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

Percentage Accurate: 99.8% → 99.8%

Time: 12.7s

Alternatives: 14

Speedup: 8.3×

• 27.8% 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: 99.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 92.8% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left|x \cdot \left(2 + \left(x \cdot x\right) \cdot 0.6666666666666666\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{0.2}{\sqrt{\pi}}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.0001)
   (* (pow PI -0.5) (fabs (* x (+ 2.0 (* (* x x) 0.6666666666666666)))))
   (fabs (* x (* (* x (* x (* x x))) (/ 0.2 (sqrt PI)))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.0001) {
		tmp = pow(((double) M_PI), -0.5) * fabs((x * (2.0 + ((x * x) * 0.6666666666666666))));
	} else {
		tmp = fabs((x * ((x * (x * (x * x))) * (0.2 / sqrt(((double) M_PI))))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.0001) {
		tmp = Math.pow(Math.PI, -0.5) * Math.abs((x * (2.0 + ((x * x) * 0.6666666666666666))));
	} else {
		tmp = Math.abs((x * ((x * (x * (x * x))) * (0.2 / Math.sqrt(Math.PI)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.0001:
		tmp = math.pow(math.pi, -0.5) * math.fabs((x * (2.0 + ((x * x) * 0.6666666666666666))))
	else:
		tmp = math.fabs((x * ((x * (x * (x * x))) * (0.2 / math.sqrt(math.pi)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.0001)
		tmp = Float64((pi ^ -0.5) * abs(Float64(x * Float64(2.0 + Float64(Float64(x * x) * 0.6666666666666666)))));
	else
		tmp = abs(Float64(x * Float64(Float64(x * Float64(x * Float64(x * x))) * Float64(0.2 / sqrt(pi)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.0001)
		tmp = (pi ^ -0.5) * abs((x * (2.0 + ((x * x) * 0.6666666666666666))));
	else
		tmp = abs((x * ((x * (x * (x * x))) * (0.2 / sqrt(pi)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.0001], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[Abs[N[(x * N[(2.0 + N[(N[(x * x), $MachinePrecision] * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(x * N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.2 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000005e-4

   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%

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

      1. associate-*r/ N/A

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

      2. div-inv N/A

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

      3. fabs-mul N/A

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

      4. fabs-div N/A

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

      5. metadata-eval N/A

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

      6. rem-sqrt-square N/A

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

      7. add-sqr-sqrt N/A

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

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left|\left|x\right| \cdot \left(2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)\right)\right|\right), \color{blue}{\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right)}\right) \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\left|\left|x\right| \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)\right)\right| \cdot {\pi}^{-0.5}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\left|\left|x\right| \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right|\right)}, \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]
   8. Step-by-step derivation

      1. fabs-mul N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left|\left|x\right|\right| \cdot \left|2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right|\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}, \frac{-1}{2}\right)\right) \]

      2. fabs-fabs N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left|x\right| \cdot \left|2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right|\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

      3. fabs-mul N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left|x \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right|\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}, \frac{-1}{2}\right)\right) \]

      4. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\left(x \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}, \frac{-1}{2}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left({x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \left({x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{5}, \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{5}, \mathsf{*.f64}\left(\frac{1}{21}, \left({x}^{2}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{5}, \mathsf{*.f64}\left(\frac{1}{21}, \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

      17. *-lowering-*.f64 99.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{5}, \mathsf{*.f64}\left(\frac{1}{21}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

   9. Simplified 99.8%

      \[\leadsto \color{blue}{\left|x \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + 0.047619047619047616 \cdot \left(x \cdot x\right)\right)\right)\right)\right|} \cdot {\pi}^{-0.5} \]

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\color{blue}{\left(x \cdot \left(2 + \frac{2}{3} \cdot {x}^{2}\right)\right)}\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \left(2 + \frac{2}{3} \cdot {x}^{2}\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{2}{3} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left({x}^{2} \cdot \frac{2}{3}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{2}{3}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{2}{3}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

       6. *-lowering-*.f64 99.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{2}{3}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

   12. Simplified 99.8%

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

   if 1.00000000000000005e-4 < (fabs.f64 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%

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

      1. *-commutative N/A

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

      2. fabs-mul N/A

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

      3. fabs-fabs N/A

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

      4. mul-fabs N/A

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

      5. fabs-lowering-fabs.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right), x\right)\right) \]

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \color{blue}{\left(\frac{1}{5} \cdot {x}^{2}\right)}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\frac{1}{5}, \left({x}^{2}\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\frac{1}{5}, \left(x \cdot x\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

      3. *-lowering-*.f64 78.5%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\frac{1}{5}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   9. Simplified 78.5%

      \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \color{blue}{0.2 \cdot \left(x \cdot x\right)}\right)}{\sqrt{\pi}} \cdot x\right| \]

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{5} \cdot {x}^{4}\right)}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]
   11. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{5} \cdot {x}^{\left(2 \cdot 2\right)}\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{5} \cdot \left({x}^{2} \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(\frac{1}{5} \cdot {x}^{2}\right) \cdot {x}^{2}\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({x}^{2} \cdot \left(\frac{1}{5} \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1}{5} \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{5} \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{5} \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \frac{1}{5}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{5}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{5}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

