Jmat.Real.dawson

Average Error: 30.0 → 0.0

Time: 21.5s

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\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2559388.4813569574616849422454833984375 \lor \neg \left(x \le 669.4641872238848918641451746225357055664\right):\\ \;\;\;\;0.2514179000665375252054900556686334311962 \cdot \frac{1}{{x}^{3}} + \left(0.1529819634592932686700805788859724998474 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{1}{\frac{\left({\left(x \cdot x\right)}^{4} \cdot \left(5.064034000000000243502107366566633572802 \cdot 10^{-4} + \sqrt{\left(x \cdot x\right) \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}} \cdot \sqrt{\left(x \cdot x\right) \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}}\right) + \left(1 + 0.1049934946999999951788851149103720672429 \cdot \left(x \cdot x\right)\right)\right) + {x}^{4} \cdot \left(0.04240606040000000076517494562722276896238 + \left(x \cdot x\right) \cdot 0.007264418199999999985194687468492702464573\right)}{\left(\left(x \cdot \left({\left(x \cdot x\right)}^{3} \cdot {x}^{3}\right)\right) \cdot \left(8.327945000000000442749725770852364803432 \cdot 10^{-4} + \left(x \cdot x\right) \cdot \left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right)\right) + \left(\left(x \cdot x\right) \cdot \left(0.7715471018999999763821051601553335785866 + 0.2909738639000000182122107617033179849386 \cdot \left(x \cdot x\right)\right) + 1\right)\right) + {x}^{6} \cdot \left(0.06945557609999999937322456844412954524159 + \left(x \cdot x\right) \cdot 0.01400054419999999938406531896362139377743\right)}}}\\ \end{array}\]

C

double f(double x) {
        double r308572 = 1.0;
        double r308573 = 0.1049934947;
        double r308574 = x;
        double r308575 = r308574 * r308574;
        double r308576 = r308573 * r308575;
        double r308577 = r308572 + r308576;
        double r308578 = 0.0424060604;
        double r308579 = r308575 * r308575;
        double r308580 = r308578 * r308579;
        double r308581 = r308577 + r308580;
        double r308582 = 0.0072644182;
        double r308583 = r308579 * r308575;
        double r308584 = r308582 * r308583;
        double r308585 = r308581 + r308584;
        double r308586 = 0.0005064034;
        double r308587 = r308583 * r308575;
        double r308588 = r308586 * r308587;
        double r308589 = r308585 + r308588;
        double r308590 = 0.0001789971;
        double r308591 = r308587 * r308575;
        double r308592 = r308590 * r308591;
        double r308593 = r308589 + r308592;
        double r308594 = 0.7715471019;
        double r308595 = r308594 * r308575;
        double r308596 = r308572 + r308595;
        double r308597 = 0.2909738639;
        double r308598 = r308597 * r308579;
        double r308599 = r308596 + r308598;
        double r308600 = 0.0694555761;
        double r308601 = r308600 * r308583;
        double r308602 = r308599 + r308601;
        double r308603 = 0.0140005442;
        double r308604 = r308603 * r308587;
        double r308605 = r308602 + r308604;
        double r308606 = 0.0008327945;
        double r308607 = r308606 * r308591;
        double r308608 = r308605 + r308607;
        double r308609 = 2.0;
        double r308610 = r308609 * r308590;
        double r308611 = r308591 * r308575;
        double r308612 = r308610 * r308611;
        double r308613 = r308608 + r308612;
        double r308614 = r308593 / r308613;
        double r308615 = r308614 * r308574;
        return r308615;
}

