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

Percentage Accurate: 99.8% → 99.9%

Time: 12.2s

Alternatives: 7

Speedup: 3.5×

• 28.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_1))
      (* (/ 1.0 21.0) (* (* t_1 (fabs x)) (fabs x))))))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = (t_0 * fabs(x)) * fabs(x);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * fabs(x)) * fabs(x))))));
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = (t_0 * Math.abs(x)) * Math.abs(x);
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * Math.abs(x)) * Math.abs(x))))));
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = (t_0 * math.fabs(x)) * math.fabs(x)
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * math.fabs(x)) * math.fabs(x))))))

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(Float64(t_0 * abs(x)) * abs(x))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_0)) + Float64(Float64(1.0 / 5.0) * t_1)) + Float64(Float64(1.0 / 21.0) * Float64(Float64(t_1 * abs(x)) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = (t_0 * abs(x)) * abs(x);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * abs(x)) * abs(x))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_1))
      (* (/ 1.0 21.0) (* (* t_1 (fabs x)) (fabs x))))))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = (t_0 * fabs(x)) * fabs(x);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * fabs(x)) * fabs(x))))));
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = (t_0 * Math.abs(x)) * Math.abs(x);
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * Math.abs(x)) * Math.abs(x))))));
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = (t_0 * math.fabs(x)) * math.fabs(x)
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * math.fabs(x)) * math.fabs(x))))))

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(Float64(t_0 * abs(x)) * abs(x))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_0)) + Float64(Float64(1.0 / 5.0) * t_1)) + Float64(Float64(1.0 / 21.0) * Float64(Float64(t_1 * abs(x)) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = (t_0 * abs(x)) * abs(x);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * abs(x)) * abs(x))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Alternative 1: 99.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{\pi}} \cdot \left|x \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (sqrt (/ 1.0 PI))
  (fabs
   (*
    x
    (+
     2.0
     (+
      (* 0.047619047619047616 (pow x 6.0))
      (+ (* 0.2 (pow x 4.0)) (* 0.6666666666666666 (pow x 2.0)))))))))

C

double code(double x) {
	return sqrt((1.0 / ((double) M_PI))) * fabs((x * (2.0 + ((0.047619047619047616 * pow(x, 6.0)) + ((0.2 * pow(x, 4.0)) + (0.6666666666666666 * pow(x, 2.0)))))));
}

Java

public static double code(double x) {
	return Math.sqrt((1.0 / Math.PI)) * Math.abs((x * (2.0 + ((0.047619047619047616 * Math.pow(x, 6.0)) + ((0.2 * Math.pow(x, 4.0)) + (0.6666666666666666 * Math.pow(x, 2.0)))))));
}

Python

def code(x):
	return math.sqrt((1.0 / math.pi)) * math.fabs((x * (2.0 + ((0.047619047619047616 * math.pow(x, 6.0)) + ((0.2 * math.pow(x, 4.0)) + (0.6666666666666666 * math.pow(x, 2.0)))))))

Julia

function code(x)
	return Float64(sqrt(Float64(1.0 / pi)) * abs(Float64(x * Float64(2.0 + Float64(Float64(0.047619047619047616 * (x ^ 6.0)) + Float64(Float64(0.2 * (x ^ 4.0)) + Float64(0.6666666666666666 * (x ^ 2.0))))))))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((1.0 / pi)) * abs((x * (2.0 + ((0.047619047619047616 * (x ^ 6.0)) + ((0.2 * (x ^ 4.0)) + (0.6666666666666666 * (x ^ 2.0)))))));
end

Wolfram

code[x_] := N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[Abs[N[(x * N[(2.0 + N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. pow1 99.8%

      \[\leadsto \color{blue}{{\left(\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|\right)}^{1}} \]

   2. mul-fabs 99.8%

      \[\leadsto {\color{blue}{\left(\left|x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|\right)}}^{1} \]

   3. +-commutative 99.8%

      \[\leadsto {\left(\left|x \cdot \frac{\color{blue}{\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}}{\sqrt{\pi}}\right|\right)}^{1} \]

   4. pow2 99.8%

      \[\leadsto {\left(\left|x \cdot \frac{\mathsf{fma}\left(0.6666666666666666, \color{blue}{{x}^{2}}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right|\right)}^{1} \]

6. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{{\left(\left|x \cdot \frac{\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right|\right)}^{1}} \]
7. Step-by-step derivation

   1. unpow1 99.8%

      \[\leadsto \color{blue}{\left|x \cdot \frac{\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right|} \]

   2. associate-*r/ 99.5%

      \[\leadsto \left|\color{blue}{\frac{x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)}{\sqrt{\pi}}}\right| \]

   3. fabs-div 99.5%

      \[\leadsto \color{blue}{\frac{\left|x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right|}{\left|\sqrt{\pi}\right|}} \]

   4. rem-square-sqrt 99.6%

      \[\leadsto \frac{\left|x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right|}{\left|\color{blue}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}\right|} \]

   5. fabs-sqr 99.6%

      \[\leadsto \frac{\left|x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right|}{\color{blue}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}} \]

   6. rem-square-sqrt 99.5%

      \[\leadsto \frac{\left|x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right|}{\color{blue}{\sqrt{\pi}}} \]

   7. +-commutative 99.5%

      \[\leadsto \frac{\left|x \cdot \color{blue}{\left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}\right|}{\sqrt{\pi}} \]

8. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{\left|x \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)\right|}{\sqrt{\pi}}} \]

9. Taylor expanded in x around 0 99.9%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left|x \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|} \]
10. Add Preprocessing

Alternative 2: 98.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.001:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.001)
   (fabs (* x (/ 2.0 (sqrt PI))))
   (fabs
    (* 0.047619047619047616 (* (sqrt (/ 1.0 PI)) (* (pow x 6.0) (fabs x)))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.001) {
		tmp = fabs((x * (2.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs((0.047619047619047616 * (sqrt((1.0 / ((double) M_PI))) * (pow(x, 6.0) * fabs(x)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.001) {
		tmp = Math.abs((x * (2.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs((0.047619047619047616 * (Math.sqrt((1.0 / Math.PI)) * (Math.pow(x, 6.0) * Math.abs(x)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.001:
		tmp = math.fabs((x * (2.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs((0.047619047619047616 * (math.sqrt((1.0 / math.pi)) * (math.pow(x, 6.0) * math.fabs(x)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.001)
		tmp = abs(Float64(x * Float64(2.0 / sqrt(pi))));
	else
		tmp = abs(Float64(0.047619047619047616 * Float64(sqrt(Float64(1.0 / pi)) * Float64((x ^ 6.0) * abs(x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.001)
		tmp = abs((x * (2.0 / sqrt(pi))));
	else
		tmp = abs((0.047619047619047616 * (sqrt((1.0 / pi)) * ((x ^ 6.0) * abs(x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.001], N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[Power[x, 6.0], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1e-3

   1. Initial program 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.3%

      \[\leadsto \left|\color{blue}{2 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)}\right| \]
   6. Step-by-step derivation

      1. *-commutative 99.3%

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

      2. associate-*l* 99.3%

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

   7. Simplified 99.3%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot 2\right)}\right| \]
   8. Step-by-step derivation

      1. *-commutative 99.3%

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

      2. sqrt-div 99.3%

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

      3. metadata-eval 99.3%

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

      4. un-div-inv 98.9%

         \[\leadsto \left|\color{blue}{\frac{\left|x\right| \cdot 2}{\sqrt{\pi}}}\right| \]

      5. add-sqr-sqrt 50.1%

         \[\leadsto \left|\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot 2}{\sqrt{\pi}}\right| \]

      6. fabs-sqr 50.1%

         \[\leadsto \left|\frac{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot 2}{\sqrt{\pi}}\right| \]

      7. add-sqr-sqrt 98.9%

         \[\leadsto \left|\frac{\color{blue}{x} \cdot 2}{\sqrt{\pi}}\right| \]

   9. Applied egg-rr 98.9%

      \[\leadsto \left|\color{blue}{\frac{x \cdot 2}{\sqrt{\pi}}}\right| \]
   10. Step-by-step derivation

       1. associate-/l* 99.3%

          \[\leadsto \left|\color{blue}{x \cdot \frac{2}{\sqrt{\pi}}}\right| \]