       11. *-lowering-*.f64 78.5%

           \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{5}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   12. Simplified 78.5%

       \[\leadsto \left|\frac{\color{blue}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.2\right)}}{\sqrt{\pi}} \cdot x\right| \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{5}}{\sqrt{\mathsf{PI}\left(\right)}}\right), x\right)\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \frac{\frac{1}{5}}{\sqrt{\mathsf{PI}\left(\right)}}\right), x\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \left(\frac{\frac{1}{5}}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right), x\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), \left(\frac{\frac{1}{5}}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right), x\right)\right) \]

       5. cube-unmult N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(x \cdot {x}^{3}\right), \left(\frac{\frac{1}{5}}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right), x\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{3}\right)\right), \left(\frac{\frac{1}{5}}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right), x\right)\right) \]

       7. cube-unmult N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot x\right)\right)\right), \left(\frac{\frac{1}{5}}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right), x\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right), \left(\frac{\frac{1}{5}}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right), x\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{1}{5}}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right), x\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{5}, \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right)\right), x\right)\right) \]

       11. sqrt-lowering-sqrt.f64 N/A

           \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{5}, \mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right)\right)\right), x\right)\right) \]

       12. PI-lowering-PI.f64 78.5%

           \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{5}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right)\right), x\right)\right) \]

   14. Applied egg-rr 78.5%

       \[\leadsto \left|\color{blue}{\left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{0.2}{\sqrt{\pi}}\right)} \cdot x\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0001:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left|x \cdot \left(2 + \left(x \cdot x\right) \cdot 0.6666666666666666\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \frac{0.2}{\sqrt{\pi}}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \left|x \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)\right| \cdot {\pi}^{-0.5} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fabs
   (*
    x
    (+
     2.0
     (*
      (* x x)
      (+
       0.6666666666666666
       (* (* x x) (+ 0.2 (* (* x x) 0.047619047619047616))))))))
  (pow PI -0.5)))

C

double code(double x) {
	return fabs((x * (2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * (0.2 + ((x * x) * 0.047619047619047616)))))))) * pow(((double) M_PI), -0.5);
}

Java

public static double code(double x) {
	return Math.abs((x * (2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * (0.2 + ((x * x) * 0.047619047619047616)))))))) * Math.pow(Math.PI, -0.5);
}

Python

def code(x):
	return math.fabs((x * (2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * (0.2 + ((x * x) * 0.047619047619047616)))))))) * math.pow(math.pi, -0.5)

Julia

function code(x)
	return Float64(abs(Float64(x * Float64(2.0 + Float64(Float64(x * x) * Float64(0.6666666666666666 + Float64(Float64(x * x) * Float64(0.2 + Float64(Float64(x * x) * 0.047619047619047616)))))))) * (pi ^ -0.5))
end

MATLAB

function tmp = code(x)
	tmp = abs((x * (2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * (0.2 + ((x * x) * 0.047619047619047616)))))))) * (pi ^ -0.5);
end

Wolfram

code[x_] := N[(N[Abs[N[(x * N[(2.0 + N[(N[(x * x), $MachinePrecision] * N[(0.6666666666666666 + N[(N[(x * x), $MachinePrecision] * N[(0.2 + N[(N[(x * x), $MachinePrecision] * 0.047619047619047616), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[Pi, -0.5], $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%

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

   1. associate-*r/ N/A

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

   2. div-inv N/A

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

   3. fabs-mul N/A

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

   4. fabs-div N/A

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

   5. metadata-eval N/A

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

   6. rem-sqrt-square N/A

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

   7. add-sqr-sqrt N/A

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

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left|\left|x\right| \cdot \left(2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)\right)\right|\right), \color{blue}{\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right)}\right) \]

6. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\left|\left|x\right| \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)\right)\right| \cdot {\pi}^{-0.5}} \]

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\left|\left|x\right| \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right|\right)}, \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]
8. Step-by-step derivation

   1. fabs-mul N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left|\left|x\right|\right| \cdot \left|2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right|\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}, \frac{-1}{2}\right)\right) \]

   2. fabs-fabs N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left|x\right| \cdot \left|2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right|\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

   3. fabs-mul N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left|x \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right|\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}, \frac{-1}{2}\right)\right) \]

   4. fabs-lowering-fabs.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\left(x \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}, \frac{-1}{2}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left({x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \left({x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{5}, \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{5}, \mathsf{*.f64}\left(\frac{1}{21}, \left({x}^{2}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{5}, \mathsf{*.f64}\left(\frac{1}{21}, \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

   17. *-lowering-*.f64 99.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{5}, \mathsf{*.f64}\left(\frac{1}{21}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

9. Simplified 99.8%

   \[\leadsto \color{blue}{\left|x \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + 0.047619047619047616 \cdot \left(x \cdot x\right)\right)\right)\right)\right|} \cdot {\pi}^{-0.5} \]

