Average Error: 28.8 → 0.0
Time: 3.1m
Precision: 64
\[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1345151761911.458:\\ \;\;\;\;\frac{0.5 + \frac{0.2514179000665375}{x \cdot x}}{x} + \frac{0.15298196345929327}{{x}^{5}}\\ \mathbf{elif}\;x \le 1189.8967229595753:\\ \;\;\;\;x \cdot \frac{\frac{1}{\sqrt{\frac{\mathsf{fma}\left(\left(0.0001789971 \cdot 2\right), \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right), \left(\mathsf{fma}\left(\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) \cdot 0.0008327945\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(\left(x \cdot x\right), 0.7715471019, 1\right) + \mathsf{fma}\left(0.2909738639, \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \left(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)\right)\right)\right)\right)\right)\right)\right)}{\mathsf{fma}\left(0.0001789971, \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\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(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\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)}}}}{\sqrt{\frac{\mathsf{fma}\left(\left(0.0001789971 \cdot 2\right), \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right), \left(\mathsf{fma}\left(\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) \cdot 0.0008327945\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(\left(x \cdot x\right), 0.7715471019, 1\right) + \mathsf{fma}\left(0.2909738639, \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \left(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)\right)\right)\right)\right)\right)\right)\right)}{\mathsf{fma}\left(0.0001789971, \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\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(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\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)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{0.2514179000665375}{x \cdot x}}{x} + \frac{0.15298196345929327}{{x}^{5}}\\ \end{array}\]
\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 -1345151761911.458:\\
\;\;\;\;\frac{0.5 + \frac{0.2514179000665375}{x \cdot x}}{x} + \frac{0.15298196345929327}{{x}^{5}}\\

\mathbf{elif}\;x \le 1189.8967229595753:\\
\;\;\;\;x \cdot \frac{\frac{1}{\sqrt{\frac{\mathsf{fma}\left(\left(0.0001789971 \cdot 2\right), \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right), \left(\mathsf{fma}\left(\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) \cdot 0.0008327945\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(\left(x \cdot x\right), 0.7715471019, 1\right) + \mathsf{fma}\left(0.2909738639, \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \left(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)\right)\right)\right)\right)\right)\right)\right)}{\mathsf{fma}\left(0.0001789971, \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\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(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\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)}}}}{\sqrt{\frac{\mathsf{fma}\left(\left(0.0001789971 \cdot 2\right), \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right), \left(\mathsf{fma}\left(\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) \cdot 0.0008327945\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(\left(x \cdot x\right), 0.7715471019, 1\right) + \mathsf{fma}\left(0.2909738639, \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \left(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)\right)\right)\right)\right)\right)\right)\right)}{\mathsf{fma}\left(0.0001789971, \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\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(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\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)}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{0.5 + \frac{0.2514179000665375}{x \cdot x}}{x} + \frac{0.15298196345929327}{{x}^{5}}\\