↓

double f(double x) {
        double r308616 = x;
        double r308617 = -2559388.4813569575;
        bool r308618 = r308616 <= r308617;
        double r308619 = 669.4641872238849;
        bool r308620 = r308616 <= r308619;
        double r308621 = !r308620;
        bool r308622 = r308618 || r308621;
        double r308623 = 0.2514179000665375;
        double r308624 = 1.0;
        double r308625 = 3.0;
        double r308626 = pow(r308616, r308625);
        double r308627 = r308624 / r308626;
        double r308628 = r308623 * r308627;
        double r308629 = 0.15298196345929327;
        double r308630 = 5.0;
        double r308631 = pow(r308616, r308630);
        double r308632 = r308624 / r308631;
        double r308633 = r308629 * r308632;
        double r308634 = 0.5;
        double r308635 = r308624 / r308616;
        double r308636 = r308634 * r308635;
        double r308637 = r308633 + r308636;
        double r308638 = r308628 + r308637;
        double r308639 = r308616 * r308616;
        double r308640 = 4.0;
        double r308641 = pow(r308639, r308640);
        double r308642 = 0.0005064034;
        double r308643 = 0.0001789971;
        double r308644 = r308639 * r308643;
        double r308645 = sqrt(r308644);
        double r308646 = r308645 * r308645;
        double r308647 = r308642 + r308646;
        double r308648 = r308641 * r308647;
        double r308649 = 1.0;
        double r308650 = 0.1049934947;
        double r308651 = r308650 * r308639;
        double r308652 = r308649 + r308651;
        double r308653 = r308648 + r308652;
        double r308654 = pow(r308616, r308640);
        double r308655 = 0.0424060604;
        double r308656 = 0.0072644182;
        double r308657 = r308639 * r308656;
        double r308658 = r308655 + r308657;
        double r308659 = r308654 * r308658;
        double r308660 = r308653 + r308659;
        double r308661 = pow(r308639, r308625);
        double r308662 = r308661 * r308626;
        double r308663 = r308616 * r308662;
        double r308664 = 0.0008327945;
        double r308665 = 2.0;
        double r308666 = r308665 * r308643;
        double r308667 = r308639 * r308666;
        double r308668 = r308664 + r308667;
        double r308669 = r308663 * r308668;
        double r308670 = 0.7715471019;
        double r308671 = 0.2909738639;
        double r308672 = r308671 * r308639;
        double r308673 = r308670 + r308672;
        double r308674 = r308639 * r308673;
        double r308675 = r308674 + r308649;
        double r308676 = r308669 + r308675;
        double r308677 = 6.0;
        double r308678 = pow(r308616, r308677);
        double r308679 = 0.0694555761;
        double r308680 = 0.0140005442;
        double r308681 = r308639 * r308680;
        double r308682 = r308679 + r308681;
        double r308683 = r308678 * r308682;
        double r308684 = r308676 + r308683;
        double r308685 = r308660 / r308684;
        double r308686 = r308624 / r308685;
        double r308687 = r308616 / r308686;
        double r308688 = r308622 ? r308638 : r308687;
        return r308688;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -2559388.4813569575 or 669.4641872238849 < x

   1. Initial program 59.5

      \[\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 59.5

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

   3. Taylor expanded around inf 0.0

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

   if -2559388.4813569575 < x < 669.4641872238849

   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

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

   4. Applied add-sqr-sqrt 0.0

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

   6. Applied clear-num 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2559388.4813569574616849422454833984375 \lor \neg \left(x \le 669.4641872238848918641451746225357055664\right):\\ \;\;\;\;0.2514179000665375252054900556686334311962 \cdot \frac{1}{{x}^{3}} + \left(0.1529819634592932686700805788859724998474 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{1}{\frac{\left({\left(x \cdot x\right)}^{4} \cdot \left(5.064034000000000243502107366566633572802 \cdot 10^{-4} + \sqrt{\left(x \cdot x\right) \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}} \cdot \sqrt{\left(x \cdot x\right) \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}}\right) + \left(1 + 0.1049934946999999951788851149103720672429 \cdot \left(x \cdot x\right)\right)\right) + {x}^{4} \cdot \left(0.04240606040000000076517494562722276896238 + \left(x \cdot x\right) \cdot 0.007264418199999999985194687468492702464573\right)}{\left(\left(x \cdot \left({\left(x \cdot x\right)}^{3} \cdot {x}^{3}\right)\right) \cdot \left(8.327945000000000442749725770852364803432 \cdot 10^{-4} + \left(x \cdot x\right) \cdot \left(2 \cdot 1.789971000000000009994005623070734145585 \cdot 10^{-4}\right)\right) + \left(\left(x \cdot x\right) \cdot \left(0.7715471018999999763821051601553335785866 + 0.2909738639000000182122107617033179849386 \cdot \left(x \cdot x\right)\right) + 1\right)\right) + {x}^{6} \cdot \left(0.06945557609999999937322456844412954524159 + \left(x \cdot x\right) \cdot 0.01400054419999999938406531896362139377743\right)}}}\\ \end{array}\]

Reproduce

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