Jmat.Real.dawson

Average Error: 30.0 → 0.0

Time: 10.8s

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\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -4293930135.755392551422119140625 \lor \neg \left(x \le 585.2479772194044471689267084002494812012\right):\\ \;\;\;\;1 \cdot \mathsf{fma}\left(0.2514179000665375252054900556686334311962, \frac{1}{{x}^{3}}, \mathsf{fma}\left(0.1529819634592932686700805788859724998474, \frac{1}{{x}^{5}}, \frac{0.5}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{x}{\frac{\mathsf{fma}\left({x}^{6}, \mathsf{fma}\left(x, x \cdot 0.01400054419999999938406531896362139377743, 0.06945557609999999937322456844412954524159\right), \mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 2, 1.789971000000000009994005623070734145585 \cdot 10^{-4}, 8.327945000000000442749725770852364803432 \cdot 10^{-4}\right), x \cdot \left({\left(x \cdot x\right)}^{3} \cdot {x}^{3}\right), \mathsf{fma}\left(0.2909738639000000182122107617033179849386 \cdot x, {x}^{3}, \mathsf{fma}\left(0.7715471018999999763821051601553335785866 \cdot x, x, 1\right)\right)\right)\right)}{\mathsf{fma}\left({x}^{4}, \mathsf{fma}\left(x, x \cdot 0.007264418199999999985194687468492702464573, 0.04240606040000000076517494562722276896238\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}, 5.064034000000000243502107366566633572802 \cdot 10^{-4}\right), {\left(x \cdot x\right)}^{4}, \mathsf{fma}\left(0.1049934946999999951788851149103720672429 \cdot x, x, 1\right)\right)\right)}}\\ \end{array}\]

C

double f(double x) {
        double r226639 = 1.0;
        double r226640 = 0.1049934947;
        double r226641 = x;
        double r226642 = r226641 * r226641;
        double r226643 = r226640 * r226642;
        double r226644 = r226639 + r226643;
        double r226645 = 0.0424060604;
        double r226646 = r226642 * r226642;
        double r226647 = r226645 * r226646;
        double r226648 = r226644 + r226647;
        double r226649 = 0.0072644182;
        double r226650 = r226646 * r226642;
        double r226651 = r226649 * r226650;
        double r226652 = r226648 + r226651;
        double r226653 = 0.0005064034;
        double r226654 = r226650 * r226642;
        double r226655 = r226653 * r226654;
        double r226656 = r226652 + r226655;
        double r226657 = 0.0001789971;
        double r226658 = r226654 * r226642;
        double r226659 = r226657 * r226658;
        double r226660 = r226656 + r226659;
        double r226661 = 0.7715471019;
        double r226662 = r226661 * r226642;
        double r226663 = r226639 + r226662;
        double r226664 = 0.2909738639;
        double r226665 = r226664 * r226646;
        double r226666 = r226663 + r226665;
        double r226667 = 0.0694555761;
        double r226668 = r226667 * r226650;
        double r226669 = r226666 + r226668;
        double r226670 = 0.0140005442;
        double r226671 = r226670 * r226654;
        double r226672 = r226669 + r226671;
        double r226673 = 0.0008327945;
        double r226674 = r226673 * r226658;
        double r226675 = r226672 + r226674;
        double r226676 = 2.0;
        double r226677 = r226676 * r226657;
        double r226678 = r226658 * r226642;
        double r226679 = r226677 * r226678;
        double r226680 = r226675 + r226679;
        double r226681 = r226660 / r226680;
        double r226682 = r226681 * r226641;
        return r226682;
}

