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

Percentage Accurate: 99.8% → 99.9%

Time: 10.4s

Alternatives: 6

Speedup: 2.6×

• 29.1% 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, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[Abs[x], $MachinePrecision] * N[Abs[N[(N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / 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 99.9%

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

Alternative 2: 98.7% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((x * 2.0) + (0.047619047619047616 * ((x * x) * ((x * x) * (Math.abs(x) * (x * x))))))));
}

Python

def code(x):
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((x * 2.0) + (0.047619047619047616 * ((x * x) * ((x * x) * (math.fabs(x) * (x * x))))))))

Julia

function code(x)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(x * 2.0) + Float64(0.047619047619047616 * Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(abs(x) * Float64(x * x))))))))
end

MATLAB

function tmp = code(x)
	tmp = abs(((1.0 / sqrt(pi)) * ((x * 2.0) + (0.047619047619047616 * ((x * x) * ((x * x) * (abs(x) * (x * x))))))));
end

Wolfram

code[x_] := N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(x * 2.0), $MachinePrecision] + N[(0.047619047619047616 * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * N[(x * x), $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 99.9%

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

5. Taylor expanded in x around 0 99.6%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\color{blue}{2 \cdot \left|x\right|} + 0.047619047619047616 \cdot \left(\left(\left(\left|x\right| \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right)\right)\right| \]
6. Step-by-step derivation

   1. rem-square-sqrt 30.7%

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

   2. fabs-sqr 30.7%

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

   3. rem-square-sqrt 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(x \cdot 2 + 0.047619047619047616 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right| \]
9. Add Preprocessing

Alternative 3: 34.3% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.2)
   (* (pow PI -0.5) (+ (* x 2.0) (* 0.6666666666666666 (pow x 3.0))))
   (/ (* 0.047619047619047616 (pow x 7.0)) (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = pow(((double) M_PI), -0.5) * ((x * 2.0) + (0.6666666666666666 * pow(x, 3.0)));
	} else {
		tmp = (0.047619047619047616 * pow(x, 7.0)) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = Math.pow(Math.PI, -0.5) * ((x * 2.0) + (0.6666666666666666 * Math.pow(x, 3.0)));
	} else {
		tmp = (0.047619047619047616 * Math.pow(x, 7.0)) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.2:
		tmp = math.pow(math.pi, -0.5) * ((x * 2.0) + (0.6666666666666666 * math.pow(x, 3.0)))
	else:
		tmp = (0.047619047619047616 * math.pow(x, 7.0)) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.2)
		tmp = Float64((pi ^ -0.5) * Float64(Float64(x * 2.0) + Float64(0.6666666666666666 * (x ^ 3.0))));
	else
		tmp = Float64(Float64(0.047619047619047616 * (x ^ 7.0)) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.2)
		tmp = (pi ^ -0.5) * ((x * 2.0) + (0.6666666666666666 * (x ^ 3.0)));
	else
		tmp = (0.047619047619047616 * (x ^ 7.0)) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.2], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(N[(x * 2.0), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   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 99.9%

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

   5. Taylor expanded in x around 0 88.6%

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

      1. +-commutative 88.6%

         \[\leadsto \left|\color{blue}{2 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right) + 0.6666666666666666 \cdot \left(\left({x}^{2} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

      2. fma-define 88.6%

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

      3. *-commutative 88.6%

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

      4. rem-square-sqrt 31.0%

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

      5. fabs-sqr 31.0%

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

      6. rem-square-sqrt 88.6%

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

      7. associate-*r* 88.6%

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

      8. *-commutative 88.6%

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

      9. rem-square-sqrt 31.2%

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

      10. fabs-sqr 31.2%

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

      11. rem-square-sqrt 88.6%

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

      12. pow-plus 88.6%

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

      13. metadata-eval 88.6%

          \[\leadsto \left|\mathsf{fma}\left(2, x \cdot \sqrt{\frac{1}{\pi}}, \sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{\color{blue}{3}}\right)\right)\right| \]

   7. Simplified 88.6%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(2, x \cdot \sqrt{\frac{1}{\pi}}, \sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right)\right)}\right| \]

   8. Taylor expanded in x around 0 88.6%

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

   10. Simplified 32.6%

       \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right)} \]

   if 2.2000000000000002 < 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 99.9%