10. Final simplification 99.8%

    \[\leadsto \left|x \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)\right| \cdot {\pi}^{-0.5} \]
11. Add Preprocessing

Alternative 4: 99.8% accurate, 8.3× speedup

Math

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

FPCore

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

C

double code(double x) {
	return fabs((x * ((2.0 + ((x * x) * (0.6666666666666666 + (x * (x * (0.2 + (x * (x * 0.047619047619047616)))))))) / sqrt(((double) M_PI)))));
}

Java

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

Python

def code(x):
	return math.fabs((x * ((2.0 + ((x * x) * (0.6666666666666666 + (x * (x * (0.2 + (x * (x * 0.047619047619047616)))))))) / math.sqrt(math.pi))))

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Abs[N[(x * N[(N[(2.0 + N[(N[(x * x), $MachinePrecision] * N[(0.6666666666666666 + N[(x * N[(x * N[(0.2 + N[(x * N[(x * 0.047619047619047616), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 99.8%

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

   1. *-commutative N/A

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

   2. fabs-mul N/A

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

   3. fabs-fabs N/A

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

   4. mul-fabs N/A

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

   5. fabs-lowering-fabs.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right), x\right)\right) \]

6. Applied egg-rr 99.8%

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

   1. associate-*l* N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{5}, \left(x \cdot \left(x \cdot \frac{1}{21}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{5}, \left(\left(x \cdot \frac{1}{21}\right) \cdot x\right)\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{5}, \mathsf{*.f64}\left(\left(x \cdot \frac{1}{21}\right), x\right)\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   4. *-lowering-*.f64 99.8%

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{5}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{21}\right), x\right)\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 5: 99.4% accurate, 8.3× speedup

Math

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

FPCore

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

C

double code(double x) {
	return fabs(((2.0 + ((x * x) * (0.6666666666666666 + (x * (x * (0.2 + ((x * x) * 0.047619047619047616))))))) * (x / sqrt(((double) M_PI)))));
}

Java

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

Python

def code(x):
	return math.fabs(((2.0 + ((x * x) * (0.6666666666666666 + (x * (x * (0.2 + ((x * x) * 0.047619047619047616))))))) * (x / math.sqrt(math.pi))))

Julia

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

MATLAB

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

Wolfram

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

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

   1. *-commutative N/A

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

   2. fabs-mul N/A

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

   3. fabs-fabs N/A

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

   4. mul-fabs N/A

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

   5. fabs-lowering-fabs.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right), x\right)\right) \]

6. Applied egg-rr 99.8%

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

   1. associate-*l/ N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\left(2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + x \cdot \left(x \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)\right)\right) \cdot x}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\left(2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + x \cdot \left(x \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)\right)\right) \cdot \frac{x}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + x \cdot \left(x \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)\right)\right), \left(\frac{x}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

8. Applied egg-rr 99.4%

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

Alternative 6: 99.1% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \left|x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   x
   (*
    (pow PI -0.5)
    (+
     2.0
     (*
      (* x x)
      (+ 0.6666666666666666 (* (* x x) (* (* x x) 0.047619047619047616)))))))))

C

double code(double x) {
	return fabs((x * (pow(((double) M_PI), -0.5) * (2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * ((x * x) * 0.047619047619047616))))))));
}

Java

public static double code(double x) {
	return Math.abs((x * (Math.pow(Math.PI, -0.5) * (2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * ((x * x) * 0.047619047619047616))))))));
}

Python

def code(x):
	return math.fabs((x * (math.pow(math.pi, -0.5) * (2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * ((x * x) * 0.047619047619047616))))))))

Julia

function code(x)
	return abs(Float64(x * Float64((pi ^ -0.5) * Float64(2.0 + Float64(Float64(x * x) * Float64(0.6666666666666666 + Float64(Float64(x * x) * Float64(Float64(x * x) * 0.047619047619047616))))))))
end

MATLAB

function tmp = code(x)
	tmp = abs((x * ((pi ^ -0.5) * (2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * ((x * x) * 0.047619047619047616))))))));
end

Wolfram

code[x_] := N[Abs[N[(x * N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(2.0 + N[(N[(x * x), $MachinePrecision] * N[(0.6666666666666666 + N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.047619047619047616), $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%

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

   1. *-commutative N/A

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

   2. fabs-mul N/A

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

   3. fabs-fabs N/A

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

   4. mul-fabs N/A

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

   5. fabs-lowering-fabs.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right), x\right)\right) \]

6. Applied egg-rr 99.8%

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

7. Taylor expanded in x around inf

   \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{21} \cdot {x}^{3}\right)}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]
8. Step-by-step derivation

   1. unpow3 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \left(\frac{1}{21} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \left(\frac{1}{21} \cdot \left({x}^{2} \cdot x\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{21} \cdot {x}^{2}\right) \cdot x\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{21}, \left({x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{21}, \left(x \cdot x\right)\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   8. *-lowering-*.f64 99.1%

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{21}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

9. Simplified 99.1%

   \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + x \cdot \color{blue}{\left(x \cdot \left(0.047619047619047616 \cdot \left(x \cdot x\right)\right)\right)}\right)}{\sqrt{\pi}} \cdot x\right| \]
10. Step-by-step derivation

