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

Percentage Accurate: 99.8% → 99.8%

Time: 18.9s

Alternatives: 11

Speedup: 3.0×

• 29.2% 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, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * 0.047619047619047616), $MachinePrecision] + N[(N[Abs[x], $MachinePrecision] * N[(0.2 * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[Abs[x], $MachinePrecision] * N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 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. Add Preprocessing
3. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

4. Applied egg-rr 99.8%

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

Alternative 2: 99.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|x\right| \cdot \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.2, 0.6666666666666666\right), 2\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left|x\right| \cdot \frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.047619047619047616, 0.2\right), 0.6666666666666666\right)}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.4)
   (*
    (fabs x)
    (/ (fma x (* x (fma x (* x 0.2) 0.6666666666666666)) 2.0) (sqrt PI)))
   (*
    (fabs x)
    (/
     (*
      (* x x)
      (fma (* x x) (fma x (* x 0.047619047619047616) 0.2) 0.6666666666666666))
     (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.4) {
		tmp = fabs(x) * (fma(x, (x * fma(x, (x * 0.2), 0.6666666666666666)), 2.0) / sqrt(((double) M_PI)));
	} else {
		tmp = fabs(x) * (((x * x) * fma((x * x), fma(x, (x * 0.047619047619047616), 0.2), 0.6666666666666666)) / sqrt(((double) M_PI)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.4)
		tmp = Float64(abs(x) * Float64(fma(x, Float64(x * fma(x, Float64(x * 0.2), 0.6666666666666666)), 2.0) / sqrt(pi)));
	else
		tmp = Float64(abs(x) * Float64(Float64(Float64(x * x) * fma(Float64(x * x), fma(x, Float64(x * 0.047619047619047616), 0.2), 0.6666666666666666)) / sqrt(pi)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.4], N[(N[Abs[x], $MachinePrecision] * N[(N[(x * N[(x * N[(x * N[(x * 0.2), $MachinePrecision] + 0.6666666666666666), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.047619047619047616), $MachinePrecision] + 0.2), $MachinePrecision] + 0.6666666666666666), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.40000000000000002

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

   4. Applied egg-rr 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 99.4

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

   9. Simplified 99.4%

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

   if 0.40000000000000002 < (fabs.f64 x)

   1. Initial program 99.8%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

   4. Applied egg-rr 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around inf

      \[\leadsto \left|x\right| \cdot \frac{\color{blue}{{x}^{6} \cdot \left(\frac{1}{21} + \left(\frac{\frac{2}{3}}{{x}^{4}} + \frac{1}{5} \cdot \frac{1}{{x}^{2}}\right)\right)}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   9. Simplified 99.1%

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

Alternative 3: 99.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|x\right| \cdot \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.2, 0.6666666666666666\right), 2\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left|x\right| \cdot \frac{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot 0.047619047619047616, 0.2\right)\right)}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.4)
   (*
    (fabs x)
    (/ (fma x (* x (fma x (* x 0.2) 0.6666666666666666)) 2.0) (sqrt PI)))
   (*
    (fabs x)
    (/
     (* (* x x) (* (* x x) (fma x (* x 0.047619047619047616) 0.2)))
     (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.4) {
		tmp = fabs(x) * (fma(x, (x * fma(x, (x * 0.2), 0.6666666666666666)), 2.0) / sqrt(((double) M_PI)));
	} else {
		tmp = fabs(x) * (((x * x) * ((x * x) * fma(x, (x * 0.047619047619047616), 0.2))) / sqrt(((double) M_PI)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.4)
		tmp = Float64(abs(x) * Float64(fma(x, Float64(x * fma(x, Float64(x * 0.2), 0.6666666666666666)), 2.0) / sqrt(pi)));
	else
		tmp = Float64(abs(x) * Float64(Float64(Float64(x * x) * Float64(Float64(x * x) * fma(x, Float64(x * 0.047619047619047616), 0.2))) / sqrt(pi)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.4], N[(N[Abs[x], $MachinePrecision] * N[(N[(x * N[(x * N[(x * N[(x * 0.2), $MachinePrecision] + 0.6666666666666666), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.047619047619047616), $MachinePrecision] + 0.2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.40000000000000002

