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

Percentage Accurate: 99.8% → 99.8%

Time: 11.8s

Alternatives: 10

Speedup: 3.6×

• 28.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \left|x\right| \cdot \left|{\pi}^{-0.5} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fabs x)
  (fabs
   (*
    (pow PI -0.5)
    (+
     2.0
     (+
      (* 0.047619047619047616 (pow x 6.0))
      (+ (* 0.2 (pow x 4.0)) (* 0.6666666666666666 (* x x)))))))))

C

double code(double x) {
	return fabs(x) * fabs((pow(((double) M_PI), -0.5) * (2.0 + ((0.047619047619047616 * pow(x, 6.0)) + ((0.2 * pow(x, 4.0)) + (0.6666666666666666 * (x * x)))))));
}

Java

public static double code(double x) {
	return Math.abs(x) * Math.abs((Math.pow(Math.PI, -0.5) * (2.0 + ((0.047619047619047616 * Math.pow(x, 6.0)) + ((0.2 * Math.pow(x, 4.0)) + (0.6666666666666666 * (x * x)))))));
}

Python

def code(x):
	return math.fabs(x) * math.fabs((math.pow(math.pi, -0.5) * (2.0 + ((0.047619047619047616 * math.pow(x, 6.0)) + ((0.2 * math.pow(x, 4.0)) + (0.6666666666666666 * (x * x)))))))

Julia

function code(x)
	return Float64(abs(x) * abs(Float64((pi ^ -0.5) * Float64(2.0 + Float64(Float64(0.047619047619047616 * (x ^ 6.0)) + Float64(Float64(0.2 * (x ^ 4.0)) + Float64(0.6666666666666666 * Float64(x * x))))))))
end

MATLAB

function tmp = code(x)
	tmp = abs(x) * abs(((pi ^ -0.5) * (2.0 + ((0.047619047619047616 * (x ^ 6.0)) + ((0.2 * (x ^ 4.0)) + (0.6666666666666666 * (x * x)))))));
end

Wolfram

code[x_] := N[(N[Abs[x], $MachinePrecision] * N[Abs[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(2.0 + N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 99.8%

   \[\leadsto \color{blue}{\left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot {x}^{2}\right)\right)\right)\right|} \]
6. Step-by-step derivation

   1. pow2 99.8%

      \[\leadsto \left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right| \]

7. Applied egg-rr 99.8%

   \[\leadsto \left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right| \]
8. Step-by-step derivation

   1. *-un-lft-identity 99.8%

      \[\leadsto \left|x\right| \cdot \left|\color{blue}{\left(1 \cdot \sqrt{\frac{1}{\pi}}\right)} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)\right| \]

   2. pow1/2 99.8%

      \[\leadsto \left|x\right| \cdot \left|\left(1 \cdot \color{blue}{{\left(\frac{1}{\pi}\right)}^{0.5}}\right) \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)\right| \]

   3. inv-pow 99.8%

      \[\leadsto \left|x\right| \cdot \left|\left(1 \cdot {\color{blue}{\left({\pi}^{-1}\right)}}^{0.5}\right) \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)\right| \]

   4. pow-pow 99.8%

      \[\leadsto \left|x\right| \cdot \left|\left(1 \cdot \color{blue}{{\pi}^{\left(-1 \cdot 0.5\right)}}\right) \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)\right| \]

   5. metadata-eval 99.8%

      \[\leadsto \left|x\right| \cdot \left|\left(1 \cdot {\pi}^{\color{blue}{-0.5}}\right) \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)\right| \]

9. Applied egg-rr 99.8%

   \[\leadsto \left|x\right| \cdot \left|\color{blue}{\left(1 \cdot {\pi}^{-0.5}\right)} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)\right| \]
10. Step-by-step derivation

    1. *-lft-identity 99.8%

       \[\leadsto \left|x\right| \cdot \left|\color{blue}{{\pi}^{-0.5}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)\right| \]

11. Simplified 99.8%

    \[\leadsto \left|x\right| \cdot \left|\color{blue}{{\pi}^{-0.5}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)\right| \]
12. Add Preprocessing

Alternative 2: 66.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0005:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{6}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.0005)
   (fabs
    (*
     (sqrt (/ 1.0 PI))
     (* (fabs x) (fma 0.6666666666666666 (pow x 2.0) 2.0))))
   (* x (* 0.047619047619047616 (* (pow PI -0.5) (pow x 6.0))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.0005) {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * (fabs(x) * fma(0.6666666666666666, pow(x, 2.0), 2.0))));
	} else {
		tmp = x * (0.047619047619047616 * (pow(((double) M_PI), -0.5) * pow(x, 6.0)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.0005)
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(abs(x) * fma(0.6666666666666666, (x ^ 2.0), 2.0))));
	else
		tmp = Float64(x * Float64(0.047619047619047616 * Float64((pi ^ -0.5) * (x ^ 6.0))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.0005], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(x * N[(0.047619047619047616 * N[(N[Power[Pi, -0.5], $MachinePrecision] * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 99.8%