    1. clear-num N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{\frac{\sqrt{\mathsf{PI}\left(\right)}}{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + x \cdot \left(x \cdot \left(\frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)}}\right), x\right)\right) \]

    2. associate-/r/ N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + x \cdot \left(x \cdot \left(\frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right), x\right)\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right), \left(2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + x \cdot \left(x \cdot \left(\frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right), x\right)\right) \]

    4. pow1/2 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{{\mathsf{PI}\left(\right)}^{\frac{1}{2}}}\right), \left(2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + x \cdot \left(x \cdot \left(\frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right), x\right)\right) \]

    5. pow-flip N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\mathsf{PI}\left(\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right), \left(2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + x \cdot \left(x \cdot \left(\frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right), x\right)\right) \]

    6. metadata-eval N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\mathsf{PI}\left(\right)}^{\frac{-1}{2}}\right), \left(2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + x \cdot \left(x \cdot \left(\frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right), x\right)\right) \]

    7. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI}\left(\right), \frac{-1}{2}\right), \left(2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + x \cdot \left(x \cdot \left(\frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right), x\right)\right) \]

    8. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \left(2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + x \cdot \left(x \cdot \left(\frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right), x\right)\right) \]

    9. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(2, \left(\left(x \cdot x\right) \cdot \left(\frac{2}{3} + x \cdot \left(x \cdot \left(\frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

    10. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{2}{3} + x \cdot \left(x \cdot \left(\frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{2}{3} + x \cdot \left(x \cdot \left(\frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

    12. +-lowering-+.f64 N/A

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \left(x \cdot \left(x \cdot \left(\frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

    13. associate-*r* N/A

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

    14. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

    15. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right), x\right)\right) \]

    16. *-commutative N/A

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)\right)\right)\right)\right), x\right)\right) \]

    17. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{21}\right)\right)\right)\right)\right)\right), x\right)\right) \]

    18. *-lowering-*.f64 99.1%

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{21}\right)\right)\right)\right)\right)\right), x\right)\right) \]

11. Applied egg-rr 99.1%

    \[\leadsto \left|\color{blue}{\left({\pi}^{-0.5} \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)\right)} \cdot x\right| \]

12. Final simplification 99.1%

    \[\leadsto \left|x \cdot \left({\pi}^{-0.5} \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)\right)\right| \]
13. Add Preprocessing

Alternative 7: 99.1% accurate, 8.4× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return Math.abs((x * ((2.0 + (x * (x * (0.6666666666666666 + (x * (x * (x * (x * 0.047619047619047616)))))))) / Math.sqrt(Math.PI))));
}

Python

def code(x):
	return math.fabs((x * ((2.0 + (x * (x * (0.6666666666666666 + (x * (x * (x * (x * 0.047619047619047616)))))))) / math.sqrt(math.pi))))

Julia

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

MATLAB

function tmp = code(x)
	tmp = abs((x * ((2.0 + (x * (x * (0.6666666666666666 + (x * (x * (x * (x * 0.047619047619047616)))))))) / sqrt(pi))));
end

Wolfram

code[x_] := N[Abs[N[(x * N[(N[(2.0 + N[(x * N[(x * N[(0.6666666666666666 + N[(x * N[(x * N[(x * N[(x * 0.047619047619047616), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 99.8%

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

   1. *-commutative N/A

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

   2. fabs-mul N/A

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

   3. fabs-fabs N/A

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

   4. mul-fabs N/A

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

   5. fabs-lowering-fabs.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right), x\right)\right) \]

6. Applied egg-rr 99.8%

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

7. Taylor expanded in x around inf

   \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{21} \cdot {x}^{3}\right)}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]
8. Step-by-step derivation

   1. unpow3 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \left(\frac{1}{21} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \left(\frac{1}{21} \cdot \left({x}^{2} \cdot x\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{21} \cdot {x}^{2}\right) \cdot x\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{21}, \left({x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{21}, \left(x \cdot x\right)\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   8. *-lowering-*.f64 99.1%

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{21}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

9. Simplified 99.1%

   \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + x \cdot \color{blue}{\left(x \cdot \left(0.047619047619047616 \cdot \left(x \cdot x\right)\right)\right)}\right)}{\sqrt{\pi}} \cdot x\right| \]

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{1}{21} \cdot {x}^{4}\right)\right)}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]
11. Step-by-step derivation

    1. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left({x}^{2} \cdot \left(\frac{2}{3} + \frac{1}{21} \cdot {x}^{4}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

    2. unpow2 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\left(x \cdot x\right) \cdot \left(\frac{2}{3} + \frac{1}{21} \cdot {x}^{4}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

    3. associate-*l* N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(x \cdot \left(x \cdot \left(\frac{2}{3} + \frac{1}{21} \cdot {x}^{4}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{2}{3} + \frac{1}{21} \cdot {x}^{4}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

    5. +-commutative N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{21} \cdot {x}^{4} + \frac{2}{3}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