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

   4. Applied egg-rr 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 99.4

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

   9. Simplified 99.4%

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

   if 0.40000000000000002 < (fabs.f64 x)

   1. Initial program 99.8%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

   4. Applied egg-rr 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around inf

      \[\leadsto \left|x\right| \cdot \frac{\color{blue}{{x}^{6} \cdot \left(\frac{1}{21} + \frac{1}{5} \cdot \frac{1}{{x}^{2}}\right)}}{\sqrt{\mathsf{PI}\left(\right)}} \]
   8. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \left|x\right| \cdot \frac{\color{blue}{\frac{1}{21} \cdot {x}^{6} + \left(\frac{1}{5} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{6}}}{\sqrt{\mathsf{PI}\left(\right)}} \]

      2. metadata-eval N/A

         \[\leadsto \left|x\right| \cdot \frac{\frac{1}{21} \cdot {x}^{\color{blue}{\left(5 + 1\right)}} + \left(\frac{1}{5} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{6}}{\sqrt{\mathsf{PI}\left(\right)}} \]

      3. metadata-eval N/A

         \[\leadsto \left|x\right| \cdot \frac{\frac{1}{21} \cdot {x}^{\left(\color{blue}{\left(4 + 1\right)} + 1\right)} + \left(\frac{1}{5} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{6}}{\sqrt{\mathsf{PI}\left(\right)}} \]

      4. pow-plus N/A

         \[\leadsto \left|x\right| \cdot \frac{\frac{1}{21} \cdot \color{blue}{\left({x}^{\left(4 + 1\right)} \cdot x\right)} + \left(\frac{1}{5} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{6}}{\sqrt{\mathsf{PI}\left(\right)}} \]

      5. pow-plus N/A

         \[\leadsto \left|x\right| \cdot \frac{\frac{1}{21} \cdot \left(\color{blue}{\left({x}^{4} \cdot x\right)} \cdot x\right) + \left(\frac{1}{5} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{6}}{\sqrt{\mathsf{PI}\left(\right)}} \]

      6. associate-*r* N/A

         \[\leadsto \left|x\right| \cdot \frac{\frac{1}{21} \cdot \color{blue}{\left({x}^{4} \cdot \left(x \cdot x\right)\right)} + \left(\frac{1}{5} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{6}}{\sqrt{\mathsf{PI}\left(\right)}} \]

      7. unpow2 N/A

         \[\leadsto \left|x\right| \cdot \frac{\frac{1}{21} \cdot \left({x}^{4} \cdot \color{blue}{{x}^{2}}\right) + \left(\frac{1}{5} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{6}}{\sqrt{\mathsf{PI}\left(\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left|x\right| \cdot \frac{\frac{1}{21} \cdot \color{blue}{\left({x}^{2} \cdot {x}^{4}\right)} + \left(\frac{1}{5} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{6}}{\sqrt{\mathsf{PI}\left(\right)}} \]

      9. associate-*l* N/A

         \[\leadsto \left|x\right| \cdot \frac{\color{blue}{\left(\frac{1}{21} \cdot {x}^{2}\right) \cdot {x}^{4}} + \left(\frac{1}{5} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{6}}{\sqrt{\mathsf{PI}\left(\right)}} \]

      10. associate-*l* N/A

          \[\leadsto \left|x\right| \cdot \frac{\left(\frac{1}{21} \cdot {x}^{2}\right) \cdot {x}^{4} + \color{blue}{\frac{1}{5} \cdot \left(\frac{1}{{x}^{2}} \cdot {x}^{6}\right)}}{\sqrt{\mathsf{PI}\left(\right)}} \]

      11. associate-*l/ N/A

          \[\leadsto \left|x\right| \cdot \frac{\left(\frac{1}{21} \cdot {x}^{2}\right) \cdot {x}^{4} + \frac{1}{5} \cdot \color{blue}{\frac{1 \cdot {x}^{6}}{{x}^{2}}}}{\sqrt{\mathsf{PI}\left(\right)}} \]