   11. Simplified 99.3%

       \[\leadsto \left|\color{blue}{x \cdot \frac{2}{\sqrt{\pi}}}\right| \]

   if 1e-3 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 99.4%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.001:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{6} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (/ (fabs x) (sqrt PI))
   (+
    (* 0.047619047619047616 (pow x 6.0))
    (fma 0.6666666666666666 (* x x) 2.0)))))

C

double code(double x) {
	return fabs(((fabs(x) / sqrt(((double) M_PI))) * ((0.047619047619047616 * pow(x, 6.0)) + fma(0.6666666666666666, (x * x), 2.0))));
}

Julia

function code(x)
	return abs(Float64(Float64(abs(x) / sqrt(pi)) * Float64(Float64(0.047619047619047616 * (x ^ 6.0)) + fma(0.6666666666666666, Float64(x * x), 2.0))))
end

Wolfram

code[x_] := N[Abs[N[(N[(N[Abs[x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.5%

   \[\leadsto \color{blue}{\left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right|} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 99.3%

   \[\leadsto \left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(\color{blue}{0.047619047619047616 \cdot {x}^{6}} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]
6. Add Preprocessing

Alternative 4: 35.3% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (* x (+ 2.0 (fma 0.047619047619047616 (pow x 6.0) (* 0.2 (pow x 4.0)))))
  (sqrt PI)))

C

double code(double x) {
	return (x * (2.0 + fma(0.047619047619047616, pow(x, 6.0), (0.2 * pow(x, 4.0))))) / sqrt(((double) M_PI));
}

Julia

function code(x)
	return Float64(Float64(x * Float64(2.0 + fma(0.047619047619047616, (x ^ 6.0), Float64(0.2 * (x ^ 4.0))))) / sqrt(pi))
end

Wolfram

code[x_] := N[(N[(x * N[(2.0 + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision] + N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. pow1 99.8%

      \[\leadsto \color{blue}{{\left(\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|\right)}^{1}} \]

   2. mul-fabs 99.8%

      \[\leadsto {\color{blue}{\left(\left|x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|\right)}}^{1} \]

   3. +-commutative 99.8%

      \[\leadsto {\left(\left|x \cdot \frac{\color{blue}{\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}}{\sqrt{\pi}}\right|\right)}^{1} \]

   4. pow2 99.8%

      \[\leadsto {\left(\left|x \cdot \frac{\mathsf{fma}\left(0.6666666666666666, \color{blue}{{x}^{2}}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right|\right)}^{1} \]

6. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{{\left(\left|x \cdot \frac{\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right|\right)}^{1}} \]
7. Step-by-step derivation

   1. unpow1 99.8%

      \[\leadsto \color{blue}{\left|x \cdot \frac{\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)}{\sqrt{\pi}}\right|} \]

   2. associate-*r/ 99.5%

      \[\leadsto \left|\color{blue}{\frac{x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)}{\sqrt{\pi}}}\right| \]

   3. fabs-div 99.5%

      \[\leadsto \color{blue}{\frac{\left|x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right|}{\left|\sqrt{\pi}\right|}} \]

   4. rem-square-sqrt 99.6%

      \[\leadsto \frac{\left|x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right|}{\left|\color{blue}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}\right|} \]

   5. fabs-sqr 99.6%

      \[\leadsto \frac{\left|x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right|}{\color{blue}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}} \]

   6. rem-square-sqrt 99.5%

      \[\leadsto \frac{\left|x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right|}{\color{blue}{\sqrt{\pi}}} \]

   7. +-commutative 99.5%

      \[\leadsto \frac{\left|x \cdot \color{blue}{\left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}\right|}{\sqrt{\pi}} \]

8. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{\left|x \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)\right|}{\sqrt{\pi}}} \]

9. Taylor expanded in x around 0 99.2%

   \[\leadsto \frac{\left|x \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \color{blue}{2}\right)\right|}{\sqrt{\pi}} \]
10. Step-by-step derivation

    1. *-un-lft-identity 99.2%

       \[\leadsto \color{blue}{1 \cdot \frac{\left|x \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2\right)\right|}{\sqrt{\pi}}} \]

    2. add-sqr-sqrt 33.1%

       \[\leadsto 1 \cdot \frac{\left|\color{blue}{\sqrt{x \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2\right)} \cdot \sqrt{x \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2\right)}}\right|}{\sqrt{\pi}} \]