\end{array}
double f(double x) {
        double r27756047 = 1.0;
        double r27756048 = 0.1049934947;
        double r27756049 = x;
        double r27756050 = r27756049 * r27756049;
        double r27756051 = r27756048 * r27756050;
        double r27756052 = r27756047 + r27756051;
        double r27756053 = 0.0424060604;
        double r27756054 = r27756050 * r27756050;
        double r27756055 = r27756053 * r27756054;
        double r27756056 = r27756052 + r27756055;
        double r27756057 = 0.0072644182;
        double r27756058 = r27756054 * r27756050;
        double r27756059 = r27756057 * r27756058;
        double r27756060 = r27756056 + r27756059;
        double r27756061 = 0.0005064034;
        double r27756062 = r27756058 * r27756050;
        double r27756063 = r27756061 * r27756062;
        double r27756064 = r27756060 + r27756063;
        double r27756065 = 0.0001789971;
        double r27756066 = r27756062 * r27756050;
        double r27756067 = r27756065 * r27756066;
        double r27756068 = r27756064 + r27756067;
        double r27756069 = 0.7715471019;
        double r27756070 = r27756069 * r27756050;
        double r27756071 = r27756047 + r27756070;
        double r27756072 = 0.2909738639;
        double r27756073 = r27756072 * r27756054;
        double r27756074 = r27756071 + r27756073;
        double r27756075 = 0.0694555761;
        double r27756076 = r27756075 * r27756058;
        double r27756077 = r27756074 + r27756076;
        double r27756078 = 0.0140005442;
        double r27756079 = r27756078 * r27756062;
        double r27756080 = r27756077 + r27756079;
        double r27756081 = 0.0008327945;
        double r27756082 = r27756081 * r27756066;
        double r27756083 = r27756080 + r27756082;
        double r27756084 = 2.0;
        double r27756085 = r27756084 * r27756065;
        double r27756086 = r27756066 * r27756050;
        double r27756087 = r27756085 * r27756086;
        double r27756088 = r27756083 + r27756087;
        double r27756089 = r27756068 / r27756088;
        double r27756090 = r27756089 * r27756049;
        return r27756090;
}
↓
double f(double x) {
        double r27756091 = x;
        double r27756092 = -1345151761911.458;
        bool r27756093 = r27756091 <= r27756092;
        double r27756094 = 0.5;
        double r27756095 = 0.2514179000665375;
        double r27756096 = r27756091 * r27756091;
        double r27756097 = r27756095 / r27756096;
        double r27756098 = r27756094 + r27756097;
        double r27756099 = r27756098 / r27756091;
        double r27756100 = 0.15298196345929327;
        double r27756101 = 5.0;
        double r27756102 = pow(r27756091, r27756101);
        double r27756103 = r27756100 / r27756102;
        double r27756104 = r27756099 + r27756103;
        double r27756105 = 1189.8967229595753;
        bool r27756106 = r27756091 <= r27756105;
        double r27756107 = 1.0;
        double r27756108 = 0.0001789971;
        double r27756109 = 2.0;
        double r27756110 = r27756108 * r27756109;
        double r27756111 = r27756096 * r27756096;
        double r27756112 = r27756111 * r27756096;
        double r27756113 = r27756112 * r27756112;
        double r27756114 = r27756111 * r27756111;
        double r27756115 = 0.0008327945;
        double r27756116 = r27756114 * r27756115;
        double r27756117 = 0.0140005442;
        double r27756118 = 0.7715471019;
        double r27756119 = fma(r27756096, r27756118, r27756107);
        double r27756120 = 0.2909738639;
        double r27756121 = 0.0694555761;
        double r27756122 = r27756121 * r27756112;
        double r27756123 = fma(r27756120, r27756111, r27756122);
        double r27756124 = r27756119 + r27756123;
        double r27756125 = fma(r27756114, r27756117, r27756124);
        double r27756126 = fma(r27756116, r27756096, r27756125);
        double r27756127 = fma(r27756110, r27756113, r27756126);
        double r27756128 = r27756111 * r27756112;
        double r27756129 = 0.0005064034;
        double r27756130 = 0.0072644182;
        double r27756131 = 0.0424060604;
        double r27756132 = 0.1049934947;
        double r27756133 = fma(r27756132, r27756096, r27756107);
        double r27756134 = fma(r27756111, r27756131, r27756133);
        double r27756135 = fma(r27756130, r27756112, r27756134);
        double r27756136 = fma(r27756129, r27756114, r27756135);
        double r27756137 = fma(r27756108, r27756128, r27756136);
        double r27756138 = r27756127 / r27756137;
        double r27756139 = sqrt(r27756138);
        double r27756140 = r27756107 / r27756139;
        double r27756141 = r27756140 / r27756139;
        double r27756142 = r27756091 * r27756141;
        double r27756143 = r27756106 ? r27756142 : r27756104;
        double r27756144 = r27756093 ? r27756104 : r27756143;
        return r27756144;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -1345151761911.458 or 1189.8967229595753 < x

    1. Initial program 59.0

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

    2. Simplified59.0

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

    3. 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. Simplified0.0

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

    if -1345151761911.458 < x < 1189.8967229595753

    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. Simplified0.0

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

    4. Applied clear-num0.0

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

    6. Applied add-sqr-sqrt0.0

      \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{\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)}{\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 \sqrt{\frac{\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)}{\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 x\]

    7. Applied associate-/r*0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1345151761911.458:\\ \;\;\;\;\frac{0.5 + \frac{0.2514179000665375}{x \cdot x}}{x} + \frac{0.15298196345929327}{{x}^{5}}\\ \mathbf{elif}\;x \le 1189.8967229595753:\\ \;\;\;\;x \cdot \frac{\frac{1}{\sqrt{\frac{\mathsf{fma}\left(\left(0.0001789971 \cdot 2\right), \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right), \left(\mathsf{fma}\left(\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) \cdot 0.0008327945\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(\left(x \cdot x\right), 0.7715471019, 1\right) + \mathsf{fma}\left(0.2909738639, \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \left(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)\right)\right)\right)\right)\right)\right)\right)}{\mathsf{fma}\left(0.0001789971, \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\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(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\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)}}}}{\sqrt{\frac{\mathsf{fma}\left(\left(0.0001789971 \cdot 2\right), \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right), \left(\mathsf{fma}\left(\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) \cdot 0.0008327945\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(\left(x \cdot x\right), 0.7715471019, 1\right) + \mathsf{fma}\left(0.2909738639, \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \left(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)\right)\right)\right)\right)\right)\right)\right)}{\mathsf{fma}\left(0.0001789971, \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\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(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\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)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 + \frac{0.2514179000665375}{x \cdot x}}{x} + \frac{0.15298196345929327}{{x}^{5}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 +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))