      12. *-lft-identity N/A

          \[\leadsto \left|x\right| \cdot \frac{\left(\frac{1}{21} \cdot {x}^{2}\right) \cdot {x}^{4} + \frac{1}{5} \cdot \frac{\color{blue}{{x}^{6}}}{{x}^{2}}}{\sqrt{\mathsf{PI}\left(\right)}} \]

      13. *-rgt-identity N/A

          \[\leadsto \left|x\right| \cdot \frac{\left(\frac{1}{21} \cdot {x}^{2}\right) \cdot {x}^{4} + \frac{1}{5} \cdot \frac{{x}^{6}}{\color{blue}{{x}^{2} \cdot 1}}}{\sqrt{\mathsf{PI}\left(\right)}} \]

      14. *-inverses N/A

          \[\leadsto \left|x\right| \cdot \frac{\left(\frac{1}{21} \cdot {x}^{2}\right) \cdot {x}^{4} + \frac{1}{5} \cdot \frac{{x}^{6}}{{x}^{2} \cdot \color{blue}{\frac{x}{x}}}}{\sqrt{\mathsf{PI}\left(\right)}} \]

      15. associate-/l* N/A

          \[\leadsto \left|x\right| \cdot \frac{\left(\frac{1}{21} \cdot {x}^{2}\right) \cdot {x}^{4} + \frac{1}{5} \cdot \frac{{x}^{6}}{\color{blue}{\frac{{x}^{2} \cdot x}{x}}}}{\sqrt{\mathsf{PI}\left(\right)}} \]

      16. unpow2 N/A

          \[\leadsto \left|x\right| \cdot \frac{\left(\frac{1}{21} \cdot {x}^{2}\right) \cdot {x}^{4} + \frac{1}{5} \cdot \frac{{x}^{6}}{\frac{\color{blue}{\left(x \cdot x\right)} \cdot x}{x}}}{\sqrt{\mathsf{PI}\left(\right)}} \]

      17. unpow3 N/A

          \[\leadsto \left|x\right| \cdot \frac{\left(\frac{1}{21} \cdot {x}^{2}\right) \cdot {x}^{4} + \frac{1}{5} \cdot \frac{{x}^{6}}{\frac{\color{blue}{{x}^{3}}}{x}}}{\sqrt{\mathsf{PI}\left(\right)}} \]

      18. associate-/r/ N/A

          \[\leadsto \left|x\right| \cdot \frac{\left(\frac{1}{21} \cdot {x}^{2}\right) \cdot {x}^{4} + \frac{1}{5} \cdot \color{blue}{\left(\frac{{x}^{6}}{{x}^{3}} \cdot x\right)}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   9. Simplified 98.9%

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

Alternative 4: 99.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|x\right| \cdot \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.2, 0.6666666666666666\right), 2\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left|x\right| \cdot \frac{\left(x \cdot x\right) \cdot \left(0.047619047619047616 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.4)
   (*
    (fabs x)
    (/ (fma x (* x (fma x (* x 0.2) 0.6666666666666666)) 2.0) (sqrt PI)))
   (*
    (fabs x)
    (/ (* (* x x) (* 0.047619047619047616 (* x (* x (* x x))))) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.4) {
		tmp = fabs(x) * (fma(x, (x * fma(x, (x * 0.2), 0.6666666666666666)), 2.0) / sqrt(((double) M_PI)));
	} else {
		tmp = fabs(x) * (((x * x) * (0.047619047619047616 * (x * (x * (x * x))))) / sqrt(((double) M_PI)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.4)
		tmp = Float64(abs(x) * Float64(fma(x, Float64(x * fma(x, Float64(x * 0.2), 0.6666666666666666)), 2.0) / sqrt(pi)));
	else
		tmp = Float64(abs(x) * Float64(Float64(Float64(x * x) * Float64(0.047619047619047616 * Float64(x * Float64(x * Float64(x * x))))) / sqrt(pi)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.4], N[(N[Abs[x], $MachinePrecision] * N[(N[(x * N[(x * N[(x * N[(x * 0.2), $MachinePrecision] + 0.6666666666666666), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(0.047619047619047616 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.40000000000000002

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

   4. Applied egg-rr 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 99.4

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

   9. Simplified 99.4%

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

   if 0.40000000000000002 < (fabs.f64 x)

