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

Percentage Accurate: 99.8% → 99.9%

Time: 11.1s

Alternatives: 6

Speedup: 3.6×

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

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% 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 99.9%

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

5. Taylor expanded in x around 0 99.9%

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

   1. pow2 99.9%

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

7. Applied egg-rr 99.9%

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

Alternative 2: 34.6% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.05:\\ \;\;\;\;x \cdot \frac{2 + 0.6666666666666666 \cdot {x}^{2}}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.05)
   (* x (/ (+ 2.0 (* 0.6666666666666666 (pow x 2.0))) (sqrt PI)))
   (* 0.047619047619047616 (/ (pow x 7.0) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.05) {
		tmp = x * ((2.0 + (0.6666666666666666 * pow(x, 2.0))) / sqrt(((double) M_PI)));
	} else {
		tmp = 0.047619047619047616 * (pow(x, 7.0) / sqrt(((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.05) {
		tmp = x * ((2.0 + (0.6666666666666666 * Math.pow(x, 2.0))) / Math.sqrt(Math.PI));
	} else {
		tmp = 0.047619047619047616 * (Math.pow(x, 7.0) / Math.sqrt(Math.PI));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.05:
		tmp = x * ((2.0 + (0.6666666666666666 * math.pow(x, 2.0))) / math.sqrt(math.pi))
	else:
		tmp = 0.047619047619047616 * (math.pow(x, 7.0) / math.sqrt(math.pi))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.05)
		tmp = Float64(x * Float64(Float64(2.0 + Float64(0.6666666666666666 * (x ^ 2.0))) / sqrt(pi)));
	else
		tmp = Float64(0.047619047619047616 * Float64((x ^ 7.0) / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.05)
		tmp = x * ((2.0 + (0.6666666666666666 * (x ^ 2.0))) / sqrt(pi));
	else
		tmp = 0.047619047619047616 * ((x ^ 7.0) / sqrt(pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.05], N[(x * N[(N[(2.0 + N[(0.6666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[Power[x, 7.0], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.050000000000000003

   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 52.6%

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

      3. fabs-sqr 52.6%

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

      4. add-sqr-sqrt 55.0%

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

      5. add-sqr-sqrt 54.1%

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

      6. fabs-sqr 54.1%

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

      7. add-sqr-sqrt 55.0%

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

      8. fma-define 55.0%

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

      9. pow2 55.0%

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

   7. Applied egg-rr 55.0%

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

      1. unpow1 55.0%

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

   9. Simplified 55.0%

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

   10. Taylor expanded in x around 0 55.0%

       \[\leadsto x \cdot \frac{\color{blue}{2 + 0.6666666666666666 \cdot {x}^{2}}}{\sqrt{\pi}} \]

   if 0.050000000000000003 < (fabs.f64 x)

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 99.1%

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

      1. add-sqr-sqrt 99.1%

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

      2. fabs-sqr 99.1%

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

      3. add-sqr-sqrt 99.1%

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

      4. *-commutative 99.1%

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

      5. inv-pow 99.1%

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

      6. sqrt-pow1 99.1%

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

      7. metadata-eval 99.1%

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

      8. *-commutative 99.1%

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

      9. add-sqr-sqrt 0.0%

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

      10. fabs-sqr 0.0%

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

      11. add-sqr-sqrt 0.1%

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

   7. Applied egg-rr 0.1%

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

      1. *-commutative 0.1%

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

      2. associate-*l* 0.1%

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

      3. *-commutative 0.1%

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

   9. Simplified 0.1%

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

       1. expm1-log1p-u 0.0%

          \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right)\right)} \]

       2. expm1-undefine 0.0%

          \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right)} - 1\right)} \]

       3. *-commutative 0.0%

          \[\leadsto {\pi}^{-0.5} \cdot \left(e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \color{blue}{\left({x}^{6} \cdot x\right)}\right)} - 1\right) \]

