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

Percentage Accurate: 99.8% → 99.8%

Time: 12.8s

Alternatives: 12

Speedup: 1.5×

• 28.6% 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, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| \cdot \left(x \cdot x\right)\\ t_1 := \left|x\right| \cdot \left(\left|x\right| \cdot t_0\right)\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + 0.6666666666666666 \cdot t_0\right) + 0.2 \cdot t_1\right) + 0.047619047619047616 \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot t_1\right)\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) (* x x))) (t_1 (* (fabs x) (* (fabs x) t_0))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (+ (* 2.0 (fabs x)) (* 0.6666666666666666 t_0)) (* 0.2 t_1))
      (* 0.047619047619047616 (* (fabs x) (* (fabs x) t_1))))))))

C

double code(double x) {
	double t_0 = fabs(x) * (x * x);
	double t_1 = fabs(x) * (fabs(x) * t_0);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + (0.6666666666666666 * t_0)) + (0.2 * t_1)) + (0.047619047619047616 * (fabs(x) * (fabs(x) * t_1))))));
}

Java

public static double code(double x) {
	double t_0 = Math.abs(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)) + (0.6666666666666666 * t_0)) + (0.2 * t_1)) + (0.047619047619047616 * (Math.abs(x) * (Math.abs(x) * t_1))))));
}

Python

def code(x):
	t_0 = math.fabs(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)) + (0.6666666666666666 * t_0)) + (0.2 * t_1)) + (0.047619047619047616 * (math.fabs(x) * (math.fabs(x) * t_1))))))

Julia

function code(x)
	t_0 = Float64(abs(x) * Float64(x * x))
	t_1 = Float64(abs(x) * Float64(abs(x) * t_0))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(0.6666666666666666 * t_0)) + Float64(0.2 * t_1)) + Float64(0.047619047619047616 * Float64(abs(x) * Float64(abs(x) * t_1))))))
end

MATLAB

function tmp = code(x)
	t_0 = abs(x) * (x * x);
	t_1 = abs(x) * (abs(x) * t_0);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + (0.6666666666666666 * t_0)) + (0.2 * t_1)) + (0.047619047619047616 * (abs(x) * (abs(x) * t_1))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * t$95$0), $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[(0.6666666666666666 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.2 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * 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%

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

Alternative 2: 99.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| \cdot \left(x \cdot x\right)\\ t_1 := \left(x \cdot x\right) \cdot t_0\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot t_0\right) + 0.2 \cdot t_1\right) + 0.047619047619047616 \cdot \left(\left(x \cdot x\right) \cdot t_1\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) (* x x))) (t_1 (* (* x x) t_0)))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (fma 2.0 (fabs x) (* 0.6666666666666666 t_0)) (* 0.2 t_1))
      (* 0.047619047619047616 (* (* x x) t_1)))))))

C

double code(double x) {
	double t_0 = fabs(x) * (x * x);
	double t_1 = (x * x) * t_0;
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((fma(2.0, fabs(x), (0.6666666666666666 * t_0)) + (0.2 * t_1)) + (0.047619047619047616 * ((x * x) * t_1)))));
}

Julia

function code(x)
	t_0 = Float64(abs(x) * Float64(x * x))
	t_1 = Float64(Float64(x * x) * t_0)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(fma(2.0, abs(x), Float64(0.6666666666666666 * t_0)) + Float64(0.2 * t_1)) + Float64(0.047619047619047616 * Float64(Float64(x * x) * t_1)))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision] + N[(0.6666666666666666 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.2 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[(N[(x * x), $MachinePrecision] * t$95$1), $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%

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

5. Final simplification 99.8%

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

Alternative 3: 68.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, 2 + 0.6666666666666666 \cdot {x}^{2}\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

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

C

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

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 2.0)
		tmp = Float64(x * Float64(fma(0.2, (x ^ 4.0), Float64(2.0 + Float64(0.6666666666666666 * (x ^ 2.0)))) / sqrt(pi)));
	else
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(abs(x) * fma(0.2, (x ^ 4.0), Float64(0.047619047619047616 * (x ^ 6.0))))));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2

   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 99.2%

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

   5. Taylor expanded in x around 0 98.1%

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

      1. div-inv 98.8%

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

      2. add-sqr-sqrt 48.6%

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

      3. fabs-sqr 48.6%

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

      4. add-sqr-sqrt 50.7%

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

      5. add-sqr-sqrt 50.7%

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

      6. fabs-sqr 50.7%

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

      7. add-sqr-sqrt 50.7%

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

      8. clear-num 50.7%

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

      9. fma-def 50.7%

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

      10. pow2 50.7%

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

   7. Applied egg-rr 50.7%

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

      1. fma-udef 50.7%

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

   9. Applied egg-rr 50.7%

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

   if 2 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 99.6%

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

   7. Simplified 99.6%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, 2 + 0.6666666666666666 \cdot {x}^{2}\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.4%

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

   1. fma-udef 99.4%

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

6. Applied egg-rr 99.4%

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

7. Final simplification 99.4%

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

Alternative 5: 68.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, 2 + 0.6666666666666666 \cdot {x}^{2}\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \left(\frac{21}{{x}^{6}} + \frac{-88.2}{{x}^{8}}\right)\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 2.0)
   (*
    x
    (/
     (fma 0.2 (pow x 4.0) (+ 2.0 (* 0.6666666666666666 (pow x 2.0))))
     (sqrt PI)))
   (/
    (fabs x)
    (fabs (* (sqrt PI) (+ (/ 21.0 (pow x 6.0)) (/ -88.2 (pow x 8.0))))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 2.0) {
		tmp = x * (fma(0.2, pow(x, 4.0), (2.0 + (0.6666666666666666 * pow(x, 2.0)))) / sqrt(((double) M_PI)));
	} else {
		tmp = fabs(x) / fabs((sqrt(((double) M_PI)) * ((21.0 / pow(x, 6.0)) + (-88.2 / pow(x, 8.0)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 2.0)
		tmp = Float64(x * Float64(fma(0.2, (x ^ 4.0), Float64(2.0 + Float64(0.6666666666666666 * (x ^ 2.0)))) / sqrt(pi)));
	else
		tmp = Float64(abs(x) / abs(Float64(sqrt(pi) * Float64(Float64(21.0 / (x ^ 6.0)) + Float64(-88.2 / (x ^ 8.0))))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 2.0], N[(x * N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision] + N[(2.0 + N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x], $MachinePrecision] / N[Abs[N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[(21.0 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(-88.2 / N[Power[x, 8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2

   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 99.2%

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

   5. Taylor expanded in x around 0 98.1%

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

      1. div-inv 98.8%

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

      2. add-sqr-sqrt 48.6%

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

      3. fabs-sqr 48.6%

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

      4. add-sqr-sqrt 50.7%

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

      5. add-sqr-sqrt 50.7%

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

      6. fabs-sqr 50.7%

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

      7. add-sqr-sqrt 50.7%

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

      8. clear-num 50.7%

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

      9. fma-def 50.7%

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

      10. pow2 50.7%

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

   7. Applied egg-rr 50.7%

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

      1. fma-udef 50.7%

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

   9. Applied egg-rr 50.7%

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

   if 2 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 99.5%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{-88.2 \cdot \left(\frac{1}{{x}^{8}} \cdot \sqrt{\pi}\right) + 21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}\right|} \]
   6. Step-by-step derivation

      1. +-commutative 99.5%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right) + -88.2 \cdot \left(\frac{1}{{x}^{8}} \cdot \sqrt{\pi}\right)}\right|} \]

