Jmat.Real.erfi, branch x less than or equal to 0.5

Average Error: 0.2 → 0.2

Time: 11.8s

Precision: binary64

Cost: 85888

\[x \leq 0.5\]

Math

\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]

↓

\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(0.047619047619047616 \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right) + \left(\left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.2 + \left(\left|x\right| \cdot 2 + \frac{2 \cdot {\left(\left|x\right|\right)}^{3}}{3}\right)\right)\right)\right|\]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (/ 1.0 (sqrt PI))
   (+
    (+
     (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) (* (* (fabs x) (fabs x)) (fabs x))))
     (*
      (/ 1.0 5.0)
      (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x))))
    (*
     (/ 1.0 21.0)
     (*
      (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x))
      (fabs x)))))))

↓

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (/ 1.0 (sqrt PI))
   (+
    (*
     0.047619047619047616
     (* (fabs x) (* (fabs x) (* (fabs x) (* (fabs x) (* (fabs x) (* x x)))))))
    (+
     (* (* (fabs x) (* (fabs x) (* (fabs x) (* x x)))) 0.2)
     (+ (* (fabs x) 2.0) (/ (* 2.0 (pow (fabs x) 3.0)) 3.0)))))))

C

double code(double x) {
	return fabs((1.0 / sqrt((double) M_PI)) * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * ((fabs(x) * fabs(x)) * fabs(x)))) + ((1.0 / 5.0) * ((((fabs(x) * fabs(x)) * fabs(x)) * fabs(x)) * fabs(x)))) + ((1.0 / 21.0) * ((((((fabs(x) * fabs(x)) * fabs(x)) * fabs(x)) * fabs(x)) * fabs(x)) * fabs(x)))));
}

↓

double code(double x) {
	return fabs((1.0 / sqrt((double) M_PI)) * ((0.047619047619047616 * (fabs(x) * (fabs(x) * (fabs(x) * (fabs(x) * (fabs(x) * (x * x))))))) + (((fabs(x) * (fabs(x) * (fabs(x) * (x * x)))) * 0.2) + ((fabs(x) * 2.0) + ((2.0 * pow(fabs(x), 3.0)) / 3.0)))));
}

Error

Bits error versus x

Alternatives

Alternative 1
Error	1.4
Cost	144960

\[\left|\sqrt[3]{\frac{\left(\left|x\right| \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) + 0.2 \cdot {\left(\left|x\right|\right)}^{5}\right) + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}}{\sqrt{\pi}}} \cdot \left(\sqrt[3]{\frac{\left(\left|x\right| \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) + 0.2 \cdot {\left(\left|x\right|\right)}^{5}\right) + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}}{\sqrt{\pi}}} \cdot \sqrt[3]{\frac{\left(\left|x\right| \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) + 0.2 \cdot {\left(\left|x\right|\right)}^{5}\right) + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}}{\sqrt{\pi}}}\right)\right|\]

Alternative 2
Error	41.6
Cost	128320

\[\left|\frac{21 \cdot \left({\left(\left|x\right| \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)}^{3} + {\left(0.2 \cdot {\left(\left|x\right|\right)}^{5}\right)}^{3}\right) + {\left(\left|x\right|\right)}^{7} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left({\left(\left|x\right|\right)}^{5} \cdot \left(0.2 \cdot {\left(\left|x\right|\right)}^{5} - \left|x\right| \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)\right)}{\sqrt{\pi} \cdot \left(21 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left({\left(\left|x\right|\right)}^{5} \cdot \left(0.2 \cdot {\left(\left|x\right|\right)}^{5} - \left|x\right| \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)}\right|\]

Alternative 3
Error	0.7
Cost	98752

\[\left|\sqrt{\frac{\left(\left|x\right| \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) + 0.2 \cdot {\left(\left|x\right|\right)}^{5}\right) + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}}{\sqrt{\pi}}} \cdot \sqrt{\frac{\left(\left|x\right| \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) + 0.2 \cdot {\left(\left|x\right|\right)}^{5}\right) + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}}{\sqrt{\pi}}}\right|\]

Alternative 4
Error	0.2
Cost	85888

\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(0.047619047619047616 \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right) + \left(0.2 \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right)\right) + \left(2 \cdot \left|x\right| + \frac{2 \cdot {\left(\left|x\right|\right)}^{3}}{3}\right)\right)\right)\right|\]

Alternative 5
Error	0.2
Cost	85760

\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(0.047619047619047616 \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right) + \left(0.2 \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right)\right) + \left(2 \cdot \left|x\right| + 0.6666666666666666 \cdot {\left(\left|x\right|\right)}^{3}\right)\right)\right)\right|\]

Alternative 6
Error	29.1
Cost	81088

\[\left|\frac{21 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right) - \left(x \cdot x\right) \cdot \left(0.04 \cdot {\left(\left|x\right|\right)}^{8}\right)\right) + {\left(\left|x\right|\right)}^{7} \cdot \left(\left|x\right| \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) - 0.2 \cdot {\left(\left|x\right|\right)}^{5}\right)}{\sqrt{\pi} \cdot \left(21 \cdot \left(\left|x\right| \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) - 0.2 \cdot {\left(\left|x\right|\right)}^{5}\right)\right)}\right|\]

