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

Percentage Accurate: 99.8% → 99.8%

Time: 4.4s

Alternatives: 11

Speedup: 1.6×

• 28.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, 0.047619047619047616, \mathsf{fma}\left(0.2 \cdot \left|x\right|, \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (/ 1.0 (sqrt PI))
   (fma
    (pow (fabs x) 7.0)
    0.047619047619047616
    (fma
     (* 0.2 (fabs x))
     (* (* (* x x) x) x)
     (* (fabs x) (fma (* x x) 0.6666666666666666 2.0)))))))

C

double code(double x) {
	return fabs(((1.0 / sqrt(((double) M_PI))) * fma(pow(fabs(x), 7.0), 0.047619047619047616, fma((0.2 * fabs(x)), (((x * x) * x) * x), (fabs(x) * fma((x * x), 0.6666666666666666, 2.0))))));
}

Julia

function code(x)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * fma((abs(x) ^ 7.0), 0.047619047619047616, fma(Float64(0.2 * abs(x)), Float64(Float64(Float64(x * x) * x) * x), Float64(abs(x) * fma(Float64(x * x), 0.6666666666666666, 2.0))))))
end

Wolfram

code[x_] := N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision] * 0.047619047619047616 + N[(N[(0.2 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] + N[(N[Abs[x], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.6666666666666666 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)\right)}\right| \]

4. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, 0.047619047619047616, \mathsf{fma}\left(0.2 \cdot \left|x\right|, \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)}\right| \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ \frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.2 \cdot \left(x \cdot x\right), x \cdot x, \left(t\_0 \cdot t\_0\right) \cdot 0.047619047619047616\right), \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right| \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

Alternative 3: 98.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;\frac{1}{\sqrt{\pi}} \cdot \left|x + x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|{x}^{7} \cdot 0.047619047619047616\right|}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.9)
   (* (/ 1.0 (sqrt PI)) (fabs (+ x x)))
   (/ (fabs (* (pow x 7.0) 0.047619047619047616)) (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = (1.0 / sqrt(((double) M_PI))) * fabs((x + x));
	} else {
		tmp = fabs((pow(x, 7.0) * 0.047619047619047616)) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = (1.0 / Math.sqrt(Math.PI)) * Math.abs((x + x));
	} else {
		tmp = Math.abs((Math.pow(x, 7.0) * 0.047619047619047616)) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.9:
		tmp = (1.0 / math.sqrt(math.pi)) * math.fabs((x + x))
	else:
		tmp = math.fabs((math.pow(x, 7.0) * 0.047619047619047616)) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.9)
		tmp = Float64(Float64(1.0 / sqrt(pi)) * abs(Float64(x + x)));
	else
		tmp = Float64(abs(Float64((x ^ 7.0) * 0.047619047619047616)) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.9)
		tmp = (1.0 / sqrt(pi)) * abs((x + x));
	else
		tmp = abs(((x ^ 7.0) * 0.047619047619047616)) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.9], N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[(N[Power[x, 7.0], $MachinePrecision] * 0.047619047619047616), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8999999999999999

   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. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 68.8%

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

   if 1.8999999999999999 < 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. Applied rewrites 99.4%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 35.7%

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

   7. Taylor expanded in x around 0

      \[\leadsto \frac{\left|\frac{1}{21} \cdot \color{blue}{{x}^{7}}\right|}{\sqrt{\pi}} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\left|\frac{1}{21} \cdot {x}^{7}\right|}{\sqrt{\pi}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left|{x}^{7} \cdot \frac{1}{\color{blue}{21}}\right|}{\sqrt{\pi}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\left|{x}^{7} \cdot \frac{1}{\color{blue}{21}}\right|}{\sqrt{\pi}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{\left|{x}^{7} \cdot \frac{1}{21}\right|}{\sqrt{\pi}} \]

