\;x \le 9551.679161702108:\\ \;\;\;\;\mathsf{log1p}\left(\left(\mathsf{expm1}\left(\left(\frac{x \cdot \mathsf{fma}\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.0001789971\right), \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), \left(\mathsf{fma}\left(0.0005064034, \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), \left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), 0.0072644182, \left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), 0.0424060604, \left(\mathsf{fma}\left(\left(x \cdot x\right), 0.1049934947, 1\right)\right)\right)\right)\right)\right)\right)\right)\right)}{\mathsf{fma}\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.0001789971\right) \cdot 2\right), \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), \left(\left(0.0140005442 + \left(x \cdot x\right) \cdot 0.0008327945\right) \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) + \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), 0.2909738639, \left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), 0.0694555761, \left(\mathsf{fma}\left(\left(x \cdot x\right), 0.7715471019, 1\right)\right)\right)\right)\right)\right)\right)}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{0.2514179000665375}{x \cdot x}}{x} + \frac{0.15298196345929327}{{x}^{5}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.dawson"
  (* (/ (+ (+ (+ (+ (+ 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))