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

      1. associate-*r* 99.8%

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

      2. unpow2 99.8%

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

      3. sqr-abs 99.8%

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

      4. unpow3 99.8%

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

      5. *-commutative 99.8%

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

      6. associate-*r* 99.8%

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

      7. distribute-rgt-in 99.8%

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

      8. *-commutative 99.8%

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

   7. Simplified 99.8%

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

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

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 99.2%

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

      1. *-commutative 99.2%

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

      2. *-commutative 99.2%

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

      3. associate-*l* 99.2%

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

      4. *-commutative 99.2%

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

      5. *-commutative 99.2%

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

   7. Simplified 99.2%

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

   8. Taylor expanded in x around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. unpow-1 99.2%

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

      3. metadata-eval 99.2%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\sqrt{{\pi}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right| \]

      4. pow-sqr 99.2%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\sqrt{\color{blue}{{\pi}^{-0.5} \cdot {\pi}^{-0.5}}} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right| \]

      5. rem-sqrt-square 99.2%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\color{blue}{\left|{\pi}^{-0.5}\right|} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right| \]

      6. rem-square-sqrt 99.2%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\left|\color{blue}{\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}}\right| \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right| \]

      7. fabs-sqr 99.2%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\color{blue}{\left(\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}\right)} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right| \]

      8. rem-square-sqrt 99.2%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\color{blue}{{\pi}^{-0.5}} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right| \]

      9. *-commutative 99.2%

         \[\leadsto \left|0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot \color{blue}{\left(\left|x\right| \cdot {x}^{6}\right)}\right)\right| \]

      10. associate-*r* 99.2%

          \[\leadsto \left|\color{blue}{\left(0.047619047619047616 \cdot {\pi}^{-0.5}\right) \cdot \left(\left|x\right| \cdot {x}^{6}\right)}\right| \]

      11. *-commutative 99.2%

          \[\leadsto \left|\color{blue}{\left({\pi}^{-0.5} \cdot 0.047619047619047616\right)} \cdot \left(\left|x\right| \cdot {x}^{6}\right)\right| \]

      12. rem-square-sqrt 99.2%

          \[\leadsto \left|\color{blue}{\sqrt{\left({\pi}^{-0.5} \cdot 0.047619047619047616\right) \cdot \left(\left|x\right| \cdot {x}^{6}\right)} \cdot \sqrt{\left({\pi}^{-0.5} \cdot 0.047619047619047616\right) \cdot \left(\left|x\right| \cdot {x}^{6}\right)}}\right| \]

      13. fabs-sqr 99.2%

          \[\leadsto \color{blue}{\sqrt{\left({\pi}^{-0.5} \cdot 0.047619047619047616\right) \cdot \left(\left|x\right| \cdot {x}^{6}\right)} \cdot \sqrt{\left({\pi}^{-0.5} \cdot 0.047619047619047616\right) \cdot \left(\left|x\right| \cdot {x}^{6}\right)}} \]

   10. Simplified 99.2%

       \[\leadsto \color{blue}{\left|x\right| \cdot \left({x}^{6} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right)} \]
   11. Step-by-step derivation

       1. pow1 99.2%

          \[\leadsto \color{blue}{{\left(\left|x\right| \cdot \left({x}^{6} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right)\right)}^{1}} \]

       2. add-sqr-sqrt 0.0%

          \[\leadsto {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \left({x}^{6} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right)\right)}^{1} \]

       3. fabs-sqr 0.0%

          \[\leadsto {\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left({x}^{6} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right)\right)}^{1} \]

       4. add-sqr-sqrt 0.1%

          \[\leadsto {\left(\color{blue}{x} \cdot \left({x}^{6} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right)\right)}^{1} \]

   12. Applied egg-rr 0.1%

       \[\leadsto \color{blue}{{\left(x \cdot \left({x}^{6} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right)\right)}^{1}} \]
   13. Step-by-step derivation

       1. unpow1 0.1%

          \[\leadsto \color{blue}{x \cdot \left({x}^{6} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right)} \]

       2. *-commutative 0.1%

          \[\leadsto x \cdot \color{blue}{\left(\left(0.047619047619047616 \cdot {\pi}^{-0.5}\right) \cdot {x}^{6}\right)} \]

       3. associate-*l* 0.1%

          \[\leadsto x \cdot \color{blue}{\left(0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{6}\right)\right)} \]

   14. Simplified 0.1%

       \[\leadsto \color{blue}{x \cdot \left(0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{6}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.8% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 99.8%

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

   1. pow2 99.8%

      \[\leadsto \left|x\right| \cdot \left|\sqrt{\frac{1}{\pi}} \cdot \left(2 + \left(0.047619047619047616 \cdot {x}^{6} + \left(0.2 \cdot {x}^{4} + 0.6666666666666666 \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right| \]

7. Applied egg-rr 99.8%

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

Alternative 4: 99.2% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around inf 99.6%

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

Alternative 5: 34.3% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around inf 99.6%

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

   1. pow1 99.6%

      \[\leadsto \color{blue}{{\left(\left|x\right| \cdot \left|\frac{0.047619047619047616 \cdot {x}^{6} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|\right)}^{1}} \]

   2. add-sqr-sqrt 35.6%

      \[\leadsto {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \left|\frac{0.047619047619047616 \cdot {x}^{6} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|\right)}^{1} \]

   3. fabs-sqr 35.6%

      \[\leadsto {\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left|\frac{0.047619047619047616 \cdot {x}^{6} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|\right)}^{1} \]

   4. add-sqr-sqrt 37.3%

      \[\leadsto {\left(\color{blue}{x} \cdot \left|\frac{0.047619047619047616 \cdot {x}^{6} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|\right)}^{1} \]

