Jmat.Real.dawson

Average Error: 29.3 → 0.0

Time: 1.1m

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 -12082511.11669040657579898834228515625:\\ \;\;\;\;\frac{0.2514179000665375252054900556686334311962}{\left(x \cdot x\right) \cdot x} + \left(\frac{0.5}{x} + \frac{0.1529819634592932686700805788859724998474}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}\right)\\ \mathbf{elif}\;x \le 22327.93475145058255293406546115875244141:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.007264418199999999985194687468492702464573, \left(x \cdot x\right) \cdot \left(x \cdot x\right), \mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right), 0.04240606040000000076517494562722276896238, \mathsf{fma}\left(0.1049934946999999951788851149103720672429, x \cdot x, 1\right)\right)\right) + \mathsf{fma}\left(x \cdot x, 1.789971000000000009994005623070734145585 \cdot 10^{-4}, 5.064034000000000243502107366566633572802 \cdot 10^{-4}\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}{\mathsf{fma}\left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}, \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right), \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right), \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 8.327945000000000442749725770852364803432 \cdot 10^{-4} + \left(x \cdot x\right) \cdot 0.01400054419999999938406531896362139377743, \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right), 0.06945557609999999937322456844412954524159, \mathsf{fma}\left(0.2909738639000000182122107617033179849386, \left(x \cdot x\right) \cdot \left(x \cdot x\right), \mathsf{fma}\left(x \cdot x, 0.7715471018999999763821051601553335785866, 1\right)\right)\right)\right)\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{0.2514179000665375252054900556686334311962}{\left(x \cdot x\right) \cdot x} + \left(\frac{0.5}{x} + \frac{0.1529819634592932686700805788859724998474}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}\right)\\ \end{array}\]

C

double f(double x) {
        double r5653306 = 1.0;
        double r5653307 = 0.1049934947;
        double r5653308 = x;
        double r5653309 = r5653308 * r5653308;
        double r5653310 = r5653307 * r5653309;
        double r5653311 = r5653306 + r5653310;
        double r5653312 = 0.0424060604;
        double r5653313 = r5653309 * r5653309;
        double r5653314 = r5653312 * r5653313;
        double r5653315 = r5653311 + r5653314;
        double r5653316 = 0.0072644182;
        double r5653317 = r5653313 * r5653309;
        double r5653318 = r5653316 * r5653317;
        double r5653319 = r5653315 + r5653318;
        double r5653320 = 0.0005064034;
        double r5653321 = r5653317 * r5653309;
        double r5653322 = r5653320 * r5653321;
        double r5653323 = r5653319 + r5653322;
        double r5653324 = 0.0001789971;
        double r5653325 = r5653321 * r5653309;
        double r5653326 = r5653324 * r5653325;
        double r5653327 = r5653323 + r5653326;
        double r5653328 = 0.7715471019;
        double r5653329 = r5653328 * r5653309;
        double r5653330 = r5653306 + r5653329;
        double r5653331 = 0.2909738639;
        double r5653332 = r5653331 * r5653313;
        double r5653333 = r5653330 + r5653332;
        double r5653334 = 0.0694555761;
        double r5653335 = r5653334 * r5653317;
        double r5653336 = r5653333 + r5653335;
        double r5653337 = 0.0140005442;
        double r5653338 = r5653337 * r5653321;
        double r5653339 = r5653336 + r5653338;
        double r5653340 = 0.0008327945;
        double r5653341 = r5653340 * r5653325;
        double r5653342 = r5653339 + r5653341;
        double r5653343 = 2.0;
        double r5653344 = r5653343 * r5653324;
        double r5653345 = r5653325 * r5653309;
        double r5653346 = r5653344 * r5653345;
        double r5653347 = r5653342 + r5653346;
        double r5653348 = r5653327 / r5653347;
        double r5653349 = r5653348 * r5653308;
        return r5653349;
}

