Jmat.Real.dawson

Average Error: 28.8 → 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 -4.936575955309851 \cdot 10^{+24}:\\ \;\;\;\;\frac{0.5}{x} + \left(\frac{0.2514179000665375}{\left(x \cdot x\right) \cdot x} + \frac{0.15298196345929327}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}\right)\\ \mathbf{elif}\;x \le 782.3997553574673:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\left(0.0001789971 \cdot 2\right), \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(\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(\left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), \left(x \cdot x\right), \left(\mathsf{fma}\left(0.2909738639, \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \left(\mathsf{fma}\left(\left(x \cdot x\right), 0.7715471019, 1\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)}{\mathsf{fma}\left(\left(0.0001789971 \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(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(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), \left(x \cdot x\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\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} + \left(\frac{0.2514179000665375}{\left(x \cdot x\right) \cdot x} + \frac{0.15298196345929327}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}\right)\\ \end{array}\]

C

double f(double x) {
        double r7561218 = 1.0;
        double r7561219 = 0.1049934947;
        double r7561220 = x;
        double r7561221 = r7561220 * r7561220;
        double r7561222 = r7561219 * r7561221;
        double r7561223 = r7561218 + r7561222;
        double r7561224 = 0.0424060604;
        double r7561225 = r7561221 * r7561221;
        double r7561226 = r7561224 * r7561225;
        double r7561227 = r7561223 + r7561226;
        double r7561228 = 0.0072644182;
        double r7561229 = r7561225 * r7561221;
        double r7561230 = r7561228 * r7561229;
        double r7561231 = r7561227 + r7561230;
        double r7561232 = 0.0005064034;
        double r7561233 = r7561229 * r7561221;
        double r7561234 = r7561232 * r7561233;
        double r7561235 = r7561231 + r7561234;
        double r7561236 = 0.0001789971;
        double r7561237 = r7561233 * r7561221;
        double r7561238 = r7561236 * r7561237;
        double r7561239 = r7561235 + r7561238;
        double r7561240 = 0.7715471019;
        double r7561241 = r7561240 * r7561221;
        double r7561242 = r7561218 + r7561241;
        double r7561243 = 0.2909738639;
        double r7561244 = r7561243 * r7561225;
        double r7561245 = r7561242 + r7561244;
        double r7561246 = 0.0694555761;
        double r7561247 = r7561246 * r7561229;
        double r7561248 = r7561245 + r7561247;
        double r7561249 = 0.0140005442;
        double r7561250 = r7561249 * r7561233;
        double r7561251 = r7561248 + r7561250;
        double r7561252 = 0.0008327945;
        double r7561253 = r7561252 * r7561237;
        double r7561254 = r7561251 + r7561253;
        double r7561255 = 2.0;
        double r7561256 = r7561255 * r7561236;
        double r7561257 = r7561237 * r7561221;
        double r7561258 = r7561256 * r7561257;
        double r7561259 = r7561254 + r7561258;
        double r7561260 = r7561239 / r7561259;
        double r7561261 = r7561260 * r7561220;
        return r7561261;
}

↓

double f(double x) {
        double r7561262 = x;
        double r7561263 = -4.936575955309851e+24;
        bool r7561264 = r7561262 <= r7561263;
        double r7561265 = 0.5;
        double r7561266 = r7561265 / r7561262;
        double r7561267 = 0.2514179000665375;
        double r7561268 = r7561262 * r7561262;
        double r7561269 = r7561268 * r7561262;
        double r7561270 = r7561267 / r7561269;
        double r7561271 = 0.15298196345929327;
        double r7561272 = r7561268 * r7561269;
        double r7561273 = r7561271 / r7561272;
        double r7561274 = r7561270 + r7561273;
        double r7561275 = r7561266 + r7561274;
        double r7561276 = 782.3997553574673;
        bool r7561277 = r7561262 <= r7561276;
        double r7561278 = 1.0;
        double r7561279 = 0.0001789971;
        double r7561280 = 2.0;
        double r7561281 = r7561279 * r7561280;
        double r7561282 = r7561268 * r7561268;
        double r7561283 = r7561282 * r7561282;
        double r7561284 = r7561282 * r7561283;
        double r7561285 = 0.0008327945;
        double r7561286 = r7561285 * r7561283;
        double r7561287 = 0.0140005442;
        double r7561288 = 0.0694555761;
        double r7561289 = r7561288 * r7561282;
        double r7561290 = 0.2909738639;
        double r7561291 = 0.7715471019;
        double r7561292 = fma(r7561268, r7561291, r7561278);
        double r7561293 = fma(r7561290, r7561282, r7561292);
        double r7561294 = fma(r7561289, r7561268, r7561293);
        double r7561295 = fma(r7561283, r7561287, r7561294);
        double r7561296 = fma(r7561286, r7561268, r7561295);
        double r7561297 = fma(r7561281, r7561284, r7561296);
        double r7561298 = r7561279 * r7561283;
        double r7561299 = 0.0005064034;
        double r7561300 = 0.0072644182;
        double r7561301 = r7561300 * r7561282;
        double r7561302 = 0.0424060604;
        double r7561303 = 0.1049934947;
        double r7561304 = fma(r7561303, r7561268, r7561278);
        double r7561305 = fma(r7561282, r7561302, r7561304);
        double r7561306 = fma(r7561301, r7561268, r7561305);
        double r7561307 = fma(r7561299, r7561283, r7561306);
        double r7561308 = fma(r7561298, r7561268, r7561307);
        double r7561309 = r7561297 / r7561308;
        double r7561310 = r7561278 / r7561309;
        double r7561311 = r7561310 * r7561262;
        double r7561312 = r7561277 ? r7561311 : r7561275;
        double r7561313 = r7561264 ? r7561275 : r7561312;
        return r7561313;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -4.936575955309851e+24 or 782.3997553574673 < x

   1. Initial program 60.5

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

   3. Applied *-un-lft-identity 60.5

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

   4. Applied associate-/l* 60.5

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

   5. Simplified 60.5

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

   7. Applied expm1-log1p-u 60.5

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

   8. Taylor expanded around inf 0.0

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

   9. Simplified 0.0

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

   if -4.936575955309851e+24 < x < 782.3997553574673

   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. Using strategy rm

   3. Applied *-un-lft-identity 0.0

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

   4. Applied associate-/l* 0.0

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

   5. Simplified 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.936575955309851 \cdot 10^{+24}:\\ \;\;\;\;\frac{0.5}{x} + \left(\frac{0.2514179000665375}{\left(x \cdot x\right) \cdot x} + \frac{0.15298196345929327}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}\right)\\ \mathbf{elif}\;x \le 782.3997553574673:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\left(0.0001789971 \cdot 2\right), \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(\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(\left(0.0694555761 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), \left(x \cdot x\right), \left(\mathsf{fma}\left(0.2909738639, \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \left(\mathsf{fma}\left(\left(x \cdot x\right), 0.7715471019, 1\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)\right)}{\mathsf{fma}\left(\left(0.0001789971 \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(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(0.0072644182 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right), \left(x \cdot x\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\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} + \left(\frac{0.2514179000665375}{\left(x \cdot x\right) \cdot x} + \frac{0.15298196345929327}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 +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))