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

Percentage Accurate: 99.8% → 99.9%

Time: 11.7s

Alternatives: 10

Speedup: 3.0×

• 29.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 99.9%

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

6. Final simplification 99.9%

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

Alternative 3: 99.1% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0005:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(0.6666666666666666 \cdot {x}^{2}\right) + x \cdot 2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{x}^{6} \cdot \frac{x \cdot 0.047619047619047616 + \frac{0.2}{x}}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.0005)
   (fabs
    (*
     (sqrt (/ 1.0 PI))
     (+ (* x (* 0.6666666666666666 (pow x 2.0))) (* x 2.0))))
   (fabs
    (* (pow x 6.0) (/ (+ (* x 0.047619047619047616) (/ 0.2 x)) (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.0005) {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * ((x * (0.6666666666666666 * pow(x, 2.0))) + (x * 2.0))));
	} else {
		tmp = fabs((pow(x, 6.0) * (((x * 0.047619047619047616) + (0.2 / x)) / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.0005) {
		tmp = Math.abs((Math.sqrt((1.0 / Math.PI)) * ((x * (0.6666666666666666 * Math.pow(x, 2.0))) + (x * 2.0))));
	} else {
		tmp = Math.abs((Math.pow(x, 6.0) * (((x * 0.047619047619047616) + (0.2 / x)) / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.0005:
		tmp = math.fabs((math.sqrt((1.0 / math.pi)) * ((x * (0.6666666666666666 * math.pow(x, 2.0))) + (x * 2.0))))
	else:
		tmp = math.fabs((math.pow(x, 6.0) * (((x * 0.047619047619047616) + (0.2 / x)) / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.0005)
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(x * Float64(0.6666666666666666 * (x ^ 2.0))) + Float64(x * 2.0))));
	else
		tmp = abs(Float64((x ^ 6.0) * Float64(Float64(Float64(x * 0.047619047619047616) + Float64(0.2 / x)) / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.0005)
		tmp = abs((sqrt((1.0 / pi)) * ((x * (0.6666666666666666 * (x ^ 2.0))) + (x * 2.0))));
	else
		tmp = abs(((x ^ 6.0) * (((x * 0.047619047619047616) + (0.2 / x)) / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.0005], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(x * N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Power[x, 6.0], $MachinePrecision] * N[(N[(N[(x * 0.047619047619047616), $MachinePrecision] + N[(0.2 / x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 5.0000000000000001e-4

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.8%

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

      1. associate-*r* 99.8%

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

      2. *-commutative 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*r* 99.8%

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

      5. distribute-rgt-in 99.8%

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

      6. *-commutative 99.8%

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

      7. associate-*r* 99.8%

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

      8. distribute-rgt-in 99.8%

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

      9. fma-undefine 99.8%

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

   7. Simplified 99.8%

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

      1. add-sqr-sqrt 53.4%

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

      2. fabs-sqr 53.4%

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

      3. add-sqr-sqrt 99.8%

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

      4. fma-undefine 99.8%

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

      5. distribute-rgt-in 99.8%

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

   9. Applied egg-rr 99.8%

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

   if 5.0000000000000001e-4 < (fabs.f64 x)