      2. associate-*r* 99.6%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\left(21 \cdot \frac{1}{{x}^{6}}\right) \cdot \sqrt{\pi}} + -88.2 \cdot \left(\frac{1}{{x}^{8}} \cdot \sqrt{\pi}\right)\right|} \]

      3. associate-*r* 99.6%

         \[\leadsto \frac{\left|x\right|}{\left|\left(21 \cdot \frac{1}{{x}^{6}}\right) \cdot \sqrt{\pi} + \color{blue}{\left(-88.2 \cdot \frac{1}{{x}^{8}}\right) \cdot \sqrt{\pi}}\right|} \]

      4. distribute-rgt-out 99.6%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot \left(21 \cdot \frac{1}{{x}^{6}} + -88.2 \cdot \frac{1}{{x}^{8}}\right)}\right|} \]

      5. associate-*r/ 99.6%

         \[\leadsto \frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \left(\color{blue}{\frac{21 \cdot 1}{{x}^{6}}} + -88.2 \cdot \frac{1}{{x}^{8}}\right)\right|} \]

      6. metadata-eval 99.6%

         \[\leadsto \frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \left(\frac{\color{blue}{21}}{{x}^{6}} + -88.2 \cdot \frac{1}{{x}^{8}}\right)\right|} \]

      7. associate-*r/ 99.6%

         \[\leadsto \frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \left(\frac{21}{{x}^{6}} + \color{blue}{\frac{-88.2 \cdot 1}{{x}^{8}}}\right)\right|} \]

      8. metadata-eval 99.6%

         \[\leadsto \frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \left(\frac{21}{{x}^{6}} + \frac{\color{blue}{-88.2}}{{x}^{8}}\right)\right|} \]

   7. Simplified 99.6%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot \left(\frac{21}{{x}^{6}} + \frac{-88.2}{{x}^{8}}\right)}\right|} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, 2 + 0.6666666666666666 \cdot {x}^{2}\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \left(\frac{21}{{x}^{6}} + \frac{-88.2}{{x}^{8}}\right)\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 35.1% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 2.0)
		tmp = Float64(x * Float64(fma(0.2, (x ^ 4.0), Float64(2.0 + Float64(0.6666666666666666 * (x ^ 2.0)))) / sqrt(pi)));
	else
		tmp = Float64(Float64(0.047619047619047616 / sqrt(pi)) * Float64(x / (x ^ -6.0)));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2

   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 99.2%

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

   5. Taylor expanded in x around 0 98.1%

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

      1. div-inv 98.8%

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

      2. add-sqr-sqrt 48.6%

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

      3. fabs-sqr 48.6%

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

      4. add-sqr-sqrt 50.7%

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

      5. add-sqr-sqrt 50.7%

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

      6. fabs-sqr 50.7%

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

      7. add-sqr-sqrt 50.7%

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

      8. clear-num 50.7%

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

      9. fma-def 50.7%

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

      10. pow2 50.7%

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

   7. Applied egg-rr 50.7%

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

      1. fma-udef 50.7%

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

   9. Applied egg-rr 50.7%

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

   if 2 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 99.2%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}\right|} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

         \[\leadsto \frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{\left|21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)\right|} \]

      2. fabs-sqr 0.0%

         \[\leadsto \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\left|21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)\right|} \]

      3. add-sqr-sqrt 0.0%

         \[\leadsto \frac{\sqrt{x} \cdot \sqrt{x}}{\left|\color{blue}{\sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)} \cdot \sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}}\right|} \]

      4. fabs-sqr 0.0%

         \[\leadsto \frac{\sqrt{x} \cdot \sqrt{x}}{\color{blue}{\sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)} \cdot \sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}}} \]

      5. add-sqr-sqrt 0.1%

         \[\leadsto \frac{\color{blue}{x}}{\sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)} \cdot \sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}} \]

      6. *-un-lft-identity 0.1%

         \[\leadsto \frac{\color{blue}{1 \cdot x}}{\sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)} \cdot \sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}} \]

      7. add-sqr-sqrt 0.1%

         \[\leadsto \frac{1 \cdot x}{\color{blue}{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}} \]

      8. times-frac 0.1%

         \[\leadsto \color{blue}{\frac{1}{21} \cdot \frac{x}{\frac{1}{{x}^{6}} \cdot \sqrt{\pi}}} \]

      9. metadata-eval 0.1%

         \[\leadsto \color{blue}{0.047619047619047616} \cdot \frac{x}{\frac{1}{{x}^{6}} \cdot \sqrt{\pi}} \]

      10. *-commutative 0.1%

          \[\leadsto 0.047619047619047616 \cdot \frac{x}{\color{blue}{\sqrt{\pi} \cdot \frac{1}{{x}^{6}}}} \]

      11. pow-flip 0.1%

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

      12. metadata-eval 0.1%

          \[\leadsto 0.047619047619047616 \cdot \frac{x}{\sqrt{\pi} \cdot {x}^{\color{blue}{-6}}} \]

   7. Applied egg-rr 0.1%

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

      1. expm1-log1p-u 0.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot \frac{x}{\sqrt{\pi} \cdot {x}^{-6}}\right)\right)} \]

      2. expm1-udef 0.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \frac{x}{\sqrt{\pi} \cdot {x}^{-6}}\right)} - 1} \]

      3. associate-*r/ 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{0.047619047619047616 \cdot x}{\sqrt{\pi} \cdot {x}^{-6}}}\right)} - 1 \]

      4. times-frac 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{0.047619047619047616}{\sqrt{\pi}} \cdot \frac{x}{{x}^{-6}}}\right)} - 1 \]

   9. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.047619047619047616}{\sqrt{\pi}} \cdot \frac{x}{{x}^{-6}}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 0.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.047619047619047616}{\sqrt{\pi}} \cdot \frac{x}{{x}^{-6}}\right)\right)} \]

       2. expm1-log1p 0.1%

          \[\leadsto \color{blue}{\frac{0.047619047619047616}{\sqrt{\pi}} \cdot \frac{x}{{x}^{-6}}} \]

   11. Simplified 0.1%

       \[\leadsto \color{blue}{\frac{0.047619047619047616}{\sqrt{\pi}} \cdot \frac{x}{{x}^{-6}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, 2 + 0.6666666666666666 \cdot {x}^{2}\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.047619047619047616}{\sqrt{\pi}} \cdot \frac{x}{{x}^{-6}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.0% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \mathbf{if}\;\left|x\right| \leq 2:\\ \;\;\;\;0.6666666666666666 \cdot \left(t_0 \cdot {x}^{3}\right) + 2 \cdot \left(x \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.047619047619047616}{\sqrt{\pi}} \cdot \frac{x}{{x}^{-6}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (if (<= (fabs x) 2.0)
     (+ (* 0.6666666666666666 (* t_0 (pow x 3.0))) (* 2.0 (* x t_0)))
     (* (/ 0.047619047619047616 (sqrt PI)) (/ x (pow x -6.0))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 / ((double) M_PI)));
	double tmp;
	if (fabs(x) <= 2.0) {
		tmp = (0.6666666666666666 * (t_0 * pow(x, 3.0))) + (2.0 * (x * t_0));
	} else {
		tmp = (0.047619047619047616 / sqrt(((double) M_PI))) * (x / pow(x, -6.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 / Math.PI));
	double tmp;
	if (Math.abs(x) <= 2.0) {
		tmp = (0.6666666666666666 * (t_0 * Math.pow(x, 3.0))) + (2.0 * (x * t_0));
	} else {
		tmp = (0.047619047619047616 / Math.sqrt(Math.PI)) * (x / Math.pow(x, -6.0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 / math.pi))
	tmp = 0
	if math.fabs(x) <= 2.0:
		tmp = (0.6666666666666666 * (t_0 * math.pow(x, 3.0))) + (2.0 * (x * t_0))
	else:
		tmp = (0.047619047619047616 / math.sqrt(math.pi)) * (x / math.pow(x, -6.0))
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / pi))
	tmp = 0.0
	if (abs(x) <= 2.0)
		tmp = Float64(Float64(0.6666666666666666 * Float64(t_0 * (x ^ 3.0))) + Float64(2.0 * Float64(x * t_0)));
	else
		tmp = Float64(Float64(0.047619047619047616 / sqrt(pi)) * Float64(x / (x ^ -6.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 / pi));
	tmp = 0.0;
	if (abs(x) <= 2.0)
		tmp = (0.6666666666666666 * (t_0 * (x ^ 3.0))) + (2.0 * (x * t_0));
	else
		tmp = (0.047619047619047616 / sqrt(pi)) * (x / (x ^ -6.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 2.0], N[(N[(0.6666666666666666 * N[(t$95$0 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.047619047619047616 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(x / N[Power[x, -6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2