      5. metadata-eval 35.7

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

   9. Applied rewrites 35.7%

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

Alternative 4: 89.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, 0.047619047619047616, x + x\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs (* (/ 1.0 (sqrt PI)) (fma (pow x 7.0) 0.047619047619047616 (+ x x)))))

C

double code(double x) {
	return fabs(((1.0 / sqrt(((double) M_PI))) * fma(pow(x, 7.0), 0.047619047619047616, (x + x))));
}

Julia

function code(x)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * fma((x ^ 7.0), 0.047619047619047616, Float64(x + x))))
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)\right)}\right| \]

4. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, 0.047619047619047616, \mathsf{fma}\left(0.2 \cdot \left|x\right|, \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)}\right| \]

5. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \color{blue}{2 \cdot \left|x\right|}\right)\right| \]
6. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + 2 \cdot \left|x\right|\right)\right| \]

   2. *-commutative N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left({\left(\left|x\right|\right)}^{7} \cdot \frac{1}{21} + 2 \cdot \left|\color{blue}{x}\right|\right)\right| \]

   3. lower-fma.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, \frac{1}{\color{blue}{21}}, 2 \cdot \left|x\right|\right)\right| \]

   4. lift-pow.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

   5. rem-sqrt-square-rev N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\sqrt{x \cdot x}\right)}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

   6. pow2 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\sqrt{{x}^{2}}\right)}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

   7. sqrt-pow1 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left({x}^{\left(\frac{2}{2}\right)}\right)}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

   8. metadata-eval N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left({x}^{1}\right)}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

   9. unpow1 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

   10. metadata-eval N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

   11. count-2-rev N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, \left|x\right| + \left|x\right|\right)\right| \]

   12. lower-+.f64 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, \left|x\right| + \left|x\right|\right)\right| \]

   13. rem-sqrt-square-rev N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, \sqrt{x \cdot x} + \left|x\right|\right)\right| \]

   14. pow2 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, \sqrt{{x}^{2}} + \left|x\right|\right)\right| \]

   15. sqrt-pow1 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, {x}^{\left(\frac{2}{2}\right)} + \left|x\right|\right)\right| \]

   16. metadata-eval N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, {x}^{1} + \left|x\right|\right)\right| \]

   17. unpow1 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, x + \left|x\right|\right)\right| \]

   18. rem-sqrt-square-rev N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, x + \sqrt{x \cdot x}\right)\right| \]

   19. pow2 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, x + \sqrt{{x}^{2}}\right)\right| \]

   20. sqrt-pow1 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, x + {x}^{\left(\frac{2}{2}\right)}\right)\right| \]

   21. metadata-eval N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, x + {x}^{1}\right)\right| \]

   22. unpow1 98.8

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, 0.047619047619047616, x + x\right)\right| \]

7. Applied rewrites 98.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \color{blue}{0.047619047619047616}, x + x\right)\right| \]
8. Add Preprocessing

Alternative 5: 89.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.8:\\ \;\;\;\;\frac{1}{\sqrt{\pi}} \cdot \left|x + x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left(0.2 \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right|}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.8)
   (* (/ 1.0 (sqrt PI)) (fabs (+ x x)))
   (/ (fabs (* (* (* 0.2 x) x) (* (* x x) x))) (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (x <= 1.8) {
		tmp = (1.0 / sqrt(((double) M_PI))) * fabs((x + x));
	} else {
		tmp = fabs((((0.2 * x) * x) * ((x * x) * x))) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.8) {
		tmp = (1.0 / Math.sqrt(Math.PI)) * Math.abs((x + x));
	} else {
		tmp = Math.abs((((0.2 * x) * x) * ((x * x) * x))) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.8:
		tmp = (1.0 / math.sqrt(math.pi)) * math.fabs((x + x))
	else:
		tmp = math.fabs((((0.2 * x) * x) * ((x * x) * x))) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.8)
		tmp = Float64(Float64(1.0 / sqrt(pi)) * abs(Float64(x + x)));
	else
		tmp = Float64(abs(Float64(Float64(Float64(0.2 * x) * x) * Float64(Float64(x * x) * x))) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.8)
		tmp = (1.0 / sqrt(pi)) * abs((x + x));
	else
		tmp = abs((((0.2 * x) * x) * ((x * x) * x))) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.8], N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[(N[(N[(0.2 * x), $MachinePrecision] * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.80000000000000004