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*r* 99.4%

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

      3. associate-*r* 99.4%

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

      4. distribute-rgt-out 99.4%

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

   7. Simplified 99.4%

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

      1. pow1 99.4%

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

   9. Applied egg-rr 99.4%

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

       1. unpow1 99.4%

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

       2. fma-undefine 99.4%

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

       3. +-commutative 99.4%

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

       4. *-commutative 99.4%

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

       5. associate-*r/ 99.4%

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

       6. associate-*l/ 99.4%

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

       7. associate-/r/ 99.4%

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

       8. unpow2 99.4%

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

       9. associate-/l* 99.4%

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

       10. *-inverses 99.4%

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

       11. *-rgt-identity 99.4%

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

   11. Simplified 99.4%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0005:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(0.6666666666666666 \cdot {x}^{2}\right) + x \cdot 2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{x}^{6} \cdot \frac{x \cdot 0.047619047619047616 + \frac{0.2}{x}}{\sqrt{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.6% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0005:\\ \;\;\;\;x \cdot \frac{2 + 0.2 \cdot {x}^{4}}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left|{x}^{6} \cdot \frac{x \cdot 0.047619047619047616 + \frac{0.2}{x}}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.0005)
   (* x (/ (+ 2.0 (* 0.2 (pow x 4.0))) (sqrt PI)))
   (fabs
    (* (pow x 6.0) (/ (+ (* x 0.047619047619047616) (/ 0.2 x)) (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.0005) {
		tmp = x * ((2.0 + (0.2 * pow(x, 4.0))) / sqrt(((double) M_PI)));
	} else {
		tmp = fabs((pow(x, 6.0) * (((x * 0.047619047619047616) + (0.2 / x)) / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.0005) {
		tmp = x * ((2.0 + (0.2 * Math.pow(x, 4.0))) / Math.sqrt(Math.PI));
	} else {
		tmp = Math.abs((Math.pow(x, 6.0) * (((x * 0.047619047619047616) + (0.2 / x)) / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.0005:
		tmp = x * ((2.0 + (0.2 * math.pow(x, 4.0))) / math.sqrt(math.pi))
	else:
		tmp = math.fabs((math.pow(x, 6.0) * (((x * 0.047619047619047616) + (0.2 / x)) / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.0005)
		tmp = Float64(x * Float64(Float64(2.0 + Float64(0.2 * (x ^ 4.0))) / sqrt(pi)));
	else
		tmp = abs(Float64((x ^ 6.0) * Float64(Float64(Float64(x * 0.047619047619047616) + Float64(0.2 / x)) / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.0005)
		tmp = x * ((2.0 + (0.2 * (x ^ 4.0))) / sqrt(pi));
	else
		tmp = abs(((x ^ 6.0) * (((x * 0.047619047619047616) + (0.2 / x)) / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.0005], N[(x * N[(N[(2.0 + N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[Power[x, 6.0], $MachinePrecision] * N[(N[(N[(x * 0.047619047619047616), $MachinePrecision] + N[(0.2 / x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 5.0000000000000001e-4

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.9%

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

   6. Taylor expanded in x around 0 99.5%

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

      1. add-sqr-sqrt 53.3%

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

      2. fabs-sqr 53.3%

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

      3. add-sqr-sqrt 52.9%

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

      4. fabs-sqr 52.9%

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

      5. add-sqr-sqrt 53.3%

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

      6. add-sqr-sqrt 55.5%

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

      7. clear-num 55.5%

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

      8. un-div-inv 55.1%

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

   8. Applied egg-rr 55.1%

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

      1. associate-/r/ 55.1%

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

      2. associate-*l/ 55.1%

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

      3. associate-/l* 55.5%

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

   10. Simplified 55.5%

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

   11. Taylor expanded in x around 0 55.5%

       \[\leadsto x \cdot \frac{\color{blue}{2 + 0.2 \cdot {x}^{4}}}{\sqrt{\pi}} \]

   if 5.0000000000000001e-4 < (fabs.f64 x)