Alternative 7
Error	0.2
Cost	79552

\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right)\right)\right) + 0.047619047619047616 \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right|\]

Alternative 8
Error	0.2
Cost	79552

\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(0.047619047619047616 \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right) + \left(0.2 \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right)\right) + \left(2 \cdot \left|x\right| + \left|x\right| \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right|\]

Alternative 9
Error	1.6
Cost	78784

\[\left|\sqrt[3]{\sqrt{\frac{1}{\pi}}} \cdot \left(\left(\left(\left|x\right| \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) + 0.2 \cdot {\left(\left|x\right|\right)}^{5}\right) + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right) \cdot \left(\sqrt[3]{\sqrt{\frac{1}{\pi}}} \cdot \sqrt[3]{\sqrt{\frac{1}{\pi}}}\right)\right)\right|\]

Alternative 10
Error	1.0
Cost	78528

\[\left|\left(\left(\left(\left|x\right| \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) + 0.2 \cdot {\left(\left|x\right|\right)}^{5}\right) + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right) \cdot \left|\frac{\sqrt[3]{1}}{\sqrt[3]{\pi}}\right|\right) \cdot \sqrt{\frac{\sqrt[3]{1}}{\sqrt[3]{\pi}}}\right|\]

Alternative 11
Error	28.6
Cost	73984

\[\left|\frac{21 \cdot \left(5 \cdot \left(\left(x \cdot x\right) \cdot 4 - {\left(\left|x\right|\right)}^{6} \cdot 0.4444444444444444\right) + {\left(\left|x\right|\right)}^{5} \cdot \left(\left|x\right| \cdot \left(2 + \left(x \cdot x\right) \cdot -0.6666666666666666\right)\right)\right) + {\left(\left|x\right|\right)}^{7} \cdot \left(5 \cdot \left(\left|x\right| \cdot \left(2 + \left(x \cdot x\right) \cdot -0.6666666666666666\right)\right)\right)}{\sqrt{\pi} \cdot \left(\left(\left|x\right| \cdot \left(2 + \left(x \cdot x\right) \cdot -0.6666666666666666\right)\right) \cdot 105\right)}\right|\]

Alternative 12
Error	1.0
Cost	72256

\[\left|\left(\left(\left(\left|x\right| \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) + 0.2 \cdot {\left(\left|x\right|\right)}^{5}\right) + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right) \cdot \sqrt{\frac{1}{\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}}}\right) \cdot \sqrt{\frac{1}{\sqrt[3]{\pi}}}\right|\]

Alternative 13
Error	0.6
Cost	72000

\[\left|\frac{\left|x\right| \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) + 0.2 \cdot {\left(\left|x\right|\right)}^{5}}{\sqrt{\pi}} + {\left(\left|x\right|\right)}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}\right|\]

Alternative 14
Error	0.2
Cost	65728

\[\left|\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \left(\left(\left(\left|x\right| \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) + 0.2 \cdot {\left(\left|x\right|\right)}^{5}\right) + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right) \cdot \frac{\sqrt[3]{1}}{\sqrt{\pi}}\right)\right|\]

Alternative 15
Error	0.5
Cost	65728

\[\left|\sqrt{\frac{1}{\sqrt{\pi}}} \cdot \left(\left(\left(\left|x\right| \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) + 0.2 \cdot {\left(\left|x\right|\right)}^{5}\right) + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right) \cdot \sqrt{\frac{1}{\sqrt{\pi}}}\right)\right|\]

Alternative 16
Error	0.6
Cost	65600

\[\left|\frac{\left|x\right| \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) + 0.2 \cdot {\left(\left|x\right|\right)}^{5}}{\sqrt{\pi}} + {\left(\left|x\right|\right)}^{7} \cdot \left(\sqrt{0.047619047619047616} \cdot \frac{\sqrt{0.047619047619047616}}{\sqrt{\pi}}\right)\right|\]

Alternative 17
Error	0.3
Cost	65600

\[\left|\frac{1}{\sqrt{\sqrt{\pi}}} \cdot \frac{\left(\left|x\right| \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) + 0.2 \cdot {\left(\left|x\right|\right)}^{5}\right) + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}}{\sqrt{\sqrt{\pi}}}\right|\]

Alternative 18
Error	0.6
Cost	65472

\[\left|\frac{\left|x\right| \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) + 0.2 \cdot {\left(\left|x\right|\right)}^{5}}{\sqrt{\pi}} + {\left(\left|x\right|\right)}^{7} \cdot e^{\log \left(\frac{0.047619047619047616}{\sqrt{\pi}}\right)}\right|\]