   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 99.2%

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

   5. Taylor expanded in x around 0 98.1%

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

      1. div-inv 98.8%

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

      2. add-sqr-sqrt 48.6%

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

      3. fabs-sqr 48.6%

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

      4. add-sqr-sqrt 50.7%

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

      5. add-sqr-sqrt 50.7%

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

      6. fabs-sqr 50.7%

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

      7. add-sqr-sqrt 50.7%

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

      8. clear-num 50.7%

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

      9. fma-def 50.7%

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

      10. pow2 50.7%

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

   7. Applied egg-rr 50.7%

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

   8. Taylor expanded in x around 0 50.6%

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

   if 2 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 99.2%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}\right|} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

         \[\leadsto \frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{\left|21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)\right|} \]

      2. fabs-sqr 0.0%

         \[\leadsto \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\left|21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)\right|} \]

      3. add-sqr-sqrt 0.0%

         \[\leadsto \frac{\sqrt{x} \cdot \sqrt{x}}{\left|\color{blue}{\sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)} \cdot \sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}}\right|} \]

      4. fabs-sqr 0.0%

         \[\leadsto \frac{\sqrt{x} \cdot \sqrt{x}}{\color{blue}{\sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)} \cdot \sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}}} \]

      5. add-sqr-sqrt 0.1%

         \[\leadsto \frac{\color{blue}{x}}{\sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)} \cdot \sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}} \]

      6. *-un-lft-identity 0.1%

         \[\leadsto \frac{\color{blue}{1 \cdot x}}{\sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)} \cdot \sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}} \]

      7. add-sqr-sqrt 0.1%

         \[\leadsto \frac{1 \cdot x}{\color{blue}{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}} \]

      8. times-frac 0.1%

         \[\leadsto \color{blue}{\frac{1}{21} \cdot \frac{x}{\frac{1}{{x}^{6}} \cdot \sqrt{\pi}}} \]

      9. metadata-eval 0.1%

         \[\leadsto \color{blue}{0.047619047619047616} \cdot \frac{x}{\frac{1}{{x}^{6}} \cdot \sqrt{\pi}} \]

      10. *-commutative 0.1%

          \[\leadsto 0.047619047619047616 \cdot \frac{x}{\color{blue}{\sqrt{\pi} \cdot \frac{1}{{x}^{6}}}} \]

      11. pow-flip 0.1%

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

      12. metadata-eval 0.1%

          \[\leadsto 0.047619047619047616 \cdot \frac{x}{\sqrt{\pi} \cdot {x}^{\color{blue}{-6}}} \]

   7. Applied egg-rr 0.1%

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

      1. expm1-log1p-u 0.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot \frac{x}{\sqrt{\pi} \cdot {x}^{-6}}\right)\right)} \]

      2. expm1-udef 0.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \frac{x}{\sqrt{\pi} \cdot {x}^{-6}}\right)} - 1} \]

      3. associate-*r/ 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{0.047619047619047616 \cdot x}{\sqrt{\pi} \cdot {x}^{-6}}}\right)} - 1 \]

      4. times-frac 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{0.047619047619047616}{\sqrt{\pi}} \cdot \frac{x}{{x}^{-6}}}\right)} - 1 \]

   9. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.047619047619047616}{\sqrt{\pi}} \cdot \frac{x}{{x}^{-6}}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 0.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.047619047619047616}{\sqrt{\pi}} \cdot \frac{x}{{x}^{-6}}\right)\right)} \]

       2. expm1-log1p 0.1%

          \[\leadsto \color{blue}{\frac{0.047619047619047616}{\sqrt{\pi}} \cdot \frac{x}{{x}^{-6}}} \]

   11. Simplified 0.1%

       \[\leadsto \color{blue}{\frac{0.047619047619047616}{\sqrt{\pi}} \cdot \frac{x}{{x}^{-6}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2:\\ \;\;\;\;0.6666666666666666 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x}^{3}\right) + 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.047619047619047616}{\sqrt{\pi}} \cdot \frac{x}{{x}^{-6}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 35.0% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.047619047619047616}{\sqrt{\pi}} \cdot \frac{x}{{x}^{-6}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 2.0)
   (* (sqrt (/ 1.0 PI)) (+ (* 2.0 x) (* 0.6666666666666666 (pow x 3.0))))
   (* (/ 0.047619047619047616 (sqrt PI)) (/ x (pow x -6.0)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 2.0) {
		tmp = sqrt((1.0 / ((double) M_PI))) * ((2.0 * x) + (0.6666666666666666 * pow(x, 3.0)));
	} else {
		tmp = (0.047619047619047616 / sqrt(((double) M_PI))) * (x / pow(x, -6.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 2.0) {
		tmp = Math.sqrt((1.0 / Math.PI)) * ((2.0 * x) + (0.6666666666666666 * Math.pow(x, 3.0)));
	} else {
		tmp = (0.047619047619047616 / Math.sqrt(Math.PI)) * (x / Math.pow(x, -6.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 2.0:
		tmp = math.sqrt((1.0 / math.pi)) * ((2.0 * x) + (0.6666666666666666 * math.pow(x, 3.0)))
	else:
		tmp = (0.047619047619047616 / math.sqrt(math.pi)) * (x / math.pow(x, -6.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 2.0)
		tmp = Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(2.0 * x) + Float64(0.6666666666666666 * (x ^ 3.0))));
	else
		tmp = Float64(Float64(0.047619047619047616 / sqrt(pi)) * Float64(x / (x ^ -6.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 2.0)
		tmp = sqrt((1.0 / pi)) * ((2.0 * x) + (0.6666666666666666 * (x ^ 3.0)));
	else
		tmp = (0.047619047619047616 / sqrt(pi)) * (x / (x ^ -6.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 2.0], N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(2.0 * x), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.047619047619047616 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(x / N[Power[x, -6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2

