Jmat.Real.dawson

Average Error: 28.9 → 0.0

Time: 1.5m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -985.3534887239557065186090767383575439453 \lor \neg \left(x \le 641.7354232188947662507416680455207824707\right):\\ \;\;\;\;\frac{0.1529819634592932686700805788859724998474}{{x}^{5}} + \left(\frac{0.5}{x} + \frac{0.2514179000665375252054900556686334311962}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\mathsf{fma}\left({x}^{10}, 1.789971000000000009994005623070734145585 \cdot 10^{-4}, \mathsf{fma}\left({x}^{8}, 5.064034000000000243502107366566633572802 \cdot 10^{-4}, \mathsf{fma}\left(0.007264418199999999985194687468492702464573, {x}^{6}, \mathsf{fma}\left({x}^{4}, 0.04240606040000000076517494562722276896238, \sqrt[3]{{\left(\mathsf{fma}\left({x}^{2}, 0.1049934946999999951788851149103720672429, 1\right)\right)}^{3}}\right)\right)\right)\right)}{\mathsf{fma}\left({x}^{12} \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}, 2, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.01400054419999999938406531896362139377743, {x}^{6}, 8.327945000000000442749725770852364803432 \cdot 10^{-4} \cdot {x}^{8}\right), \mathsf{fma}\left(0.06945557609999999937322456844412954524159, {x}^{6}, \mathsf{fma}\left(0.2909738639000000182122107617033179849386, {x}^{4}, \mathsf{fma}\left(x \cdot x, 0.7715471018999999763821051601553335785866, 1\right)\right)\right)\right)\right)}\right)}^{3}} \cdot x\\ \end{array}\]

C

double f(double x) {
        double r169155 = 1.0;
        double r169156 = 0.1049934947;
        double r169157 = x;
        double r169158 = r169157 * r169157;
        double r169159 = r169156 * r169158;
        double r169160 = r169155 + r169159;
        double r169161 = 0.0424060604;
        double r169162 = r169158 * r169158;
        double r169163 = r169161 * r169162;
        double r169164 = r169160 + r169163;
        double r169165 = 0.0072644182;
        double r169166 = r169162 * r169158;
        double r169167 = r169165 * r169166;
        double r169168 = r169164 + r169167;
        double r169169 = 0.0005064034;
        double r169170 = r169166 * r169158;
        double r169171 = r169169 * r169170;
        double r169172 = r169168 + r169171;
        double r169173 = 0.0001789971;
        double r169174 = r169170 * r169158;
        double r169175 = r169173 * r169174;
        double r169176 = r169172 + r169175;
        double r169177 = 0.7715471019;
        double r169178 = r169177 * r169158;
        double r169179 = r169155 + r169178;
        double r169180 = 0.2909738639;
        double r169181 = r169180 * r169162;
        double r169182 = r169179 + r169181;
        double r169183 = 0.0694555761;
        double r169184 = r169183 * r169166;
        double r169185 = r169182 + r169184;
        double r169186 = 0.0140005442;
        double r169187 = r169186 * r169170;
        double r169188 = r169185 + r169187;
        double r169189 = 0.0008327945;
        double r169190 = r169189 * r169174;
        double r169191 = r169188 + r169190;
        double r169192 = 2.0;
        double r169193 = r169192 * r169173;
        double r169194 = r169174 * r169158;
        double r169195 = r169193 * r169194;
        double r169196 = r169191 + r169195;
        double r169197 = r169176 / r169196;
        double r169198 = r169197 * r169157;
        return r169198;
}

