Jmat.Real.dawson

Average Error: 29.3 → 0.0

Time: 2.0m

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 -7072094.3689390718936920166015625:\\ \;\;\;\;\frac{0.1529819634592932686700805788859724998474}{{x}^{5}} + \left(\frac{0.5}{x} + \frac{0.2514179000665375252054900556686334311962}{x \cdot \left(x \cdot x\right)}\right)\\ \mathbf{elif}\;x \le 22327.93475145058255293406546115875244141:\\ \;\;\;\;x \cdot \left(\frac{\left(\left(1 + \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 0.007264418199999999985194687468492702464573\right) + 0.1049934946999999951788851149103720672429 \cdot \left(x \cdot x\right)\right) + \left(\left(5.064034000000000243502107366566633572802 \cdot 10^{-4} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + 1.789971000000000009994005623070734145585 \cdot 10^{-4} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) + 0.04240606040000000076517494562722276896238\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{\sqrt{\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(\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) + \left(0.7715471018999999763821051601553335785866 + \left(\left(x \cdot x\right) \cdot 0.2909738639000000182122107617033179849386 + 0.06945557609999999937322456844412954524159 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right) + 1}} \cdot \frac{1}{\sqrt{\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(\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) + \left(0.7715471018999999763821051601553335785866 + \left(\left(x \cdot x\right) \cdot 0.2909738639000000182122107617033179849386 + 0.06945557609999999937322456844412954524159 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right) + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.1529819634592932686700805788859724998474}{{x}^{5}} + \left(\frac{0.5}{x} + \frac{0.2514179000665375252054900556686334311962}{x \cdot \left(x \cdot x\right)}\right)\\ \end{array}\]

C

double f(double x) {
        double r8352557 = 1.0;
        double r8352558 = 0.1049934947;
        double r8352559 = x;
        double r8352560 = r8352559 * r8352559;
        double r8352561 = r8352558 * r8352560;
        double r8352562 = r8352557 + r8352561;
        double r8352563 = 0.0424060604;
        double r8352564 = r8352560 * r8352560;
        double r8352565 = r8352563 * r8352564;
        double r8352566 = r8352562 + r8352565;
        double r8352567 = 0.0072644182;
        double r8352568 = r8352564 * r8352560;
        double r8352569 = r8352567 * r8352568;
        double r8352570 = r8352566 + r8352569;
        double r8352571 = 0.0005064034;
        double r8352572 = r8352568 * r8352560;
        double r8352573 = r8352571 * r8352572;
        double r8352574 = r8352570 + r8352573;
        double r8352575 = 0.0001789971;
        double r8352576 = r8352572 * r8352560;
        double r8352577 = r8352575 * r8352576;
        double r8352578 = r8352574 + r8352577;
        double r8352579 = 0.7715471019;
        double r8352580 = r8352579 * r8352560;
        double r8352581 = r8352557 + r8352580;
        double r8352582 = 0.2909738639;
        double r8352583 = r8352582 * r8352564;
        double r8352584 = r8352581 + r8352583;
        double r8352585 = 0.0694555761;
        double r8352586 = r8352585 * r8352568;
        double r8352587 = r8352584 + r8352586;
        double r8352588 = 0.0140005442;
        double r8352589 = r8352588 * r8352572;
        double r8352590 = r8352587 + r8352589;
        double r8352591 = 0.0008327945;
        double r8352592 = r8352591 * r8352576;
        double r8352593 = r8352590 + r8352592;
        double r8352594 = 2.0;
        double r8352595 = r8352594 * r8352575;
        double r8352596 = r8352576 * r8352560;
        double r8352597 = r8352595 * r8352596;
        double r8352598 = r8352593 + r8352597;
        double r8352599 = r8352578 / r8352598;
        double r8352600 = r8352599 * r8352559;
        return r8352600;
}