   1. Initial program 99.8%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

   4. Applied egg-rr 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

         \[\leadsto \left|x\right| \cdot \frac{\frac{1}{21} \cdot {x}^{\left(\color{blue}{\left(4 + 1\right)} + 1\right)}}{\sqrt{\mathsf{PI}\left(\right)}} \]

      3. pow-plus N/A

         \[\leadsto \left|x\right| \cdot \frac{\frac{1}{21} \cdot \color{blue}{\left({x}^{\left(4 + 1\right)} \cdot x\right)}}{\sqrt{\mathsf{PI}\left(\right)}} \]

      4. pow-plus N/A

         \[\leadsto \left|x\right| \cdot \frac{\frac{1}{21} \cdot \left(\color{blue}{\left({x}^{4} \cdot x\right)} \cdot x\right)}{\sqrt{\mathsf{PI}\left(\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \left|x\right| \cdot \frac{\frac{1}{21} \cdot \color{blue}{\left({x}^{4} \cdot \left(x \cdot x\right)\right)}}{\sqrt{\mathsf{PI}\left(\right)}} \]

      6. unpow2 N/A

         \[\leadsto \left|x\right| \cdot \frac{\frac{1}{21} \cdot \left({x}^{4} \cdot \color{blue}{{x}^{2}}\right)}{\sqrt{\mathsf{PI}\left(\right)}} \]

      7. *-commutative N/A

         \[\leadsto \left|x\right| \cdot \frac{\color{blue}{\left({x}^{4} \cdot {x}^{2}\right) \cdot \frac{1}{21}}}{\sqrt{\mathsf{PI}\left(\right)}} \]

      8. associate-*r* N/A

         \[\leadsto \left|x\right| \cdot \frac{\color{blue}{{x}^{4} \cdot \left({x}^{2} \cdot \frac{1}{21}\right)}}{\sqrt{\mathsf{PI}\left(\right)}} \]

      9. *-commutative N/A

         \[\leadsto \left|x\right| \cdot \frac{{x}^{4} \cdot \color{blue}{\left(\frac{1}{21} \cdot {x}^{2}\right)}}{\sqrt{\mathsf{PI}\left(\right)}} \]

      10. metadata-eval N/A

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

      11. pow-sqr N/A

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

      12. associate-*r* N/A

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

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

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

      19. associate-*r* N/A

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

      20. unpow2 N/A

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

      21. unpow3 N/A

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

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

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

      23. unpow3 N/A

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

      24. unpow2 N/A

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

      25. associate-*r* N/A

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

      26. *-commutative N/A

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

   9. Simplified 97.6%

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

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

       3. associate-*r* N/A

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

       4. cube-unmult N/A

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

       5. associate-*r* N/A

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

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

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

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

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

       8. cube-unmult N/A

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

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

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

       10. *-lowering-*.f64 97.7

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

   11. Applied egg-rr 97.7%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|x\right| \cdot \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.2, 0.6666666666666666\right), 2\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left|x\right| \cdot \frac{\left(x \cdot x\right) \cdot \left(0.047619047619047616 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|x\right| \cdot \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.2, 0.6666666666666666\right), 2\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left|x\right| \cdot \left(x \cdot \frac{x \cdot \left(x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.047619047619047616\right)\right)}{\sqrt{\pi}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.4)
   (*
    (fabs x)
    (/ (fma x (* x (fma x (* x 0.2) 0.6666666666666666)) 2.0) (sqrt PI)))
   (*
    (fabs x)
    (* x (/ (* x (* x (* (* x (* x x)) 0.047619047619047616))) (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.4) {
		tmp = fabs(x) * (fma(x, (x * fma(x, (x * 0.2), 0.6666666666666666)), 2.0) / sqrt(((double) M_PI)));
	} else {
		tmp = fabs(x) * (x * ((x * (x * ((x * (x * x)) * 0.047619047619047616))) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.4)
		tmp = Float64(abs(x) * Float64(fma(x, Float64(x * fma(x, Float64(x * 0.2), 0.6666666666666666)), 2.0) / sqrt(pi)));
	else
		tmp = Float64(abs(x) * Float64(x * Float64(Float64(x * Float64(x * Float64(Float64(x * Float64(x * x)) * 0.047619047619047616))) / sqrt(pi))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.4], N[(N[Abs[x], $MachinePrecision] * N[(N[(x * N[(x * N[(x * N[(x * 0.2), $MachinePrecision] + 0.6666666666666666), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[x], $MachinePrecision] * N[(x * N[(N[(x * N[(x * N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.047619047619047616), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.40000000000000002