↓

double f(double x) {
        double r169199 = x;
        double r169200 = -985.3534887239557;
        bool r169201 = r169199 <= r169200;
        double r169202 = 641.7354232188948;
        bool r169203 = r169199 <= r169202;
        double r169204 = !r169203;
        bool r169205 = r169201 || r169204;
        double r169206 = 0.15298196345929327;
        double r169207 = 5.0;
        double r169208 = pow(r169199, r169207);
        double r169209 = r169206 / r169208;
        double r169210 = 0.5;
        double r169211 = r169210 / r169199;
        double r169212 = 0.2514179000665375;
        double r169213 = 3.0;
        double r169214 = pow(r169199, r169213);
        double r169215 = r169212 / r169214;
        double r169216 = r169211 + r169215;
        double r169217 = r169209 + r169216;
        double r169218 = 10.0;
        double r169219 = pow(r169199, r169218);
        double r169220 = 0.0001789971;
        double r169221 = 8.0;
        double r169222 = pow(r169199, r169221);
        double r169223 = 0.0005064034;
        double r169224 = 0.0072644182;
        double r169225 = 6.0;
        double r169226 = pow(r169199, r169225);
        double r169227 = 4.0;
        double r169228 = pow(r169199, r169227);
        double r169229 = 0.0424060604;
        double r169230 = 2.0;
        double r169231 = pow(r169199, r169230);
        double r169232 = 0.1049934947;
        double r169233 = 1.0;
        double r169234 = fma(r169231, r169232, r169233);
        double r169235 = pow(r169234, r169213);
        double r169236 = cbrt(r169235);
        double r169237 = fma(r169228, r169229, r169236);
        double r169238 = fma(r169224, r169226, r169237);
        double r169239 = fma(r169222, r169223, r169238);
        double r169240 = fma(r169219, r169220, r169239);
        double r169241 = 12.0;
        double r169242 = pow(r169199, r169241);
        double r169243 = r169242 * r169220;
        double r169244 = 2.0;
        double r169245 = r169199 * r169199;
        double r169246 = 0.0140005442;
        double r169247 = 0.0008327945;
        double r169248 = r169247 * r169222;
        double r169249 = fma(r169246, r169226, r169248);
        double r169250 = 0.0694555761;
        double r169251 = 0.2909738639;
        double r169252 = 0.7715471019;
        double r169253 = fma(r169245, r169252, r169233);
        double r169254 = fma(r169251, r169228, r169253);
        double r169255 = fma(r169250, r169226, r169254);
        double r169256 = fma(r169245, r169249, r169255);
        double r169257 = fma(r169243, r169244, r169256);
        double r169258 = r169240 / r169257;
        double r169259 = pow(r169258, r169213);
        double r169260 = cbrt(r169259);
        double r169261 = r169260 * r169199;
        double r169262 = r169205 ? r169217 : r169261;
        return r169262;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -985.3534887239557 or 641.7354232188948 < x

   1. Initial program 58.8

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

   3. Applied *-un-lft-identity 58.8

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

   4. Applied associate-*r* 58.8

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

   5. Simplified 58.8

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

   6. Taylor expanded around inf 0.0

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

   7. Simplified 0.0

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

   if -985.3534887239557 < x < 641.7354232188948

   1. Initial program 0.0

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

   3. Applied *-un-lft-identity 0.0

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

   4. Applied associate-*r* 0.0

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

   5. Simplified 0.0

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

   7. Applied add-cbrt-cube 0.0

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

   8. Applied add-cbrt-cube 0.0

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

   9. Applied cbrt-undiv 0.0

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

   10. Simplified 0.0

       \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{\mathsf{fma}\left({x}^{10}, 1.789971000000000009994005623070734145585 \cdot 10^{-4}, \mathsf{fma}\left({x}^{8}, 5.064034000000000243502107366566633572802 \cdot 10^{-4}, \mathsf{fma}\left(0.007264418199999999985194687468492702464573, {x}^{6}, \mathsf{fma}\left({x}^{4}, 0.04240606040000000076517494562722276896238, \mathsf{fma}\left(x \cdot x, 0.1049934946999999951788851149103720672429, 1\right)\right)\right)\right)\right)}{\mathsf{fma}\left({x}^{12} \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}, 2, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.01400054419999999938406531896362139377743, {x}^{6}, 8.327945000000000442749725770852364803432 \cdot 10^{-4} \cdot {x}^{8}\right), \mathsf{fma}\left(0.06945557609999999937322456844412954524159, {x}^{6}, \mathsf{fma}\left(0.2909738639000000182122107617033179849386, {x}^{4}, \mathsf{fma}\left(x \cdot x, 0.7715471018999999763821051601553335785866, 1\right)\right)\right)\right)\right)}\right)}^{3}}} \cdot x\]
   11. Using strategy rm

