Average Error: 28.6 → 0.0
Time: 15.6s
Precision: binary64
\[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\]
↓
\[\begin{array}{l} \mathbf{if}\;x \leq -771.0527622577687 \lor \neg \left(x \leq 754.544229179161\right):\\ \;\;\;\;\frac{0.2514179000665375}{{x}^{3}} + \left(\frac{0.15298196345929327}{{x}^{5}} + \frac{0.5}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{\frac{1 + \left(x \cdot \left(x \cdot 0.7715471019\right) + \left({x}^{4} \cdot 0.2909738639 + \left({x}^{6} \cdot 0.0694555761 + \left({x}^{8} \cdot 0.0140005442 + \left({x}^{10} \cdot 0.0008327945 + 0.0001789971 \cdot \left(2 \cdot {x}^{12}\right)\right)\right)\right)\right)\right)}{1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + \left({x}^{4} \cdot 0.0424060604 + \left({x}^{6} \cdot 0.0072644182 + \left({x}^{8} \cdot 0.0005064034 + {x}^{10} \cdot 0.0001789971\right)\right)\right)\right)}}\\ \end{array}\]
\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \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) + 0.0001789971 \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 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \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) + 0.0008327945 \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 0.0001789971\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 \leq -771.0527622577687 \lor \neg \left(x \leq 754.544229179161\right):\\
\;\;\;\;\frac{0.2514179000665375}{{x}^{3}} + \left(\frac{0.15298196345929327}{{x}^{5}} + \frac{0.5}{x}\right)\\

\mathbf{else}:\\
\;\;\;\;x \cdot \frac{1}{\frac{1 + \left(x \cdot \left(x \cdot 0.7715471019\right) + \left({x}^{4} \cdot 0.2909738639 + \left({x}^{6} \cdot 0.0694555761 + \left({x}^{8} \cdot 0.0140005442 + \left({x}^{10} \cdot 0.0008327945 + 0.0001789971 \cdot \left(2 \cdot {x}^{12}\right)\right)\right)\right)\right)\right)}{1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + \left({x}^{4} \cdot 0.0424060604 + \left({x}^{6} \cdot 0.0072644182 + \left({x}^{8} \cdot 0.0005064034 + {x}^{10} \cdot 0.0001789971\right)\right)\right)\right)}}\\

\end{array}
double code(double x) {
	return ((double) ((((double) (((double) (((double) (((double) (((double) (1.0 + ((double) (0.1049934947 * ((double) (x * x)))))) + ((double) (0.0424060604 * ((double) (((double) (x * x)) * ((double) (x * x)))))))) + ((double) (0.0072644182 * ((double) (((double) (((double) (x * x)) * ((double) (x * x)))) * ((double) (x * x)))))))) + ((double) (0.0005064034 * ((double) (((double) (((double) (((double) (x * x)) * ((double) (x * x)))) * ((double) (x * x)))) * ((double) (x * x)))))))) + ((double) (0.0001789971 * ((double) (((double) (((double) (((double) (((double) (x * x)) * ((double) (x * x)))) * ((double) (x * x)))) * ((double) (x * x)))) * ((double) (x * x)))))))) / ((double) (((double) (((double) (((double) (((double) (((double) (1.0 + ((double) (0.7715471019 * ((double) (x * x)))))) + ((double) (0.2909738639 * ((double) (((double) (x * x)) * ((double) (x * x)))))))) + ((double) (0.0694555761 * ((double) (((double) (((double) (x * x)) * ((double) (x * x)))) * ((double) (x * x)))))))) + ((double) (0.0140005442 * ((double) (((double) (((double) (((double) (x * x)) * ((double) (x * x)))) * ((double) (x * x)))) * ((double) (x * x)))))))) + ((double) (0.0008327945 * ((double) (((double) (((double) (((double) (((double) (x * x)) * ((double) (x * x)))) * ((double) (x * x)))) * ((double) (x * x)))) * ((double) (x * x)))))))) + ((double) (((double) (2.0 * 0.0001789971)) * ((double) (((double) (((double) (((double) (((double) (((double) (x * x)) * ((double) (x * x)))) * ((double) (x * x)))) * ((double) (x * x)))) * ((double) (x * x)))) * ((double) (x * x))))))))) * x));
}
↓
double code(double x) {
	double VAR;
	if (((x <= -771.0527622577687) || !(x <= 754.544229179161))) {
		VAR = ((double) ((0.2514179000665375 / ((double) pow(x, 3.0))) + ((double) ((0.15298196345929327 / ((double) pow(x, 5.0))) + (0.5 / x)))));
	} else {
		VAR = ((double) (x * (1.0 / (((double) (1.0 + ((double) (((double) (x * ((double) (x * 0.7715471019)))) + ((double) (((double) (((double) pow(x, 4.0)) * 0.2909738639)) + ((double) (((double) (((double) pow(x, 6.0)) * 0.0694555761)) + ((double) (((double) (((double) pow(x, 8.0)) * 0.0140005442)) + ((double) (((double) (((double) pow(x, 10.0)) * 0.0008327945)) + ((double) (0.0001789971 * ((double) (2.0 * ((double) pow(x, 12.0)))))))))))))))))) / ((double) (1.0 + ((double) (((double) (0.1049934947 * ((double) (x * x)))) + ((double) (((double) (((double) pow(x, 4.0)) * 0.0424060604)) + ((double) (((double) (((double) pow(x, 6.0)) * 0.0072644182)) + ((double) (((double) (((double) pow(x, 8.0)) * 0.0005064034)) + ((double) (((double) pow(x, 10.0)) * 0.0001789971))))))))))))))));
	}
	return VAR;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -771.052762257768677 or 754.54422917916099 < x