    6. metadata-eval N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{21} \cdot {x}^{\left(2 \cdot 2\right)} + \frac{2}{3}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

    7. pow-sqr N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{21} \cdot \left({x}^{2} \cdot {x}^{2}\right) + \frac{2}{3}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

    8. associate-*l* N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(x \cdot \left(\left(\frac{1}{21} \cdot {x}^{2}\right) \cdot {x}^{2} + \frac{2}{3}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

    9. *-commutative N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{21} \cdot {x}^{2}\right) + \frac{2}{3}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

    10. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \left(\frac{1}{21} \cdot {x}^{2}\right) + \frac{2}{3}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

    11. +-commutative N/A

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

    12. +-lowering-+.f64 N/A

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{3}, \left({x}^{2} \cdot \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

    13. unpow2 N/A

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{3}, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

    14. associate-*l* N/A

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{3}, \left(x \cdot \left(x \cdot \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

    15. *-commutative N/A

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{3}, \left(x \cdot \left(\left(\frac{1}{21} \cdot {x}^{2}\right) \cdot x\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

    16. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{21} \cdot {x}^{2}\right) \cdot x\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

    17. *-commutative N/A

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

    18. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

    19. *-commutative N/A

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2} \cdot \frac{1}{21}\right)\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

    20. unpow2 N/A

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

    21. associate-*l* N/A

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot \frac{1}{21}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

    22. *-commutative N/A

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{21} \cdot x\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

    23. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{21} \cdot x\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

    24. *-commutative N/A

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{21}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

    25. *-lowering-*.f64 99.1%

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{21}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

12. Simplified 99.1%

    \[\leadsto \left|\frac{\color{blue}{2 + x \cdot \left(x \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.047619047619047616\right)\right)\right)\right)\right)}}{\sqrt{\pi}} \cdot x\right| \]

13. Final simplification 99.1%

    \[\leadsto \left|x \cdot \frac{2 + x \cdot \left(x \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.047619047619047616\right)\right)\right)\right)\right)}{\sqrt{\pi}}\right| \]
14. Add Preprocessing

Alternative 8: 98.7% accurate, 8.4× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return Math.abs(((x / Math.sqrt(Math.PI)) * (2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * (x * (x * 0.047619047619047616))))))));
}

Python

def code(x):
	return math.fabs(((x / math.sqrt(math.pi)) * (2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * (x * (x * 0.047619047619047616))))))))

Julia

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

MATLAB

function tmp = code(x)
	tmp = abs(((x / sqrt(pi)) * (2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * (x * (x * 0.047619047619047616))))))));
end

Wolfram

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

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

   1. *-commutative N/A

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

   2. fabs-mul N/A

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

   3. fabs-fabs N/A

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

   4. mul-fabs N/A

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

   5. fabs-lowering-fabs.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right), x\right)\right) \]

6. Applied egg-rr 99.8%

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

7. Taylor expanded in x around inf

   \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{21} \cdot {x}^{3}\right)}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]
8. Step-by-step derivation

   1. unpow3 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \left(\frac{1}{21} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \left(\frac{1}{21} \cdot \left({x}^{2} \cdot x\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{21} \cdot {x}^{2}\right) \cdot x\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{21}, \left({x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{21}, \left(x \cdot x\right)\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   8. *-lowering-*.f64 99.1%

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{21}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

9. Simplified 99.1%

   \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + x \cdot \color{blue}{\left(x \cdot \left(0.047619047619047616 \cdot \left(x \cdot x\right)\right)\right)}\right)}{\sqrt{\pi}} \cdot x\right| \]
10. Step-by-step derivation

    1. associate-*l/ N/A

       \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\left(2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + x \cdot \left(x \cdot \left(\frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot x}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right) \]

    2. associate-/l* N/A

       \[\leadsto \mathsf{fabs.f64}\left(\left(\left(2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + x \cdot \left(x \cdot \left(\frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot \frac{x}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + x \cdot \left(x \cdot \left(\frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right), \left(\frac{x}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

    4. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \left(\left(x \cdot x\right) \cdot \left(\frac{2}{3} + x \cdot \left(x \cdot \left(\frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right), \left(\frac{x}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{2}{3} + x \cdot \left(x \cdot \left(\frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right), \left(\frac{x}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{2}{3} + x \cdot \left(x \cdot \left(\frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right), \left(\frac{x}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

    7. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \left(x \cdot \left(x \cdot \left(\frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right), \left(\frac{x}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

    8. associate-*r* N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right), \left(\frac{x}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

    9. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right), \left(\frac{x}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

    10. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right), \left(\frac{x}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

    11. *-commutative N/A

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)\right)\right)\right), \left(\frac{x}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

    12. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{21}\right)\right)\right)\right)\right), \left(\frac{x}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

    13. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{21}\right)\right)\right)\right)\right), \left(\frac{x}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

    14. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{21}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

    15. sqrt-lowering-sqrt.f64 N/A

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{21}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right)\right)\right)\right) \]

    16. PI-lowering-PI.f64 98.6%

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{21}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

