Jmat.Real.dawson

Average Error: 29.3 → 0.0

Time: 27.3s

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 -62102401015.90998077392578125 \lor \neg \left(x \le 719.129990331679209702997468411922454834\right):\\ \;\;\;\;\frac{0.1529819634592932686700805788859724998474}{{x}^{5}} + \left(\frac{0.2514179000665375252054900556686334311962}{{x}^{3}} + \frac{0.5}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left({\left(x \cdot x\right)}^{4}, \mathsf{fma}\left(x, x \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}, 5.064034000000000243502107366566633572802 \cdot 10^{-4}\right), \mathsf{fma}\left(0.007264418199999999985194687468492702464573, {x}^{6}, \mathsf{fma}\left(0.04240606040000000076517494562722276896238, {x}^{4}, \mathsf{fma}\left(x \cdot x, 0.1049934946999999951788851149103720672429, 1\right)\right)\right)\right)}{\mathsf{fma}\left(2, {\left(x \cdot x\right)}^{6} \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}, \mathsf{fma}\left(\mathsf{fma}\left(8.327945000000000442749725770852364803432 \cdot 10^{-4}, x \cdot x, 0.01400054419999999938406531896362139377743\right), {\left(x \cdot x\right)}^{4}, \mathsf{fma}\left({x}^{6}, 0.06945557609999999937322456844412954524159, \mathsf{fma}\left({x}^{4}, 0.2909738639000000182122107617033179849386, \mathsf{fma}\left(x, x \cdot 0.7715471018999999763821051601553335785866, 1\right)\right)\right)\right)\right)}} \cdot \left(x \cdot \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}, x, 5.064034000000000243502107366566633572802 \cdot 10^{-4}\right), {\left({x}^{2}\right)}^{4}, \mathsf{fma}\left({x}^{6}, 0.007264418199999999985194687468492702464573, \mathsf{fma}\left({x}^{4}, 0.04240606040000000076517494562722276896238, \mathsf{fma}\left(0.1049934946999999951788851149103720672429, {x}^{2}, 1\right)\right)\right)\right)}}{\sqrt{\mathsf{fma}\left(2, 1.789971000000000009994005623070734145585 \cdot 10^{-4} \cdot {\left({x}^{2}\right)}^{6}, \mathsf{fma}\left({\left({x}^{2}\right)}^{4}, \mathsf{fma}\left(8.327945000000000442749725770852364803432 \cdot 10^{-4}, x \cdot x, 0.01400054419999999938406531896362139377743\right), \mathsf{fma}\left(0.06945557609999999937322456844412954524159, {x}^{6}, \mathsf{fma}\left(0.2909738639000000182122107617033179849386, {x}^{4}, \mathsf{fma}\left({x}^{2}, 0.7715471018999999763821051601553335785866, 1\right)\right)\right)\right)\right)}}\right)\\ \end{array}\]

C

double f(double x) {
        double r188289 = 1.0;
        double r188290 = 0.1049934947;
        double r188291 = x;
        double r188292 = r188291 * r188291;
        double r188293 = r188290 * r188292;
        double r188294 = r188289 + r188293;
        double r188295 = 0.0424060604;
        double r188296 = r188292 * r188292;
        double r188297 = r188295 * r188296;
        double r188298 = r188294 + r188297;
        double r188299 = 0.0072644182;
        double r188300 = r188296 * r188292;
        double r188301 = r188299 * r188300;
        double r188302 = r188298 + r188301;
        double r188303 = 0.0005064034;
        double r188304 = r188300 * r188292;
        double r188305 = r188303 * r188304;
        double r188306 = r188302 + r188305;
        double r188307 = 0.0001789971;
        double r188308 = r188304 * r188292;
        double r188309 = r188307 * r188308;
        double r188310 = r188306 + r188309;
        double r188311 = 0.7715471019;
        double r188312 = r188311 * r188292;
        double r188313 = r188289 + r188312;
        double r188314 = 0.2909738639;
        double r188315 = r188314 * r188296;
        double r188316 = r188313 + r188315;
        double r188317 = 0.0694555761;
        double r188318 = r188317 * r188300;
        double r188319 = r188316 + r188318;
        double r188320 = 0.0140005442;
        double r188321 = r188320 * r188304;
        double r188322 = r188319 + r188321;
        double r188323 = 0.0008327945;
        double r188324 = r188323 * r188308;
        double r188325 = r188322 + r188324;
        double r188326 = 2.0;
        double r188327 = r188326 * r188307;
        double r188328 = r188308 * r188292;
        double r188329 = r188327 * r188328;
        double r188330 = r188325 + r188329;
        double r188331 = r188310 / r188330;
        double r188332 = r188331 * r188291;
        return r188332;
}