Alternative 19
Error	40.0
Cost	59136

\[\left|\sqrt[3]{{\left(\sqrt{\frac{1}{\pi}} \cdot \left(\left(\left|x\right| \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) + 0.2 \cdot {\left(\left|x\right|\right)}^{5}\right) + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)}^{3}}\right|\]

Alternative 20
Error	40.0
Cost	59008

\[\left|\sqrt[3]{{\left(\frac{\left(\left|x\right| \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) + 0.2 \cdot {\left(\left|x\right|\right)}^{5}\right) + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}}{\sqrt{\pi}}\right)}^{3}}\right|\]

Alternative 21
Error	0.6
Cost	52800

\[\left|\frac{1}{\frac{\sqrt{\pi}}{\left|x\right| \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) + 0.2 \cdot {\left(\left|x\right|\right)}^{5}}} + {\left(\left|x\right|\right)}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\right|\]

Alternative 22
Error	0.6
Cost	52800

\[\left|\frac{\left|x\right| \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) + 0.2 \cdot {\left(\left|x\right|\right)}^{5}}{\sqrt{\pi}} + {\left(\left|x\right|\right)}^{7} \cdot \left(\frac{1}{\sqrt{\pi}} \cdot 0.047619047619047616\right)\right|\]

Alternative 23
Error	0.6
Cost	52672

\[\left|{\left(\left|x\right|\right)}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}} + \frac{\left|x\right| \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) + 0.2 \cdot {\left(\left|x\right|\right)}^{5}}{\sqrt{\pi}}\right|\]

Alternative 24
Error	1.0
Cost	52544

\[\left|\left(2 \cdot \left|x\right| + 0.2 \cdot {\left(\left|x\right|\right)}^{5}\right) \cdot \sqrt{\frac{1}{\pi}} + {\left(\left|x\right|\right)}^{7} \cdot \left(\frac{1}{\sqrt{\pi}} \cdot 0.047619047619047616\right)\right|\]

Alternative 25
Error	0.2
Cost	46272

\[\left|\sqrt{\frac{1}{\pi}} \cdot \left(\left(\left|x\right| \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) + 0.2 \cdot {\left(\left|x\right|\right)}^{5}\right) + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right|\]

Alternative 26
Error	0.6
Cost	46144

\[\left|\frac{\left(\left|x\right| \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) + 0.2 \cdot {\left(\left|x\right|\right)}^{5}\right) + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}}{\sqrt{\pi}}\right|\]

Alternative 27
Error	56.3
Cost	45888

\[\left|{\left(\left|x\right|\right)}^{7} \cdot \left(\frac{1}{\sqrt{\pi}} \cdot 0.047619047619047616\right) + 0.6666666666666666 \cdot \left({\left(\left|x\right|\right)}^{3} \cdot \sqrt{\frac{1}{\pi}}\right)\right|\]

Alternative 28
Error	1.0
Cost	45888

\[\left|\sqrt{\frac{1}{\pi}} \cdot \left(\left(2 \cdot \left|x\right| + 0.2 \cdot {\left(\left|x\right|\right)}^{5}\right) + 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right)\right|\]

Alternative 29
Error	56.2
Cost	45760

\[\left|{\left(\left|x\right|\right)}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}} + 0.6666666666666666 \cdot \left({\left(\left|x\right|\right)}^{3} \cdot \sqrt{\frac{1}{\pi}}\right)\right|\]

Alternative 30
Error	0.1
Cost	33344

\[\left|\left|x\right| \cdot \frac{0.047619047619047616 \cdot {x}^{6} + \left(2 + \left(0.6666666666666666 \cdot \left(x \cdot x\right) + 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right|\]

Alternative 31
Error	60.6
Cost	64

\[1\]

Alternative 32
Error	60.7
Cost	64

\[0\]

Alternative 33
Error	62.6
Cost	64

\[-1\]

Derivation

1. Initial program 0.2

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
2. Using strategy rm

3. Applied associate-*l/_binary64_1726 0.2

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \color{blue}{\frac{2 \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)}{3}}\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]

4. Simplified 0.2

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{\color{blue}{2 \cdot {\left(\left|x\right|\right)}^{3}}}{3}\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]

5. Simplified 0.2

   \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(0.047619047619047616 \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right) + \left(0.2 \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right)\right) + \left(2 \cdot \left|x\right| + \frac{2 \cdot {\left(\left|x\right|\right)}^{3}}{3}\right)\right)\right)\right|}\]

6. Final simplification 0.2

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(0.047619047619047616 \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right) + \left(\left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right)\right) \cdot 0.2 + \left(\left|x\right| \cdot 2 + \frac{2 \cdot {\left(\left|x\right|\right)}^{3}}{3}\right)\right)\right)\right|\]

Reproduce

herbie shell --seed 2021022 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x less than or equal to 0.5"
  :precision binary64
  :pre (<= x 0.5)
  (fabs (* (/ 1.0 (sqrt PI)) (+ (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) (* (* (fabs x) (fabs x)) (fabs x)))) (* (/ 1.0 5.0) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))) (* (/ 1.0 21.0) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)))))))