       4. pow-plus 0.0%

          \[\leadsto {\pi}^{-0.5} \cdot \left(e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \color{blue}{{x}^{\left(6 + 1\right)}}\right)} - 1\right) \]

       5. metadata-eval 0.0%

          \[\leadsto {\pi}^{-0.5} \cdot \left(e^{\mathsf{log1p}\left(0.047619047619047616 \cdot {x}^{\color{blue}{7}}\right)} - 1\right) \]

   11. Applied egg-rr 0.0%

       \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(0.047619047619047616 \cdot {x}^{7}\right)} - 1\right)} \]
   12. Step-by-step derivation

       1. expm1-define 0.0%

          \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot {x}^{7}\right)\right)} \]

   13. Simplified 0.0%

       \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot {x}^{7}\right)\right)} \]
   14. Step-by-step derivation

       1. pow-to-exp 0.0%

          \[\leadsto \color{blue}{e^{\log \pi \cdot -0.5}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot {x}^{7}\right)\right) \]

       2. add-exp-log 0.0%

          \[\leadsto e^{\log \pi \cdot -0.5} \cdot \color{blue}{e^{\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot {x}^{7}\right)\right)\right)}} \]

       3. prod-exp 0.0%

          \[\leadsto \color{blue}{e^{\log \pi \cdot -0.5 + \log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot {x}^{7}\right)\right)\right)}} \]

       4. rem-log-exp 0.0%

          \[\leadsto e^{\color{blue}{\log \left(e^{\log \pi \cdot -0.5}\right)} + \log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot {x}^{7}\right)\right)\right)} \]

       5. pow-to-exp 0.0%

          \[\leadsto e^{\log \color{blue}{\left({\pi}^{-0.5}\right)} + \log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot {x}^{7}\right)\right)\right)} \]

       6. metadata-eval 0.0%

          \[\leadsto e^{\log \left({\pi}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right) + \log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot {x}^{7}\right)\right)\right)} \]

       7. sqrt-pow2 0.0%

          \[\leadsto e^{\log \color{blue}{\left({\left(\sqrt{\pi}\right)}^{-1}\right)} + \log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot {x}^{7}\right)\right)\right)} \]

       8. inv-pow 0.0%

          \[\leadsto e^{\log \color{blue}{\left(\frac{1}{\sqrt{\pi}}\right)} + \log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot {x}^{7}\right)\right)\right)} \]

       9. expm1-log1p-u 0.0%

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

       10. log-prod 0.0%

           \[\leadsto e^{\log \left(\frac{1}{\sqrt{\pi}}\right) + \color{blue}{\left(\log 0.047619047619047616 + \log \left({x}^{7}\right)\right)}} \]

       11. metadata-eval 0.0%

           \[\leadsto e^{\log \left(\frac{1}{\sqrt{\pi}}\right) + \left(\log 0.047619047619047616 + \log \left({x}^{\color{blue}{\left(5 + 2\right)}}\right)\right)} \]

       12. pow-prod-up 0.0%

           \[\leadsto e^{\log \left(\frac{1}{\sqrt{\pi}}\right) + \left(\log 0.047619047619047616 + \log \color{blue}{\left({x}^{5} \cdot {x}^{2}\right)}\right)} \]

       13. log-prod 0.0%

           \[\leadsto e^{\log \left(\frac{1}{\sqrt{\pi}}\right) + \color{blue}{\log \left(0.047619047619047616 \cdot \left({x}^{5} \cdot {x}^{2}\right)\right)}} \]

       14. pow2 0.0%

           \[\leadsto e^{\log \left(\frac{1}{\sqrt{\pi}}\right) + \log \left(0.047619047619047616 \cdot \left({x}^{5} \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)} \]

   15. Applied egg-rr 0.1%

       \[\leadsto \color{blue}{\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}} \]
   16. Step-by-step derivation