   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. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 68.8%

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

   if 1.80000000000000004 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)\right)}\right| \]

   4. Applied rewrites 99.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, 0.047619047619047616, \mathsf{fma}\left(0.2 \cdot \left|x\right|, \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)}\right| \]

   5. Taylor expanded in x around inf

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\frac{1}{5} \cdot \color{blue}{\left({x}^{4} \cdot \left|x\right|\right)}\right)\right| \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\frac{1}{5} \cdot \left({x}^{4} \cdot \left|\color{blue}{x}\right|\right)\right)\right| \]

      2. unpow1 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\frac{1}{5} \cdot \left({\left({x}^{1}\right)}^{4} \cdot \left|x\right|\right)\right)\right| \]

      3. metadata-eval N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\frac{1}{5} \cdot \left({\left({x}^{\left(\frac{2}{2}\right)}\right)}^{4} \cdot \left|x\right|\right)\right)\right| \]

      4. sqrt-pow1 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\frac{1}{5} \cdot \left({\left(\sqrt{{x}^{2}}\right)}^{4} \cdot \left|x\right|\right)\right)\right| \]

      5. pow2 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\frac{1}{5} \cdot \left({\left(\sqrt{x \cdot x}\right)}^{4} \cdot \left|x\right|\right)\right)\right| \]

      6. rem-sqrt-square-rev N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\frac{1}{5} \cdot \left({\left(\left|x\right|\right)}^{4} \cdot \left|x\right|\right)\right)\right| \]

      7. metadata-eval N/A

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

      8. pow-plus N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\frac{1}{5} \cdot \left(\left({\left(\left|x\right|\right)}^{3} \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

      9. pow3 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   7. Applied rewrites 30.6%

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

   9. Applied rewrites 30.6%

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

Alternative 6: 83.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \frac{1}{\sqrt{\pi}}\\ t_2 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathbf{if}\;\left|t\_1 \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_2\right) + \frac{1}{21} \cdot \left(\left(t\_2 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \leq 0.0001:\\ \;\;\;\;t\_1 \cdot \left|x + x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(0.6666666666666666 \cdot x\right) \cdot \left(x \cdot x\right)\right|}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (/ 1.0 (sqrt PI)))
        (t_2 (* (* t_0 (fabs x)) (fabs x))))
   (if (<=
        (fabs
         (*
          t_1
          (+
           (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_2))
           (* (/ 1.0 21.0) (* (* t_2 (fabs x)) (fabs x))))))
        0.0001)
     (* t_1 (fabs (+ x x)))
     (/ (fabs (* (* 0.6666666666666666 x) (* x x))) (sqrt PI)))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = 1.0 / sqrt(((double) M_PI));
	double t_2 = (t_0 * fabs(x)) * fabs(x);
	double tmp;
	if (fabs((t_1 * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_2)) + ((1.0 / 21.0) * ((t_2 * fabs(x)) * fabs(x)))))) <= 0.0001) {
		tmp = t_1 * fabs((x + x));
	} else {
		tmp = fabs(((0.6666666666666666 * x) * (x * x))) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = 1.0 / Math.sqrt(Math.PI);
	double t_2 = (t_0 * Math.abs(x)) * Math.abs(x);
	double tmp;
	if (Math.abs((t_1 * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_2)) + ((1.0 / 21.0) * ((t_2 * Math.abs(x)) * Math.abs(x)))))) <= 0.0001) {
		tmp = t_1 * Math.abs((x + x));
	} else {
		tmp = Math.abs(((0.6666666666666666 * x) * (x * x))) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = 1.0 / math.sqrt(math.pi)
	t_2 = (t_0 * math.fabs(x)) * math.fabs(x)
	tmp = 0
	if math.fabs((t_1 * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_2)) + ((1.0 / 21.0) * ((t_2 * math.fabs(x)) * math.fabs(x)))))) <= 0.0001:
		tmp = t_1 * math.fabs((x + x))
	else:
		tmp = math.fabs(((0.6666666666666666 * x) * (x * x))) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(1.0 / sqrt(pi))
	t_2 = Float64(Float64(t_0 * abs(x)) * abs(x))
	tmp = 0.0
	if (abs(Float64(t_1 * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_0)) + Float64(Float64(1.0 / 5.0) * t_2)) + Float64(Float64(1.0 / 21.0) * Float64(Float64(t_2 * abs(x)) * abs(x)))))) <= 0.0001)
		tmp = Float64(t_1 * abs(Float64(x + x)));
	else
		tmp = Float64(abs(Float64(Float64(0.6666666666666666 * x) * Float64(x * x))) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = 1.0 / sqrt(pi);
	t_2 = (t_0 * abs(x)) * abs(x);
	tmp = 0.0;
	if (abs((t_1 * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_2)) + ((1.0 / 21.0) * ((t_2 * abs(x)) * abs(x)))))) <= 0.0001)
		tmp = t_1 * abs((x + x));
	else
		tmp = abs(((0.6666666666666666 * x) * (x * x))) / sqrt(pi);
	end
	tmp_2 = tmp;
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[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[N[(t$95$1 * 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$2), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$2 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0001], N[(t$95$1 * N[Abs[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[(N[(0.6666666666666666 * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 (PI.f64))) (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))))) < 1.00000000000000005e-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. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 99.7%

