Jmat.Real.dawson

Average Error: 29.3 → 0.0

Time: 8.7s

Precision: 64

Math

\[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.042406060400000001 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.00726441819999999999 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 5.0640340000000002 \cdot 10^{-4} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 1.789971 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.77154710189999998 \cdot \left(x \cdot x\right)\right) + 0.29097386390000002 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.069455576099999999 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.014000544199999999 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 8.32794500000000044 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 1.789971 \cdot 10^{-4}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -9452824402.3195496 \lor \neg \left(x \le 805.830708573443758\right):\\ \;\;\;\;\mathsf{fma}\left(0.25141790006653753, \frac{1}{{x}^{3}}, \mathsf{fma}\left(0.1529819634592933, \frac{1}{{x}^{5}}, \frac{0.5}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x \cdot \left(-\left(\left(\mathsf{fma}\left(0.042406060400000001, {x}^{4}, \mathsf{fma}\left(0.1049934947 \cdot x, x, 1\right)\right) + \mathsf{fma}\left(0.00726441819999999999, {x}^{6}, 5.0640340000000002 \cdot 10^{-4} \cdot {x}^{8}\right)\right) + \left(1.789971 \cdot 10^{-4} \cdot {x}^{9}\right) \cdot x\right)\right)}{-\left(\left(\mathsf{fma}\left(0.069455576099999999, {x}^{6}, \mathsf{fma}\left(0.29097386390000002, {x}^{4}, \mathsf{fma}\left(0.77154710189999998 \cdot x, x, 1\right)\right)\right) + \mathsf{fma}\left(0.014000544199999999, {x}^{8}, \left({x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right)\right) \cdot 8.32794500000000044 \cdot 10^{-4}\right)\right) + \left({x}^{12} \cdot 1.789971 \cdot 10^{-4}\right) \cdot 2\right)}\right)}^{1}\\ \end{array}\]

C

double code(double x) {
	return (((((((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);
}

↓

double code(double x) {
	double VAR;
	if (((x <= -9452824402.31955) || !(x <= 805.8307085734438))) {
		VAR = fma(0.2514179000665375, (1.0 / pow(x, 3.0)), fma(0.15298196345929327, (1.0 / pow(x, 5.0)), (0.5 / x)));
	} else {
		VAR = pow(((x * -((fma(0.0424060604, pow(x, 4.0), fma((0.1049934947 * x), x, 1.0)) + fma(0.0072644182, pow(x, 6.0), (0.0005064034 * pow(x, 8.0)))) + ((0.0001789971 * pow(x, 9.0)) * x))) / -((fma(0.0694555761, pow(x, 6.0), fma(0.2909738639, pow(x, 4.0), fma((0.7715471019 * x), x, 1.0))) + fma(0.0140005442, pow(x, 8.0), ((pow(x, 2.0) * (pow(x, 2.0) * (pow(x, 2.0) * (x * pow(x, 3.0))))) * 0.0008327945))) + ((pow(x, 12.0) * 0.0001789971) * 2.0))), 1.0);
	}
	return VAR;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -9452824402.31955 or 805.8307085734438 < x

   1. Initial program 60.3

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.042406060400000001 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.00726441819999999999 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 5.0640340000000002 \cdot 10^{-4} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 1.789971 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.77154710189999998 \cdot \left(x \cdot x\right)\right) + 0.29097386390000002 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.069455576099999999 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.014000544199999999 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 8.32794500000000044 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 1.789971 \cdot 10^{-4}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\]
   2. Using strategy rm

   3. Applied frac-2neg 60.3

      \[\leadsto \color{blue}{\frac{-\left(\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.042406060400000001 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.00726441819999999999 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 5.0640340000000002 \cdot 10^{-4} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 1.789971 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)}{-\left(\left(\left(\left(\left(\left(1 + 0.77154710189999998 \cdot \left(x \cdot x\right)\right) + 0.29097386390000002 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.069455576099999999 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.014000544199999999 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 8.32794500000000044 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 1.789971 \cdot 10^{-4}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)}} \cdot x\]

