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

Percentage Accurate: 99.8% → 99.8%

Time: 13.1s

Alternatives: 10

Speedup: 3.0×

• 28.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.5× 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: 99.9% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 99.8%

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

6. Final simplification 99.8%

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

Alternative 3: 67.6% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(x \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left({x}^{6} \cdot \left(0.047619047619047616 + \frac{0.2}{{x}^{2}}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.6)
   (fabs (* x (/ 2.0 (sqrt PI))))
   (fabs
    (*
     (* x (sqrt (/ 1.0 PI)))
     (* (pow x 6.0) (+ 0.047619047619047616 (/ 0.2 (pow x 2.0))))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.6) {
		tmp = fabs((x * (2.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs(((x * sqrt((1.0 / ((double) M_PI)))) * (pow(x, 6.0) * (0.047619047619047616 + (0.2 / pow(x, 2.0))))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.6) {
		tmp = Math.abs((x * (2.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs(((x * Math.sqrt((1.0 / Math.PI))) * (Math.pow(x, 6.0) * (0.047619047619047616 + (0.2 / Math.pow(x, 2.0))))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.6:
		tmp = math.fabs((x * (2.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs(((x * math.sqrt((1.0 / math.pi))) * (math.pow(x, 6.0) * (0.047619047619047616 + (0.2 / math.pow(x, 2.0))))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.6)
		tmp = abs(Float64(x * Float64(2.0 / sqrt(pi))));
	else
		tmp = abs(Float64(Float64(x * sqrt(Float64(1.0 / pi))) * Float64((x ^ 6.0) * Float64(0.047619047619047616 + Float64(0.2 / (x ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.6)
		tmp = abs((x * (2.0 / sqrt(pi))));
	else
		tmp = abs(((x * sqrt((1.0 / pi))) * ((x ^ 6.0) * (0.047619047619047616 + (0.2 / (x ^ 2.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.6], N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x * N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Power[x, 6.0], $MachinePrecision] * N[(0.047619047619047616 + N[(0.2 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6000000000000001

   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%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. *-commutative 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-*l* 99.7%

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

      4. associate-+r+ 99.7%

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

      5. +-commutative 99.7%

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

      6. fma-define 99.7%

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

      7. rem-square-sqrt 34.7%

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

      8. fabs-sqr 34.7%

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

      9. rem-square-sqrt 99.7%

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

      10. +-commutative 99.7%

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

      11. fma-define 99.7%

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

      12. rem-square-sqrt 34.7%

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

      13. fabs-sqr 34.7%

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

      14. rem-square-sqrt 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0 70.8%

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

      1. associate-*r* 70.8%

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

   10. Simplified 70.8%

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

       1. associate-*l* 70.8%

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

       2. sqrt-div 70.8%

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

       3. metadata-eval 70.8%

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

       4. div-inv 70.3%

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

       5. clear-num 70.3%

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

       6. un-div-inv 70.3%

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

   12. Applied egg-rr 70.3%

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

       1. associate-/r/ 70.8%

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

   14. Simplified 70.8%

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

   if 1.6000000000000001 < 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%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. *-commutative 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-*l* 99.7%