    3. fabs-sqr 33.1%

       \[\leadsto 1 \cdot \frac{\color{blue}{\sqrt{x \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2\right)} \cdot \sqrt{x \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2\right)}}}{\sqrt{\pi}} \]

    4. add-sqr-sqrt 34.7%

       \[\leadsto 1 \cdot \frac{\color{blue}{x \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2\right)}}{\sqrt{\pi}} \]

11. Applied egg-rr 34.7%

    \[\leadsto \color{blue}{1 \cdot \frac{x \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2\right)}{\sqrt{\pi}}} \]
12. Step-by-step derivation

    1. *-lft-identity 34.7%

       \[\leadsto \color{blue}{\frac{x \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2\right)}{\sqrt{\pi}}} \]

    2. remove-double-neg 34.7%

       \[\leadsto \frac{\color{blue}{-\left(-x \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2\right)\right)}}{\sqrt{\pi}} \]

    3. *-commutative 34.7%

       \[\leadsto \frac{-\left(-\color{blue}{\left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2\right) \cdot x}\right)}{\sqrt{\pi}} \]

    4. distribute-rgt-neg-out 34.7%

       \[\leadsto \frac{-\color{blue}{\left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2\right) \cdot \left(-x\right)}}{\sqrt{\pi}} \]

    5. *-commutative 34.7%

       \[\leadsto \frac{-\color{blue}{\left(-x\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2\right)}}{\sqrt{\pi}} \]

    6. distribute-lft-neg-in 34.7%

       \[\leadsto \frac{\color{blue}{\left(-\left(-x\right)\right) \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2\right)}}{\sqrt{\pi}} \]

    7. remove-double-neg 34.7%

       \[\leadsto \frac{\color{blue}{x} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + 2\right)}{\sqrt{\pi}} \]

    8. +-commutative 34.7%

       \[\leadsto \frac{x \cdot \color{blue}{\left(2 + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)}}{\sqrt{\pi}} \]

    9. fma-undefine 34.7%

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

    10. +-commutative 34.7%

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

    11. fma-define 34.7%

        \[\leadsto \frac{x \cdot \left(2 + \color{blue}{\mathsf{fma}\left(0.047619047619047616, {x}^{6}, 0.2 \cdot {x}^{4}\right)}\right)}{\sqrt{\pi}} \]

13. Simplified 34.7%

    \[\leadsto \color{blue}{\frac{x \cdot \left(2 + \mathsf{fma}\left(0.047619047619047616, {x}^{6}, 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}} \]
14. Add Preprocessing

Alternative 5: 98.6% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.001:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.001)
   (fabs (* x (/ 2.0 (sqrt PI))))
   (fabs (* (pow x 7.0) (/ 0.047619047619047616 (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.001) {
		tmp = fabs((x * (2.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs((pow(x, 7.0) * (0.047619047619047616 / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.001) {
		tmp = Math.abs((x * (2.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs((Math.pow(x, 7.0) * (0.047619047619047616 / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.001:
		tmp = math.fabs((x * (2.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs((math.pow(x, 7.0) * (0.047619047619047616 / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.001)
		tmp = abs(Float64(x * Float64(2.0 / sqrt(pi))));
	else
		tmp = abs(Float64((x ^ 7.0) * Float64(0.047619047619047616 / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.001)
		tmp = abs((x * (2.0 / sqrt(pi))));
	else
		tmp = abs(((x ^ 7.0) * (0.047619047619047616 / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.001], N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Power[x, 7.0], $MachinePrecision] * N[(0.047619047619047616 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1e-3

   1. Initial program 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.3%

      \[\leadsto \left|\color{blue}{2 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)}\right| \]
   6. Step-by-step derivation

      1. *-commutative 99.3%

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

      2. associate-*l* 99.3%

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

   7. Simplified 99.3%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot 2\right)}\right| \]
   8. Step-by-step derivation

      1. *-commutative 99.3%

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

      2. sqrt-div 99.3%

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

      3. metadata-eval 99.3%

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

      4. un-div-inv 98.9%

         \[\leadsto \left|\color{blue}{\frac{\left|x\right| \cdot 2}{\sqrt{\pi}}}\right| \]