↓

double f(double x) {
        double r8352601 = x;
        double r8352602 = -7072094.368939072;
        bool r8352603 = r8352601 <= r8352602;
        double r8352604 = 0.15298196345929327;
        double r8352605 = 5.0;
        double r8352606 = pow(r8352601, r8352605);
        double r8352607 = r8352604 / r8352606;
        double r8352608 = 0.5;
        double r8352609 = r8352608 / r8352601;
        double r8352610 = 0.2514179000665375;
        double r8352611 = r8352601 * r8352601;
        double r8352612 = r8352601 * r8352611;
        double r8352613 = r8352610 / r8352612;
        double r8352614 = r8352609 + r8352613;
        double r8352615 = r8352607 + r8352614;
        double r8352616 = 22327.934751450583;
        bool r8352617 = r8352601 <= r8352616;
        double r8352618 = 1.0;
        double r8352619 = r8352611 * r8352611;
        double r8352620 = r8352611 * r8352619;
        double r8352621 = 0.0072644182;
        double r8352622 = r8352620 * r8352621;
        double r8352623 = r8352618 + r8352622;
        double r8352624 = 0.1049934947;
        double r8352625 = r8352624 * r8352611;
        double r8352626 = r8352623 + r8352625;
        double r8352627 = 0.0005064034;
        double r8352628 = r8352627 * r8352619;
        double r8352629 = 0.0001789971;
        double r8352630 = r8352629 * r8352620;
        double r8352631 = r8352628 + r8352630;
        double r8352632 = 0.0424060604;
        double r8352633 = r8352631 + r8352632;
        double r8352634 = r8352633 * r8352619;
        double r8352635 = r8352626 + r8352634;
        double r8352636 = r8352619 * r8352619;
        double r8352637 = 2.0;
        double r8352638 = r8352637 * r8352611;
        double r8352639 = r8352638 * r8352629;
        double r8352640 = 0.0008327945;
        double r8352641 = r8352639 + r8352640;
        double r8352642 = r8352641 * r8352611;
        double r8352643 = 0.0140005442;
        double r8352644 = r8352642 + r8352643;
        double r8352645 = r8352636 * r8352644;
        double r8352646 = 0.7715471019;
        double r8352647 = 0.2909738639;
        double r8352648 = r8352611 * r8352647;
        double r8352649 = 0.0694555761;
        double r8352650 = r8352649 * r8352619;
        double r8352651 = r8352648 + r8352650;
        double r8352652 = r8352646 + r8352651;
        double r8352653 = r8352652 * r8352611;
        double r8352654 = r8352645 + r8352653;
        double r8352655 = r8352654 + r8352618;
        double r8352656 = sqrt(r8352655);
        double r8352657 = r8352635 / r8352656;
        double r8352658 = 1.0;
        double r8352659 = r8352658 / r8352656;
        double r8352660 = r8352657 * r8352659;
        double r8352661 = r8352601 * r8352660;
        double r8352662 = r8352617 ? r8352661 : r8352615;
        double r8352663 = r8352603 ? r8352615 : r8352662;
        return r8352663;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -7072094.368939072 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 59.3

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

   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

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

   if -7072094.368939072 < 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

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

   4. Applied add-sqr-sqrt 0.0

      \[\leadsto \frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4} + 5.064034000000000243502107366566633572802 \cdot 10^{-4} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.04240606040000000076517494562722276896238\right) + \left(\left(x \cdot x\right) \cdot 0.1049934946999999951788851149103720672429 + \left(0.007264418199999999985194687468492702464573 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 1\right)\right)}{\color{blue}{\sqrt{1 + \left(\left(x \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot 0.2909738639000000182122107617033179849386 + 0.06945557609999999937322456844412954524159 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.7715471018999999763821051601553335785866\right) + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(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{1 + \left(\left(x \cdot x\right) \cdot \left(\left(\left(x \cdot x\right) \cdot 0.2909738639000000182122107617033179849386 + 0.06945557609999999937322456844412954524159 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.7715471018999999763821051601553335785866\right) + \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(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 *-un-lft-identity 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7072094.3689390718936920166015625:\\ \;\;\;\;\frac{0.1529819634592932686700805788859724998474}{{x}^{5}} + \left(\frac{0.5}{x} + \frac{0.2514179000665375252054900556686334311962}{x \cdot \left(x \cdot x\right)}\right)\\ \mathbf{elif}\;x \le 22327.93475145058255293406546115875244141:\\ \;\;\;\;x \cdot \left(\frac{\left(\left(1 + \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot 0.007264418199999999985194687468492702464573\right) + 0.1049934946999999951788851149103720672429 \cdot \left(x \cdot x\right)\right) + \left(\left(5.064034000000000243502107366566633572802 \cdot 10^{-4} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) + 1.789971000000000009994005623070734145585 \cdot 10^{-4} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) + 0.04240606040000000076517494562722276896238\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{\sqrt{\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(\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) + \left(0.7715471018999999763821051601553335785866 + \left(\left(x \cdot x\right) \cdot 0.2909738639000000182122107617033179849386 + 0.06945557609999999937322456844412954524159 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right) + 1}} \cdot \frac{1}{\sqrt{\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \left(\left(\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) + \left(0.7715471018999999763821051601553335785866 + \left(\left(x \cdot x\right) \cdot 0.2909738639000000182122107617033179849386 + 0.06945557609999999937322456844412954524159 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right) \cdot \left(x \cdot x\right)\right) + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.1529819634592932686700805788859724998474}{{x}^{5}} + \left(\frac{0.5}{x} + \frac{0.2514179000665375252054900556686334311962}{x \cdot \left(x \cdot x\right)}\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))