   5. add-sqr-sqrt 36.7%

      \[\leadsto {\left(x \cdot \left|\color{blue}{\sqrt{\frac{0.047619047619047616 \cdot {x}^{6} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \cdot \sqrt{\frac{0.047619047619047616 \cdot {x}^{6} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}}}\right|\right)}^{1} \]

   6. fabs-sqr 36.7%

      \[\leadsto {\left(x \cdot \color{blue}{\left(\sqrt{\frac{0.047619047619047616 \cdot {x}^{6} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}} \cdot \sqrt{\frac{0.047619047619047616 \cdot {x}^{6} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}}\right)}\right)}^{1} \]

   7. add-sqr-sqrt 37.3%

      \[\leadsto {\left(x \cdot \color{blue}{\frac{0.047619047619047616 \cdot {x}^{6} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}}\right)}^{1} \]

   8. *-commutative 37.3%

      \[\leadsto {\left(x \cdot \frac{\color{blue}{{x}^{6} \cdot 0.047619047619047616} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right)}^{1} \]

   9. fma-define 37.3%

      \[\leadsto {\left(x \cdot \frac{\color{blue}{\mathsf{fma}\left({x}^{6}, 0.047619047619047616, \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)}}{\sqrt{\pi}}\right)}^{1} \]

   10. pow2 37.3%

       \[\leadsto {\left(x \cdot \frac{\mathsf{fma}\left({x}^{6}, 0.047619047619047616, \mathsf{fma}\left(0.6666666666666666, \color{blue}{{x}^{2}}, 2\right)\right)}{\sqrt{\pi}}\right)}^{1} \]

7. Applied egg-rr 37.3%

   \[\leadsto \color{blue}{{\left(x \cdot \frac{\mathsf{fma}\left({x}^{6}, 0.047619047619047616, \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)}{\sqrt{\pi}}\right)}^{1}} \]
8. Step-by-step derivation

   1. unpow1 37.3%

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

   2. remove-double-neg 37.3%

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

   3. distribute-lft-neg-in 37.3%

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

   4. *-commutative 37.3%

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

   5. associate-*l/ 37.1%

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

   6. associate-/l* 37.1%

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

   7. distribute-frac-neg 37.1%

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

   8. distribute-frac-neg2 37.1%

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

   9. distribute-rgt-neg-in 37.1%

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

   10. fma-undefine 37.1%

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

   11. *-commutative 37.1%

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

   12. fma-define 37.1%

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

   13. distribute-frac-neg2 37.1%

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

   14. remove-double-neg 37.1%

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

9. Simplified 37.1%

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

Alternative 6: 34.3% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0005:\\ \;\;\;\;x \cdot \frac{\mathsf{fma}\left(0.2, {x}^{4}, 2\right)}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{6}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.0005)
   (* x (/ (fma 0.2 (pow x 4.0) 2.0) (sqrt PI)))
   (* x (* 0.047619047619047616 (* (pow PI -0.5) (pow x 6.0))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.0005) {
		tmp = x * (fma(0.2, pow(x, 4.0), 2.0) / sqrt(((double) M_PI)));
	} else {
		tmp = x * (0.047619047619047616 * (pow(((double) M_PI), -0.5) * pow(x, 6.0)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.0005)
		tmp = Float64(x * Float64(fma(0.2, (x ^ 4.0), 2.0) / sqrt(pi)));
	else
		tmp = Float64(x * Float64(0.047619047619047616 * Float64((pi ^ -0.5) * (x ^ 6.0))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.0005], N[(x * N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.047619047619047616 * N[(N[Power[Pi, -0.5], $MachinePrecision] * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 99.8%

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

   6. Taylor expanded in x around 0 99.3%

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

      1. pow1 99.3%

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

      2. add-sqr-sqrt 49.9%

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

      3. fabs-sqr 49.9%

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

      4. add-sqr-sqrt 52.2%

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

      5. add-sqr-sqrt 51.3%

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

      6. fabs-sqr 51.3%

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

      7. add-sqr-sqrt 52.2%

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

      8. fma-define 52.2%

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

   8. Applied egg-rr 52.2%

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

      1. unpow1 52.2%

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

   10. Simplified 52.2%

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

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

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 99.2%

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

      1. *-commutative 99.2%

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

      2. *-commutative 99.2%

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

      3. associate-*l* 99.2%

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

      4. *-commutative 99.2%

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

      5. *-commutative 99.2%

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

   7. Simplified 99.2%

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

   8. Taylor expanded in x around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. unpow-1 99.2%

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

      3. metadata-eval 99.2%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\sqrt{{\pi}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right| \]

      4. pow-sqr 99.2%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\sqrt{\color{blue}{{\pi}^{-0.5} \cdot {\pi}^{-0.5}}} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right| \]

      5. rem-sqrt-square 99.2%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\color{blue}{\left|{\pi}^{-0.5}\right|} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right| \]

      6. rem-square-sqrt 99.2%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\left|\color{blue}{\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}}\right| \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right| \]

      7. fabs-sqr 99.2%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\color{blue}{\left(\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}\right)} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right| \]

      8. rem-square-sqrt 99.2%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\color{blue}{{\pi}^{-0.5}} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right| \]