       1. associate-/l* 0.1%

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

   17. Simplified 0.1%

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

Alternative 3: 34.7% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return x * ((2.0 + ((0.047619047619047616 * Math.pow(x, 6.0)) + (0.6666666666666666 * Math.pow(x, 2.0)))) / Math.sqrt(Math.PI));
}

Python

def code(x):
	return x * ((2.0 + ((0.047619047619047616 * math.pow(x, 6.0)) + (0.6666666666666666 * math.pow(x, 2.0)))) / math.sqrt(math.pi))

Julia

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

MATLAB

function tmp = code(x)
	tmp = x * ((2.0 + ((0.047619047619047616 * (x ^ 6.0)) + (0.6666666666666666 * (x ^ 2.0)))) / sqrt(pi));
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around inf 99.4%

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

   1. pow1 99.4%

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

   2. add-sqr-sqrt 36.0%

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

   3. fabs-sqr 36.0%

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

   4. add-sqr-sqrt 37.6%

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

   5. add-sqr-sqrt 37.0%

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

   6. fabs-sqr 37.0%

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

   7. add-sqr-sqrt 37.6%

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

   8. fma-define 37.6%

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

   9. pow2 37.6%

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

7. Applied egg-rr 37.6%

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

   1. unpow1 37.6%

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

9. Simplified 37.6%

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

    1. fma-undefine 37.6%

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

    2. fma-undefine 37.6%

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

    3. associate-+r+ 37.6%

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

11. Applied egg-rr 37.6%

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

12. Final simplification 37.6%

    \[\leadsto x \cdot \frac{2 + \left(0.047619047619047616 \cdot {x}^{6} + 0.6666666666666666 \cdot {x}^{2}\right)}{\sqrt{\pi}} \]
13. Add Preprocessing

Alternative 4: 34.6% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \sqrt{\frac{4}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (* x (sqrt (/ 4.0 PI)))
   (* 0.047619047619047616 (/ (pow x 7.0) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = x * sqrt((4.0 / ((double) M_PI)));
	} else {
		tmp = 0.047619047619047616 * (pow(x, 7.0) / sqrt(((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = x * Math.sqrt((4.0 / Math.PI));
	} else {
		tmp = 0.047619047619047616 * (Math.pow(x, 7.0) / Math.sqrt(Math.PI));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = x * math.sqrt((4.0 / math.pi))
	else:
		tmp = 0.047619047619047616 * (math.pow(x, 7.0) / math.sqrt(math.pi))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = Float64(x * sqrt(Float64(4.0 / pi)));
	else
		tmp = Float64(0.047619047619047616 * Float64((x ^ 7.0) / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = x * sqrt((4.0 / pi));
	else
		tmp = 0.047619047619047616 * ((x ^ 7.0) / sqrt(pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[(x * N[Sqrt[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[Power[x, 7.0], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.8%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 69.8%

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

      1. *-commutative 69.8%

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

      2. associate-*r* 69.8%

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

   7. Simplified 69.8%

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

      1. add-sqr-sqrt 69.2%

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

      2. fabs-sqr 69.2%

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

      3. add-sqr-sqrt 69.8%

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

      4. *-commutative 69.8%

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

      5. inv-pow 69.8%

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

      6. sqrt-pow1 69.8%

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

      7. metadata-eval 69.8%

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

      8. *-commutative 69.8%

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

      9. add-sqr-sqrt 35.9%

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

      10. fabs-sqr 35.9%

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

      11. add-sqr-sqrt 37.7%

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

   9. Applied egg-rr 37.7%

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

       1. *-commutative 37.7%

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

       2. *-commutative 37.7%

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

       3. associate-*r* 37.7%

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

       4. add-sqr-sqrt 36.0%

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

       5. sqrt-unprod 50.7%

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

       6. swap-sqr 50.7%

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

       7. metadata-eval 50.7%

          \[\leadsto \sqrt{\color{blue}{4} \cdot \left(\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(x \cdot {\pi}^{-0.5}\right)\right)} \]

