Jmat.Real.dawson

Average Error: 29.3 → 0.0

Time: 32.6s

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 -314168.2807482867501676082611083984375 \lor \neg \left(x \le 662.7162576505425022332929074764251708984\right):\\ \;\;\;\;\frac{0.1529819634592932686700805788859724998474}{{x}^{5}} + \left(\frac{0.5}{x} + \frac{0.2514179000665375252054900556686334311962}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(0.1049934946999999951788851149103720672429 \cdot \left(x \cdot x\right) + \left({x}^{6} \cdot 0.007264418199999999985194687468492702464573 + 1\right)\right) + \left(\left(5.064034000000000243502107366566633572802 \cdot 10^{-4} \cdot {x}^{4} + {x}^{6} \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right) + 0.04240606040000000076517494562722276896238\right) \cdot {x}^{4}}}{\sqrt{\left(\left({x}^{4} \cdot \left(0.2909738639000000182122107617033179849386 + x \cdot \left(x \cdot 0.06945557609999999937322456844412954524159\right)\right) + 1\right) + \left(0.01400054419999999938406531896362139377743 + \left(x \cdot x\right) \cdot \left(\left(2 \cdot \left(x \cdot x\right)\right) \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4} + 8.327945000000000442749725770852364803432 \cdot 10^{-4}\right)\right) \cdot {\left(x \cdot x\right)}^{4}\right) + \left(x \cdot x\right) \cdot 0.7715471018999999763821051601553335785866}} \cdot \left(\sqrt{\frac{\left({x}^{4} \cdot \left(5.064034000000000243502107366566633572802 \cdot 10^{-4} \cdot {x}^{4} + \left({x}^{6} \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4} + 0.04240606040000000076517494562722276896238\right)\right) + 0.1049934946999999951788851149103720672429 \cdot \left(x \cdot x\right)\right) + \left({x}^{6} \cdot 0.007264418199999999985194687468492702464573 + 1\right)}{\left(0.01400054419999999938406531896362139377743 + \left(x \cdot x\right) \cdot \left(\left(2 \cdot \left(x \cdot x\right)\right) \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4} + 8.327945000000000442749725770852364803432 \cdot 10^{-4}\right)\right) \cdot {\left(x \cdot x\right)}^{4} + \left({x}^{4} \cdot \left(0.2909738639000000182122107617033179849386 + \left(x \cdot x\right) \cdot 0.06945557609999999937322456844412954524159\right) + \left(\left(x \cdot 0.7715471018999999763821051601553335785866\right) \cdot x + 1\right)\right)}} \cdot x\right)\\ \end{array}\]

C

double f(double x) {
        double r178402 = 1.0;
        double r178403 = 0.1049934947;
        double r178404 = x;
        double r178405 = r178404 * r178404;
        double r178406 = r178403 * r178405;
        double r178407 = r178402 + r178406;
        double r178408 = 0.0424060604;
        double r178409 = r178405 * r178405;
        double r178410 = r178408 * r178409;
        double r178411 = r178407 + r178410;
        double r178412 = 0.0072644182;
        double r178413 = r178409 * r178405;
        double r178414 = r178412 * r178413;
        double r178415 = r178411 + r178414;
        double r178416 = 0.0005064034;
        double r178417 = r178413 * r178405;
        double r178418 = r178416 * r178417;
        double r178419 = r178415 + r178418;
        double r178420 = 0.0001789971;
        double r178421 = r178417 * r178405;
        double r178422 = r178420 * r178421;
        double r178423 = r178419 + r178422;
        double r178424 = 0.7715471019;
        double r178425 = r178424 * r178405;
        double r178426 = r178402 + r178425;
        double r178427 = 0.2909738639;
        double r178428 = r178427 * r178409;
        double r178429 = r178426 + r178428;
        double r178430 = 0.0694555761;
        double r178431 = r178430 * r178413;
        double r178432 = r178429 + r178431;
        double r178433 = 0.0140005442;
        double r178434 = r178433 * r178417;
        double r178435 = r178432 + r178434;
        double r178436 = 0.0008327945;
        double r178437 = r178436 * r178421;
        double r178438 = r178435 + r178437;
        double r178439 = 2.0;
        double r178440 = r178439 * r178420;
        double r178441 = r178421 * r178405;
        double r178442 = r178440 * r178441;
        double r178443 = r178438 + r178442;
        double r178444 = r178423 / r178443;
        double r178445 = r178444 * r178404;
        return r178445;
}