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

      4. associate-+r+ 99.7%

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

      5. +-commutative 99.7%

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

      6. fma-define 99.7%

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

      7. rem-square-sqrt 34.7%

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

      8. fabs-sqr 34.7%

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

      9. rem-square-sqrt 99.7%

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

      10. +-commutative 99.7%

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

      11. fma-define 99.7%

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

      12. rem-square-sqrt 34.7%

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

      13. fabs-sqr 34.7%

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

      14. rem-square-sqrt 99.7%

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

   7. Simplified 99.7%

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

      1. fma-undefine 99.7%

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in x around inf 32.4%

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

       1. associate-*r/ 32.4%

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

       2. metadata-eval 32.4%

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

   12. Simplified 32.4%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(x \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left({x}^{6} \cdot \left(0.047619047619047616 + \frac{0.2}{{x}^{2}}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.6% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \frac{0.2}{{x}^{2}}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.6)
   (fabs (* x (/ 2.0 (sqrt PI))))
   (fabs
    (*
     (pow x 7.0)
     (* (sqrt (/ 1.0 PI)) (+ 0.047619047619047616 (/ 0.2 (pow x 2.0))))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.6) {
		tmp = fabs((x * (2.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs((pow(x, 7.0) * (sqrt((1.0 / ((double) M_PI))) * (0.047619047619047616 + (0.2 / pow(x, 2.0))))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.6) {
		tmp = Math.abs((x * (2.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs((Math.pow(x, 7.0) * (Math.sqrt((1.0 / Math.PI)) * (0.047619047619047616 + (0.2 / Math.pow(x, 2.0))))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.6:
		tmp = math.fabs((x * (2.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs((math.pow(x, 7.0) * (math.sqrt((1.0 / math.pi)) * (0.047619047619047616 + (0.2 / math.pow(x, 2.0))))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.6)
		tmp = abs(Float64(x * Float64(2.0 / sqrt(pi))));
	else
		tmp = abs(Float64((x ^ 7.0) * Float64(sqrt(Float64(1.0 / pi)) * Float64(0.047619047619047616 + Float64(0.2 / (x ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.6)
		tmp = abs((x * (2.0 / sqrt(pi))));
	else
		tmp = abs(((x ^ 7.0) * (sqrt((1.0 / pi)) * (0.047619047619047616 + (0.2 / (x ^ 2.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.6], N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Power[x, 7.0], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(0.047619047619047616 + N[(0.2 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6000000000000001

   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%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. *-commutative 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-*l* 99.7%

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

      4. associate-+r+ 99.7%

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

      5. +-commutative 99.7%

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

      6. fma-define 99.7%

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

      7. rem-square-sqrt 34.7%

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

      8. fabs-sqr 34.7%

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

      9. rem-square-sqrt 99.7%

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

      10. +-commutative 99.7%

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

      11. fma-define 99.7%

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

      12. rem-square-sqrt 34.7%

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

      13. fabs-sqr 34.7%

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

      14. rem-square-sqrt 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0 70.8%

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

      1. associate-*r* 70.8%

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

   10. Simplified 70.8%

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

       1. associate-*l* 70.8%

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

       2. sqrt-div 70.8%

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

       3. metadata-eval 70.8%

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

       4. div-inv 70.3%

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

       5. clear-num 70.3%

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

       6. un-div-inv 70.3%

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

   12. Applied egg-rr 70.3%

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

       1. associate-/r/ 70.8%

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

   14. Simplified 70.8%

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

   if 1.6000000000000001 < 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%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. *-commutative 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-*l* 99.7%

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

      4. associate-+r+ 99.7%

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

      5. +-commutative 99.7%

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

      6. fma-define 99.7%

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

      7. rem-square-sqrt 34.7%

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

      8. fabs-sqr 34.7%

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

      9. rem-square-sqrt 99.7%

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

      10. +-commutative 99.7%

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

      11. fma-define 99.7%

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

      12. rem-square-sqrt 34.7%

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

      13. fabs-sqr 34.7%

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

      14. rem-square-sqrt 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around inf 32.4%

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

      1. associate-*r* 32.4%

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

      2. distribute-rgt-out 32.4%

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

      3. associate-*r/ 32.4%

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

      4. metadata-eval 32.4%

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

   10. Simplified 32.4%

       \[\leadsto \left|\color{blue}{{x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \frac{0.2}{{x}^{2}}\right)\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{x}^{7} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 + \frac{0.2}{{x}^{2}}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Abs[N[(N[(2.0 + N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * N[Sqrt[N[(1.0 / Pi), $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 99.4%