       8. swap-sqr 50.5%

          \[\leadsto \sqrt{4 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)\right)}} \]

       9. pow2 50.5%

          \[\leadsto \sqrt{4 \cdot \left(\color{blue}{{x}^{2}} \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)\right)} \]

       10. pow-prod-up 50.6%

           \[\leadsto \sqrt{4 \cdot \left({x}^{2} \cdot \color{blue}{{\pi}^{\left(-0.5 + -0.5\right)}}\right)} \]

       11. metadata-eval 50.6%

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

   11. Applied egg-rr 50.6%

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

       1. associate-*r* 50.7%

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

       2. metadata-eval 50.7%

          \[\leadsto \sqrt{\left(4 \cdot {x}^{2}\right) \cdot {\pi}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       3. pow-sqr 50.5%

          \[\leadsto \sqrt{\left(4 \cdot {x}^{2}\right) \cdot \color{blue}{\left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)}} \]

       4. *-commutative 50.5%

          \[\leadsto \sqrt{\color{blue}{\left({x}^{2} \cdot 4\right)} \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)} \]

       5. associate-*l* 50.5%

          \[\leadsto \sqrt{\color{blue}{{x}^{2} \cdot \left(4 \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)\right)}} \]

       6. pow-sqr 50.7%

          \[\leadsto \sqrt{{x}^{2} \cdot \left(4 \cdot \color{blue}{{\pi}^{\left(2 \cdot -0.5\right)}}\right)} \]

       7. metadata-eval 50.7%

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

       8. unpow-1 50.7%

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

       9. associate-*r/ 50.7%

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

       10. metadata-eval 50.7%

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

   13. Simplified 50.7%

       \[\leadsto \color{blue}{\sqrt{{x}^{2} \cdot \frac{4}{\pi}}} \]
   14. Step-by-step derivation

       1. *-commutative 50.7%

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

       2. sqrt-prod 50.7%

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

       3. sqrt-pow1 37.7%

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

       4. metadata-eval 37.7%

          \[\leadsto \sqrt{\frac{4}{\pi}} \cdot {x}^{\color{blue}{1}} \]

       5. pow1 37.7%

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

   15. Applied egg-rr 37.7%

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

   if 1.8500000000000001 < x

   1. Initial program 99.8%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 35.4%

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

      1. add-sqr-sqrt 35.4%

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

      2. fabs-sqr 35.4%

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

      3. add-sqr-sqrt 35.4%

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

      4. *-commutative 35.4%

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

      5. inv-pow 35.4%

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

      6. sqrt-pow1 35.4%

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

      7. metadata-eval 35.4%

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

      8. *-commutative 35.4%

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

      9. add-sqr-sqrt 2.1%

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

      10. fabs-sqr 2.1%

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

      11. add-sqr-sqrt 3.9%

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

   7. Applied egg-rr 3.9%

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

      1. *-commutative 3.9%

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

      2. associate-*l* 3.9%

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

      3. *-commutative 3.9%

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

   9. Simplified 3.9%

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

       1. expm1-log1p-u 3.9%

          \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right)\right)} \]

       2. expm1-undefine 3.9%

          \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right)} - 1\right)} \]

       3. *-commutative 3.9%

          \[\leadsto {\pi}^{-0.5} \cdot \left(e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \color{blue}{\left({x}^{6} \cdot x\right)}\right)} - 1\right) \]

       4. pow-plus 3.9%

          \[\leadsto {\pi}^{-0.5} \cdot \left(e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \color{blue}{{x}^{\left(6 + 1\right)}}\right)} - 1\right) \]

       5. metadata-eval 3.9%

          \[\leadsto {\pi}^{-0.5} \cdot \left(e^{\mathsf{log1p}\left(0.047619047619047616 \cdot {x}^{\color{blue}{7}}\right)} - 1\right) \]

