Jmat.Real.dawson

Average Error: 29.8 → 0.0

Time: 58.9s

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\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1019.119841081965660123387351632118225098 \lor \neg \left(x \le 57958238.795458965003490447998046875\right):\\ \;\;\;\;\frac{0.5}{x} + \left(\frac{0.2514179000665375252054900556686334311962}{{x}^{3}} + \frac{0.1529819634592932686700805788859724998474}{{x}^{5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(1.789971000000000009994005623070734145585 \cdot 10^{-4}, {x}^{6} \cdot {x}^{4}, \mathsf{fma}\left({x}^{8}, 5.064034000000000243502107366566633572802 \cdot 10^{-4}, \mathsf{fma}\left(0.007264418199999999985194687468492702464573, {x}^{6}, \mathsf{fma}\left({x}^{4}, 0.04240606040000000076517494562722276896238, \mathsf{fma}\left(0.1049934946999999951788851149103720672429, x \cdot x, 1\right)\right)\right)\right)\right)}{\mathsf{fma}\left(1.789971000000000009994005623070734145585 \cdot 10^{-4}, 2 \cdot {\left({x}^{4}\right)}^{3}, \mathsf{fma}\left({x}^{6} \cdot {x}^{4}, 8.327945000000000442749725770852364803432 \cdot 10^{-4}, \mathsf{fma}\left(0.01400054419999999938406531896362139377743, {x}^{8}, \mathsf{fma}\left(0.06945557609999999937322456844412954524159, {x}^{6}, \mathsf{fma}\left(0.2909738639000000182122107617033179849386, {x}^{4}, \mathsf{fma}\left(0.7715471018999999763821051601553335785866 \cdot x, x, 1\right)\right)\right)\right)\right)\right)}\right)\right) \cdot x\\ \end{array}\]

C

double f(double x) {
        double r104597 = 1.0;
        double r104598 = 0.1049934947;
        double r104599 = x;
        double r104600 = r104599 * r104599;
        double r104601 = r104598 * r104600;
        double r104602 = r104597 + r104601;
        double r104603 = 0.0424060604;
        double r104604 = r104600 * r104600;
        double r104605 = r104603 * r104604;
        double r104606 = r104602 + r104605;
        double r104607 = 0.0072644182;
        double r104608 = r104604 * r104600;
        double r104609 = r104607 * r104608;
        double r104610 = r104606 + r104609;
        double r104611 = 0.0005064034;
        double r104612 = r104608 * r104600;
        double r104613 = r104611 * r104612;
        double r104614 = r104610 + r104613;
        double r104615 = 0.0001789971;
        double r104616 = r104612 * r104600;
        double r104617 = r104615 * r104616;
        double r104618 = r104614 + r104617;
        double r104619 = 0.7715471019;
        double r104620 = r104619 * r104600;
        double r104621 = r104597 + r104620;
        double r104622 = 0.2909738639;
        double r104623 = r104622 * r104604;
        double r104624 = r104621 + r104623;
        double r104625 = 0.0694555761;
        double r104626 = r104625 * r104608;
        double r104627 = r104624 + r104626;
        double r104628 = 0.0140005442;
        double r104629 = r104628 * r104612;
        double r104630 = r104627 + r104629;
        double r104631 = 0.0008327945;
        double r104632 = r104631 * r104616;
        double r104633 = r104630 + r104632;
        double r104634 = 2.0;
        double r104635 = r104634 * r104615;
        double r104636 = r104616 * r104600;
        double r104637 = r104635 * r104636;
        double r104638 = r104633 + r104637;
        double r104639 = r104618 / r104638;
        double r104640 = r104639 * r104599;
        return r104640;
}