   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 99.2%

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

   5. Taylor expanded in x around 0 98.1%

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

      1. div-inv 98.8%

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

      2. add-sqr-sqrt 48.6%

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

      3. fabs-sqr 48.6%

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

      4. add-sqr-sqrt 50.7%

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

      5. add-sqr-sqrt 50.7%

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

      6. fabs-sqr 50.7%

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

      7. add-sqr-sqrt 50.7%

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

      8. clear-num 50.7%

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

      9. fma-def 50.7%

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

      10. pow2 50.7%

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

   7. Applied egg-rr 50.7%

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

   8. Taylor expanded in x around 0 50.6%

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

      1. +-commutative 50.6%

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

      2. associate-*r* 50.6%

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

      3. *-commutative 50.6%

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

      4. associate-*r* 50.6%

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

      5. distribute-rgt-out 50.6%

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

      6. *-commutative 50.6%

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

   10. Simplified 50.6%

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

   if 2 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 99.2%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}\right|} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 0.0%

         \[\leadsto \frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{\left|21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)\right|} \]

      2. fabs-sqr 0.0%

         \[\leadsto \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\left|21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)\right|} \]

      3. add-sqr-sqrt 0.0%

         \[\leadsto \frac{\sqrt{x} \cdot \sqrt{x}}{\left|\color{blue}{\sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)} \cdot \sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}}\right|} \]

      4. fabs-sqr 0.0%

         \[\leadsto \frac{\sqrt{x} \cdot \sqrt{x}}{\color{blue}{\sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)} \cdot \sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}}} \]

      5. add-sqr-sqrt 0.1%

         \[\leadsto \frac{\color{blue}{x}}{\sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)} \cdot \sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}} \]

      6. *-un-lft-identity 0.1%

         \[\leadsto \frac{\color{blue}{1 \cdot x}}{\sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)} \cdot \sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}} \]

      7. add-sqr-sqrt 0.1%

         \[\leadsto \frac{1 \cdot x}{\color{blue}{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}} \]

      8. times-frac 0.1%

         \[\leadsto \color{blue}{\frac{1}{21} \cdot \frac{x}{\frac{1}{{x}^{6}} \cdot \sqrt{\pi}}} \]

      9. metadata-eval 0.1%

         \[\leadsto \color{blue}{0.047619047619047616} \cdot \frac{x}{\frac{1}{{x}^{6}} \cdot \sqrt{\pi}} \]

      10. *-commutative 0.1%

          \[\leadsto 0.047619047619047616 \cdot \frac{x}{\color{blue}{\sqrt{\pi} \cdot \frac{1}{{x}^{6}}}} \]

      11. pow-flip 0.1%

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

      12. metadata-eval 0.1%

          \[\leadsto 0.047619047619047616 \cdot \frac{x}{\sqrt{\pi} \cdot {x}^{\color{blue}{-6}}} \]

   7. Applied egg-rr 0.1%

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

      1. expm1-log1p-u 0.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot \frac{x}{\sqrt{\pi} \cdot {x}^{-6}}\right)\right)} \]

      2. expm1-udef 0.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \frac{x}{\sqrt{\pi} \cdot {x}^{-6}}\right)} - 1} \]

      3. associate-*r/ 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{0.047619047619047616 \cdot x}{\sqrt{\pi} \cdot {x}^{-6}}}\right)} - 1 \]

      4. times-frac 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{0.047619047619047616}{\sqrt{\pi}} \cdot \frac{x}{{x}^{-6}}}\right)} - 1 \]

   9. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.047619047619047616}{\sqrt{\pi}} \cdot \frac{x}{{x}^{-6}}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 0.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.047619047619047616}{\sqrt{\pi}} \cdot \frac{x}{{x}^{-6}}\right)\right)} \]

       2. expm1-log1p 0.1%

          \[\leadsto \color{blue}{\frac{0.047619047619047616}{\sqrt{\pi}} \cdot \frac{x}{{x}^{-6}}} \]

   11. Simplified 0.1%

       \[\leadsto \color{blue}{\frac{0.047619047619047616}{\sqrt{\pi}} \cdot \frac{x}{{x}^{-6}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.047619047619047616}{\sqrt{\pi}} \cdot \frac{x}{{x}^{-6}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 34.9% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \frac{x}{\sqrt{\pi} \cdot {x}^{-6}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (* x (/ 2.0 (sqrt PI)))
   (* 0.047619047619047616 (/ x (* (sqrt PI) (pow x -6.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = 0.047619047619047616 * (x / (sqrt(((double) M_PI)) * pow(x, -6.0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = x * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = 0.047619047619047616 * (x / (Math.sqrt(Math.PI) * Math.pow(x, -6.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = 0.047619047619047616 * (x / (math.sqrt(math.pi) * math.pow(x, -6.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = Float64(x * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64(0.047619047619047616 * Float64(x / Float64(sqrt(pi) * (x ^ -6.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = x * (2.0 / sqrt(pi));
	else
		tmp = 0.047619047619047616 * (x / (sqrt(pi) * (x ^ -6.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(x / N[(N[Sqrt[Pi], $MachinePrecision] * N[Power[x, -6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 68.6%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{0.5 \cdot \sqrt{\pi}}\right|} \]
   6. Step-by-step derivation

      1. *-commutative 68.6%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

   7. Simplified 68.6%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

   8. Taylor expanded in x around 0 68.6%

      \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|0.5 \cdot \sqrt{\pi}\right|}} \]
   9. Step-by-step derivation

      1. fabs-neg 68.6%

         \[\leadsto \frac{\color{blue}{\left|-x\right|}}{\left|0.5 \cdot \sqrt{\pi}\right|} \]

      2. *-commutative 68.6%

         \[\leadsto \frac{\left|-x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

      3. fabs-div 68.6%

         \[\leadsto \color{blue}{\left|\frac{-x}{\sqrt{\pi} \cdot 0.5}\right|} \]

      4. neg-mul-1 68.6%

         \[\leadsto \left|\frac{\color{blue}{-1 \cdot x}}{\sqrt{\pi} \cdot 0.5}\right| \]

      5. *-commutative 68.6%

         \[\leadsto \left|\frac{-1 \cdot x}{\color{blue}{0.5 \cdot \sqrt{\pi}}}\right| \]

      6. times-frac 68.6%

         \[\leadsto \left|\color{blue}{\frac{-1}{0.5} \cdot \frac{x}{\sqrt{\pi}}}\right| \]

      7. metadata-eval 68.6%

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

      8. metadata-eval 68.6%

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

      9. distribute-lft-neg-in 68.6%

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

      10. fabs-neg 68.6%

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

      11. rem-square-sqrt 32.8%

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

      12. fabs-sqr 32.8%

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

      13. rem-square-sqrt 34.3%

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

      14. *-commutative 34.3%

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

      15. metadata-eval 34.3%

          \[\leadsto \frac{x}{\sqrt{\pi}} \cdot \color{blue}{\frac{1}{0.5}} \]

      16. times-frac 34.3%

          \[\leadsto \color{blue}{\frac{x \cdot 1}{\sqrt{\pi} \cdot 0.5}} \]

      17. *-rgt-identity 34.3%

          \[\leadsto \frac{\color{blue}{x}}{\sqrt{\pi} \cdot 0.5} \]

      18. *-commutative 34.3%

          \[\leadsto \frac{x}{\color{blue}{0.5 \cdot \sqrt{\pi}}} \]

      19. associate-/r* 34.3%

          \[\leadsto \color{blue}{\frac{\frac{x}{0.5}}{\sqrt{\pi}}} \]

   10. Simplified 34.3%

       \[\leadsto \color{blue}{\frac{\frac{x}{0.5}}{\sqrt{\pi}}} \]
   11. Step-by-step derivation

       1. associate-/l/ 34.3%

          \[\leadsto \color{blue}{\frac{x}{\sqrt{\pi} \cdot 0.5}} \]