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

   if 1.00000000000000005e-4 < (fabs.f64 (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 (PI.f64))) (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (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. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 69.5%

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

      1. metadata-eval N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. unpow1 N/A

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

      6. metadata-eval N/A

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

      7. sqrt-pow1 N/A

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

      8. pow2 N/A

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

      9. rem-sqrt-square-rev N/A

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

      10. unpow1 N/A

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

      11. metadata-eval N/A

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

      12. sqrt-pow1 N/A

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

      13. pow2 N/A

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

      14. rem-sqrt-square-rev N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

   8. Applied rewrites 69.5%

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

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\left|\frac{2}{3} \cdot \color{blue}{{x}^{3}}\right|}{\sqrt{\pi}} \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \frac{\left|\frac{2}{3} \cdot {x}^{3}\right|}{\sqrt{\pi}} \]

       2. unpow1 N/A

          \[\leadsto \frac{\left|\frac{2}{3} \cdot {x}^{3}\right|}{\sqrt{\pi}} \]

       3. metadata-eval N/A

          \[\leadsto \frac{\left|\frac{2}{3} \cdot {x}^{3}\right|}{\sqrt{\pi}} \]

       4. sqrt-pow1 N/A

          \[\leadsto \frac{\left|\frac{2}{3} \cdot {x}^{3}\right|}{\sqrt{\pi}} \]

       5. pow2 N/A

          \[\leadsto \frac{\left|\frac{2}{3} \cdot {x}^{3}\right|}{\sqrt{\pi}} \]

       6. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\left|\frac{2}{3} \cdot {x}^{3}\right|}{\sqrt{\pi}} \]

       7. *-commutative N/A

          \[\leadsto \frac{\left|\frac{2}{3} \cdot {x}^{3}\right|}{\sqrt{\pi}} \]

       8. rem-sqrt-square-rev N/A

          \[\leadsto \frac{\left|\frac{2}{3} \cdot {x}^{3}\right|}{\sqrt{\pi}} \]

       9. pow2 N/A

          \[\leadsto \frac{\left|\frac{2}{3} \cdot {x}^{3}\right|}{\sqrt{\pi}} \]