   4. Simplified 60.4

      \[\leadsto \frac{\color{blue}{\left(\left(-\mathsf{fma}\left(0.042406060400000001, {x}^{4}, \mathsf{fma}\left(0.1049934947 \cdot x, x, 1\right)\right)\right) + \left(-\mathsf{fma}\left(0.00726441819999999999, {x}^{6}, 5.0640340000000002 \cdot 10^{-4} \cdot {x}^{8}\right)\right)\right) - \left(1.789971 \cdot 10^{-4} \cdot {x}^{9}\right) \cdot x}}{-\left(\left(\left(\left(\left(\left(1 + 0.77154710189999998 \cdot \left(x \cdot x\right)\right) + 0.29097386390000002 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.069455576099999999 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.014000544199999999 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 8.32794500000000044 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 1.789971 \cdot 10^{-4}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)} \cdot x\]

   5. Simplified 60.3

      \[\leadsto \frac{\left(\left(-\mathsf{fma}\left(0.042406060400000001, {x}^{4}, \mathsf{fma}\left(0.1049934947 \cdot x, x, 1\right)\right)\right) + \left(-\mathsf{fma}\left(0.00726441819999999999, {x}^{6}, 5.0640340000000002 \cdot 10^{-4} \cdot {x}^{8}\right)\right)\right) - \left(1.789971 \cdot 10^{-4} \cdot {x}^{9}\right) \cdot x}{\color{blue}{\left(\left(-\mathsf{fma}\left(0.069455576099999999, {x}^{6}, \mathsf{fma}\left(0.29097386390000002, {x}^{4}, \mathsf{fma}\left(0.77154710189999998 \cdot x, x, 1\right)\right)\right)\right) + \left(-\mathsf{fma}\left(0.014000544199999999, {x}^{8}, \left({x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right)\right) \cdot 8.32794500000000044 \cdot 10^{-4}\right)\right)\right) - \left({x}^{12} \cdot 1.789971 \cdot 10^{-4}\right) \cdot 2}} \cdot x\]

   6. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{0.25141790006653753 \cdot \frac{1}{{x}^{3}} + \left(0.1529819634592933 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{x}\right)}\]

   7. Simplified 0.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.25141790006653753, \frac{1}{{x}^{3}}, \mathsf{fma}\left(0.1529819634592933, \frac{1}{{x}^{5}}, \frac{0.5}{x}\right)\right)}\]

   if -9452824402.31955 < x < 805.8307085734438

   1. Initial program 0.0

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.042406060400000001 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.00726441819999999999 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 5.0640340000000002 \cdot 10^{-4} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 1.789971 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.77154710189999998 \cdot \left(x \cdot x\right)\right) + 0.29097386390000002 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.069455576099999999 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.014000544199999999 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 8.32794500000000044 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 1.789971 \cdot 10^{-4}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\]
   2. Using strategy rm

   3. Applied frac-2neg 0.0

      \[\leadsto \color{blue}{\frac{-\left(\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.042406060400000001 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.00726441819999999999 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 5.0640340000000002 \cdot 10^{-4} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 1.789971 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)}{-\left(\left(\left(\left(\left(\left(1 + 0.77154710189999998 \cdot \left(x \cdot x\right)\right) + 0.29097386390000002 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.069455576099999999 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.014000544199999999 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 8.32794500000000044 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 1.789971 \cdot 10^{-4}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)}} \cdot x\]