   12. Applied add-cbrt-cube 0.0

       \[\leadsto \sqrt[3]{{\left(\frac{\mathsf{fma}\left({x}^{10}, 1.789971000000000009994005623070734145585 \cdot 10^{-4}, \mathsf{fma}\left({x}^{8}, 5.064034000000000243502107366566633572802 \cdot 10^{-4}, \mathsf{fma}\left(0.007264418199999999985194687468492702464573, {x}^{6}, \mathsf{fma}\left({x}^{4}, 0.04240606040000000076517494562722276896238, \color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(x \cdot x, 0.1049934946999999951788851149103720672429, 1\right) \cdot \mathsf{fma}\left(x \cdot x, 0.1049934946999999951788851149103720672429, 1\right)\right) \cdot \mathsf{fma}\left(x \cdot x, 0.1049934946999999951788851149103720672429, 1\right)}}\right)\right)\right)\right)}{\mathsf{fma}\left({x}^{12} \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}, 2, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.01400054419999999938406531896362139377743, {x}^{6}, 8.327945000000000442749725770852364803432 \cdot 10^{-4} \cdot {x}^{8}\right), \mathsf{fma}\left(0.06945557609999999937322456844412954524159, {x}^{6}, \mathsf{fma}\left(0.2909738639000000182122107617033179849386, {x}^{4}, \mathsf{fma}\left(x \cdot x, 0.7715471018999999763821051601553335785866, 1\right)\right)\right)\right)\right)}\right)}^{3}} \cdot x\]

   13. Simplified 0.0

       \[\leadsto \sqrt[3]{{\left(\frac{\mathsf{fma}\left({x}^{10}, 1.789971000000000009994005623070734145585 \cdot 10^{-4}, \mathsf{fma}\left({x}^{8}, 5.064034000000000243502107366566633572802 \cdot 10^{-4}, \mathsf{fma}\left(0.007264418199999999985194687468492702464573, {x}^{6}, \mathsf{fma}\left({x}^{4}, 0.04240606040000000076517494562722276896238, \sqrt[3]{\color{blue}{{\left(\mathsf{fma}\left({x}^{2}, 0.1049934946999999951788851149103720672429, 1\right)\right)}^{3}}}\right)\right)\right)\right)}{\mathsf{fma}\left({x}^{12} \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}, 2, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.01400054419999999938406531896362139377743, {x}^{6}, 8.327945000000000442749725770852364803432 \cdot 10^{-4} \cdot {x}^{8}\right), \mathsf{fma}\left(0.06945557609999999937322456844412954524159, {x}^{6}, \mathsf{fma}\left(0.2909738639000000182122107617033179849386, {x}^{4}, \mathsf{fma}\left(x \cdot x, 0.7715471018999999763821051601553335785866, 1\right)\right)\right)\right)\right)}\right)}^{3}} \cdot x\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -985.3534887239557065186090767383575439453 \lor \neg \left(x \le 641.7354232188947662507416680455207824707\right):\\ \;\;\;\;\frac{0.1529819634592932686700805788859724998474}{{x}^{5}} + \left(\frac{0.5}{x} + \frac{0.2514179000665375252054900556686334311962}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\mathsf{fma}\left({x}^{10}, 1.789971000000000009994005623070734145585 \cdot 10^{-4}, \mathsf{fma}\left({x}^{8}, 5.064034000000000243502107366566633572802 \cdot 10^{-4}, \mathsf{fma}\left(0.007264418199999999985194687468492702464573, {x}^{6}, \mathsf{fma}\left({x}^{4}, 0.04240606040000000076517494562722276896238, \sqrt[3]{{\left(\mathsf{fma}\left({x}^{2}, 0.1049934946999999951788851149103720672429, 1\right)\right)}^{3}}\right)\right)\right)\right)}{\mathsf{fma}\left({x}^{12} \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}, 2, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.01400054419999999938406531896362139377743, {x}^{6}, 8.327945000000000442749725770852364803432 \cdot 10^{-4} \cdot {x}^{8}\right), \mathsf{fma}\left(0.06945557609999999937322456844412954524159, {x}^{6}, \mathsf{fma}\left(0.2909738639000000182122107617033179849386, {x}^{4}, \mathsf{fma}\left(x \cdot x, 0.7715471018999999763821051601553335785866, 1\right)\right)\right)\right)\right)}\right)}^{3}} \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.dawson"
  :precision binary64
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