      9. *-commutative 99.2%

         \[\leadsto \left|0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot \color{blue}{\left(\left|x\right| \cdot {x}^{6}\right)}\right)\right| \]

      10. associate-*r* 99.2%

          \[\leadsto \left|\color{blue}{\left(0.047619047619047616 \cdot {\pi}^{-0.5}\right) \cdot \left(\left|x\right| \cdot {x}^{6}\right)}\right| \]

      11. *-commutative 99.2%

          \[\leadsto \left|\color{blue}{\left({\pi}^{-0.5} \cdot 0.047619047619047616\right)} \cdot \left(\left|x\right| \cdot {x}^{6}\right)\right| \]

      12. rem-square-sqrt 99.2%

          \[\leadsto \left|\color{blue}{\sqrt{\left({\pi}^{-0.5} \cdot 0.047619047619047616\right) \cdot \left(\left|x\right| \cdot {x}^{6}\right)} \cdot \sqrt{\left({\pi}^{-0.5} \cdot 0.047619047619047616\right) \cdot \left(\left|x\right| \cdot {x}^{6}\right)}}\right| \]

      13. fabs-sqr 99.2%

          \[\leadsto \color{blue}{\sqrt{\left({\pi}^{-0.5} \cdot 0.047619047619047616\right) \cdot \left(\left|x\right| \cdot {x}^{6}\right)} \cdot \sqrt{\left({\pi}^{-0.5} \cdot 0.047619047619047616\right) \cdot \left(\left|x\right| \cdot {x}^{6}\right)}} \]

   10. Simplified 99.2%

       \[\leadsto \color{blue}{\left|x\right| \cdot \left({x}^{6} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right)} \]
   11. Step-by-step derivation

       1. pow1 99.2%

          \[\leadsto \color{blue}{{\left(\left|x\right| \cdot \left({x}^{6} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right)\right)}^{1}} \]

       2. add-sqr-sqrt 0.0%

          \[\leadsto {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \left({x}^{6} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right)\right)}^{1} \]

       3. fabs-sqr 0.0%

          \[\leadsto {\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left({x}^{6} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right)\right)}^{1} \]

       4. add-sqr-sqrt 0.1%

          \[\leadsto {\left(\color{blue}{x} \cdot \left({x}^{6} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right)\right)}^{1} \]

   12. Applied egg-rr 0.1%

       \[\leadsto \color{blue}{{\left(x \cdot \left({x}^{6} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right)\right)}^{1}} \]
   13. Step-by-step derivation

       1. unpow1 0.1%

          \[\leadsto \color{blue}{x \cdot \left({x}^{6} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right)} \]

       2. *-commutative 0.1%

          \[\leadsto x \cdot \color{blue}{\left(\left(0.047619047619047616 \cdot {\pi}^{-0.5}\right) \cdot {x}^{6}\right)} \]

       3. associate-*l* 0.1%

          \[\leadsto x \cdot \color{blue}{\left(0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{6}\right)\right)} \]

   14. Simplified 0.1%

       \[\leadsto \color{blue}{x \cdot \left(0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{6}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 34.3% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0005:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{6}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.0005)
   (* x (/ 2.0 (sqrt PI)))
   (* x (* 0.047619047619047616 (* (pow PI -0.5) (pow x 6.0))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.0005) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = x * (0.047619047619047616 * (pow(((double) M_PI), -0.5) * pow(x, 6.0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.0005) {
		tmp = x * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = x * (0.047619047619047616 * (Math.pow(Math.PI, -0.5) * Math.pow(x, 6.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.0005:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = x * (0.047619047619047616 * (math.pow(math.pi, -0.5) * math.pow(x, 6.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.0005)
		tmp = Float64(x * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64(x * Float64(0.047619047619047616 * Float64((pi ^ -0.5) * (x ^ 6.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.0005)
		tmp = x * (2.0 / sqrt(pi));
	else
		tmp = x * (0.047619047619047616 * ((pi ^ -0.5) * (x ^ 6.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.0005], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.047619047619047616 * N[(N[Power[Pi, -0.5], $MachinePrecision] * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in x around 0 99.3%

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

      1. associate-*r* 99.3%

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

      2. *-commutative 99.3%

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

      3. unpow-1 99.3%

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

      4. metadata-eval 99.3%

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

      5. pow-sqr 99.3%

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

      6. rem-sqrt-square 99.3%

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

      7. rem-square-sqrt 99.3%

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

      8. fabs-sqr 99.3%

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

      9. rem-square-sqrt 99.3%

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

      10. associate-*l* 99.3%

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

      11. fabs-mul 99.3%

          \[\leadsto \color{blue}{\left|\left|x\right| \cdot 2\right| \cdot \left|{\pi}^{-0.5}\right|} \]

      12. fabs-mul 99.3%

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

      13. fabs-fabs 99.3%

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

      14. metadata-eval 99.3%

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

      15. rem-square-sqrt 99.3%

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

      16. fabs-sqr 99.3%

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

   10. Simplified 99.3%

       \[\leadsto \color{blue}{\left|x\right| \cdot \left(2 \cdot {\pi}^{-0.5}\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 99.3%