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*r* 99.4%

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

      3. associate-*r* 99.4%

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

      4. distribute-rgt-out 99.4%

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

   7. Simplified 99.4%

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

      1. pow1 99.4%

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

   9. Applied egg-rr 99.4%

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

       1. unpow1 99.4%

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

       2. fma-undefine 99.4%

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

       3. +-commutative 99.4%

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

       4. *-commutative 99.4%

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

       5. associate-*r/ 99.4%

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

       6. associate-*l/ 99.4%

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

       7. associate-/r/ 99.4%

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

       8. unpow2 99.4%

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

       9. associate-/l* 99.4%

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

       10. *-inverses 99.4%

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

       11. *-rgt-identity 99.4%

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

   11. Simplified 99.4%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0005:\\ \;\;\;\;x \cdot \frac{2 + 0.2 \cdot {x}^{4}}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left|{x}^{6} \cdot \frac{x \cdot 0.047619047619047616 + \frac{0.2}{x}}{\sqrt{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.4% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0005:\\ \;\;\;\;x \cdot \frac{2 + 0.2 \cdot {x}^{4}}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left|{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.0005)
   (* x (/ (+ 2.0 (* 0.2 (pow x 4.0))) (sqrt PI)))
   (fabs (* (pow x 7.0) (/ 0.047619047619047616 (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.0005) {
		tmp = x * ((2.0 + (0.2 * pow(x, 4.0))) / sqrt(((double) M_PI)));
	} else {
		tmp = fabs((pow(x, 7.0) * (0.047619047619047616 / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.0005) {
		tmp = x * ((2.0 + (0.2 * Math.pow(x, 4.0))) / Math.sqrt(Math.PI));
	} else {
		tmp = Math.abs((Math.pow(x, 7.0) * (0.047619047619047616 / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.0005:
		tmp = x * ((2.0 + (0.2 * math.pow(x, 4.0))) / math.sqrt(math.pi))
	else:
		tmp = math.fabs((math.pow(x, 7.0) * (0.047619047619047616 / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.0005)
		tmp = Float64(x * Float64(Float64(2.0 + Float64(0.2 * (x ^ 4.0))) / sqrt(pi)));
	else
		tmp = abs(Float64((x ^ 7.0) * Float64(0.047619047619047616 / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.0005)
		tmp = x * ((2.0 + (0.2 * (x ^ 4.0))) / sqrt(pi));
	else
		tmp = abs(((x ^ 7.0) * (0.047619047619047616 / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.0005], N[(x * N[(N[(2.0 + N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[Power[x, 7.0], $MachinePrecision] * N[(0.047619047619047616 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 5.0000000000000001e-4

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.9%

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

   6. Taylor expanded in x around 0 99.5%

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

      1. add-sqr-sqrt 53.3%

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

      2. fabs-sqr 53.3%

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

      3. add-sqr-sqrt 52.9%

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

      4. fabs-sqr 52.9%

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

      5. add-sqr-sqrt 53.3%

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

      6. add-sqr-sqrt 55.5%

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

      7. clear-num 55.5%

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

      8. un-div-inv 55.1%

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

   8. Applied egg-rr 55.1%

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

      1. associate-/r/ 55.1%

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

      2. associate-*l/ 55.1%

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

      3. associate-/l* 55.5%

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

   10. Simplified 55.5%

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

   11. Taylor expanded in x around 0 55.5%

       \[\leadsto x \cdot \frac{\color{blue}{2 + 0.2 \cdot {x}^{4}}}{\sqrt{\pi}} \]

   if 5.0000000000000001e-4 < (fabs.f64 x)

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 98.5%

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

      1. associate-*r* 98.5%

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

      2. sqrt-div 98.5%

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

      3. metadata-eval 98.5%

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

      4. un-div-inv 98.5%

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

      5. *-commutative 98.5%

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

      6. add-sqr-sqrt 0.0%

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

      7. fabs-sqr 0.0%

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

      8. add-sqr-sqrt 98.5%

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

   7. Applied egg-rr 98.5%

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

      1. *-commutative 98.5%

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

      2. associate-/l* 98.5%

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

      3. *-commutative 98.5%

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

      4. pow-plus 98.6%

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

      5. metadata-eval 98.6%

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

   9. Simplified 98.6%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0005:\\ \;\;\;\;x \cdot \frac{2 + 0.2 \cdot {x}^{4}}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left|{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 33.5% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 99.9%