      5. add-sqr-sqrt 50.1%

         \[\leadsto \left|\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot 2}{\sqrt{\pi}}\right| \]

      6. fabs-sqr 50.1%

         \[\leadsto \left|\frac{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot 2}{\sqrt{\pi}}\right| \]

      7. add-sqr-sqrt 98.9%

         \[\leadsto \left|\frac{\color{blue}{x} \cdot 2}{\sqrt{\pi}}\right| \]

   9. Applied egg-rr 98.9%

      \[\leadsto \left|\color{blue}{\frac{x \cdot 2}{\sqrt{\pi}}}\right| \]
   10. Step-by-step derivation

       1. associate-/l* 99.3%

          \[\leadsto \left|\color{blue}{x \cdot \frac{2}{\sqrt{\pi}}}\right| \]

   11. Simplified 99.3%

       \[\leadsto \left|\color{blue}{x \cdot \frac{2}{\sqrt{\pi}}}\right| \]

   if 1e-3 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 99.4%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
   6. Step-by-step derivation

      1. associate-*r* 99.4%

         \[\leadsto \left|\color{blue}{\left(0.047619047619047616 \cdot \left({x}^{6} \cdot \left|x\right|\right)\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]

      2. sqrt-div 99.4%

         \[\leadsto \left|\left(0.047619047619047616 \cdot \left({x}^{6} \cdot \left|x\right|\right)\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right| \]

      3. metadata-eval 99.4%

         \[\leadsto \left|\left(0.047619047619047616 \cdot \left({x}^{6} \cdot \left|x\right|\right)\right) \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right| \]

      4. un-div-inv 99.4%

         \[\leadsto \left|\color{blue}{\frac{0.047619047619047616 \cdot \left({x}^{6} \cdot \left|x\right|\right)}{\sqrt{\pi}}}\right| \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto \left|\frac{0.047619047619047616 \cdot \left({x}^{6} \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}{\sqrt{\pi}}\right| \]

      6. fabs-sqr 0.0%

         \[\leadsto \left|\frac{0.047619047619047616 \cdot \left({x}^{6} \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right)}{\sqrt{\pi}}\right| \]

      7. add-sqr-sqrt 99.4%

         \[\leadsto \left|\frac{0.047619047619047616 \cdot \left({x}^{6} \cdot \color{blue}{x}\right)}{\sqrt{\pi}}\right| \]

      8. *-commutative 99.4%

         \[\leadsto \left|\frac{0.047619047619047616 \cdot \color{blue}{\left(x \cdot {x}^{6}\right)}}{\sqrt{\pi}}\right| \]

   7. Applied egg-rr 99.4%

      \[\leadsto \left|\color{blue}{\frac{0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)}{\sqrt{\pi}}}\right| \]
   8. Step-by-step derivation

      1. *-commutative 99.4%

         \[\leadsto \left|\frac{\color{blue}{\left(x \cdot {x}^{6}\right) \cdot 0.047619047619047616}}{\sqrt{\pi}}\right| \]

      2. associate-/l* 99.4%

         \[\leadsto \left|\color{blue}{\left(x \cdot {x}^{6}\right) \cdot \frac{0.047619047619047616}{\sqrt{\pi}}}\right| \]

      3. *-commutative 99.4%

         \[\leadsto \left|\color{blue}{\left({x}^{6} \cdot x\right)} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\right| \]

      4. pow-plus 99.4%

         \[\leadsto \left|\color{blue}{{x}^{\left(6 + 1\right)}} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\right| \]

      5. metadata-eval 99.4%

         \[\leadsto \left|{x}^{\color{blue}{7}} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\right| \]