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

   5. Taylor expanded in x around inf 39.9%

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

      1. associate-*r* 39.9%

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

      2. *-commutative 39.9%

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

      3. rem-square-sqrt 1.7%

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

      4. fabs-sqr 1.7%

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

      5. rem-square-sqrt 39.9%

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

      6. pow-plus 39.9%

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

      7. metadata-eval 39.9%

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

   7. Simplified 39.9%

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

      1. add-sqr-sqrt 3.5%

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

      2. fabs-sqr 3.5%

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

      3. add-sqr-sqrt 3.6%

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

      4. *-commutative 3.6%

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

      5. sqrt-div 3.6%

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

      6. metadata-eval 3.6%

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

      7. un-div-inv 3.6%

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

   9. Applied egg-rr 3.6%

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

4. Final simplification 32.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 34.3% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.2)
   (* x (* (sqrt (/ 1.0 PI)) (+ 2.0 (* 0.6666666666666666 (* x x)))))
   (/ (* 0.047619047619047616 (pow x 7.0)) (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = x * (sqrt((1.0 / ((double) M_PI))) * (2.0 + (0.6666666666666666 * (x * x))));
	} else {
		tmp = (0.047619047619047616 * pow(x, 7.0)) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = x * (Math.sqrt((1.0 / Math.PI)) * (2.0 + (0.6666666666666666 * (x * x))));
	} else {
		tmp = (0.047619047619047616 * Math.pow(x, 7.0)) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.2:
		tmp = x * (math.sqrt((1.0 / math.pi)) * (2.0 + (0.6666666666666666 * (x * x))))
	else:
		tmp = (0.047619047619047616 * math.pow(x, 7.0)) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.2)
		tmp = Float64(x * Float64(sqrt(Float64(1.0 / pi)) * Float64(2.0 + Float64(0.6666666666666666 * Float64(x * x)))));
	else
		tmp = Float64(Float64(0.047619047619047616 * (x ^ 7.0)) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.2)
		tmp = x * (sqrt((1.0 / pi)) * (2.0 + (0.6666666666666666 * (x * x))));
	else
		tmp = (0.047619047619047616 * (x ^ 7.0)) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.2], N[(x * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(2.0 + N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   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 99.9%

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

   5. Taylor expanded in x around 0 88.6%

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

      1. +-commutative 88.6%

         \[\leadsto \left|\color{blue}{2 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right) + 0.6666666666666666 \cdot \left(\left({x}^{2} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

      2. fma-define 88.6%

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

      3. *-commutative 88.6%

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

      4. rem-square-sqrt 31.0%

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

      5. fabs-sqr 31.0%

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

      6. rem-square-sqrt 88.6%

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

      7. associate-*r* 88.6%

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

      8. *-commutative 88.6%

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

      9. rem-square-sqrt 31.2%

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

      10. fabs-sqr 31.2%

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

      11. rem-square-sqrt 88.6%

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

      12. pow-plus 88.6%

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

      13. metadata-eval 88.6%

          \[\leadsto \left|\mathsf{fma}\left(2, x \cdot \sqrt{\frac{1}{\pi}}, \sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{\color{blue}{3}}\right)\right)\right| \]

   7. Simplified 88.6%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(2, x \cdot \sqrt{\frac{1}{\pi}}, \sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right)\right)}\right| \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 31.0%

         \[\leadsto \left|\color{blue}{\sqrt{\mathsf{fma}\left(2, x \cdot \sqrt{\frac{1}{\pi}}, \sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right)\right)} \cdot \sqrt{\mathsf{fma}\left(2, x \cdot \sqrt{\frac{1}{\pi}}, \sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right)\right)}}\right| \]

      2. fabs-sqr 31.0%

         \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(2, x \cdot \sqrt{\frac{1}{\pi}}, \sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right)\right)} \cdot \sqrt{\mathsf{fma}\left(2, x \cdot \sqrt{\frac{1}{\pi}}, \sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right)\right)}} \]

      3. add-sqr-sqrt 32.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(2, x \cdot \sqrt{\frac{1}{\pi}}, \sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right)\right)} \]

      4. fma-undefine 32.6%

         \[\leadsto \color{blue}{2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right)} \]

      5. associate-*r* 32.6%

         \[\leadsto \color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}} + \sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right) \]

      6. *-commutative 32.6%

         \[\leadsto \color{blue}{\left(x \cdot 2\right)} \cdot \sqrt{\frac{1}{\pi}} + \sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right) \]