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

5. Taylor expanded in x around 0 99.6%

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

   1. *-commutative 99.6%

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

   2. *-commutative 99.6%

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

   3. associate-*l* 99.7%

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

   4. associate-+r+ 99.7%

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

   5. +-commutative 99.7%

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

   6. fma-define 99.7%

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

   7. rem-square-sqrt 34.7%

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

   8. fabs-sqr 34.7%

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

   9. rem-square-sqrt 99.7%

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

   10. +-commutative 99.7%

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

   11. fma-define 99.7%

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

   12. rem-square-sqrt 34.7%

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

   13. fabs-sqr 34.7%

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

   14. rem-square-sqrt 99.7%

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

7. Simplified 99.7%

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

   1. fma-undefine 99.7%

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

9. Applied egg-rr 99.7%

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

    1. fma-undefine 99.7%

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

    2. associate-+r+ 99.7%

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

11. Applied egg-rr 99.7%

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

12. Final simplification 99.7%

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

Alternative 6: 98.6% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 99.6%

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

   1. *-commutative 99.6%

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

   2. *-commutative 99.6%

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

   3. associate-*l* 99.7%

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

   4. associate-+r+ 99.7%

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

   5. +-commutative 99.7%

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

   6. fma-define 99.7%

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

   7. rem-square-sqrt 34.7%

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

   8. fabs-sqr 34.7%

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

   9. rem-square-sqrt 99.7%

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

   10. +-commutative 99.7%

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

   11. fma-define 99.7%

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

   12. rem-square-sqrt 34.7%

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

   13. fabs-sqr 34.7%

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

   14. rem-square-sqrt 99.7%

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

7. Simplified 99.7%

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

   1. fma-undefine 99.7%

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

9. Applied egg-rr 99.7%

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

    1. fma-undefine 99.7%

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

    2. associate-+r+ 99.7%

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

11. Applied egg-rr 99.7%

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

    1. sqrt-div 34.1%

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

    2. metadata-eval 34.1%

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

    3. div-inv 34.1%

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

13. Applied egg-rr 99.2%

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

14. Final simplification 99.2%

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

Alternative 7: 67.6% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. *-commutative 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-*l* 99.7%

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

      4. associate-+r+ 99.7%

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

      5. +-commutative 99.7%

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

      6. fma-define 99.7%

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

      7. rem-square-sqrt 34.7%

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

      8. fabs-sqr 34.7%

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

      9. rem-square-sqrt 99.7%

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

      10. +-commutative 99.7%

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

      11. fma-define 99.7%

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

      12. rem-square-sqrt 34.7%

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

      13. fabs-sqr 34.7%

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

      14. rem-square-sqrt 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0 70.8%

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

      1. associate-*r* 70.8%

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

   10. Simplified 70.8%

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

       1. associate-*l* 70.8%

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

       2. sqrt-div 70.8%

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

       3. metadata-eval 70.8%

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

       4. div-inv 70.3%

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

       5. clear-num 70.3%

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

       6. un-div-inv 70.3%

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

   12. Applied egg-rr 70.3%

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

       1. associate-/r/ 70.8%

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

   14. Simplified 70.8%

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

   if 1.8500000000000001 < x

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. *-commutative 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-*l* 99.7%