11. Applied egg-rr 98.6%

    \[\leadsto \left|\color{blue}{\left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right) \cdot \frac{x}{\sqrt{\pi}}}\right| \]
12. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

    2. associate-*r* N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(\frac{1}{21} \cdot x\right) \cdot x\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left(\frac{1}{21} \cdot x\right), x\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left(x \cdot \frac{1}{21}\right), x\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

    5. *-lowering-*.f64 98.6%

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \frac{1}{21}\right), x\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

13. Applied egg-rr 98.6%

    \[\leadsto \left|\left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \color{blue}{\left(\left(x \cdot 0.047619047619047616\right) \cdot x\right)}\right)\right) \cdot \frac{x}{\sqrt{\pi}}\right| \]

14. Final simplification 98.6%

    \[\leadsto \left|\frac{x}{\sqrt{\pi}} \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 0.047619047619047616\right)\right)\right)\right)\right| \]
15. Add Preprocessing

Alternative 9: 93.4% accurate, 8.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   x
   (/ (+ 2.0 (* (* x x) (+ 0.6666666666666666 (* (* x x) 0.2)))) (sqrt PI)))))

C

double code(double x) {
	return fabs((x * ((2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * 0.2)))) / sqrt(((double) M_PI)))));
}

Java

public static double code(double x) {
	return Math.abs((x * ((2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * 0.2)))) / Math.sqrt(Math.PI))));
}

Python

def code(x):
	return math.fabs((x * ((2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * 0.2)))) / math.sqrt(math.pi))))

Julia

function code(x)
	return abs(Float64(x * Float64(Float64(2.0 + Float64(Float64(x * x) * Float64(0.6666666666666666 + Float64(Float64(x * x) * 0.2)))) / sqrt(pi))))
end

MATLAB

function tmp = code(x)
	tmp = abs((x * ((2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * 0.2)))) / sqrt(pi))));
end

Wolfram

code[x_] := N[Abs[N[(x * N[(N[(2.0 + N[(N[(x * x), $MachinePrecision] * N[(0.6666666666666666 + N[(N[(x * x), $MachinePrecision] * 0.2), $MachinePrecision]), $MachinePrecision]), $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 99.8%

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

   1. *-commutative N/A

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

   2. fabs-mul N/A

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

   3. fabs-fabs N/A

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

   4. mul-fabs N/A

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

   5. fabs-lowering-fabs.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right), x\right)\right) \]

6. Applied egg-rr 99.8%

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

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \color{blue}{\left(\frac{1}{5} \cdot {x}^{2}\right)}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]
8. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\frac{1}{5}, \left({x}^{2}\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\frac{1}{5}, \left(x \cdot x\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   3. *-lowering-*.f64 92.7%

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\frac{1}{5}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

9. Simplified 92.7%

   \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \color{blue}{0.2 \cdot \left(x \cdot x\right)}\right)}{\sqrt{\pi}} \cdot x\right| \]

10. Final simplification 92.7%

    \[\leadsto \left|x \cdot \frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot 0.2\right)}{\sqrt{\pi}}\right| \]
11. Add Preprocessing

Alternative 10: 92.9% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \left|x \cdot \frac{2 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.2\right)}{\sqrt{\pi}}\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs (* x (/ (+ 2.0 (* (* x x) (* (* x x) 0.2))) (sqrt PI)))))

C

double code(double x) {
	return fabs((x * ((2.0 + ((x * x) * ((x * x) * 0.2))) / sqrt(((double) M_PI)))));
}

Java

public static double code(double x) {
	return Math.abs((x * ((2.0 + ((x * x) * ((x * x) * 0.2))) / Math.sqrt(Math.PI))));
}

Python

def code(x):
	return math.fabs((x * ((2.0 + ((x * x) * ((x * x) * 0.2))) / math.sqrt(math.pi))))

Julia

function code(x)
	return abs(Float64(x * Float64(Float64(2.0 + Float64(Float64(x * x) * Float64(Float64(x * x) * 0.2))) / sqrt(pi))))
end

MATLAB

function tmp = code(x)
	tmp = abs((x * ((2.0 + ((x * x) * ((x * x) * 0.2))) / sqrt(pi))));
end

Wolfram

code[x_] := N[Abs[N[(x * N[(N[(2.0 + N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.2), $MachinePrecision]), $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 99.8%

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

   1. *-commutative N/A

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

   2. fabs-mul N/A

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

   3. fabs-fabs N/A

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

   4. mul-fabs N/A

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

   5. fabs-lowering-fabs.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right), x\right)\right) \]

6. Applied egg-rr 99.8%

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

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \color{blue}{\left(\frac{1}{5} \cdot {x}^{2}\right)}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]
8. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\frac{1}{5}, \left({x}^{2}\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\frac{1}{5}, \left(x \cdot x\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   3. *-lowering-*.f64 92.7%

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\frac{1}{5}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

9. Simplified 92.7%

   \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \color{blue}{0.2 \cdot \left(x \cdot x\right)}\right)}{\sqrt{\pi}} \cdot x\right| \]