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

6. Taylor expanded in x around 0 99.5%

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

   1. add-sqr-sqrt 37.9%

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

   2. fabs-sqr 37.9%

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

   3. add-sqr-sqrt 37.6%

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

   4. fabs-sqr 37.6%

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

   5. add-sqr-sqrt 37.9%

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

   6. add-sqr-sqrt 39.5%

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

   7. clear-num 39.5%

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

   8. un-div-inv 39.2%

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

8. Applied egg-rr 39.2%

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

   1. associate-/r/ 39.2%

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

   2. associate-*l/ 39.2%

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

   3. associate-/l* 39.5%

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

10. Simplified 39.5%

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

11. Taylor expanded in x around 0 39.5%

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

12. Final simplification 39.5%

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

Alternative 7: 33.5% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;x \cdot \frac{2 + 0.2 \cdot {x}^{4}}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{0.047619047619047616 \cdot {x}^{6}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.4)
   (* x (/ (+ 2.0 (* 0.2 (pow x 4.0))) (sqrt PI)))
   (* x (/ (* 0.047619047619047616 (pow x 6.0)) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 2.4) {
		tmp = x * ((2.0 + (0.2 * pow(x, 4.0))) / sqrt(((double) M_PI)));
	} else {
		tmp = x * ((0.047619047619047616 * pow(x, 6.0)) / sqrt(((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.4) {
		tmp = x * ((2.0 + (0.2 * Math.pow(x, 4.0))) / Math.sqrt(Math.PI));
	} else {
		tmp = x * ((0.047619047619047616 * Math.pow(x, 6.0)) / Math.sqrt(Math.PI));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.4:
		tmp = x * ((2.0 + (0.2 * math.pow(x, 4.0))) / math.sqrt(math.pi))
	else:
		tmp = x * ((0.047619047619047616 * math.pow(x, 6.0)) / math.sqrt(math.pi))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.4)
		tmp = Float64(x * Float64(Float64(2.0 + Float64(0.2 * (x ^ 4.0))) / sqrt(pi)));
	else
		tmp = Float64(x * Float64(Float64(0.047619047619047616 * (x ^ 6.0)) / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.4)
		tmp = x * ((2.0 + (0.2 * (x ^ 4.0))) / sqrt(pi));
	else
		tmp = x * ((0.047619047619047616 * (x ^ 6.0)) / sqrt(pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.4], N[(x * N[(N[(2.0 + N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999991

   1. Initial program 99.8%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.9%

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

   6. Taylor expanded in x around 0 99.5%

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

      1. add-sqr-sqrt 37.9%

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

      2. fabs-sqr 37.9%

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

      3. add-sqr-sqrt 37.6%

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

      4. fabs-sqr 37.6%

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

      5. add-sqr-sqrt 37.9%

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

      6. add-sqr-sqrt 39.5%

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

      7. clear-num 39.5%

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

      8. un-div-inv 39.2%

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

   8. Applied egg-rr 39.2%

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

      1. associate-/r/ 39.2%

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

      2. associate-*l/ 39.2%

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

      3. associate-/l* 39.5%

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

   10. Simplified 39.5%

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

   11. Taylor expanded in x around 0 39.5%

       \[\leadsto x \cdot \frac{\color{blue}{2 + 0.2 \cdot {x}^{4}}}{\sqrt{\pi}} \]

   if 2.39999999999999991 < x

   1. Initial program 99.8%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.9%

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

   6. Taylor expanded in x around 0 99.5%

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

      1. add-sqr-sqrt 37.9%

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

      2. fabs-sqr 37.9%

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

      3. add-sqr-sqrt 37.6%

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

      4. fabs-sqr 37.6%

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

      5. add-sqr-sqrt 37.9%

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

      6. add-sqr-sqrt 39.5%

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

      7. clear-num 39.5%

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

      8. un-div-inv 39.2%

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

   8. Applied egg-rr 39.2%

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

      1. associate-/r/ 39.2%

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

      2. associate-*l/ 39.2%

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

      3. associate-/l* 39.5%

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

   10. Simplified 39.5%

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

   11. Taylor expanded in x around inf 4.0%

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

4. Final simplification 39.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;x \cdot \frac{2 + 0.2 \cdot {x}^{4}}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{0.047619047619047616 \cdot {x}^{6}}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 33.6% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x}^{7}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (* x (/ 2.0 (sqrt PI)))
   (* 0.047619047619047616 (* (sqrt (/ 1.0 PI)) (pow x 7.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = 0.047619047619047616 * (sqrt((1.0 / ((double) M_PI))) * pow(x, 7.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = x * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = 0.047619047619047616 * (Math.sqrt((1.0 / Math.PI)) * Math.pow(x, 7.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = 0.047619047619047616 * (math.sqrt((1.0 / math.pi)) * math.pow(x, 7.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = Float64(x * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64(0.047619047619047616 * Float64(sqrt(Float64(1.0 / pi)) * (x ^ 7.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = x * (2.0 / sqrt(pi));
	else
		tmp = 0.047619047619047616 * (sqrt((1.0 / pi)) * (x ^ 7.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.8%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.9%