   9. Simplified 99.4%

      \[\leadsto \left|\color{blue}{{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 68.5% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \sqrt{\frac{{x}^{14}}{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (fabs (* x (/ 2.0 (sqrt PI))))
   (fabs (* 0.047619047619047616 (sqrt (/ (pow x 14.0) PI))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = fabs((x * (2.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs((0.047619047619047616 * sqrt((pow(x, 14.0) / ((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = Math.abs((x * (2.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs((0.047619047619047616 * Math.sqrt((Math.pow(x, 14.0) / Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = math.fabs((x * (2.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs((0.047619047619047616 * math.sqrt((math.pow(x, 14.0) / math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = abs(Float64(x * Float64(2.0 / sqrt(pi))));
	else
		tmp = abs(Float64(0.047619047619047616 * sqrt(Float64((x ^ 14.0) / pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = abs((x * (2.0 / sqrt(pi))));
	else
		tmp = abs((0.047619047619047616 * sqrt(((x ^ 14.0) / pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[Sqrt[N[(N[Power[x, 14.0], $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 67.8%

      \[\leadsto \left|\color{blue}{2 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)}\right| \]
   6. Step-by-step derivation

      1. *-commutative 67.8%

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

      2. associate-*l* 67.8%

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

   7. Simplified 67.8%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot 2\right)}\right| \]
   8. Step-by-step derivation

      1. *-commutative 67.8%

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

      2. sqrt-div 67.8%

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

      3. metadata-eval 67.8%

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

      4. un-div-inv 67.5%

         \[\leadsto \left|\color{blue}{\frac{\left|x\right| \cdot 2}{\sqrt{\pi}}}\right| \]

      5. add-sqr-sqrt 33.2%

         \[\leadsto \left|\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot 2}{\sqrt{\pi}}\right| \]

      6. fabs-sqr 33.2%

         \[\leadsto \left|\frac{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot 2}{\sqrt{\pi}}\right| \]

      7. add-sqr-sqrt 67.5%

         \[\leadsto \left|\frac{\color{blue}{x} \cdot 2}{\sqrt{\pi}}\right| \]

   9. Applied egg-rr 67.5%

      \[\leadsto \left|\color{blue}{\frac{x \cdot 2}{\sqrt{\pi}}}\right| \]
   10. Step-by-step derivation

       1. associate-/l* 67.8%

          \[\leadsto \left|\color{blue}{x \cdot \frac{2}{\sqrt{\pi}}}\right| \]

   11. Simplified 67.8%

       \[\leadsto \left|\color{blue}{x \cdot \frac{2}{\sqrt{\pi}}}\right| \]

   if 1.8500000000000001 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 37.2%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \left(\left({x}^{6} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
   6. Step-by-step derivation

      1. pow1 37.2%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\color{blue}{{\left({x}^{6} \cdot \left|x\right|\right)}^{1}} \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      2. add-sqr-sqrt 1.8%

         \[\leadsto \left|0.047619047619047616 \cdot \left({\left({x}^{6} \cdot \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{1} \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      3. fabs-sqr 1.8%

         \[\leadsto \left|0.047619047619047616 \cdot \left({\left({x}^{6} \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right)}^{1} \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      4. add-sqr-sqrt 37.2%

         \[\leadsto \left|0.047619047619047616 \cdot \left({\left({x}^{6} \cdot \color{blue}{x}\right)}^{1} \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      5. *-commutative 37.2%

         \[\leadsto \left|0.047619047619047616 \cdot \left({\color{blue}{\left(x \cdot {x}^{6}\right)}}^{1} \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

   7. Applied egg-rr 37.2%

      \[\leadsto \left|0.047619047619047616 \cdot \left(\color{blue}{{\left(x \cdot {x}^{6}\right)}^{1}} \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]
   8. Step-by-step derivation

      1. unpow1 37.2%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\color{blue}{\left(x \cdot {x}^{6}\right)} \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      2. *-commutative 37.2%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\color{blue}{\left({x}^{6} \cdot x\right)} \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      3. pow-plus 37.2%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\color{blue}{{x}^{\left(6 + 1\right)}} \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      4. metadata-eval 37.2%

         \[\leadsto \left|0.047619047619047616 \cdot \left({x}^{\color{blue}{7}} \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

   9. Simplified 37.2%

      \[\leadsto \left|0.047619047619047616 \cdot \left(\color{blue}{{x}^{7}} \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 3.5%

          \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\left(\sqrt{{x}^{7} \cdot \sqrt{\frac{1}{\pi}}} \cdot \sqrt{{x}^{7} \cdot \sqrt{\frac{1}{\pi}}}\right)}\right| \]

       2. sqrt-unprod 34.7%

          \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\sqrt{\left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)}}\right| \]