    1. Initial program Error: 58.8 bits

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\]

    2. SimplifiedError: 58.8 bits

      \[\leadsto \color{blue}{x \cdot \frac{1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + \left(0.0424060604 \cdot {x}^{4} + \left(0.0072644182 \cdot {x}^{6} + \left(0.0005064034 \cdot {x}^{8} + 0.0001789971 \cdot {x}^{10}\right)\right)\right)\right)}{1 + \left(\left(x \cdot \left(x \cdot 0.7715471019\right) + {x}^{4} \cdot 0.2909738639\right) + \left({x}^{6} \cdot 0.0694555761 + \left({x}^{8} \cdot 0.0140005442 + \left({x}^{10} \cdot 0.0008327945 + 0.0001789971 \cdot \left(2 \cdot {x}^{12}\right)\right)\right)\right)\right)}}\]

    3. Taylor expanded around inf Error: 0.0 bits

      \[\leadsto \color{blue}{0.2514179000665375 \cdot \frac{1}{{x}^{3}} + \left(0.15298196345929327 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{x}\right)}\]

    4. SimplifiedError: 0.0 bits

      \[\leadsto \color{blue}{\frac{0.2514179000665375}{{x}^{3}} + \left(\frac{0.15298196345929327}{{x}^{5}} + \frac{0.5}{x}\right)}\]

    if -771.052762257768677 < x < 754.54422917916099

    1. Initial program Error: 0.0 bits

      \[\frac{\left(\left(\left(\left(1 + 0.1049934947 \cdot \left(x \cdot x\right)\right) + 0.0424060604 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0072644182 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0005064034 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0001789971 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)}{\left(\left(\left(\left(\left(1 + 0.7715471019 \cdot \left(x \cdot x\right)\right) + 0.2909738639 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0694555761 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0140005442 \cdot \left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.0008327945 \cdot \left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + \left(2 \cdot 0.0001789971\right) \cdot \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)} \cdot x\]

    2. SimplifiedError: 0.0 bits

      \[\leadsto \color{blue}{x \cdot \frac{1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + \left(0.0424060604 \cdot {x}^{4} + \left(0.0072644182 \cdot {x}^{6} + \left(0.0005064034 \cdot {x}^{8} + 0.0001789971 \cdot {x}^{10}\right)\right)\right)\right)}{1 + \left(\left(x \cdot \left(x \cdot 0.7715471019\right) + {x}^{4} \cdot 0.2909738639\right) + \left({x}^{6} \cdot 0.0694555761 + \left({x}^{8} \cdot 0.0140005442 + \left({x}^{10} \cdot 0.0008327945 + 0.0001789971 \cdot \left(2 \cdot {x}^{12}\right)\right)\right)\right)\right)}}\]
    3. Using strategy rm

    4. Applied clear-numError: 0.0 bits

      \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{1 + \left(\left(x \cdot \left(x \cdot 0.7715471019\right) + {x}^{4} \cdot 0.2909738639\right) + \left({x}^{6} \cdot 0.0694555761 + \left({x}^{8} \cdot 0.0140005442 + \left({x}^{10} \cdot 0.0008327945 + 0.0001789971 \cdot \left(2 \cdot {x}^{12}\right)\right)\right)\right)\right)}{1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + \left(0.0424060604 \cdot {x}^{4} + \left(0.0072644182 \cdot {x}^{6} + \left(0.0005064034 \cdot {x}^{8} + 0.0001789971 \cdot {x}^{10}\right)\right)\right)\right)}}}\]

    5. SimplifiedError: 0.0 bits

      \[\leadsto x \cdot \frac{1}{\color{blue}{\frac{1 + \left(x \cdot \left(x \cdot 0.7715471019\right) + \left({x}^{4} \cdot 0.2909738639 + \left({x}^{6} \cdot 0.0694555761 + \left({x}^{8} \cdot 0.0140005442 + \left({x}^{10} \cdot 0.0008327945 + 0.0001789971 \cdot \left(2 \cdot {x}^{12}\right)\right)\right)\right)\right)\right)}{1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + \left(0.0424060604 \cdot {x}^{4} + \left(0.0072644182 \cdot {x}^{6} + \left(0.0005064034 \cdot {x}^{8} + 0.0001789971 \cdot {x}^{10}\right)\right)\right)\right)}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplificationError: 0.0 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -771.0527622577687 \lor \neg \left(x \leq 754.544229179161\right):\\ \;\;\;\;\frac{0.2514179000665375}{{x}^{3}} + \left(\frac{0.15298196345929327}{{x}^{5}} + \frac{0.5}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{\frac{1 + \left(x \cdot \left(x \cdot 0.7715471019\right) + \left({x}^{4} \cdot 0.2909738639 + \left({x}^{6} \cdot 0.0694555761 + \left({x}^{8} \cdot 0.0140005442 + \left({x}^{10} \cdot 0.0008327945 + 0.0001789971 \cdot \left(2 \cdot {x}^{12}\right)\right)\right)\right)\right)\right)}{1 + \left(0.1049934947 \cdot \left(x \cdot x\right) + \left({x}^{4} \cdot 0.0424060604 + \left({x}^{6} \cdot 0.0072644182 + \left({x}^{8} \cdot 0.0005064034 + {x}^{10} \cdot 0.0001789971\right)\right)\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020200 
(FPCore (x)
  :name "Jmat.Real.dawson"
  :precision binary64
  (* (/ (+ (+ (+ (+ (+ 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))