      7. associate-*r* 32.6%

         \[\leadsto \color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{1}{\pi}}\right)} + \sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right) \]

      8. +-commutative 32.6%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right) + x \cdot \left(2 \cdot \sqrt{\frac{1}{\pi}}\right)} \]

   9. Applied egg-rr 32.4%

      \[\leadsto \color{blue}{\frac{0.6666666666666666 \cdot {x}^{3}}{\sqrt{\pi}} + 2 \cdot \frac{x}{\sqrt{\pi}}} \]

   10. Taylor expanded in x around 0 32.6%

       \[\leadsto \color{blue}{x \cdot \left(0.6666666666666666 \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \sqrt{\frac{1}{\pi}}\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 32.6%

          \[\leadsto x \cdot \left(\color{blue}{\left(0.6666666666666666 \cdot {x}^{2}\right) \cdot \sqrt{\frac{1}{\pi}}} + 2 \cdot \sqrt{\frac{1}{\pi}}\right) \]

       2. distribute-rgt-out 32.6%

          \[\leadsto x \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{2} + 2\right)\right)} \]

   12. Simplified 32.6%

       \[\leadsto \color{blue}{x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{2} + 2\right)\right)} \]
   13. Step-by-step derivation

       1. unpow2 32.6%

          \[\leadsto x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot \color{blue}{\left(x \cdot x\right)} + 2\right)\right) \]

   14. Applied egg-rr 32.6%

       \[\leadsto x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot \color{blue}{\left(x \cdot x\right)} + 2\right)\right) \]

   if 2.2000000000000002 < 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 99.9%

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

   5. Taylor expanded in x around inf 39.9%

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

      1. associate-*r* 39.9%

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

      2. *-commutative 39.9%

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

      3. rem-square-sqrt 1.7%

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

      4. fabs-sqr 1.7%

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

      5. rem-square-sqrt 39.9%

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

      6. pow-plus 39.9%

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

      7. metadata-eval 39.9%

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

   7. Simplified 39.9%

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

      1. add-sqr-sqrt 3.5%

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

      2. fabs-sqr 3.5%

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

      3. add-sqr-sqrt 3.6%

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

      4. *-commutative 3.6%

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

      5. sqrt-div 3.6%

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

      6. metadata-eval 3.6%

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

      7. un-div-inv 3.6%

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

   9. Applied egg-rr 3.6%

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

4. Final simplification 32.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 34.3% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (* (sqrt (/ 1.0 PI)) (+ 2.0 (* 0.6666666666666666 (* x x))))))

C

double code(double x) {
	return x * (sqrt((1.0 / ((double) M_PI))) * (2.0 + (0.6666666666666666 * (x * x))));
}

Java

public static double code(double x) {
	return x * (Math.sqrt((1.0 / Math.PI)) * (2.0 + (0.6666666666666666 * (x * x))));
}

Python

def code(x):
	return x * (math.sqrt((1.0 / math.pi)) * (2.0 + (0.6666666666666666 * (x * x))))

Julia

function code(x)
	return Float64(x * Float64(sqrt(Float64(1.0 / pi)) * Float64(2.0 + Float64(0.6666666666666666 * Float64(x * x)))))
end

MATLAB

function tmp = code(x)
	tmp = x * (sqrt((1.0 / pi)) * (2.0 + (0.6666666666666666 * (x * x))));
end

Wolfram

code[x_] := N[(x * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(2.0 + N[(0.6666666666666666 * N[(x * x), $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 99.9%

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

5. Taylor expanded in x around 0 88.6%

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

   1. +-commutative 88.6%

      \[\leadsto \left|\color{blue}{2 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right) + 0.6666666666666666 \cdot \left(\left({x}^{2} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

   2. fma-define 88.6%

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

   3. *-commutative 88.6%

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

   4. rem-square-sqrt 31.0%

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

   5. fabs-sqr 31.0%

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

   6. rem-square-sqrt 88.6%

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

   7. associate-*r* 88.6%

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

   8. *-commutative 88.6%

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

   9. rem-square-sqrt 31.2%

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

   10. fabs-sqr 31.2%

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

   11. rem-square-sqrt 88.6%

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

   12. pow-plus 88.6%

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

   13. metadata-eval 88.6%

       \[\leadsto \left|\mathsf{fma}\left(2, x \cdot \sqrt{\frac{1}{\pi}}, \sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{\color{blue}{3}}\right)\right)\right| \]