       10. sqrt-pow1 N/A

           \[\leadsto \frac{\left|\frac{2}{3} \cdot {x}^{3}\right|}{\sqrt{\pi}} \]

       11. metadata-eval N/A

           \[\leadsto \frac{\left|\frac{2}{3} \cdot {x}^{3}\right|}{\sqrt{\pi}} \]

       12. unpow1 N/A

           \[\leadsto \frac{\left|\frac{2}{3} \cdot {x}^{3}\right|}{\sqrt{\pi}} \]

       13. *-commutative N/A

           \[\leadsto \frac{\left|\frac{2}{3} \cdot {x}^{3}\right|}{\sqrt{\pi}} \]

       14. metadata-eval N/A

           \[\leadsto \frac{\left|\frac{2}{3} \cdot {x}^{3}\right|}{\sqrt{\pi}} \]

   11. Applied rewrites 67.9%

       \[\leadsto \frac{\left|\left(0.6666666666666666 \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right|}{\sqrt{\pi}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 68.8% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[Abs[N[(N[(x * N[(0.6666666666666666 * x), $MachinePrecision] + 2.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $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. Applied rewrites 99.4%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 89.3%

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

   1. metadata-eval N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. unpow1 N/A

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

   6. metadata-eval N/A

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

   7. sqrt-pow1 N/A

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

   8. pow2 N/A

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

   9. rem-sqrt-square-rev N/A

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

   10. unpow1 N/A

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

   11. metadata-eval N/A

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

   12. sqrt-pow1 N/A

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

   13. pow2 N/A

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

   14. rem-sqrt-square-rev N/A

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

   15. *-commutative N/A

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

   16. lower-fma.f64 N/A

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

8. Applied rewrites 89.3%

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

Alternative 8: 68.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \frac{1}{\sqrt{\pi}}\\ t_2 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathbf{if}\;\left|t\_1 \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_2\right) + \frac{1}{21} \cdot \left(\left(t\_2 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \leq 2 \cdot 10^{-26}:\\ \;\;\;\;t\_1 \cdot \left|x + x\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(x + x\right) \cdot \left(x + x\right)}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (/ 1.0 (sqrt PI)))
        (t_2 (* (* t_0 (fabs x)) (fabs x))))
   (if (<=
        (fabs
         (*
          t_1
          (+
           (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_2))
           (* (/ 1.0 21.0) (* (* t_2 (fabs x)) (fabs x))))))
        2e-26)
     (* t_1 (fabs (+ x x)))
     (sqrt (/ (* (+ x x) (+ x x)) PI)))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = 1.0 / sqrt(((double) M_PI));
	double t_2 = (t_0 * fabs(x)) * fabs(x);
	double tmp;
	if (fabs((t_1 * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_2)) + ((1.0 / 21.0) * ((t_2 * fabs(x)) * fabs(x)))))) <= 2e-26) {
		tmp = t_1 * fabs((x + x));
	} else {
		tmp = sqrt((((x + x) * (x + x)) / ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = 1.0 / Math.sqrt(Math.PI);
	double t_2 = (t_0 * Math.abs(x)) * Math.abs(x);
	double tmp;
	if (Math.abs((t_1 * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_2)) + ((1.0 / 21.0) * ((t_2 * Math.abs(x)) * Math.abs(x)))))) <= 2e-26) {
		tmp = t_1 * Math.abs((x + x));
	} else {
		tmp = Math.sqrt((((x + x) * (x + x)) / Math.PI));
	}
	return tmp;
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = 1.0 / math.sqrt(math.pi)
	t_2 = (t_0 * math.fabs(x)) * math.fabs(x)
	tmp = 0
	if math.fabs((t_1 * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_2)) + ((1.0 / 21.0) * ((t_2 * math.fabs(x)) * math.fabs(x)))))) <= 2e-26:
		tmp = t_1 * math.fabs((x + x))
	else:
		tmp = math.sqrt((((x + x) * (x + x)) / math.pi))
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(1.0 / sqrt(pi))
	t_2 = Float64(Float64(t_0 * abs(x)) * abs(x))
	tmp = 0.0
	if (abs(Float64(t_1 * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_0)) + Float64(Float64(1.0 / 5.0) * t_2)) + Float64(Float64(1.0 / 21.0) * Float64(Float64(t_2 * abs(x)) * abs(x)))))) <= 2e-26)
		tmp = Float64(t_1 * abs(Float64(x + x)));
	else
		tmp = sqrt(Float64(Float64(Float64(x + x) * Float64(x + x)) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = 1.0 / sqrt(pi);
	t_2 = (t_0 * abs(x)) * abs(x);
	tmp = 0.0;
	if (abs((t_1 * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_2)) + ((1.0 / 21.0) * ((t_2 * abs(x)) * abs(x)))))) <= 2e-26)
		tmp = t_1 * abs((x + x));
	else
		tmp = sqrt((((x + x) * (x + x)) / pi));
	end
	tmp_2 = tmp;
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[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[N[(t$95$1 * 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$2), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$2 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2e-26], N[(t$95$1 * N[Abs[N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(N[(x + x), $MachinePrecision] * N[(x + x), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 (PI.f64))) (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))))) < 2.0000000000000001e-26