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

       2. metadata-eval 99.3%

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

       3. sqrt-pow1 99.3%

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

       4. inv-pow 99.3%

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

       5. *-commutative 99.3%

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

       6. metadata-eval 99.3%

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

       7. rem-sqrt-square 40.3%

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

       8. sqrt-prod 40.3%

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

       9. sqrt-prod 40.2%

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

       10. pow2 40.2%

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

       11. *-commutative 40.2%

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

       12. associate-*l/ 40.3%

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

       13. pow2 40.3%

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

       14. *-un-lft-identity 40.3%

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

       15. sqrt-div 40.2%

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

       16. *-commutative 40.2%

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

       17. sqrt-prod 40.2%

           \[\leadsto \frac{\color{blue}{\sqrt{x \cdot x} \cdot \sqrt{4}}}{\sqrt{\pi}} \]

       18. sqrt-prod 49.7%

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

       19. add-sqr-sqrt 51.9%

           \[\leadsto \frac{\color{blue}{x} \cdot \sqrt{4}}{\sqrt{\pi}} \]

       20. metadata-eval 51.9%

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

   12. Applied egg-rr 51.9%

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

       1. associate-/l* 52.2%

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

   14. Simplified 52.2%

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

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

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 99.2%

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

      1. *-commutative 99.2%

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

      2. *-commutative 99.2%

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

      3. associate-*l* 99.2%

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

      4. *-commutative 99.2%

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

      5. *-commutative 99.2%

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

   7. Simplified 99.2%

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

   8. Taylor expanded in x around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. unpow-1 99.2%

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

      3. metadata-eval 99.2%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\sqrt{{\pi}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right| \]

      4. pow-sqr 99.2%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\sqrt{\color{blue}{{\pi}^{-0.5} \cdot {\pi}^{-0.5}}} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right| \]

      5. rem-sqrt-square 99.2%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\color{blue}{\left|{\pi}^{-0.5}\right|} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right| \]

      6. rem-square-sqrt 99.2%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\left|\color{blue}{\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}}\right| \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right| \]

      7. fabs-sqr 99.2%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\color{blue}{\left(\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}\right)} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right| \]

      8. rem-square-sqrt 99.2%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\color{blue}{{\pi}^{-0.5}} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right| \]

      9. *-commutative 99.2%

         \[\leadsto \left|0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot \color{blue}{\left(\left|x\right| \cdot {x}^{6}\right)}\right)\right| \]

      10. associate-*r* 99.2%

          \[\leadsto \left|\color{blue}{\left(0.047619047619047616 \cdot {\pi}^{-0.5}\right) \cdot \left(\left|x\right| \cdot {x}^{6}\right)}\right| \]

      11. *-commutative 99.2%

          \[\leadsto \left|\color{blue}{\left({\pi}^{-0.5} \cdot 0.047619047619047616\right)} \cdot \left(\left|x\right| \cdot {x}^{6}\right)\right| \]

      12. rem-square-sqrt 99.2%

          \[\leadsto \left|\color{blue}{\sqrt{\left({\pi}^{-0.5} \cdot 0.047619047619047616\right) \cdot \left(\left|x\right| \cdot {x}^{6}\right)} \cdot \sqrt{\left({\pi}^{-0.5} \cdot 0.047619047619047616\right) \cdot \left(\left|x\right| \cdot {x}^{6}\right)}}\right| \]

      13. fabs-sqr 99.2%

          \[\leadsto \color{blue}{\sqrt{\left({\pi}^{-0.5} \cdot 0.047619047619047616\right) \cdot \left(\left|x\right| \cdot {x}^{6}\right)} \cdot \sqrt{\left({\pi}^{-0.5} \cdot 0.047619047619047616\right) \cdot \left(\left|x\right| \cdot {x}^{6}\right)}} \]

   10. Simplified 99.2%

       \[\leadsto \color{blue}{\left|x\right| \cdot \left({x}^{6} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right)} \]
   11. Step-by-step derivation

       1. pow1 99.2%

          \[\leadsto \color{blue}{{\left(\left|x\right| \cdot \left({x}^{6} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right)\right)}^{1}} \]

       2. add-sqr-sqrt 0.0%

          \[\leadsto {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \left({x}^{6} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right)\right)}^{1} \]

       3. fabs-sqr 0.0%

          \[\leadsto {\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left({x}^{6} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right)\right)}^{1} \]

       4. add-sqr-sqrt 0.1%

          \[\leadsto {\left(\color{blue}{x} \cdot \left({x}^{6} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right)\right)}^{1} \]

   12. Applied egg-rr 0.1%

       \[\leadsto \color{blue}{{\left(x \cdot \left({x}^{6} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right)\right)}^{1}} \]
   13. Step-by-step derivation

       1. unpow1 0.1%

          \[\leadsto \color{blue}{x \cdot \left({x}^{6} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right)} \]

       2. *-commutative 0.1%

          \[\leadsto x \cdot \color{blue}{\left(\left(0.047619047619047616 \cdot {\pi}^{-0.5}\right) \cdot {x}^{6}\right)} \]

       3. associate-*l* 0.1%

          \[\leadsto x \cdot \color{blue}{\left(0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{6}\right)\right)} \]