   4. Simplified 0.0

      \[\leadsto \frac{\color{blue}{\left(\left(-\mathsf{fma}\left(0.042406060400000001, {x}^{4}, \mathsf{fma}\left(0.1049934947 \cdot x, x, 1\right)\right)\right) + \left(-\mathsf{fma}\left(0.00726441819999999999, {x}^{6}, 5.0640340000000002 \cdot 10^{-4} \cdot {x}^{8}\right)\right)\right) - \left(1.789971 \cdot 10^{-4} \cdot {x}^{9}\right) \cdot x}}{-\left(\left(\left(\left(\left(\left(1 + 0.77154710189999998 \cdot \left(x \cdot x\right)\right) + 0.29097386390000002 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.069455576099999999 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.014000544199999999 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 8.32794500000000044 \cdot 10^{-4} \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 1.789971 \cdot 10^{-4}\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)} \cdot x\]

   5. Simplified 0.0

      \[\leadsto \frac{\left(\left(-\mathsf{fma}\left(0.042406060400000001, {x}^{4}, \mathsf{fma}\left(0.1049934947 \cdot x, x, 1\right)\right)\right) + \left(-\mathsf{fma}\left(0.00726441819999999999, {x}^{6}, 5.0640340000000002 \cdot 10^{-4} \cdot {x}^{8}\right)\right)\right) - \left(1.789971 \cdot 10^{-4} \cdot {x}^{9}\right) \cdot x}{\color{blue}{\left(\left(-\mathsf{fma}\left(0.069455576099999999, {x}^{6}, \mathsf{fma}\left(0.29097386390000002, {x}^{4}, \mathsf{fma}\left(0.77154710189999998 \cdot x, x, 1\right)\right)\right)\right) + \left(-\mathsf{fma}\left(0.014000544199999999, {x}^{8}, \left({x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right)\right) \cdot 8.32794500000000044 \cdot 10^{-4}\right)\right)\right) - \left({x}^{12} \cdot 1.789971 \cdot 10^{-4}\right) \cdot 2}} \cdot x\]
   6. Using strategy rm

   7. Applied pow1 0.0

      \[\leadsto \frac{\left(\left(-\mathsf{fma}\left(0.042406060400000001, {x}^{4}, \mathsf{fma}\left(0.1049934947 \cdot x, x, 1\right)\right)\right) + \left(-\mathsf{fma}\left(0.00726441819999999999, {x}^{6}, 5.0640340000000002 \cdot 10^{-4} \cdot {x}^{8}\right)\right)\right) - \left(1.789971 \cdot 10^{-4} \cdot {x}^{9}\right) \cdot x}{\left(\left(-\mathsf{fma}\left(0.069455576099999999, {x}^{6}, \mathsf{fma}\left(0.29097386390000002, {x}^{4}, \mathsf{fma}\left(0.77154710189999998 \cdot x, x, 1\right)\right)\right)\right) + \left(-\mathsf{fma}\left(0.014000544199999999, {x}^{8}, \left({x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right)\right) \cdot 8.32794500000000044 \cdot 10^{-4}\right)\right)\right) - \left({x}^{12} \cdot 1.789971 \cdot 10^{-4}\right) \cdot 2} \cdot \color{blue}{{x}^{1}}\]

   8. Applied pow1 0.0

      \[\leadsto \color{blue}{{\left(\frac{\left(\left(-\mathsf{fma}\left(0.042406060400000001, {x}^{4}, \mathsf{fma}\left(0.1049934947 \cdot x, x, 1\right)\right)\right) + \left(-\mathsf{fma}\left(0.00726441819999999999, {x}^{6}, 5.0640340000000002 \cdot 10^{-4} \cdot {x}^{8}\right)\right)\right) - \left(1.789971 \cdot 10^{-4} \cdot {x}^{9}\right) \cdot x}{\left(\left(-\mathsf{fma}\left(0.069455576099999999, {x}^{6}, \mathsf{fma}\left(0.29097386390000002, {x}^{4}, \mathsf{fma}\left(0.77154710189999998 \cdot x, x, 1\right)\right)\right)\right) + \left(-\mathsf{fma}\left(0.014000544199999999, {x}^{8}, \left({x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right)\right) \cdot 8.32794500000000044 \cdot 10^{-4}\right)\right)\right) - \left({x}^{12} \cdot 1.789971 \cdot 10^{-4}\right) \cdot 2}\right)}^{1}} \cdot {x}^{1}\]