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

      4. associate-+r+ 99.7%

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

      5. +-commutative 99.7%

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

      6. fma-define 99.7%

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

      7. rem-square-sqrt 34.7%

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

      8. fabs-sqr 34.7%

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

      9. rem-square-sqrt 99.7%

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

      10. +-commutative 99.7%

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

      11. fma-define 99.7%

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

      12. rem-square-sqrt 34.7%

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

      13. fabs-sqr 34.7%

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

      14. rem-square-sqrt 99.7%

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

   7. Simplified 99.7%

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

      1. fma-undefine 99.7%

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in x around inf 34.1%

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

       1. sqrt-div 34.1%

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

       2. metadata-eval 34.1%

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

       3. div-inv 34.1%

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

   12. Applied egg-rr 34.1%

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

4. Final simplification 70.8%

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

Alternative 8: 67.6% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.6666666666666666 \cdot \sqrt{\frac{{x}^{6}}{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.75)
   (fabs (* x (/ 2.0 (sqrt PI))))
   (fabs (* 0.6666666666666666 (sqrt (/ (pow x 6.0) PI))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.75) {
		tmp = fabs((x * (2.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs((0.6666666666666666 * sqrt((pow(x, 6.0) / ((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.75) {
		tmp = Math.abs((x * (2.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs((0.6666666666666666 * Math.sqrt((Math.pow(x, 6.0) / Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.75:
		tmp = math.fabs((x * (2.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs((0.6666666666666666 * math.sqrt((math.pow(x, 6.0) / math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.75)
		tmp = abs(Float64(x * Float64(2.0 / sqrt(pi))));
	else
		tmp = abs(Float64(0.6666666666666666 * sqrt(Float64((x ^ 6.0) / pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.75)
		tmp = abs((x * (2.0 / sqrt(pi))));
	else
		tmp = abs((0.6666666666666666 * sqrt(((x ^ 6.0) / pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.75], N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.6666666666666666 * N[Sqrt[N[(N[Power[x, 6.0], $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.75

   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%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. *-commutative 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-*l* 99.7%

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

      4. associate-+r+ 99.7%

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

      5. +-commutative 99.7%

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

      6. fma-define 99.7%

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

      7. rem-square-sqrt 34.7%

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

      8. fabs-sqr 34.7%

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

      9. rem-square-sqrt 99.7%

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

      10. +-commutative 99.7%

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

      11. fma-define 99.7%

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

      12. rem-square-sqrt 34.7%

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

      13. fabs-sqr 34.7%

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

      14. rem-square-sqrt 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0 70.8%

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

      1. associate-*r* 70.8%

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

   10. Simplified 70.8%

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

       1. associate-*l* 70.8%

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

       2. sqrt-div 70.8%

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

       3. metadata-eval 70.8%

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

       4. div-inv 70.3%

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

       5. clear-num 70.3%

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

       6. un-div-inv 70.3%

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

   12. Applied egg-rr 70.3%

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

       1. associate-/r/ 70.8%

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

   14. Simplified 70.8%

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

   if 1.75 < 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%

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

   5. Taylor expanded in x around inf 22.5%

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

      1. *-commutative 22.5%

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

      2. unpow2 22.5%

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

      3. rem-square-sqrt 2.1%

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

      4. fabs-sqr 2.1%

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

      5. rem-square-sqrt 22.5%

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

      6. unpow3 22.5%

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

   7. Simplified 22.5%

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

      1. add-sqr-sqrt 3.5%

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

      2. sqrt-unprod 27.6%

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

      3. swap-sqr 27.6%

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

      4. add-sqr-sqrt 27.6%

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

      5. pow-prod-up 27.6%

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

      6. metadata-eval 27.6%

         \[\leadsto \left|0.6666666666666666 \cdot \sqrt{\frac{1}{\pi} \cdot {x}^{\color{blue}{6}}}\right| \]

   9. Applied egg-rr 27.6%

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

       1. associate-*l/ 27.6%

          \[\leadsto \left|0.6666666666666666 \cdot \sqrt{\color{blue}{\frac{1 \cdot {x}^{6}}{\pi}}}\right| \]

       2. *-lft-identity 27.6%

          \[\leadsto \left|0.6666666666666666 \cdot \sqrt{\frac{\color{blue}{{x}^{6}}}{\pi}}\right| \]

   11. Simplified 27.6%

       \[\leadsto \left|0.6666666666666666 \cdot \color{blue}{\sqrt{\frac{{x}^{6}}{\pi}}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.6666666666666666 \cdot \sqrt{\frac{{x}^{6}}{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.6% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (fabs (* x (/ 2.0 (sqrt PI))))
   (fabs (* (pow x 7.0) (/ 0.047619047619047616 (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = fabs((x * (2.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs((pow(x, 7.0) * (0.047619047619047616 / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = Math.abs((x * (2.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs((Math.pow(x, 7.0) * (0.047619047619047616 / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = math.fabs((x * (2.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs((math.pow(x, 7.0) * (0.047619047619047616 / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = abs(Float64(x * Float64(2.0 / sqrt(pi))));
	else
		tmp = abs(Float64((x ^ 7.0) * Float64(0.047619047619047616 / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = abs((x * (2.0 / sqrt(pi))));
	else
		tmp = abs(((x ^ 7.0) * (0.047619047619047616 / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Power[x, 7.0], $MachinePrecision] * N[(0.047619047619047616 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if 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%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. *-commutative 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-*l* 99.7%