   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. Applied rewrites 99.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 99.9%

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

   if 2.0000000000000001e-26 < (fabs.f64 (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 (PI.f64))) (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (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. Applied rewrites 99.7%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 17.4%

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

   8. Applied rewrites 57.4%

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

Alternative 9: 68.8% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(x + x), $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. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 68.8%

   \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\color{blue}{x + x}\right| \]
7. Add Preprocessing

Alternative 10: 68.4% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Abs[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $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. Applied rewrites 99.4%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 68.4%

   \[\leadsto \frac{\left|\color{blue}{x + x}\right|}{\sqrt{\pi}} \]
7. Add Preprocessing

Alternative 11: 4.8% accurate, 13.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Abs[2.0], $MachinePrecision] / N[Sqrt[Pi], $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. Applied rewrites 99.4%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 89.3%

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

   1. metadata-eval N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. unpow1 N/A

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

   6. metadata-eval N/A

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

   7. sqrt-pow1 N/A

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

   8. pow2 N/A

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

   9. rem-sqrt-square-rev N/A

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

   10. unpow1 N/A

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

   11. metadata-eval N/A

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

   12. sqrt-pow1 N/A

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

   13. pow2 N/A

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

   14. rem-sqrt-square-rev N/A

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

   15. *-commutative N/A

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

   16. lower-fma.f64 N/A

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

8. Applied rewrites 89.3%

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

9. Taylor expanded in x around 0

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

    1. metadata-eval N/A

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

    2. unpow1 N/A

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

    3. metadata-eval N/A

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

    4. sqrt-pow1 N/A

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

    5. pow2 N/A

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

    6. rem-sqrt-square-rev N/A

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

    7. *-commutative N/A

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

    8. rem-sqrt-square-rev N/A

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

    9. pow2 N/A

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

    10. sqrt-pow1 N/A

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

    11. metadata-eval N/A

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

    12. unpow1 N/A

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

    13. associate-*l* N/A

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

    14. metadata-eval N/A

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

    15. count-2-rev N/A

        \[\leadsto \frac{\left|x + x\right|}{\sqrt{\pi}} \]

    16. flip-+ N/A

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

    17. +-inverses N/A

        \[\leadsto \frac{\left|\frac{0}{x - x}\right|}{\sqrt{\pi}} \]

    18. metadata-eval N/A

        \[\leadsto \frac{\left|\frac{1 - 1}{x - x}\right|}{\sqrt{\pi}} \]

    19. metadata-eval N/A

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

    20. metadata-eval N/A

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

11. Applied rewrites 4.8%

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

Reproduce

herbie shell --seed 2025117 
(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)))))))