       2. *-un-lft-identity 34.3%

          \[\leadsto \frac{\color{blue}{1 \cdot x}}{\sqrt{\pi} \cdot 0.5} \]

       3. *-commutative 34.3%

          \[\leadsto \frac{1 \cdot x}{\color{blue}{0.5 \cdot \sqrt{\pi}}} \]

       4. times-frac 34.3%

          \[\leadsto \color{blue}{\frac{1}{0.5} \cdot \frac{x}{\sqrt{\pi}}} \]

       5. metadata-eval 34.3%

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

       6. rem-log-exp 4.3%

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

       7. *-un-lft-identity 4.3%

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

       8. log-prod 4.3%

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

       9. metadata-eval 4.3%

          \[\leadsto \color{blue}{0} + \log \left(e^{2 \cdot \frac{x}{\sqrt{\pi}}}\right) \]

       10. rem-log-exp 34.3%

           \[\leadsto 0 + \color{blue}{2 \cdot \frac{x}{\sqrt{\pi}}} \]

       11. metadata-eval 34.3%

           \[\leadsto 0 + \color{blue}{\frac{1}{0.5}} \cdot \frac{x}{\sqrt{\pi}} \]

       12. times-frac 34.3%

           \[\leadsto 0 + \color{blue}{\frac{1 \cdot x}{0.5 \cdot \sqrt{\pi}}} \]

       13. *-un-lft-identity 34.3%

           \[\leadsto 0 + \frac{\color{blue}{x}}{0.5 \cdot \sqrt{\pi}} \]

       14. *-commutative 34.3%

           \[\leadsto 0 + \frac{x}{\color{blue}{\sqrt{\pi} \cdot 0.5}} \]

   12. Applied egg-rr 34.3%

       \[\leadsto \color{blue}{0 + \frac{x}{\sqrt{\pi} \cdot 0.5}} \]
   13. Step-by-step derivation

       1. +-lft-identity 34.3%

          \[\leadsto \color{blue}{\frac{x}{\sqrt{\pi} \cdot 0.5}} \]

       2. *-rgt-identity 34.3%

          \[\leadsto \frac{\color{blue}{x \cdot 1}}{\sqrt{\pi} \cdot 0.5} \]

       3. times-frac 34.3%

          \[\leadsto \color{blue}{\frac{x}{\sqrt{\pi}} \cdot \frac{1}{0.5}} \]

       4. metadata-eval 34.3%

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

       5. associate-*l/ 34.3%

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

       6. associate-*r/ 34.5%

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

   14. Simplified 34.5%

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

   if 1.8500000000000001 < x

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around inf 35.3%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}\right|} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 1.9%

         \[\leadsto \frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{\left|21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)\right|} \]

      2. fabs-sqr 1.9%

         \[\leadsto \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\left|21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)\right|} \]

      3. add-sqr-sqrt 1.9%

         \[\leadsto \frac{\sqrt{x} \cdot \sqrt{x}}{\left|\color{blue}{\sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)} \cdot \sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}}\right|} \]

      4. fabs-sqr 1.9%

         \[\leadsto \frac{\sqrt{x} \cdot \sqrt{x}}{\color{blue}{\sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)} \cdot \sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}}} \]

      5. add-sqr-sqrt 3.8%

         \[\leadsto \frac{\color{blue}{x}}{\sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)} \cdot \sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}} \]

      6. *-un-lft-identity 3.8%

         \[\leadsto \frac{\color{blue}{1 \cdot x}}{\sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)} \cdot \sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}} \]

      7. add-sqr-sqrt 3.8%

         \[\leadsto \frac{1 \cdot x}{\color{blue}{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}} \]

      8. times-frac 3.8%

         \[\leadsto \color{blue}{\frac{1}{21} \cdot \frac{x}{\frac{1}{{x}^{6}} \cdot \sqrt{\pi}}} \]

      9. metadata-eval 3.8%

         \[\leadsto \color{blue}{0.047619047619047616} \cdot \frac{x}{\frac{1}{{x}^{6}} \cdot \sqrt{\pi}} \]

      10. *-commutative 3.8%

          \[\leadsto 0.047619047619047616 \cdot \frac{x}{\color{blue}{\sqrt{\pi} \cdot \frac{1}{{x}^{6}}}} \]

      11. pow-flip 3.8%

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

      12. metadata-eval 3.8%

          \[\leadsto 0.047619047619047616 \cdot \frac{x}{\sqrt{\pi} \cdot {x}^{\color{blue}{-6}}} \]

   7. Applied egg-rr 3.8%

      \[\leadsto \color{blue}{0.047619047619047616 \cdot \frac{x}{\sqrt{\pi} \cdot {x}^{-6}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \frac{x}{\sqrt{\pi} \cdot {x}^{-6}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 34.9% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.047619047619047616}{\sqrt{\pi}} \cdot \frac{x}{{x}^{-6}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (* x (/ 2.0 (sqrt PI)))
   (* (/ 0.047619047619047616 (sqrt PI)) (/ x (pow x -6.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = (0.047619047619047616 / sqrt(((double) M_PI))) * (x / pow(x, -6.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = x * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = (0.047619047619047616 / Math.sqrt(Math.PI)) * (x / Math.pow(x, -6.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = (0.047619047619047616 / math.sqrt(math.pi)) * (x / math.pow(x, -6.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = Float64(x * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64(Float64(0.047619047619047616 / sqrt(pi)) * Float64(x / (x ^ -6.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = x * (2.0 / sqrt(pi));
	else
		tmp = (0.047619047619047616 / sqrt(pi)) * (x / (x ^ -6.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.047619047619047616 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(x / N[Power[x, -6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 68.6%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{0.5 \cdot \sqrt{\pi}}\right|} \]
   6. Step-by-step derivation

      1. *-commutative 68.6%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

   7. Simplified 68.6%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

   8. Taylor expanded in x around 0 68.6%

      \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|0.5 \cdot \sqrt{\pi}\right|}} \]
   9. Step-by-step derivation

      1. fabs-neg 68.6%

         \[\leadsto \frac{\color{blue}{\left|-x\right|}}{\left|0.5 \cdot \sqrt{\pi}\right|} \]

      2. *-commutative 68.6%

         \[\leadsto \frac{\left|-x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

      3. fabs-div 68.6%

         \[\leadsto \color{blue}{\left|\frac{-x}{\sqrt{\pi} \cdot 0.5}\right|} \]

      4. neg-mul-1 68.6%

         \[\leadsto \left|\frac{\color{blue}{-1 \cdot x}}{\sqrt{\pi} \cdot 0.5}\right| \]

      5. *-commutative 68.6%

         \[\leadsto \left|\frac{-1 \cdot x}{\color{blue}{0.5 \cdot \sqrt{\pi}}}\right| \]

      6. times-frac 68.6%

         \[\leadsto \left|\color{blue}{\frac{-1}{0.5} \cdot \frac{x}{\sqrt{\pi}}}\right| \]

      7. metadata-eval 68.6%

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

      8. metadata-eval 68.6%

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

      9. distribute-lft-neg-in 68.6%

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

      10. fabs-neg 68.6%

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

      11. rem-square-sqrt 32.8%

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

      12. fabs-sqr 32.8%

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

      13. rem-square-sqrt 34.3%

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

      14. *-commutative 34.3%

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

      15. metadata-eval 34.3%

          \[\leadsto \frac{x}{\sqrt{\pi}} \cdot \color{blue}{\frac{1}{0.5}} \]

      16. times-frac 34.3%

          \[\leadsto \color{blue}{\frac{x \cdot 1}{\sqrt{\pi} \cdot 0.5}} \]