↓

double f(double x) {
        double r178446 = x;
        double r178447 = -314168.28074828675;
        bool r178448 = r178446 <= r178447;
        double r178449 = 662.7162576505425;
        bool r178450 = r178446 <= r178449;
        double r178451 = !r178450;
        bool r178452 = r178448 || r178451;
        double r178453 = 0.15298196345929327;
        double r178454 = 5.0;
        double r178455 = pow(r178446, r178454);
        double r178456 = r178453 / r178455;
        double r178457 = 0.5;
        double r178458 = r178457 / r178446;
        double r178459 = 0.2514179000665375;
        double r178460 = 3.0;
        double r178461 = pow(r178446, r178460);
        double r178462 = r178459 / r178461;
        double r178463 = r178458 + r178462;
        double r178464 = r178456 + r178463;
        double r178465 = 0.1049934947;
        double r178466 = r178446 * r178446;
        double r178467 = r178465 * r178466;
        double r178468 = 6.0;
        double r178469 = pow(r178446, r178468);
        double r178470 = 0.0072644182;
        double r178471 = r178469 * r178470;
        double r178472 = 1.0;
        double r178473 = r178471 + r178472;
        double r178474 = r178467 + r178473;
        double r178475 = 0.0005064034;
        double r178476 = 4.0;
        double r178477 = pow(r178446, r178476);
        double r178478 = r178475 * r178477;
        double r178479 = 0.0001789971;
        double r178480 = r178469 * r178479;
        double r178481 = r178478 + r178480;
        double r178482 = 0.0424060604;
        double r178483 = r178481 + r178482;
        double r178484 = r178483 * r178477;
        double r178485 = r178474 + r178484;
        double r178486 = sqrt(r178485);
        double r178487 = 0.2909738639;
        double r178488 = 0.0694555761;
        double r178489 = r178446 * r178488;
        double r178490 = r178446 * r178489;
        double r178491 = r178487 + r178490;
        double r178492 = r178477 * r178491;
        double r178493 = r178492 + r178472;
        double r178494 = 0.0140005442;
        double r178495 = 2.0;
        double r178496 = r178495 * r178466;
        double r178497 = r178496 * r178479;
        double r178498 = 0.0008327945;
        double r178499 = r178497 + r178498;
        double r178500 = r178466 * r178499;
        double r178501 = r178494 + r178500;
        double r178502 = pow(r178466, r178476);
        double r178503 = r178501 * r178502;
        double r178504 = r178493 + r178503;
        double r178505 = 0.7715471019;
        double r178506 = r178466 * r178505;
        double r178507 = r178504 + r178506;
        double r178508 = sqrt(r178507);
        double r178509 = r178486 / r178508;
        double r178510 = r178480 + r178482;
        double r178511 = r178478 + r178510;
        double r178512 = r178477 * r178511;
        double r178513 = r178512 + r178467;
        double r178514 = r178513 + r178473;
        double r178515 = r178466 * r178488;
        double r178516 = r178487 + r178515;
        double r178517 = r178477 * r178516;
        double r178518 = r178446 * r178505;
        double r178519 = r178518 * r178446;
        double r178520 = r178519 + r178472;
        double r178521 = r178517 + r178520;
        double r178522 = r178503 + r178521;
        double r178523 = r178514 / r178522;
        double r178524 = sqrt(r178523);
        double r178525 = r178524 * r178446;
        double r178526 = r178509 * r178525;
        double r178527 = r178452 ? r178464 : r178526;
        return r178527;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -314168.28074828675 or 662.7162576505425 < x

   1. Initial program 59.1

      \[\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 59.1

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

   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

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

   if -314168.28074828675 < x < 662.7162576505425

   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

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

   4. Applied add-sqr-sqrt 0.0

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

   5. Applied add-sqr-sqrt 0.0

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

   6. Applied times-frac 0.0

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

   7. Applied associate-*l* 0.0

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

   8. Simplified 0.0

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

   10. Applied sqrt-undiv 0.0

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

   11. Simplified 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -314168.2807482867501676082611083984375 \lor \neg \left(x \le 662.7162576505425022332929074764251708984\right):\\ \;\;\;\;\frac{0.1529819634592932686700805788859724998474}{{x}^{5}} + \left(\frac{0.5}{x} + \frac{0.2514179000665375252054900556686334311962}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(0.1049934946999999951788851149103720672429 \cdot \left(x \cdot x\right) + \left({x}^{6} \cdot 0.007264418199999999985194687468492702464573 + 1\right)\right) + \left(\left(5.064034000000000243502107366566633572802 \cdot 10^{-4} \cdot {x}^{4} + {x}^{6} \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right) + 0.04240606040000000076517494562722276896238\right) \cdot {x}^{4}}}{\sqrt{\left(\left({x}^{4} \cdot \left(0.2909738639000000182122107617033179849386 + x \cdot \left(x \cdot 0.06945557609999999937322456844412954524159\right)\right) + 1\right) + \left(0.01400054419999999938406531896362139377743 + \left(x \cdot x\right) \cdot \left(\left(2 \cdot \left(x \cdot x\right)\right) \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4} + 8.327945000000000442749725770852364803432 \cdot 10^{-4}\right)\right) \cdot {\left(x \cdot x\right)}^{4}\right) + \left(x \cdot x\right) \cdot 0.7715471018999999763821051601553335785866}} \cdot \left(\sqrt{\frac{\left({x}^{4} \cdot \left(5.064034000000000243502107366566633572802 \cdot 10^{-4} \cdot {x}^{4} + \left({x}^{6} \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4} + 0.04240606040000000076517494562722276896238\right)\right) + 0.1049934946999999951788851149103720672429 \cdot \left(x \cdot x\right)\right) + \left({x}^{6} \cdot 0.007264418199999999985194687468492702464573 + 1\right)}{\left(0.01400054419999999938406531896362139377743 + \left(x \cdot x\right) \cdot \left(\left(2 \cdot \left(x \cdot x\right)\right) \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4} + 8.327945000000000442749725770852364803432 \cdot 10^{-4}\right)\right) \cdot {\left(x \cdot x\right)}^{4} + \left({x}^{4} \cdot \left(0.2909738639000000182122107617033179849386 + \left(x \cdot x\right) \cdot 0.06945557609999999937322456844412954524159\right) + \left(\left(x \cdot 0.7715471018999999763821051601553335785866\right) \cdot x + 1\right)\right)}} \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(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))