10. Taylor expanded in x around inf

    \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \color{blue}{\left(\frac{1}{5} \cdot {x}^{4}\right)}\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]
11. Step-by-step derivation

    1. metadata-eval N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{1}{5} \cdot {x}^{\left(2 \cdot 2\right)}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

    2. pow-sqr N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{1}{5} \cdot \left({x}^{2} \cdot {x}^{2}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

    3. associate-*l* N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\left(\frac{1}{5} \cdot {x}^{2}\right) \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left({x}^{2} \cdot \left(\frac{1}{5} \cdot {x}^{2}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1}{5} \cdot {x}^{2}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

    6. unpow2 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{5} \cdot {x}^{2}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{5} \cdot {x}^{2}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \frac{1}{5}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

    9. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{5}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

    10. unpow2 N/A

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{5}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

    11. *-lowering-*.f64 92.4%

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{5}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

12. Simplified 92.4%

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

13. Final simplification 92.4%

    \[\leadsto \left|x \cdot \frac{2 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.2\right)}{\sqrt{\pi}}\right| \]
14. Add Preprocessing

Alternative 11: 89.4% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ {\pi}^{-0.5} \cdot \left|x \cdot \left(2 + \left(x \cdot x\right) \cdot 0.6666666666666666\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (pow PI -0.5) (fabs (* x (+ 2.0 (* (* x x) 0.6666666666666666))))))

C

double code(double x) {
	return pow(((double) M_PI), -0.5) * fabs((x * (2.0 + ((x * x) * 0.6666666666666666))));
}

Java

public static double code(double x) {
	return Math.pow(Math.PI, -0.5) * Math.abs((x * (2.0 + ((x * x) * 0.6666666666666666))));
}

Python

def code(x):
	return math.pow(math.pi, -0.5) * math.fabs((x * (2.0 + ((x * x) * 0.6666666666666666))))

Julia

function code(x)
	return Float64((pi ^ -0.5) * abs(Float64(x * Float64(2.0 + Float64(Float64(x * x) * 0.6666666666666666)))))
end

MATLAB

function tmp = code(x)
	tmp = (pi ^ -0.5) * abs((x * (2.0 + ((x * x) * 0.6666666666666666))));
end

Wolfram

code[x_] := N[(N[Power[Pi, -0.5], $MachinePrecision] * N[Abs[N[(x * N[(2.0 + N[(N[(x * x), $MachinePrecision] * 0.6666666666666666), $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%

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

   1. associate-*r/ N/A

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

   2. div-inv N/A

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

   3. fabs-mul N/A

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

   4. fabs-div N/A

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

   5. metadata-eval N/A

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

   6. rem-sqrt-square N/A

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

   7. add-sqr-sqrt N/A

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

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left|\left|x\right| \cdot \left(2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)\right)\right|\right), \color{blue}{\left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}}\right)}\right) \]

6. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\left|\left|x\right| \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + x \cdot \left(x \cdot \left(0.2 + \left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right)\right)\right| \cdot {\pi}^{-0.5}} \]

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\left|\left|x\right| \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right|\right)}, \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]
8. Step-by-step derivation

   1. fabs-mul N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left|\left|x\right|\right| \cdot \left|2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right|\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}, \frac{-1}{2}\right)\right) \]

   2. fabs-fabs N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left|x\right| \cdot \left|2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right|\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

   3. fabs-mul N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left|x \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right|\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}, \frac{-1}{2}\right)\right) \]

   4. fabs-lowering-fabs.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\left(x \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{PI.f64}\left(\right)}, \frac{-1}{2}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left({x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \left({x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

   14. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{5}, \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{5}, \mathsf{*.f64}\left(\frac{1}{21}, \left({x}^{2}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{5}, \mathsf{*.f64}\left(\frac{1}{21}, \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

   17. *-lowering-*.f64 99.8%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{1}{5}, \mathsf{*.f64}\left(\frac{1}{21}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

9. Simplified 99.8%

   \[\leadsto \color{blue}{\left|x \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(0.2 + 0.047619047619047616 \cdot \left(x \cdot x\right)\right)\right)\right)\right|} \cdot {\pi}^{-0.5} \]

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\color{blue}{\left(x \cdot \left(2 + \frac{2}{3} \cdot {x}^{2}\right)\right)}\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]
11. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \left(2 + \frac{2}{3} \cdot {x}^{2}\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

    2. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{2}{3} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left({x}^{2} \cdot \frac{2}{3}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{2}{3}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

    5. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{2}{3}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

    6. *-lowering-*.f64 89.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{2}{3}\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{PI.f64}\left(\right), \frac{-1}{2}\right)\right) \]

12. Simplified 89.0%

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

13. Final simplification 89.0%

    \[\leadsto {\pi}^{-0.5} \cdot \left|x \cdot \left(2 + \left(x \cdot x\right) \cdot 0.6666666666666666\right)\right| \]
14. Add Preprocessing

Alternative 12: 89.4% accurate, 8.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Abs[N[(x * N[(N[(2.0 + N[(N[(x * x), $MachinePrecision] * 0.6666666666666666), $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 99.8%