↓

double f(double x) {
        double r188333 = x;
        double r188334 = -62102401015.90998;
        bool r188335 = r188333 <= r188334;
        double r188336 = 719.1299903316792;
        bool r188337 = r188333 <= r188336;
        double r188338 = !r188337;
        bool r188339 = r188335 || r188338;
        double r188340 = 0.15298196345929327;
        double r188341 = 5.0;
        double r188342 = pow(r188333, r188341);
        double r188343 = r188340 / r188342;
        double r188344 = 0.2514179000665375;
        double r188345 = 3.0;
        double r188346 = pow(r188333, r188345);
        double r188347 = r188344 / r188346;
        double r188348 = 0.5;
        double r188349 = r188348 / r188333;
        double r188350 = r188347 + r188349;
        double r188351 = r188343 + r188350;
        double r188352 = r188333 * r188333;
        double r188353 = 4.0;
        double r188354 = pow(r188352, r188353);
        double r188355 = 0.0001789971;
        double r188356 = r188333 * r188355;
        double r188357 = 0.0005064034;
        double r188358 = fma(r188333, r188356, r188357);
        double r188359 = 0.0072644182;
        double r188360 = 6.0;
        double r188361 = pow(r188333, r188360);
        double r188362 = 0.0424060604;
        double r188363 = pow(r188333, r188353);
        double r188364 = 0.1049934947;
        double r188365 = 1.0;
        double r188366 = fma(r188352, r188364, r188365);
        double r188367 = fma(r188362, r188363, r188366);
        double r188368 = fma(r188359, r188361, r188367);
        double r188369 = fma(r188354, r188358, r188368);
        double r188370 = 2.0;
        double r188371 = pow(r188352, r188360);
        double r188372 = r188371 * r188355;
        double r188373 = 0.0008327945;
        double r188374 = 0.0140005442;
        double r188375 = fma(r188373, r188352, r188374);
        double r188376 = 0.0694555761;
        double r188377 = 0.2909738639;
        double r188378 = 0.7715471019;
        double r188379 = r188333 * r188378;
        double r188380 = fma(r188333, r188379, r188365);
        double r188381 = fma(r188363, r188377, r188380);
        double r188382 = fma(r188361, r188376, r188381);
        double r188383 = fma(r188375, r188354, r188382);
        double r188384 = fma(r188370, r188372, r188383);
        double r188385 = r188369 / r188384;
        double r188386 = sqrt(r188385);
        double r188387 = fma(r188356, r188333, r188357);
        double r188388 = 2.0;
        double r188389 = pow(r188333, r188388);
        double r188390 = pow(r188389, r188353);
        double r188391 = fma(r188364, r188389, r188365);
        double r188392 = fma(r188363, r188362, r188391);
        double r188393 = fma(r188361, r188359, r188392);
        double r188394 = fma(r188387, r188390, r188393);
        double r188395 = sqrt(r188394);
        double r188396 = pow(r188389, r188360);
        double r188397 = r188355 * r188396;
        double r188398 = fma(r188389, r188378, r188365);
        double r188399 = fma(r188377, r188363, r188398);
        double r188400 = fma(r188376, r188361, r188399);
        double r188401 = fma(r188390, r188375, r188400);
        double r188402 = fma(r188370, r188397, r188401);
        double r188403 = sqrt(r188402);
        double r188404 = r188395 / r188403;
        double r188405 = r188333 * r188404;
        double r188406 = r188386 * r188405;
        double r188407 = r188339 ? r188351 : r188406;
        return r188407;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -62102401015.90998 or 719.1299903316792 < x