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

   4. Applied egg-rr 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 99.4

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

   9. Simplified 99.4%

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

   if 0.40000000000000002 < (fabs.f64 x)

   1. Initial program 99.8%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

   4. Applied egg-rr 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

         \[\leadsto \left|x\right| \cdot \frac{\frac{1}{21} \cdot {x}^{\left(\color{blue}{\left(4 + 1\right)} + 1\right)}}{\sqrt{\mathsf{PI}\left(\right)}} \]

      3. pow-plus N/A

         \[\leadsto \left|x\right| \cdot \frac{\frac{1}{21} \cdot \color{blue}{\left({x}^{\left(4 + 1\right)} \cdot x\right)}}{\sqrt{\mathsf{PI}\left(\right)}} \]

      4. pow-plus N/A

         \[\leadsto \left|x\right| \cdot \frac{\frac{1}{21} \cdot \left(\color{blue}{\left({x}^{4} \cdot x\right)} \cdot x\right)}{\sqrt{\mathsf{PI}\left(\right)}} \]

      5. associate-*r* N/A

         \[\leadsto \left|x\right| \cdot \frac{\frac{1}{21} \cdot \color{blue}{\left({x}^{4} \cdot \left(x \cdot x\right)\right)}}{\sqrt{\mathsf{PI}\left(\right)}} \]

      6. unpow2 N/A

         \[\leadsto \left|x\right| \cdot \frac{\frac{1}{21} \cdot \left({x}^{4} \cdot \color{blue}{{x}^{2}}\right)}{\sqrt{\mathsf{PI}\left(\right)}} \]

      7. *-commutative N/A

         \[\leadsto \left|x\right| \cdot \frac{\color{blue}{\left({x}^{4} \cdot {x}^{2}\right) \cdot \frac{1}{21}}}{\sqrt{\mathsf{PI}\left(\right)}} \]

      8. associate-*r* N/A

         \[\leadsto \left|x\right| \cdot \frac{\color{blue}{{x}^{4} \cdot \left({x}^{2} \cdot \frac{1}{21}\right)}}{\sqrt{\mathsf{PI}\left(\right)}} \]

      9. *-commutative N/A

         \[\leadsto \left|x\right| \cdot \frac{{x}^{4} \cdot \color{blue}{\left(\frac{1}{21} \cdot {x}^{2}\right)}}{\sqrt{\mathsf{PI}\left(\right)}} \]

      10. metadata-eval N/A

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

      11. pow-sqr N/A

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

      12. associate-*r* N/A

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

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

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

      19. associate-*r* N/A

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

      20. unpow2 N/A

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

      21. unpow3 N/A

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

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

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

      23. unpow3 N/A

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

      24. unpow2 N/A

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

      25. associate-*r* N/A

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

      26. *-commutative N/A

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

   9. Simplified 97.6%

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

       1. associate-*l* N/A

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

       2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

       7. associate-*r* N/A

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

       8. cube-unmult N/A

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

       9. *-commutative N/A

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

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

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

       11. cube-unmult N/A

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

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

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

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

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

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

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

       15. PI-lowering-PI.f64 97.7

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

   11. Applied egg-rr 97.7%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|x\right| \cdot \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.2, 0.6666666666666666\right), 2\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left|x\right| \cdot \left(x \cdot \frac{x \cdot \left(x \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot 0.047619047619047616\right)\right)}{\sqrt{\pi}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

4. Applied egg-rr 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. unpow2 N/A

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

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

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

   5. +-commutative N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. unpow2 N/A