      17. *-rgt-identity 34.3%

          \[\leadsto \frac{\color{blue}{x}}{\sqrt{\pi} \cdot 0.5} \]

      18. *-commutative 34.3%

          \[\leadsto \frac{x}{\color{blue}{0.5 \cdot \sqrt{\pi}}} \]

      19. associate-/r* 34.3%

          \[\leadsto \color{blue}{\frac{\frac{x}{0.5}}{\sqrt{\pi}}} \]

   10. Simplified 34.3%

       \[\leadsto \color{blue}{\frac{\frac{x}{0.5}}{\sqrt{\pi}}} \]
   11. Step-by-step derivation

       1. associate-/l/ 34.3%

          \[\leadsto \color{blue}{\frac{x}{\sqrt{\pi} \cdot 0.5}} \]

       2. *-un-lft-identity 34.3%

          \[\leadsto \frac{\color{blue}{1 \cdot x}}{\sqrt{\pi} \cdot 0.5} \]

       3. *-commutative 34.3%

          \[\leadsto \frac{1 \cdot x}{\color{blue}{0.5 \cdot \sqrt{\pi}}} \]

       4. times-frac 34.3%

          \[\leadsto \color{blue}{\frac{1}{0.5} \cdot \frac{x}{\sqrt{\pi}}} \]

       5. metadata-eval 34.3%

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

       6. rem-log-exp 4.3%

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

       7. *-un-lft-identity 4.3%

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

       8. log-prod 4.3%

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

       9. metadata-eval 4.3%

          \[\leadsto \color{blue}{0} + \log \left(e^{2 \cdot \frac{x}{\sqrt{\pi}}}\right) \]

       10. rem-log-exp 34.3%

           \[\leadsto 0 + \color{blue}{2 \cdot \frac{x}{\sqrt{\pi}}} \]

       11. metadata-eval 34.3%

           \[\leadsto 0 + \color{blue}{\frac{1}{0.5}} \cdot \frac{x}{\sqrt{\pi}} \]

       12. times-frac 34.3%

           \[\leadsto 0 + \color{blue}{\frac{1 \cdot x}{0.5 \cdot \sqrt{\pi}}} \]

       13. *-un-lft-identity 34.3%

           \[\leadsto 0 + \frac{\color{blue}{x}}{0.5 \cdot \sqrt{\pi}} \]

       14. *-commutative 34.3%

           \[\leadsto 0 + \frac{x}{\color{blue}{\sqrt{\pi} \cdot 0.5}} \]

   12. Applied egg-rr 34.3%

       \[\leadsto \color{blue}{0 + \frac{x}{\sqrt{\pi} \cdot 0.5}} \]
   13. Step-by-step derivation

       1. +-lft-identity 34.3%

          \[\leadsto \color{blue}{\frac{x}{\sqrt{\pi} \cdot 0.5}} \]

       2. *-rgt-identity 34.3%

          \[\leadsto \frac{\color{blue}{x \cdot 1}}{\sqrt{\pi} \cdot 0.5} \]

       3. times-frac 34.3%

          \[\leadsto \color{blue}{\frac{x}{\sqrt{\pi}} \cdot \frac{1}{0.5}} \]

       4. metadata-eval 34.3%

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

       5. associate-*l/ 34.3%

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

       6. associate-*r/ 34.5%

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

   14. Simplified 34.5%

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

   if 1.8500000000000001 < x

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around inf 35.3%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}\right|} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 1.9%

         \[\leadsto \frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{\left|21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)\right|} \]

      2. fabs-sqr 1.9%

         \[\leadsto \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\left|21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)\right|} \]

      3. add-sqr-sqrt 1.9%

         \[\leadsto \frac{\sqrt{x} \cdot \sqrt{x}}{\left|\color{blue}{\sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)} \cdot \sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}}\right|} \]

      4. fabs-sqr 1.9%

         \[\leadsto \frac{\sqrt{x} \cdot \sqrt{x}}{\color{blue}{\sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)} \cdot \sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}}} \]

      5. add-sqr-sqrt 3.8%

         \[\leadsto \frac{\color{blue}{x}}{\sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)} \cdot \sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}} \]

      6. *-un-lft-identity 3.8%

         \[\leadsto \frac{\color{blue}{1 \cdot x}}{\sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)} \cdot \sqrt{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}} \]

      7. add-sqr-sqrt 3.8%

         \[\leadsto \frac{1 \cdot x}{\color{blue}{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}} \]

      8. times-frac 3.8%

         \[\leadsto \color{blue}{\frac{1}{21} \cdot \frac{x}{\frac{1}{{x}^{6}} \cdot \sqrt{\pi}}} \]

      9. metadata-eval 3.8%

         \[\leadsto \color{blue}{0.047619047619047616} \cdot \frac{x}{\frac{1}{{x}^{6}} \cdot \sqrt{\pi}} \]

      10. *-commutative 3.8%

          \[\leadsto 0.047619047619047616 \cdot \frac{x}{\color{blue}{\sqrt{\pi} \cdot \frac{1}{{x}^{6}}}} \]

      11. pow-flip 3.8%

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

      12. metadata-eval 3.8%

          \[\leadsto 0.047619047619047616 \cdot \frac{x}{\sqrt{\pi} \cdot {x}^{\color{blue}{-6}}} \]

   7. Applied egg-rr 3.8%

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

      1. expm1-log1p-u 3.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot \frac{x}{\sqrt{\pi} \cdot {x}^{-6}}\right)\right)} \]

      2. expm1-udef 3.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \frac{x}{\sqrt{\pi} \cdot {x}^{-6}}\right)} - 1} \]

      3. associate-*r/ 3.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{0.047619047619047616 \cdot x}{\sqrt{\pi} \cdot {x}^{-6}}}\right)} - 1 \]

      4. times-frac 3.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{0.047619047619047616}{\sqrt{\pi}} \cdot \frac{x}{{x}^{-6}}}\right)} - 1 \]

   9. Applied egg-rr 3.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0.047619047619047616}{\sqrt{\pi}} \cdot \frac{x}{{x}^{-6}}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 3.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.047619047619047616}{\sqrt{\pi}} \cdot \frac{x}{{x}^{-6}}\right)\right)} \]

       2. expm1-log1p 3.8%

          \[\leadsto \color{blue}{\frac{0.047619047619047616}{\sqrt{\pi}} \cdot \frac{x}{{x}^{-6}}} \]

   11. Simplified 3.8%

       \[\leadsto \color{blue}{\frac{0.047619047619047616}{\sqrt{\pi}} \cdot \frac{x}{{x}^{-6}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.047619047619047616}{\sqrt{\pi}} \cdot \frac{x}{{x}^{-6}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 34.9% accurate, 16.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-81}:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \left(x \cdot \frac{4}{\pi}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2e-81) (* x (/ 2.0 (sqrt PI))) (sqrt (* x (* x (/ 4.0 PI))))))

C

double code(double x) {
	double tmp;
	if (x <= 2e-81) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = sqrt((x * (x * (4.0 / ((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2e-81) {
		tmp = x * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = Math.sqrt((x * (x * (4.0 / Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2e-81:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.sqrt((x * (x * (4.0 / math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2e-81)
		tmp = Float64(x * Float64(2.0 / sqrt(pi)));
	else
		tmp = sqrt(Float64(x * Float64(x * Float64(4.0 / pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2e-81)
		tmp = x * (2.0 / sqrt(pi));
	else
		tmp = sqrt((x * (x * (4.0 / pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2e-81], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(x * N[(x * N[(4.0 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.9999999999999999e-81