7. Simplified 88.6%

   \[\leadsto \left|\color{blue}{\mathsf{fma}\left(2, x \cdot \sqrt{\frac{1}{\pi}}, \sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right)\right)}\right| \]
8. Step-by-step derivation

   1. add-sqr-sqrt 31.0%

      \[\leadsto \left|\color{blue}{\sqrt{\mathsf{fma}\left(2, x \cdot \sqrt{\frac{1}{\pi}}, \sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right)\right)} \cdot \sqrt{\mathsf{fma}\left(2, x \cdot \sqrt{\frac{1}{\pi}}, \sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right)\right)}}\right| \]

   2. fabs-sqr 31.0%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(2, x \cdot \sqrt{\frac{1}{\pi}}, \sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right)\right)} \cdot \sqrt{\mathsf{fma}\left(2, x \cdot \sqrt{\frac{1}{\pi}}, \sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right)\right)}} \]

   3. add-sqr-sqrt 32.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, x \cdot \sqrt{\frac{1}{\pi}}, \sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right)\right)} \]

   4. fma-undefine 32.6%

      \[\leadsto \color{blue}{2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right)} \]

   5. associate-*r* 32.6%

      \[\leadsto \color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}} + \sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right) \]

   6. *-commutative 32.6%

      \[\leadsto \color{blue}{\left(x \cdot 2\right)} \cdot \sqrt{\frac{1}{\pi}} + \sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right) \]

   7. associate-*r* 32.6%

      \[\leadsto \color{blue}{x \cdot \left(2 \cdot \sqrt{\frac{1}{\pi}}\right)} + \sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right) \]

   8. +-commutative 32.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3}\right) + x \cdot \left(2 \cdot \sqrt{\frac{1}{\pi}}\right)} \]

9. Applied egg-rr 32.4%

   \[\leadsto \color{blue}{\frac{0.6666666666666666 \cdot {x}^{3}}{\sqrt{\pi}} + 2 \cdot \frac{x}{\sqrt{\pi}}} \]

10. Taylor expanded in x around 0 32.6%

    \[\leadsto \color{blue}{x \cdot \left(0.6666666666666666 \cdot \left({x}^{2} \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \sqrt{\frac{1}{\pi}}\right)} \]
11. Step-by-step derivation

    1. associate-*r* 32.6%

       \[\leadsto x \cdot \left(\color{blue}{\left(0.6666666666666666 \cdot {x}^{2}\right) \cdot \sqrt{\frac{1}{\pi}}} + 2 \cdot \sqrt{\frac{1}{\pi}}\right) \]

    2. distribute-rgt-out 32.6%

       \[\leadsto x \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{2} + 2\right)\right)} \]

12. Simplified 32.6%

    \[\leadsto \color{blue}{x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{2} + 2\right)\right)} \]
13. Step-by-step derivation

    1. unpow2 32.6%

       \[\leadsto x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot \color{blue}{\left(x \cdot x\right)} + 2\right)\right) \]

14. Applied egg-rr 32.6%

    \[\leadsto x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot \color{blue}{\left(x \cdot x\right)} + 2\right)\right) \]

15. Final simplification 32.6%

    \[\leadsto x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right) \]
16. Add Preprocessing

Alternative 6: 34.2% accurate, 17.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(x * N[(2.0 / N[Sqrt[Pi], $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 99.9%

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

5. Taylor expanded in x around 0 65.3%

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

   1. associate-*r* 65.4%

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

   2. rem-square-sqrt 30.7%

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

   3. fabs-sqr 30.7%

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

   4. rem-square-sqrt 65.4%

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

   5. *-commutative 65.4%

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

7. Simplified 65.4%

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

   1. add-sqr-sqrt 30.8%

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

   2. fabs-sqr 30.8%

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

   3. add-sqr-sqrt 32.4%

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

   4. *-commutative 32.4%

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

   5. sqrt-div 32.4%

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

   6. metadata-eval 32.4%

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

   7. un-div-inv 32.4%

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

9. Applied egg-rr 32.4%

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

10. Final simplification 32.4%

    \[\leadsto x \cdot \frac{2}{\sqrt{\pi}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024129 
(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)))))))