↓

double f(double x) {
        double r226683 = x;
        double r226684 = -4293930135.7553926;
        bool r226685 = r226683 <= r226684;
        double r226686 = 585.2479772194044;
        bool r226687 = r226683 <= r226686;
        double r226688 = !r226687;
        bool r226689 = r226685 || r226688;
        double r226690 = 1.0;
        double r226691 = 0.2514179000665375;
        double r226692 = 3.0;
        double r226693 = pow(r226683, r226692);
        double r226694 = r226690 / r226693;
        double r226695 = 0.15298196345929327;
        double r226696 = 5.0;
        double r226697 = pow(r226683, r226696);
        double r226698 = r226690 / r226697;
        double r226699 = 0.5;
        double r226700 = r226699 / r226683;
        double r226701 = fma(r226695, r226698, r226700);
        double r226702 = fma(r226691, r226694, r226701);
        double r226703 = r226690 * r226702;
        double r226704 = 6.0;
        double r226705 = pow(r226683, r226704);
        double r226706 = 0.0140005442;
        double r226707 = r226683 * r226706;
        double r226708 = 0.0694555761;
        double r226709 = fma(r226683, r226707, r226708);
        double r226710 = r226683 * r226683;
        double r226711 = 2.0;
        double r226712 = r226710 * r226711;
        double r226713 = 0.0001789971;
        double r226714 = 0.0008327945;
        double r226715 = fma(r226712, r226713, r226714);
        double r226716 = pow(r226710, r226692);
        double r226717 = r226716 * r226693;
        double r226718 = r226683 * r226717;
        double r226719 = 0.2909738639;
        double r226720 = r226719 * r226683;
        double r226721 = 0.7715471019;
        double r226722 = r226721 * r226683;
        double r226723 = 1.0;
        double r226724 = fma(r226722, r226683, r226723);
        double r226725 = fma(r226720, r226693, r226724);
        double r226726 = fma(r226715, r226718, r226725);
        double r226727 = fma(r226705, r226709, r226726);
        double r226728 = 4.0;
        double r226729 = pow(r226683, r226728);
        double r226730 = 0.0072644182;
        double r226731 = r226683 * r226730;
        double r226732 = 0.0424060604;
        double r226733 = fma(r226683, r226731, r226732);
        double r226734 = r226683 * r226713;
        double r226735 = 0.0005064034;
        double r226736 = fma(r226683, r226734, r226735);
        double r226737 = pow(r226710, r226728);
        double r226738 = 0.1049934947;
        double r226739 = r226738 * r226683;
        double r226740 = fma(r226739, r226683, r226723);
        double r226741 = fma(r226736, r226737, r226740);
        double r226742 = fma(r226729, r226733, r226741);
        double r226743 = r226727 / r226742;
        double r226744 = r226683 / r226743;
        double r226745 = r226690 * r226744;
        double r226746 = r226689 ? r226703 : r226745;
        return r226746;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -4293930135.7553926 or 585.2479772194044 < x

   1. Initial program 59.9

      \[\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 59.8

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

   4. Applied div-inv 59.9

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

   5. Applied *-un-lft-identity 59.9

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

   6. Applied times-frac 59.9

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

   7. Simplified 59.9

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

   9. Applied *-un-lft-identity 59.9

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

   10. Applied *-un-lft-identity 59.9

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

   11. Applied times-frac 59.9

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

   12. Applied associate-*l* 59.9

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

   13. Simplified 59.8

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

   14. Taylor expanded around inf 0.0

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

   15. Simplified 0.0

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

   if -4293930135.7553926 < x < 585.2479772194044

   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

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

   4. Applied div-inv 0.0

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

   5. Applied *-un-lft-identity 0.0

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

   6. Applied times-frac 0.0

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

   7. Simplified 0.0

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

   9. Applied *-un-lft-identity 0.0

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

   10. Applied *-un-lft-identity 0.0

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

   11. Applied times-frac 0.0

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

   12. Applied associate-*l* 0.0

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

   13. Simplified 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4293930135.755392551422119140625 \lor \neg \left(x \le 585.2479772194044471689267084002494812012\right):\\ \;\;\;\;1 \cdot \mathsf{fma}\left(0.2514179000665375252054900556686334311962, \frac{1}{{x}^{3}}, \mathsf{fma}\left(0.1529819634592932686700805788859724998474, \frac{1}{{x}^{5}}, \frac{0.5}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \frac{x}{\frac{\mathsf{fma}\left({x}^{6}, \mathsf{fma}\left(x, x \cdot 0.01400054419999999938406531896362139377743, 0.06945557609999999937322456844412954524159\right), \mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 2, 1.789971000000000009994005623070734145585 \cdot 10^{-4}, 8.327945000000000442749725770852364803432 \cdot 10^{-4}\right), x \cdot \left({\left(x \cdot x\right)}^{3} \cdot {x}^{3}\right), \mathsf{fma}\left(0.2909738639000000182122107617033179849386 \cdot x, {x}^{3}, \mathsf{fma}\left(0.7715471018999999763821051601553335785866 \cdot x, x, 1\right)\right)\right)\right)}{\mathsf{fma}\left({x}^{4}, \mathsf{fma}\left(x, x \cdot 0.007264418199999999985194687468492702464573, 0.04240606040000000076517494562722276896238\right), \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}, 5.064034000000000243502107366566633572802 \cdot 10^{-4}\right), {\left(x \cdot x\right)}^{4}, \mathsf{fma}\left(0.1049934946999999951788851149103720672429 \cdot x, x, 1\right)\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.dawson"
  :precision binary64
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