↓

double f(double x) {
        double r104641 = x;
        double r104642 = -1019.1198410819657;
        bool r104643 = r104641 <= r104642;
        double r104644 = 57958238.795458965;
        bool r104645 = r104641 <= r104644;
        double r104646 = !r104645;
        bool r104647 = r104643 || r104646;
        double r104648 = 0.5;
        double r104649 = r104648 / r104641;
        double r104650 = 0.2514179000665375;
        double r104651 = 3.0;
        double r104652 = pow(r104641, r104651);
        double r104653 = r104650 / r104652;
        double r104654 = 0.15298196345929327;
        double r104655 = 5.0;
        double r104656 = pow(r104641, r104655);
        double r104657 = r104654 / r104656;
        double r104658 = r104653 + r104657;
        double r104659 = r104649 + r104658;
        double r104660 = 0.0001789971;
        double r104661 = 6.0;
        double r104662 = pow(r104641, r104661);
        double r104663 = 4.0;
        double r104664 = pow(r104641, r104663);
        double r104665 = r104662 * r104664;
        double r104666 = 8.0;
        double r104667 = pow(r104641, r104666);
        double r104668 = 0.0005064034;
        double r104669 = 0.0072644182;
        double r104670 = 0.0424060604;
        double r104671 = 0.1049934947;
        double r104672 = r104641 * r104641;
        double r104673 = 1.0;
        double r104674 = fma(r104671, r104672, r104673);
        double r104675 = fma(r104664, r104670, r104674);
        double r104676 = fma(r104669, r104662, r104675);
        double r104677 = fma(r104667, r104668, r104676);
        double r104678 = fma(r104660, r104665, r104677);
        double r104679 = 2.0;
        double r104680 = pow(r104664, r104651);
        double r104681 = r104679 * r104680;
        double r104682 = 0.0008327945;
        double r104683 = 0.0140005442;
        double r104684 = 0.0694555761;
        double r104685 = 0.2909738639;
        double r104686 = 0.7715471019;
        double r104687 = r104686 * r104641;
        double r104688 = fma(r104687, r104641, r104673);
        double r104689 = fma(r104685, r104664, r104688);
        double r104690 = fma(r104684, r104662, r104689);
        double r104691 = fma(r104683, r104667, r104690);
        double r104692 = fma(r104665, r104682, r104691);
        double r104693 = fma(r104660, r104681, r104692);
        double r104694 = r104678 / r104693;
        double r104695 = log1p(r104694);
        double r104696 = expm1(r104695);
        double r104697 = r104696 * r104641;
        double r104698 = r104647 ? r104659 : r104697;
        return r104698;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -1019.1198410819657 or 57958238.795458965 < 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. Taylor expanded around inf 0.0

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

   3. Simplified 0.0

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

   if -1019.1198410819657 < x < 57958238.795458965

   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. Using strategy rm

   3. Applied expm1-log1p-u 0.0

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\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)}\right)\right)} \cdot x\]

   4. Simplified 0.0

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(1.789971000000000009994005623070734145585 \cdot 10^{-4}, {x}^{6} \cdot {x}^{4}, \mathsf{fma}\left({x}^{8}, 5.064034000000000243502107366566633572802 \cdot 10^{-4}, \mathsf{fma}\left(0.007264418199999999985194687468492702464573, {x}^{6}, \mathsf{fma}\left({x}^{4}, 0.04240606040000000076517494562722276896238, \mathsf{fma}\left(0.1049934946999999951788851149103720672429, x \cdot x, 1\right)\right)\right)\right)\right)}{\mathsf{fma}\left(1.789971000000000009994005623070734145585 \cdot 10^{-4}, 2 \cdot {\left({x}^{4}\right)}^{3}, \mathsf{fma}\left({x}^{6} \cdot {x}^{4}, 8.327945000000000442749725770852364803432 \cdot 10^{-4}, \mathsf{fma}\left(0.01400054419999999938406531896362139377743, {x}^{8}, \mathsf{fma}\left(0.06945557609999999937322456844412954524159, {x}^{6}, \mathsf{fma}\left(0.2909738639000000182122107617033179849386, {x}^{4}, \mathsf{fma}\left(0.7715471018999999763821051601553335785866 \cdot x, x, 1\right)\right)\right)\right)\right)\right)}\right)}\right) \cdot x\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1019.119841081965660123387351632118225098 \lor \neg \left(x \le 57958238.795458965003490447998046875\right):\\ \;\;\;\;\frac{0.5}{x} + \left(\frac{0.2514179000665375252054900556686334311962}{{x}^{3}} + \frac{0.1529819634592932686700805788859724998474}{{x}^{5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(1.789971000000000009994005623070734145585 \cdot 10^{-4}, {x}^{6} \cdot {x}^{4}, \mathsf{fma}\left({x}^{8}, 5.064034000000000243502107366566633572802 \cdot 10^{-4}, \mathsf{fma}\left(0.007264418199999999985194687468492702464573, {x}^{6}, \mathsf{fma}\left({x}^{4}, 0.04240606040000000076517494562722276896238, \mathsf{fma}\left(0.1049934946999999951788851149103720672429, x \cdot x, 1\right)\right)\right)\right)\right)}{\mathsf{fma}\left(1.789971000000000009994005623070734145585 \cdot 10^{-4}, 2 \cdot {\left({x}^{4}\right)}^{3}, \mathsf{fma}\left({x}^{6} \cdot {x}^{4}, 8.327945000000000442749725770852364803432 \cdot 10^{-4}, \mathsf{fma}\left(0.01400054419999999938406531896362139377743, {x}^{8}, \mathsf{fma}\left(0.06945557609999999937322456844412954524159, {x}^{6}, \mathsf{fma}\left(0.2909738639000000182122107617033179849386, {x}^{4}, \mathsf{fma}\left(0.7715471018999999763821051601553335785866 \cdot x, x, 1\right)\right)\right)\right)\right)\right)}\right)\right) \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.dawson"
  :precision binary64
  (* (/ (+ (+ (+ (+ (+ 1 (* 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 (* 0.7715471019 (* 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 0.0001789971) (* (* (* (* (* (* x x) (* x x)) (* x x)) (* x x)) (* x x)) (* x x))))) x))