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

   1. *-commutative N/A

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

   2. fabs-mul N/A

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

   3. fabs-fabs N/A

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

   4. mul-fabs N/A

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

   5. fabs-lowering-fabs.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right), x\right)\right) \]

6. Applied egg-rr 99.8%

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

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(2 + \frac{2}{3} \cdot {x}^{2}\right)}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]
8. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{2}{3} \cdot {x}^{2}\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{2}{3}, \left({x}^{2}\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{2}{3}, \left(x \cdot x\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   4. *-lowering-*.f64 89.0%

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

9. Simplified 89.0%

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

10. Final simplification 89.0%

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

Alternative 13: 88.9% accurate, 8.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. *-commutative N/A

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

   2. fabs-mul N/A

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

   3. fabs-fabs N/A

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

   4. mul-fabs N/A

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

   5. fabs-lowering-fabs.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right), x\right)\right) \]

6. Applied egg-rr 99.8%

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

7. Taylor expanded in x around inf

   \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{21} \cdot {x}^{3}\right)}\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]
8. Step-by-step derivation

   1. unpow3 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \left(\frac{1}{21} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \left(\frac{1}{21} \cdot \left({x}^{2} \cdot x\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \left(\left(\frac{1}{21} \cdot {x}^{2}\right) \cdot x\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(\frac{1}{21} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{21}, \left({x}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{21}, \left(x \cdot x\right)\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

   8. *-lowering-*.f64 99.1%

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(\frac{1}{21}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]

9. Simplified 99.1%

   \[\leadsto \left|\frac{2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + x \cdot \color{blue}{\left(x \cdot \left(0.047619047619047616 \cdot \left(x \cdot x\right)\right)\right)}\right)}{\sqrt{\pi}} \cdot x\right| \]
10. Step-by-step derivation

    1. associate-*l/ N/A

       \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{\left(2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + x \cdot \left(x \cdot \left(\frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot x}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right) \]

    2. associate-/l* N/A

       \[\leadsto \mathsf{fabs.f64}\left(\left(\left(2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + x \cdot \left(x \cdot \left(\frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right) \cdot \frac{x}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + x \cdot \left(x \cdot \left(\frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right), \left(\frac{x}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

    4. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \left(\left(x \cdot x\right) \cdot \left(\frac{2}{3} + x \cdot \left(x \cdot \left(\frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right), \left(\frac{x}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{2}{3} + x \cdot \left(x \cdot \left(\frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right), \left(\frac{x}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{2}{3} + x \cdot \left(x \cdot \left(\frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right), \left(\frac{x}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

    7. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \left(x \cdot \left(x \cdot \left(\frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right), \left(\frac{x}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

    8. associate-*r* N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \left(\left(x \cdot x\right) \cdot \left(\frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right), \left(\frac{x}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

    9. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right), \left(\frac{x}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

    10. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{21} \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right), \left(\frac{x}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

    11. *-commutative N/A

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)\right)\right)\right), \left(\frac{x}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

    12. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{21}\right)\right)\right)\right)\right), \left(\frac{x}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

    13. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{21}\right)\right)\right)\right)\right), \left(\frac{x}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

    14. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{21}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \left(\sqrt{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

    15. sqrt-lowering-sqrt.f64 N/A

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{21}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{PI}\left(\right)\right)\right)\right)\right) \]

    16. PI-lowering-PI.f64 98.6%

        \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{2}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{21}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

11. Applied egg-rr 98.6%

    \[\leadsto \left|\color{blue}{\left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot 0.047619047619047616\right)\right)\right) \cdot \frac{x}{\sqrt{\pi}}}\right| \]

12. Taylor expanded in x around 0

    \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(2 + \frac{2}{3} \cdot {x}^{2}\right)}, \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]
13. Step-by-step derivation

    1. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{2}{3} \cdot {x}^{2}\right)\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

    2. *-commutative N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \left({x}^{2} \cdot \frac{2}{3}\right)\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{2}{3}\right)\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

    4. unpow2 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{2}{3}\right)\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

    5. *-lowering-*.f64 88.6%

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{2}{3}\right)\right), \mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

14. Simplified 88.6%

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

15. Final simplification 88.6%

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

Alternative 14: 68.2% 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%

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

   1. *-commutative N/A

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

   2. fabs-mul N/A

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

   3. fabs-fabs N/A

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

   4. mul-fabs N/A

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

   5. fabs-lowering-fabs.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot x\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\left(\frac{2 + \left(x \cdot x\right) \cdot \left(\frac{2}{3} + \left(x \cdot x\right) \cdot \left(\frac{1}{5} + \left(x \cdot x\right) \cdot \frac{1}{21}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right), x\right)\right) \]

6. Applied egg-rr 99.8%

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

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{2}, \mathsf{sqrt.f64}\left(\mathsf{PI.f64}\left(\right)\right)\right), x\right)\right) \]
8. Step-by-step derivation

9. Simplified 68.3%

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

10. Final simplification 68.3%

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

Reproduce

herbie shell --seed 2024150 
(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)))))))