       3. *-commutative 34.7%

          \[\leadsto \left|0.047619047619047616 \cdot \sqrt{\color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot {x}^{7}\right)} \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

       4. *-commutative 34.7%

          \[\leadsto \left|0.047619047619047616 \cdot \sqrt{\left(\sqrt{\frac{1}{\pi}} \cdot {x}^{7}\right) \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot {x}^{7}\right)}}\right| \]

       5. swap-sqr 34.7%

          \[\leadsto \left|0.047619047619047616 \cdot \sqrt{\color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left({x}^{7} \cdot {x}^{7}\right)}}\right| \]

       6. add-sqr-sqrt 34.7%

          \[\leadsto \left|0.047619047619047616 \cdot \sqrt{\color{blue}{\frac{1}{\pi}} \cdot \left({x}^{7} \cdot {x}^{7}\right)}\right| \]

       7. pow-prod-up 34.7%

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

       8. metadata-eval 34.7%

          \[\leadsto \left|0.047619047619047616 \cdot \sqrt{\frac{1}{\pi} \cdot {x}^{\color{blue}{14}}}\right| \]

   11. Applied egg-rr 34.7%

       \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\sqrt{\frac{1}{\pi} \cdot {x}^{14}}}\right| \]
   12. Step-by-step derivation

       1. associate-*l/ 34.7%

          \[\leadsto \left|0.047619047619047616 \cdot \sqrt{\color{blue}{\frac{1 \cdot {x}^{14}}{\pi}}}\right| \]

       2. *-lft-identity 34.7%

          \[\leadsto \left|0.047619047619047616 \cdot \sqrt{\frac{\color{blue}{{x}^{14}}}{\pi}}\right| \]

   13. Simplified 34.7%

       \[\leadsto \left|0.047619047619047616 \cdot \color{blue}{\sqrt{\frac{{x}^{14}}{\pi}}}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 68.5% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \left|x \cdot \frac{2}{\sqrt{\pi}}\right| \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fabs (* x (/ 2.0 (sqrt PI)))))

C

double code(double x) {
	return fabs((x * (2.0 / sqrt(((double) M_PI)))));
}

Java

public static double code(double x) {
	return Math.abs((x * (2.0 / Math.sqrt(Math.PI))));
}

Python

def code(x):
	return math.fabs((x * (2.0 / math.sqrt(math.pi))))

Julia

function code(x)
	return abs(Float64(x * Float64(2.0 / sqrt(pi))))
end

MATLAB

function tmp = code(x)
	tmp = abs((x * (2.0 / sqrt(pi))));
end

Wolfram

code[x_] := N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right) + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right|} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 67.8%

   \[\leadsto \left|\color{blue}{2 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)}\right| \]
6. Step-by-step derivation

   1. *-commutative 67.8%

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

   2. associate-*l* 67.8%

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

7. Simplified 67.8%

   \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot 2\right)}\right| \]
8. Step-by-step derivation

   1. *-commutative 67.8%

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

   2. sqrt-div 67.8%

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

   3. metadata-eval 67.8%

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

   4. un-div-inv 67.5%

      \[\leadsto \left|\color{blue}{\frac{\left|x\right| \cdot 2}{\sqrt{\pi}}}\right| \]

   5. add-sqr-sqrt 33.2%

      \[\leadsto \left|\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot 2}{\sqrt{\pi}}\right| \]

   6. fabs-sqr 33.2%

      \[\leadsto \left|\frac{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot 2}{\sqrt{\pi}}\right| \]

   7. add-sqr-sqrt 67.5%

      \[\leadsto \left|\frac{\color{blue}{x} \cdot 2}{\sqrt{\pi}}\right| \]

9. Applied egg-rr 67.5%

   \[\leadsto \left|\color{blue}{\frac{x \cdot 2}{\sqrt{\pi}}}\right| \]
10. Step-by-step derivation

    1. associate-/l* 67.8%

       \[\leadsto \left|\color{blue}{x \cdot \frac{2}{\sqrt{\pi}}}\right| \]

11. Simplified 67.8%

    \[\leadsto \left|\color{blue}{x \cdot \frac{2}{\sqrt{\pi}}}\right| \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024088 
(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)))))))