   9. Applied pow-prod-down 0.0

      \[\leadsto \color{blue}{{\left(\frac{\left(\left(-\mathsf{fma}\left(0.042406060400000001, {x}^{4}, \mathsf{fma}\left(0.1049934947 \cdot x, x, 1\right)\right)\right) + \left(-\mathsf{fma}\left(0.00726441819999999999, {x}^{6}, 5.0640340000000002 \cdot 10^{-4} \cdot {x}^{8}\right)\right)\right) - \left(1.789971 \cdot 10^{-4} \cdot {x}^{9}\right) \cdot x}{\left(\left(-\mathsf{fma}\left(0.069455576099999999, {x}^{6}, \mathsf{fma}\left(0.29097386390000002, {x}^{4}, \mathsf{fma}\left(0.77154710189999998 \cdot x, x, 1\right)\right)\right)\right) + \left(-\mathsf{fma}\left(0.014000544199999999, {x}^{8}, \left({x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right)\right) \cdot 8.32794500000000044 \cdot 10^{-4}\right)\right)\right) - \left({x}^{12} \cdot 1.789971 \cdot 10^{-4}\right) \cdot 2} \cdot x\right)}^{1}}\]

   10. Simplified 0.0

       \[\leadsto {\color{blue}{\left(\frac{x \cdot \left(-\left(\left(\mathsf{fma}\left(0.042406060400000001, {x}^{4}, \mathsf{fma}\left(0.1049934947 \cdot x, x, 1\right)\right) + \mathsf{fma}\left(0.00726441819999999999, {x}^{6}, 5.0640340000000002 \cdot 10^{-4} \cdot {x}^{8}\right)\right) + \left(1.789971 \cdot 10^{-4} \cdot {x}^{9}\right) \cdot x\right)\right)}{-\left(\left(\mathsf{fma}\left(0.069455576099999999, {x}^{6}, \mathsf{fma}\left(0.29097386390000002, {x}^{4}, \mathsf{fma}\left(0.77154710189999998 \cdot x, x, 1\right)\right)\right) + \mathsf{fma}\left(0.014000544199999999, {x}^{8}, \left({x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right)\right) \cdot 8.32794500000000044 \cdot 10^{-4}\right)\right) + \left({x}^{12} \cdot 1.789971 \cdot 10^{-4}\right) \cdot 2\right)}\right)}}^{1}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9452824402.3195496 \lor \neg \left(x \le 805.830708573443758\right):\\ \;\;\;\;\mathsf{fma}\left(0.25141790006653753, \frac{1}{{x}^{3}}, \mathsf{fma}\left(0.1529819634592933, \frac{1}{{x}^{5}}, \frac{0.5}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x \cdot \left(-\left(\left(\mathsf{fma}\left(0.042406060400000001, {x}^{4}, \mathsf{fma}\left(0.1049934947 \cdot x, x, 1\right)\right) + \mathsf{fma}\left(0.00726441819999999999, {x}^{6}, 5.0640340000000002 \cdot 10^{-4} \cdot {x}^{8}\right)\right) + \left(1.789971 \cdot 10^{-4} \cdot {x}^{9}\right) \cdot x\right)\right)}{-\left(\left(\mathsf{fma}\left(0.069455576099999999, {x}^{6}, \mathsf{fma}\left(0.29097386390000002, {x}^{4}, \mathsf{fma}\left(0.77154710189999998 \cdot x, x, 1\right)\right)\right) + \mathsf{fma}\left(0.014000544199999999, {x}^{8}, \left({x}^{2} \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(x \cdot {x}^{3}\right)\right)\right)\right) \cdot 8.32794500000000044 \cdot 10^{-4}\right)\right) + \left({x}^{12} \cdot 1.789971 \cdot 10^{-4}\right) \cdot 2\right)}\right)}^{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020105 +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))