   14. Simplified 0.1%

       \[\leadsto \color{blue}{x \cdot \left(0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{6}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 34.3% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0005:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.0005)
   (* x (/ 2.0 (sqrt PI)))
   (* (pow PI -0.5) (* 0.047619047619047616 (pow x 7.0)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.0005) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = pow(((double) M_PI), -0.5) * (0.047619047619047616 * pow(x, 7.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.0005) {
		tmp = x * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = Math.pow(Math.PI, -0.5) * (0.047619047619047616 * Math.pow(x, 7.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.0005:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.pow(math.pi, -0.5) * (0.047619047619047616 * math.pow(x, 7.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.0005)
		tmp = Float64(x * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64((pi ^ -0.5) * Float64(0.047619047619047616 * (x ^ 7.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.0005)
		tmp = x * (2.0 / sqrt(pi));
	else
		tmp = (pi ^ -0.5) * (0.047619047619047616 * (x ^ 7.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.0005], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in x around 0 99.3%

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

      1. associate-*r* 99.3%

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

      2. *-commutative 99.3%

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

      3. unpow-1 99.3%

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

      4. metadata-eval 99.3%

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

      5. pow-sqr 99.3%

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

      6. rem-sqrt-square 99.3%

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

      7. rem-square-sqrt 99.3%

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

      8. fabs-sqr 99.3%

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

      9. rem-square-sqrt 99.3%

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

      10. associate-*l* 99.3%

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

      11. fabs-mul 99.3%

          \[\leadsto \color{blue}{\left|\left|x\right| \cdot 2\right| \cdot \left|{\pi}^{-0.5}\right|} \]

      12. fabs-mul 99.3%

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

      13. fabs-fabs 99.3%

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

      14. metadata-eval 99.3%

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

      15. rem-square-sqrt 99.3%

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

      16. fabs-sqr 99.3%

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

   10. Simplified 99.3%

       \[\leadsto \color{blue}{\left|x\right| \cdot \left(2 \cdot {\pi}^{-0.5}\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 99.3%

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

       2. metadata-eval 99.3%

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

       3. sqrt-pow1 99.3%

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

       4. inv-pow 99.3%

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

       5. *-commutative 99.3%

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

       6. metadata-eval 99.3%

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

       7. rem-sqrt-square 40.3%

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

       8. sqrt-prod 40.3%

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

       9. sqrt-prod 40.2%

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

       10. pow2 40.2%

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

       11. *-commutative 40.2%

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

       12. associate-*l/ 40.3%

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

       13. pow2 40.3%

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

       14. *-un-lft-identity 40.3%

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

       15. sqrt-div 40.2%

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

       16. *-commutative 40.2%

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

       17. sqrt-prod 40.2%

           \[\leadsto \frac{\color{blue}{\sqrt{x \cdot x} \cdot \sqrt{4}}}{\sqrt{\pi}} \]

       18. sqrt-prod 49.7%

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

       19. add-sqr-sqrt 51.9%

           \[\leadsto \frac{\color{blue}{x} \cdot \sqrt{4}}{\sqrt{\pi}} \]

       20. metadata-eval 51.9%

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

   12. Applied egg-rr 51.9%

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

       1. associate-/l* 52.2%

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

   14. Simplified 52.2%

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

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

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 99.2%

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

      1. *-commutative 99.2%

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

      2. *-commutative 99.2%

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

      3. associate-*l* 99.2%

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

      4. *-commutative 99.2%

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

      5. *-commutative 99.2%

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

   7. Simplified 99.2%

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

   8. Taylor expanded in x around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. unpow-1 99.2%

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

      3. metadata-eval 99.2%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\sqrt{{\pi}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right| \]

      4. pow-sqr 99.2%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\sqrt{\color{blue}{{\pi}^{-0.5} \cdot {\pi}^{-0.5}}} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right| \]

      5. rem-sqrt-square 99.2%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\color{blue}{\left|{\pi}^{-0.5}\right|} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right| \]

      6. rem-square-sqrt 99.2%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\left|\color{blue}{\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}}\right| \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right| \]

      7. fabs-sqr 99.2%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\color{blue}{\left(\sqrt{{\pi}^{-0.5}} \cdot \sqrt{{\pi}^{-0.5}}\right)} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right| \]

      8. rem-square-sqrt 99.2%

         \[\leadsto \left|0.047619047619047616 \cdot \left(\color{blue}{{\pi}^{-0.5}} \cdot \left({x}^{6} \cdot \left|x\right|\right)\right)\right| \]

      9. *-commutative 99.2%

         \[\leadsto \left|0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot \color{blue}{\left(\left|x\right| \cdot {x}^{6}\right)}\right)\right| \]

      10. associate-*r* 99.2%

          \[\leadsto \left|\color{blue}{\left(0.047619047619047616 \cdot {\pi}^{-0.5}\right) \cdot \left(\left|x\right| \cdot {x}^{6}\right)}\right| \]

      11. *-commutative 99.2%

          \[\leadsto \left|\color{blue}{\left({\pi}^{-0.5} \cdot 0.047619047619047616\right)} \cdot \left(\left|x\right| \cdot {x}^{6}\right)\right| \]

      12. rem-square-sqrt 99.2%

          \[\leadsto \left|\color{blue}{\sqrt{\left({\pi}^{-0.5} \cdot 0.047619047619047616\right) \cdot \left(\left|x\right| \cdot {x}^{6}\right)} \cdot \sqrt{\left({\pi}^{-0.5} \cdot 0.047619047619047616\right) \cdot \left(\left|x\right| \cdot {x}^{6}\right)}}\right| \]