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 66.1%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{0.5 \cdot \sqrt{\pi}}\right|} \]
   6. Step-by-step derivation

      1. *-commutative 66.1%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

   7. Simplified 66.1%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

   8. Taylor expanded in x around 0 66.1%

      \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|0.5 \cdot \sqrt{\pi}\right|}} \]
   9. Step-by-step derivation

      1. fabs-neg 66.1%

         \[\leadsto \frac{\color{blue}{\left|-x\right|}}{\left|0.5 \cdot \sqrt{\pi}\right|} \]

      2. *-commutative 66.1%

         \[\leadsto \frac{\left|-x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

      3. fabs-div 66.1%

         \[\leadsto \color{blue}{\left|\frac{-x}{\sqrt{\pi} \cdot 0.5}\right|} \]

      4. neg-mul-1 66.1%

         \[\leadsto \left|\frac{\color{blue}{-1 \cdot x}}{\sqrt{\pi} \cdot 0.5}\right| \]

      5. *-commutative 66.1%

         \[\leadsto \left|\frac{-1 \cdot x}{\color{blue}{0.5 \cdot \sqrt{\pi}}}\right| \]

      6. times-frac 66.1%

         \[\leadsto \left|\color{blue}{\frac{-1}{0.5} \cdot \frac{x}{\sqrt{\pi}}}\right| \]

      7. metadata-eval 66.1%

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

      8. metadata-eval 66.1%

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

      9. distribute-lft-neg-in 66.1%

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

      10. fabs-neg 66.1%

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

      11. rem-square-sqrt 26.7%

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

      12. fabs-sqr 26.7%

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

      13. rem-square-sqrt 28.4%

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

      14. *-commutative 28.4%

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

      15. metadata-eval 28.4%

          \[\leadsto \frac{x}{\sqrt{\pi}} \cdot \color{blue}{\frac{1}{0.5}} \]

      16. times-frac 28.4%

          \[\leadsto \color{blue}{\frac{x \cdot 1}{\sqrt{\pi} \cdot 0.5}} \]

      17. *-rgt-identity 28.4%

          \[\leadsto \frac{\color{blue}{x}}{\sqrt{\pi} \cdot 0.5} \]

      18. *-commutative 28.4%

          \[\leadsto \frac{x}{\color{blue}{0.5 \cdot \sqrt{\pi}}} \]

      19. associate-/r* 28.4%

          \[\leadsto \color{blue}{\frac{\frac{x}{0.5}}{\sqrt{\pi}}} \]

   10. Simplified 28.4%

       \[\leadsto \color{blue}{\frac{\frac{x}{0.5}}{\sqrt{\pi}}} \]
   11. Step-by-step derivation

       1. associate-/l/ 28.4%

          \[\leadsto \color{blue}{\frac{x}{\sqrt{\pi} \cdot 0.5}} \]

       2. *-un-lft-identity 28.4%

          \[\leadsto \frac{\color{blue}{1 \cdot x}}{\sqrt{\pi} \cdot 0.5} \]

       3. *-commutative 28.4%

          \[\leadsto \frac{1 \cdot x}{\color{blue}{0.5 \cdot \sqrt{\pi}}} \]

       4. times-frac 28.4%

          \[\leadsto \color{blue}{\frac{1}{0.5} \cdot \frac{x}{\sqrt{\pi}}} \]

       5. metadata-eval 28.4%

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

       6. rem-log-exp 3.7%

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

       7. *-un-lft-identity 3.7%

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

       8. log-prod 3.7%

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

       9. metadata-eval 3.7%

          \[\leadsto \color{blue}{0} + \log \left(e^{2 \cdot \frac{x}{\sqrt{\pi}}}\right) \]

       10. rem-log-exp 28.4%

           \[\leadsto 0 + \color{blue}{2 \cdot \frac{x}{\sqrt{\pi}}} \]

       11. metadata-eval 28.4%

           \[\leadsto 0 + \color{blue}{\frac{1}{0.5}} \cdot \frac{x}{\sqrt{\pi}} \]

       12. times-frac 28.4%

           \[\leadsto 0 + \color{blue}{\frac{1 \cdot x}{0.5 \cdot \sqrt{\pi}}} \]

       13. *-un-lft-identity 28.4%

           \[\leadsto 0 + \frac{\color{blue}{x}}{0.5 \cdot \sqrt{\pi}} \]

       14. *-commutative 28.4%

           \[\leadsto 0 + \frac{x}{\color{blue}{\sqrt{\pi} \cdot 0.5}} \]

   12. Applied egg-rr 28.4%

       \[\leadsto \color{blue}{0 + \frac{x}{\sqrt{\pi} \cdot 0.5}} \]
   13. Step-by-step derivation

       1. +-lft-identity 28.4%

          \[\leadsto \color{blue}{\frac{x}{\sqrt{\pi} \cdot 0.5}} \]

       2. *-rgt-identity 28.4%

          \[\leadsto \frac{\color{blue}{x \cdot 1}}{\sqrt{\pi} \cdot 0.5} \]

       3. times-frac 28.4%

          \[\leadsto \color{blue}{\frac{x}{\sqrt{\pi}} \cdot \frac{1}{0.5}} \]

       4. metadata-eval 28.4%

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

       5. associate-*l/ 28.4%

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

       6. associate-*r/ 28.6%

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

   14. Simplified 28.6%

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

   if 1.9999999999999999e-81 < x

   1. Initial program 99.7%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in x around 0 94.4%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{0.5 \cdot \sqrt{\pi}}\right|} \]
   6. Step-by-step derivation

      1. *-commutative 94.4%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

   7. Simplified 94.4%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 94.3%

         \[\leadsto \color{blue}{\sqrt{\frac{\left|x\right|}{\left|\sqrt{\pi} \cdot 0.5\right|}} \cdot \sqrt{\frac{\left|x\right|}{\left|\sqrt{\pi} \cdot 0.5\right|}}} \]

      2. sqrt-unprod 94.4%

         \[\leadsto \color{blue}{\sqrt{\frac{\left|x\right|}{\left|\sqrt{\pi} \cdot 0.5\right|} \cdot \frac{\left|x\right|}{\left|\sqrt{\pi} \cdot 0.5\right|}}} \]

      3. div-fabs 94.4%

         \[\leadsto \sqrt{\color{blue}{\left|\frac{x}{\sqrt{\pi} \cdot 0.5}\right|} \cdot \frac{\left|x\right|}{\left|\sqrt{\pi} \cdot 0.5\right|}} \]

      4. div-fabs 94.4%

         \[\leadsto \sqrt{\left|\frac{x}{\sqrt{\pi} \cdot 0.5}\right| \cdot \color{blue}{\left|\frac{x}{\sqrt{\pi} \cdot 0.5}\right|}} \]

      5. sqr-abs 94.4%

         \[\leadsto \sqrt{\color{blue}{\frac{x}{\sqrt{\pi} \cdot 0.5} \cdot \frac{x}{\sqrt{\pi} \cdot 0.5}}} \]

      6. frac-times 94.3%

         \[\leadsto \sqrt{\color{blue}{\frac{x \cdot x}{\left(\sqrt{\pi} \cdot 0.5\right) \cdot \left(\sqrt{\pi} \cdot 0.5\right)}}} \]

      7. pow2 94.3%

         \[\leadsto \sqrt{\frac{\color{blue}{{x}^{2}}}{\left(\sqrt{\pi} \cdot 0.5\right) \cdot \left(\sqrt{\pi} \cdot 0.5\right)}} \]