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

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

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

   13. accelerator-lowering-fma.f64 N/A

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

   14. *-lowering-*.f64 99.8

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

9. Simplified 99.8%

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

    1. *-commutative N/A

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

    2. div-inv N/A

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

    3. associate-*l* N/A

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

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

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

    5. accelerator-lowering-fma.f64 N/A

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

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

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

    7. associate-*l* N/A

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

    8. accelerator-lowering-fma.f64 N/A

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

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

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

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

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

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

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

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

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

11. Applied egg-rr 99.8%

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

Alternative 7: 99.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

4. Applied egg-rr 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. unpow2 N/A

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

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

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

   5. +-commutative N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. unpow2 N/A

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

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

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. unpow2 N/A

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

   12. associate-*l* N/A

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

   13. accelerator-lowering-fma.f64 N/A

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

   14. *-lowering-*.f64 99.8

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

9. Simplified 99.8%

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

Alternative 8: 93.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|x\right| \cdot \frac{\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\left|\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{x \cdot \left(\left|x\right| \cdot 0.2\right)}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.4)
   (* (fabs x) (/ (fma 0.6666666666666666 (* x x) 2.0) (sqrt PI)))
   (fabs (* (* x (* x x)) (/ (* x (* (fabs x) 0.2)) (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.4) {
		tmp = fabs(x) * (fma(0.6666666666666666, (x * x), 2.0) / sqrt(((double) M_PI)));
	} else {
		tmp = fabs(((x * (x * x)) * ((x * (fabs(x) * 0.2)) / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.4)
		tmp = Float64(abs(x) * Float64(fma(0.6666666666666666, Float64(x * x), 2.0) / sqrt(pi)));
	else
		tmp = abs(Float64(Float64(x * Float64(x * x)) * Float64(Float64(x * Float64(abs(x) * 0.2)) / sqrt(pi))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.4], N[(N[Abs[x], $MachinePrecision] * N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(x * N[(N[Abs[x], $MachinePrecision] * 0.2), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.40000000000000002

   1. Initial program 99.9%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

   4. Applied egg-rr 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 99.2

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

   9. Simplified 99.2%

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

   if 0.40000000000000002 < (fabs.f64 x)

   1. Initial program 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \left|\frac{\color{blue}{2 \cdot \left|x\right| + {x}^{2} \cdot \left(\frac{1}{5} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{2}{3} \cdot \left|x\right|\right)}}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. distribute-rgt-out N/A

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

      12. *-commutative N/A

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

   7. Simplified 79.5%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. pow-sqr N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

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

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

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

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

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

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

      18. fabs-lowering-fabs.f64 79.5

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

   10. Simplified 79.5%

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

       1. associate-*r* N/A

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

       2. pow3 N/A

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

       3. associate-/l* N/A

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

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

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

       5. cube-unmult N/A

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

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

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

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

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

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

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

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

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

       10. *-commutative N/A

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

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

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

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

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

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

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

       14. PI-lowering-PI.f64 79.5

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

   12. Applied egg-rr 79.5%

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

Alternative 9: 93.9% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

4. Applied egg-rr 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. +-commutative N/A

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

   10. unpow2 N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

   13. accelerator-lowering-fma.f64 N/A

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

   14. *-commutative N/A

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

   15. *-lowering-*.f64 92.4

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

9. Simplified 92.4%

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

Alternative 10: 89.1% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

4. Applied egg-rr 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 88.6

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

9. Simplified 88.6%

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

Alternative 11: 67.4% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 99.4%

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

5. Taylor expanded in x around 0

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

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

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

   2. fabs-lowering-fabs.f64 66.2

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

7. Simplified 66.2%

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

   1. associate-/l* N/A

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

   2. fabs-mul N/A

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

   3. metadata-eval N/A

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

   4. fabs-div N/A

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

   5. fabs-fabs N/A

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

   6. rem-sqrt-square N/A

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

   7. add-sqr-sqrt N/A

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

   8. associate-/l* N/A

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

   9. *-commutative N/A

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

   10. associate-/l* N/A

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

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

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

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

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

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

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

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

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

   15. PI-lowering-PI.f64 66.6

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

9. Applied egg-rr 66.6%

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

Reproduce

herbie shell --seed 2024199 
(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)))))))