↓

double f(double x) {
        double r5653350 = x;
        double r5653351 = -12082511.116690407;
        bool r5653352 = r5653350 <= r5653351;
        double r5653353 = 0.2514179000665375;
        double r5653354 = r5653350 * r5653350;
        double r5653355 = r5653354 * r5653350;
        double r5653356 = r5653353 / r5653355;
        double r5653357 = 0.5;
        double r5653358 = r5653357 / r5653350;
        double r5653359 = 0.15298196345929327;
        double r5653360 = r5653354 * r5653355;
        double r5653361 = r5653359 / r5653360;
        double r5653362 = r5653358 + r5653361;
        double r5653363 = r5653356 + r5653362;
        double r5653364 = 22327.934751450583;
        bool r5653365 = r5653350 <= r5653364;
        double r5653366 = 0.0072644182;
        double r5653367 = r5653354 * r5653366;
        double r5653368 = r5653354 * r5653354;
        double r5653369 = 0.0424060604;
        double r5653370 = 0.1049934947;
        double r5653371 = 1.0;
        double r5653372 = fma(r5653370, r5653354, r5653371);
        double r5653373 = fma(r5653368, r5653369, r5653372);
        double r5653374 = fma(r5653367, r5653368, r5653373);
        double r5653375 = 0.0001789971;
        double r5653376 = 0.0005064034;
        double r5653377 = fma(r5653354, r5653375, r5653376);
        double r5653378 = r5653368 * r5653368;
        double r5653379 = r5653377 * r5653378;
        double r5653380 = r5653374 + r5653379;
        double r5653381 = 2.0;
        double r5653382 = r5653381 * r5653375;
        double r5653383 = r5653368 * r5653354;
        double r5653384 = r5653383 * r5653383;
        double r5653385 = 0.0008327945;
        double r5653386 = r5653368 * r5653385;
        double r5653387 = 0.0140005442;
        double r5653388 = r5653354 * r5653387;
        double r5653389 = r5653386 + r5653388;
        double r5653390 = 0.0694555761;
        double r5653391 = 0.2909738639;
        double r5653392 = 0.7715471019;
        double r5653393 = fma(r5653354, r5653392, r5653371);
        double r5653394 = fma(r5653391, r5653368, r5653393);
        double r5653395 = fma(r5653383, r5653390, r5653394);
        double r5653396 = fma(r5653383, r5653389, r5653395);
        double r5653397 = fma(r5653382, r5653384, r5653396);
        double r5653398 = r5653380 / r5653397;
        double r5653399 = r5653398 * r5653350;
        double r5653400 = r5653365 ? r5653399 : r5653363;
        double r5653401 = r5653352 ? r5653363 : r5653400;
        return r5653401;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -12082511.116690407 or 22327.934751450583 < x

   1. Initial program 59.4

      \[\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 59.4

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

   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

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

   if -12082511.116690407 < x < 22327.934751450583

   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

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

   4. Applied *-commutative 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -12082511.11669040657579898834228515625:\\ \;\;\;\;\frac{0.2514179000665375252054900556686334311962}{\left(x \cdot x\right) \cdot x} + \left(\frac{0.5}{x} + \frac{0.1529819634592932686700805788859724998474}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}\right)\\ \mathbf{elif}\;x \le 22327.93475145058255293406546115875244141:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.007264418199999999985194687468492702464573, \left(x \cdot x\right) \cdot \left(x \cdot x\right), \mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right), 0.04240606040000000076517494562722276896238, \mathsf{fma}\left(0.1049934946999999951788851149103720672429, x \cdot x, 1\right)\right)\right) + \mathsf{fma}\left(x \cdot x, 1.789971000000000009994005623070734145585 \cdot 10^{-4}, 5.064034000000000243502107366566633572802 \cdot 10^{-4}\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}{\mathsf{fma}\left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}, \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right), \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right), \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 8.327945000000000442749725770852364803432 \cdot 10^{-4} + \left(x \cdot x\right) \cdot 0.01400054419999999938406531896362139377743, \mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right), 0.06945557609999999937322456844412954524159, \mathsf{fma}\left(0.2909738639000000182122107617033179849386, \left(x \cdot x\right) \cdot \left(x \cdot x\right), \mathsf{fma}\left(x \cdot x, 0.7715471018999999763821051601553335785866, 1\right)\right)\right)\right)\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{0.2514179000665375252054900556686334311962}{\left(x \cdot x\right) \cdot x} + \left(\frac{0.5}{x} + \frac{0.1529819634592932686700805788859724998474}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\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))