      8. swap-sqr 94.3%

         \[\leadsto \sqrt{\frac{{x}^{2}}{\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \left(0.5 \cdot 0.5\right)}}} \]

      9. add-sqr-sqrt 94.6%

         \[\leadsto \sqrt{\frac{{x}^{2}}{\color{blue}{\pi} \cdot \left(0.5 \cdot 0.5\right)}} \]

      10. metadata-eval 94.6%

          \[\leadsto \sqrt{\frac{{x}^{2}}{\pi \cdot \color{blue}{0.25}}} \]

   9. Applied egg-rr 94.6%

      \[\leadsto \color{blue}{\sqrt{\frac{{x}^{2}}{\pi \cdot 0.25}}} \]
   10. Step-by-step derivation

       1. div-inv 94.6%

          \[\leadsto \sqrt{\color{blue}{{x}^{2} \cdot \frac{1}{\pi \cdot 0.25}}} \]

       2. unpow2 94.6%

          \[\leadsto \sqrt{\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{\pi \cdot 0.25}} \]

       3. associate-*l* 94.8%

          \[\leadsto \sqrt{\color{blue}{x \cdot \left(x \cdot \frac{1}{\pi \cdot 0.25}\right)}} \]

       4. *-commutative 94.8%

          \[\leadsto \sqrt{x \cdot \left(x \cdot \frac{1}{\color{blue}{0.25 \cdot \pi}}\right)} \]

       5. associate-/r* 94.8%

          \[\leadsto \sqrt{x \cdot \left(x \cdot \color{blue}{\frac{\frac{1}{0.25}}{\pi}}\right)} \]

       6. metadata-eval 94.8%

          \[\leadsto \sqrt{x \cdot \left(x \cdot \frac{\color{blue}{4}}{\pi}\right)} \]

   11. Applied egg-rr 94.8%

       \[\leadsto \sqrt{\color{blue}{x \cdot \left(x \cdot \frac{4}{\pi}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-81}:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \left(x \cdot \frac{4}{\pi}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 34.9% accurate, 17.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 68.6%

   \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{0.5 \cdot \sqrt{\pi}}\right|} \]
6. Step-by-step derivation

   1. *-commutative 68.6%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

7. Simplified 68.6%

   \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

8. Taylor expanded in x around 0 68.6%

   \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|0.5 \cdot \sqrt{\pi}\right|}} \]
9. Step-by-step derivation

   1. fabs-neg 68.6%

      \[\leadsto \frac{\color{blue}{\left|-x\right|}}{\left|0.5 \cdot \sqrt{\pi}\right|} \]

   2. *-commutative 68.6%

      \[\leadsto \frac{\left|-x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot 0.5}\right|} \]

   3. fabs-div 68.6%

      \[\leadsto \color{blue}{\left|\frac{-x}{\sqrt{\pi} \cdot 0.5}\right|} \]

   4. neg-mul-1 68.6%

      \[\leadsto \left|\frac{\color{blue}{-1 \cdot x}}{\sqrt{\pi} \cdot 0.5}\right| \]

   5. *-commutative 68.6%

      \[\leadsto \left|\frac{-1 \cdot x}{\color{blue}{0.5 \cdot \sqrt{\pi}}}\right| \]

   6. times-frac 68.6%

      \[\leadsto \left|\color{blue}{\frac{-1}{0.5} \cdot \frac{x}{\sqrt{\pi}}}\right| \]

   7. metadata-eval 68.6%

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

   8. metadata-eval 68.6%

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

   9. distribute-lft-neg-in 68.6%

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

   10. fabs-neg 68.6%

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

   11. rem-square-sqrt 32.8%

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

   12. fabs-sqr 32.8%

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

   13. rem-square-sqrt 34.3%

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

   14. *-commutative 34.3%

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

   15. metadata-eval 34.3%

       \[\leadsto \frac{x}{\sqrt{\pi}} \cdot \color{blue}{\frac{1}{0.5}} \]

   16. times-frac 34.3%

       \[\leadsto \color{blue}{\frac{x \cdot 1}{\sqrt{\pi} \cdot 0.5}} \]

   17. *-rgt-identity 34.3%

       \[\leadsto \frac{\color{blue}{x}}{\sqrt{\pi} \cdot 0.5} \]

   18. *-commutative 34.3%

       \[\leadsto \frac{x}{\color{blue}{0.5 \cdot \sqrt{\pi}}} \]

   19. associate-/r* 34.3%

       \[\leadsto \color{blue}{\frac{\frac{x}{0.5}}{\sqrt{\pi}}} \]

10. Simplified 34.3%

    \[\leadsto \color{blue}{\frac{\frac{x}{0.5}}{\sqrt{\pi}}} \]
11. Step-by-step derivation

    1. associate-/l/ 34.3%

       \[\leadsto \color{blue}{\frac{x}{\sqrt{\pi} \cdot 0.5}} \]

    2. *-un-lft-identity 34.3%

       \[\leadsto \frac{\color{blue}{1 \cdot x}}{\sqrt{\pi} \cdot 0.5} \]

    3. *-commutative 34.3%

       \[\leadsto \frac{1 \cdot x}{\color{blue}{0.5 \cdot \sqrt{\pi}}} \]

    4. times-frac 34.3%

       \[\leadsto \color{blue}{\frac{1}{0.5} \cdot \frac{x}{\sqrt{\pi}}} \]

    5. metadata-eval 34.3%

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

    6. rem-log-exp 4.3%

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

    7. *-un-lft-identity 4.3%

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

    8. log-prod 4.3%

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

    9. metadata-eval 4.3%

       \[\leadsto \color{blue}{0} + \log \left(e^{2 \cdot \frac{x}{\sqrt{\pi}}}\right) \]

    10. rem-log-exp 34.3%

        \[\leadsto 0 + \color{blue}{2 \cdot \frac{x}{\sqrt{\pi}}} \]

    11. metadata-eval 34.3%

        \[\leadsto 0 + \color{blue}{\frac{1}{0.5}} \cdot \frac{x}{\sqrt{\pi}} \]

    12. times-frac 34.3%

        \[\leadsto 0 + \color{blue}{\frac{1 \cdot x}{0.5 \cdot \sqrt{\pi}}} \]

    13. *-un-lft-identity 34.3%

        \[\leadsto 0 + \frac{\color{blue}{x}}{0.5 \cdot \sqrt{\pi}} \]

    14. *-commutative 34.3%

        \[\leadsto 0 + \frac{x}{\color{blue}{\sqrt{\pi} \cdot 0.5}} \]

12. Applied egg-rr 34.3%

    \[\leadsto \color{blue}{0 + \frac{x}{\sqrt{\pi} \cdot 0.5}} \]
13. Step-by-step derivation

    1. +-lft-identity 34.3%

       \[\leadsto \color{blue}{\frac{x}{\sqrt{\pi} \cdot 0.5}} \]

    2. *-rgt-identity 34.3%

       \[\leadsto \frac{\color{blue}{x \cdot 1}}{\sqrt{\pi} \cdot 0.5} \]

    3. times-frac 34.3%

       \[\leadsto \color{blue}{\frac{x}{\sqrt{\pi}} \cdot \frac{1}{0.5}} \]

    4. metadata-eval 34.3%

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

    5. associate-*l/ 34.3%

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

    6. associate-*r/ 34.5%

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

14. Simplified 34.5%

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

15. Final simplification 34.5%

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

Reproduce

herbie shell --seed 2024021 
(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)))))))