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

   6. Taylor expanded in x around 0 99.5%

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

      1. add-sqr-sqrt 37.9%

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

      2. fabs-sqr 37.9%

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

      3. add-sqr-sqrt 37.6%

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

      4. fabs-sqr 37.6%

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

      5. add-sqr-sqrt 37.9%

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

      6. add-sqr-sqrt 39.5%

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

      7. clear-num 39.5%

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

      8. un-div-inv 39.2%

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

   8. Applied egg-rr 39.2%

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

      1. associate-/r/ 39.2%

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

      2. associate-*l/ 39.2%

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

      3. associate-/l* 39.5%

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

   10. Simplified 39.5%

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

   11. Taylor expanded in x around 0 39.6%

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

   if 1.8500000000000001 < x

   1. Initial program 99.8%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.9%

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

   6. Taylor expanded in x around 0 99.5%

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

      1. add-sqr-sqrt 37.9%

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

      2. fabs-sqr 37.9%

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

      3. add-sqr-sqrt 37.6%

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

      4. fabs-sqr 37.6%

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

      5. add-sqr-sqrt 37.9%

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

      6. add-sqr-sqrt 39.5%

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

      7. clear-num 39.5%

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

      8. un-div-inv 39.2%

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

   8. Applied egg-rr 39.2%

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

      1. associate-/r/ 39.2%

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

      2. associate-*l/ 39.2%

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

      3. associate-/l* 39.5%

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

   10. Simplified 39.5%

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

   11. Taylor expanded in x around inf 4.0%

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

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {x}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 33.6% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $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 99.9%

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

   5. Taylor expanded in x around 0 99.9%

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

   6. Taylor expanded in x around 0 99.5%

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

      1. add-sqr-sqrt 37.9%

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

      2. fabs-sqr 37.9%

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

      3. add-sqr-sqrt 37.6%

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

      4. fabs-sqr 37.6%

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

      5. add-sqr-sqrt 37.9%

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

      6. add-sqr-sqrt 39.5%

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

      7. clear-num 39.5%

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

      8. un-div-inv 39.2%

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

   8. Applied egg-rr 39.2%

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

      1. associate-/r/ 39.2%

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

      2. associate-*l/ 39.2%

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

      3. associate-/l* 39.5%

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

   10. Simplified 39.5%

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

   11. Taylor expanded in x around 0 39.6%

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

   if 1.8500000000000001 < x

   1. Initial program 99.8%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.9%

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

   6. Taylor expanded in x around 0 99.5%

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

      1. add-sqr-sqrt 37.9%

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

      2. fabs-sqr 37.9%

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

      3. add-sqr-sqrt 37.6%

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

      4. fabs-sqr 37.6%

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

      5. add-sqr-sqrt 37.9%

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

      6. add-sqr-sqrt 39.5%

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

      7. clear-num 39.5%

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

      8. un-div-inv 39.2%

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

   8. Applied egg-rr 39.2%

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

      1. associate-/r/ 39.2%

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

      2. associate-*l/ 39.2%

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

      3. associate-/l* 39.5%

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

   10. Simplified 39.5%

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

   11. Taylor expanded in x around inf 4.0%

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

4. Final simplification 39.6%

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

Alternative 10: 33.6% accurate, 17.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 99.9%

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

6. Taylor expanded in x around 0 99.5%

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

   1. add-sqr-sqrt 37.9%

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

   2. fabs-sqr 37.9%

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

   3. add-sqr-sqrt 37.6%

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

   4. fabs-sqr 37.6%

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

   5. add-sqr-sqrt 37.9%

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

   6. add-sqr-sqrt 39.5%

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

   7. clear-num 39.5%

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

   8. un-div-inv 39.2%

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

8. Applied egg-rr 39.2%

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

   1. associate-/r/ 39.2%

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

   2. associate-*l/ 39.2%

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

   3. associate-/l* 39.5%

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

10. Simplified 39.5%

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

11. Taylor expanded in x around 0 39.6%

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

12. Final simplification 39.6%

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

Reproduce

herbie shell --seed 2024075 
(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)))))))