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

      4. associate-+r+ 99.7%

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

      5. +-commutative 99.7%

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

      6. fma-define 99.7%

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

      7. rem-square-sqrt 34.7%

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

      8. fabs-sqr 34.7%

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

      9. rem-square-sqrt 99.7%

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

      10. +-commutative 99.7%

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

      11. fma-define 99.7%

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

      12. rem-square-sqrt 34.7%

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

      13. fabs-sqr 34.7%

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

      14. rem-square-sqrt 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0 70.8%

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

      1. associate-*r* 70.8%

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

   10. Simplified 70.8%

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

       1. associate-*l* 70.8%

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

       2. sqrt-div 70.8%

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

       3. metadata-eval 70.8%

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

       4. div-inv 70.3%

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

       5. clear-num 70.3%

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

       6. un-div-inv 70.3%

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

   12. Applied egg-rr 70.3%

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

       1. associate-/r/ 70.8%

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

   14. Simplified 70.8%

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

   if 1.8500000000000001 < x

   1. Initial program 99.8%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. *-commutative 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-*l* 99.7%

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

      4. associate-+r+ 99.7%

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

      5. +-commutative 99.7%

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

      6. fma-define 99.7%

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

      7. rem-square-sqrt 34.7%

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

      8. fabs-sqr 34.7%

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

      9. rem-square-sqrt 99.7%

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

      10. +-commutative 99.7%

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

      11. fma-define 99.7%

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

      12. rem-square-sqrt 34.7%

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

      13. fabs-sqr 34.7%

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

      14. rem-square-sqrt 99.7%

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

   7. Simplified 99.7%

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

      1. fma-undefine 99.7%

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in x around inf 34.1%

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

       1. sqrt-div 34.1%

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

       2. metadata-eval 34.1%

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

       3. div-inv 34.1%

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

       4. clear-num 34.1%

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

       5. un-div-inv 34.1%

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

   12. Applied egg-rr 34.1%

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

       1. associate-/r/ 34.1%

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

       2. associate-*l/ 34.1%

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

       3. associate-*r* 34.1%

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

       4. pow-plus 34.1%

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

       5. metadata-eval 34.1%

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

       6. *-commutative 34.1%

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

       7. associate-/l* 34.1%

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

   14. Simplified 34.1%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.6% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.4%

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

5. Taylor expanded in x around 0 99.6%

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

   1. *-commutative 99.6%

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

   2. *-commutative 99.6%

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

   3. associate-*l* 99.7%

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

   4. associate-+r+ 99.7%

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

   5. +-commutative 99.7%

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

   6. fma-define 99.7%

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

   7. rem-square-sqrt 34.7%

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

   8. fabs-sqr 34.7%

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

   9. rem-square-sqrt 99.7%

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

   10. +-commutative 99.7%

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

   11. fma-define 99.7%

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

   12. rem-square-sqrt 34.7%

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

   13. fabs-sqr 34.7%

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

   14. rem-square-sqrt 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in x around 0 70.8%

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

   1. associate-*r* 70.8%

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

10. Simplified 70.8%

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

    1. associate-*l* 70.8%

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

    2. sqrt-div 70.8%

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

    3. metadata-eval 70.8%

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

    4. div-inv 70.3%

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

    5. clear-num 70.3%

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

    6. un-div-inv 70.3%

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

12. Applied egg-rr 70.3%

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

    1. associate-/r/ 70.8%

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

14. Simplified 70.8%

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

15. Final simplification 70.8%

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

Reproduce

herbie shell --seed 2024115 
(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)))))))