   1. Initial program 59.7

      \[\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. Simplified 59.7

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

   3. Taylor expanded around inf 0.0

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

   4. Simplified 0.0

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

   if -62102401015.90998 < x < 719.1299903316792

   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. Simplified 0.0

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

   4. Applied add-sqr-sqrt 0.0

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

   5. Applied add-sqr-sqrt 0.0

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

   6. Applied times-frac 0.0

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

   7. Applied associate-*l* 0.0

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

   8. Simplified 0.0

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

   10. Applied sqrt-undiv 0.0

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

   11. Simplified 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -62102401015.90998077392578125 \lor \neg \left(x \le 719.129990331679209702997468411922454834\right):\\ \;\;\;\;\frac{0.1529819634592932686700805788859724998474}{{x}^{5}} + \left(\frac{0.2514179000665375252054900556686334311962}{{x}^{3}} + \frac{0.5}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left({\left(x \cdot x\right)}^{4}, \mathsf{fma}\left(x, x \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}, 5.064034000000000243502107366566633572802 \cdot 10^{-4}\right), \mathsf{fma}\left(0.007264418199999999985194687468492702464573, {x}^{6}, \mathsf{fma}\left(0.04240606040000000076517494562722276896238, {x}^{4}, \mathsf{fma}\left(x \cdot x, 0.1049934946999999951788851149103720672429, 1\right)\right)\right)\right)}{\mathsf{fma}\left(2, {\left(x \cdot x\right)}^{6} \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}, \mathsf{fma}\left(\mathsf{fma}\left(8.327945000000000442749725770852364803432 \cdot 10^{-4}, x \cdot x, 0.01400054419999999938406531896362139377743\right), {\left(x \cdot x\right)}^{4}, \mathsf{fma}\left({x}^{6}, 0.06945557609999999937322456844412954524159, \mathsf{fma}\left({x}^{4}, 0.2909738639000000182122107617033179849386, \mathsf{fma}\left(x, x \cdot 0.7715471018999999763821051601553335785866, 1\right)\right)\right)\right)\right)}} \cdot \left(x \cdot \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}, x, 5.064034000000000243502107366566633572802 \cdot 10^{-4}\right), {\left({x}^{2}\right)}^{4}, \mathsf{fma}\left({x}^{6}, 0.007264418199999999985194687468492702464573, \mathsf{fma}\left({x}^{4}, 0.04240606040000000076517494562722276896238, \mathsf{fma}\left(0.1049934946999999951788851149103720672429, {x}^{2}, 1\right)\right)\right)\right)}}{\sqrt{\mathsf{fma}\left(2, 1.789971000000000009994005623070734145585 \cdot 10^{-4} \cdot {\left({x}^{2}\right)}^{6}, \mathsf{fma}\left({\left({x}^{2}\right)}^{4}, \mathsf{fma}\left(8.327945000000000442749725770852364803432 \cdot 10^{-4}, x \cdot x, 0.01400054419999999938406531896362139377743\right), \mathsf{fma}\left(0.06945557609999999937322456844412954524159, {x}^{6}, \mathsf{fma}\left(0.2909738639000000182122107617033179849386, {x}^{4}, \mathsf{fma}\left({x}^{2}, 0.7715471018999999763821051601553335785866, 1\right)\right)\right)\right)\right)}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.dawson"
  (* (/ (+ (+ (+ (+ (+ 1.0 (* 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 (* 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 0.0001789971) (* (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)) (* x x))))) x))