      13. fabs-sqr 99.2%

          \[\leadsto \color{blue}{\sqrt{\left({\pi}^{-0.5} \cdot 0.047619047619047616\right) \cdot \left(\left|x\right| \cdot {x}^{6}\right)} \cdot \sqrt{\left({\pi}^{-0.5} \cdot 0.047619047619047616\right) \cdot \left(\left|x\right| \cdot {x}^{6}\right)}} \]

   10. Simplified 99.2%

       \[\leadsto \color{blue}{\left|x\right| \cdot \left({x}^{6} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right)} \]
   11. Step-by-step derivation

       1. pow1 99.2%

          \[\leadsto \color{blue}{{\left(\left|x\right| \cdot \left({x}^{6} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right)\right)}^{1}} \]

       2. add-sqr-sqrt 0.0%

          \[\leadsto {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \left({x}^{6} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right)\right)}^{1} \]

       3. fabs-sqr 0.0%

          \[\leadsto {\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \left({x}^{6} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right)\right)}^{1} \]

       4. add-sqr-sqrt 0.1%

          \[\leadsto {\left(\color{blue}{x} \cdot \left({x}^{6} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right)\right)}^{1} \]

   12. Applied egg-rr 0.1%

       \[\leadsto \color{blue}{{\left(x \cdot \left({x}^{6} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right)\right)}^{1}} \]
   13. Step-by-step derivation

       1. unpow1 0.1%

          \[\leadsto \color{blue}{x \cdot \left({x}^{6} \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)\right)} \]

       2. associate-*r* 0.1%

          \[\leadsto \color{blue}{\left(x \cdot {x}^{6}\right) \cdot \left(0.047619047619047616 \cdot {\pi}^{-0.5}\right)} \]

       3. associate-*r* 0.1%

          \[\leadsto \color{blue}{\left(\left(x \cdot {x}^{6}\right) \cdot 0.047619047619047616\right) \cdot {\pi}^{-0.5}} \]

       4. *-commutative 0.1%

          \[\leadsto \left(\color{blue}{\left({x}^{6} \cdot x\right)} \cdot 0.047619047619047616\right) \cdot {\pi}^{-0.5} \]

       5. pow-plus 0.1%

          \[\leadsto \left(\color{blue}{{x}^{\left(6 + 1\right)}} \cdot 0.047619047619047616\right) \cdot {\pi}^{-0.5} \]

       6. metadata-eval 0.1%

          \[\leadsto \left({x}^{\color{blue}{7}} \cdot 0.047619047619047616\right) \cdot {\pi}^{-0.5} \]

   14. Simplified 0.1%

       \[\leadsto \color{blue}{\left({x}^{7} \cdot 0.047619047619047616\right) \cdot {\pi}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.0005:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 52.8% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-16}:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{4 \cdot {x}^{2}}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 1e-16)
   (* x (/ 2.0 (sqrt PI)))
   (sqrt (/ (* 4.0 (pow x 2.0)) PI))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 1e-16) {
		tmp = x * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = sqrt(((4.0 * pow(x, 2.0)) / ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 1e-16) {
		tmp = x * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = Math.sqrt(((4.0 * Math.pow(x, 2.0)) / Math.PI));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 1e-16:
		tmp = x * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.sqrt(((4.0 * math.pow(x, 2.0)) / math.pi))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 1e-16)
		tmp = Float64(x * Float64(2.0 / sqrt(pi)));
	else
		tmp = sqrt(Float64(Float64(4.0 * (x ^ 2.0)) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 1e-16)
		tmp = x * (2.0 / sqrt(pi));
	else
		tmp = sqrt(((4.0 * (x ^ 2.0)) / pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 1e-16], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(4.0 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 9.9999999999999998e-17

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l* 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around 0 99.8%

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

      1. associate-*r* 99.8%

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

      2. *-commutative 99.8%

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

      3. unpow-1 99.8%

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

      4. metadata-eval 99.8%

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

      5. pow-sqr 99.8%

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

      6. rem-sqrt-square 99.8%

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

      7. rem-square-sqrt 99.8%

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

      8. fabs-sqr 99.8%

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

      9. rem-square-sqrt 99.8%

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

      10. associate-*l* 99.8%

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

      11. fabs-mul 99.8%

          \[\leadsto \color{blue}{\left|\left|x\right| \cdot 2\right| \cdot \left|{\pi}^{-0.5}\right|} \]

      12. fabs-mul 99.8%

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

      13. fabs-fabs 99.8%

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

      14. metadata-eval 99.8%

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

      15. rem-square-sqrt 99.8%

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

      16. fabs-sqr 99.8%

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

   10. Simplified 99.8%

       \[\leadsto \color{blue}{\left|x\right| \cdot \left(2 \cdot {\pi}^{-0.5}\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 99.8%

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

       2. metadata-eval 99.8%

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

       3. sqrt-pow1 99.8%

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

       4. inv-pow 99.8%

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

       5. *-commutative 99.8%

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

       6. metadata-eval 99.8%

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

       7. rem-sqrt-square 39.1%

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

       8. sqrt-prod 39.1%

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

       9. sqrt-prod 39.0%

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

       10. pow2 39.0%

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

       11. *-commutative 39.0%

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

       12. associate-*l/ 39.0%

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

       13. pow2 39.0%

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

       14. *-un-lft-identity 39.0%

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

       15. sqrt-div 39.0%

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

       16. *-commutative 39.0%

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

       17. sqrt-prod 39.0%

           \[\leadsto \frac{\color{blue}{\sqrt{x \cdot x} \cdot \sqrt{4}}}{\sqrt{\pi}} \]

       18. sqrt-prod 50.2%

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

       19. add-sqr-sqrt 52.5%

           \[\leadsto \frac{\color{blue}{x} \cdot \sqrt{4}}{\sqrt{\pi}} \]

       20. metadata-eval 52.5%

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

   12. Applied egg-rr 52.5%

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

       1. associate-/l* 52.7%

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

   14. Simplified 52.7%

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

   if 9.9999999999999998e-17 < (fabs.f64 x)

   1. Initial program 99.8%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 10.8%

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

      1. *-commutative 10.8%

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

      2. associate-*l* 10.8%

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

   7. Simplified 10.8%

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

   8. Taylor expanded in x around 0 10.8%

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

      1. associate-*r* 10.8%

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

      2. *-commutative 10.8%

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

      3. unpow-1 10.8%

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

      4. metadata-eval 10.8%

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

      5. pow-sqr 10.8%

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

      6. rem-sqrt-square 10.8%

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

      7. rem-square-sqrt 10.8%

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

      8. fabs-sqr 10.8%

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

      9. rem-square-sqrt 10.8%

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

      10. associate-*l* 10.8%

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

      11. fabs-mul 10.8%

          \[\leadsto \color{blue}{\left|\left|x\right| \cdot 2\right| \cdot \left|{\pi}^{-0.5}\right|} \]

      12. fabs-mul 10.8%

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

      13. fabs-fabs 10.8%

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

      14. metadata-eval 10.8%

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

      15. rem-square-sqrt 10.8%

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

      16. fabs-sqr 10.8%

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

   10. Simplified 10.8%

       \[\leadsto \color{blue}{\left|x\right| \cdot \left(2 \cdot {\pi}^{-0.5}\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 10.8%

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

       2. metadata-eval 10.8%

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

       3. sqrt-pow1 10.8%

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

       4. inv-pow 10.8%

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

       5. *-commutative 10.8%

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

       6. metadata-eval 10.8%

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

       7. rem-sqrt-square 62.6%

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

       8. sqrt-prod 62.6%

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

       9. sqrt-prod 62.6%

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

       10. pow2 62.6%

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

       11. *-commutative 62.6%

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

       12. associate-*l/ 62.6%

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

       13. pow2 62.6%

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

       14. *-un-lft-identity 62.6%

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

       15. pow2 62.6%

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

   12. Applied egg-rr 62.6%

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

Alternative 10: 34.4% accurate, 17.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 72.3%

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

   1. *-commutative 72.3%

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

   2. associate-*l* 72.3%

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

7. Simplified 72.3%

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

8. Taylor expanded in x around 0 72.3%

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

   1. associate-*r* 72.3%

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

   2. *-commutative 72.3%

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

   3. unpow-1 72.3%

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

   4. metadata-eval 72.3%

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

   5. pow-sqr 72.3%

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

   6. rem-sqrt-square 72.3%

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

   7. rem-square-sqrt 72.3%

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

   8. fabs-sqr 72.3%

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

   9. rem-square-sqrt 72.3%

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

   10. associate-*l* 72.3%

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

   11. fabs-mul 72.3%

       \[\leadsto \color{blue}{\left|\left|x\right| \cdot 2\right| \cdot \left|{\pi}^{-0.5}\right|} \]

   12. fabs-mul 72.3%

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

   13. fabs-fabs 72.3%

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

   14. metadata-eval 72.3%

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

   15. rem-square-sqrt 72.3%

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

   16. fabs-sqr 72.3%

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

10. Simplified 72.3%

    \[\leadsto \color{blue}{\left|x\right| \cdot \left(2 \cdot {\pi}^{-0.5}\right)} \]
11. Step-by-step derivation

    1. associate-*r* 72.3%

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

    2. metadata-eval 72.3%

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

    3. sqrt-pow1 72.3%

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

    4. inv-pow 72.3%

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

    5. *-commutative 72.3%

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

    6. metadata-eval 72.3%

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

    7. rem-sqrt-square 46.4%

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

    8. sqrt-prod 46.4%

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

    9. sqrt-prod 46.3%

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

    10. pow2 46.3%

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

    11. *-commutative 46.3%

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

    12. associate-*l/ 46.3%

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

    13. pow2 46.3%

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

    14. *-un-lft-identity 46.3%

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

    15. sqrt-div 46.3%

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

    16. *-commutative 46.3%

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

    17. sqrt-prod 46.3%

        \[\leadsto \frac{\color{blue}{\sqrt{x \cdot x} \cdot \sqrt{4}}}{\sqrt{\pi}} \]

    18. sqrt-prod 35.4%

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

    19. add-sqr-sqrt 37.0%

        \[\leadsto \frac{\color{blue}{x} \cdot \sqrt{4}}{\sqrt{\pi}} \]

    20. metadata-eval 37.0%

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

12. Applied egg-rr 37.0%

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

    1. associate-/l* 37.2%

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

14. Simplified 37.2%

    \[\leadsto \color{blue}{x \cdot \frac{2}{\sqrt{\pi}}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024181 
(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)))))))