   11. Applied egg-rr 3.9%

       \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(0.047619047619047616 \cdot {x}^{7}\right)} - 1\right)} \]
   12. Step-by-step derivation

       1. expm1-define 3.9%

          \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot {x}^{7}\right)\right)} \]

   13. Simplified 3.9%

       \[\leadsto {\pi}^{-0.5} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot {x}^{7}\right)\right)} \]
   14. Step-by-step derivation

       1. pow-to-exp 3.9%

          \[\leadsto \color{blue}{e^{\log \pi \cdot -0.5}} \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot {x}^{7}\right)\right) \]

       2. add-exp-log 3.8%

          \[\leadsto e^{\log \pi \cdot -0.5} \cdot \color{blue}{e^{\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot {x}^{7}\right)\right)\right)}} \]

       3. prod-exp 3.8%

          \[\leadsto \color{blue}{e^{\log \pi \cdot -0.5 + \log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot {x}^{7}\right)\right)\right)}} \]

       4. rem-log-exp 3.8%

          \[\leadsto e^{\color{blue}{\log \left(e^{\log \pi \cdot -0.5}\right)} + \log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot {x}^{7}\right)\right)\right)} \]

       5. pow-to-exp 3.8%

          \[\leadsto e^{\log \color{blue}{\left({\pi}^{-0.5}\right)} + \log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot {x}^{7}\right)\right)\right)} \]

       6. metadata-eval 3.8%

          \[\leadsto e^{\log \left({\pi}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right) + \log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot {x}^{7}\right)\right)\right)} \]

       7. sqrt-pow2 3.8%

          \[\leadsto e^{\log \color{blue}{\left({\left(\sqrt{\pi}\right)}^{-1}\right)} + \log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot {x}^{7}\right)\right)\right)} \]

       8. inv-pow 3.8%

          \[\leadsto e^{\log \color{blue}{\left(\frac{1}{\sqrt{\pi}}\right)} + \log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot {x}^{7}\right)\right)\right)} \]

       9. expm1-log1p-u 3.8%

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

       10. log-prod 3.8%

           \[\leadsto e^{\log \left(\frac{1}{\sqrt{\pi}}\right) + \color{blue}{\left(\log 0.047619047619047616 + \log \left({x}^{7}\right)\right)}} \]

       11. metadata-eval 3.8%

           \[\leadsto e^{\log \left(\frac{1}{\sqrt{\pi}}\right) + \left(\log 0.047619047619047616 + \log \left({x}^{\color{blue}{\left(5 + 2\right)}}\right)\right)} \]

       12. pow-prod-up 3.8%

           \[\leadsto e^{\log \left(\frac{1}{\sqrt{\pi}}\right) + \left(\log 0.047619047619047616 + \log \color{blue}{\left({x}^{5} \cdot {x}^{2}\right)}\right)} \]

       13. log-prod 3.8%

           \[\leadsto e^{\log \left(\frac{1}{\sqrt{\pi}}\right) + \color{blue}{\log \left(0.047619047619047616 \cdot \left({x}^{5} \cdot {x}^{2}\right)\right)}} \]

       14. pow2 3.8%

           \[\leadsto e^{\log \left(\frac{1}{\sqrt{\pi}}\right) + \log \left(0.047619047619047616 \cdot \left({x}^{5} \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)} \]

   15. Applied egg-rr 3.9%

       \[\leadsto \color{blue}{\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}} \]
   16. Step-by-step derivation

       1. associate-/l* 3.9%

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

   17. Simplified 3.9%

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

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \sqrt{\frac{4}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 34.6% accurate, 16.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-36}:\\ \;\;\;\;x \cdot \sqrt{\frac{4}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(x \cdot x\right) \cdot \frac{4}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5e-36) (* x (sqrt (/ 4.0 PI))) (sqrt (* (* x x) (/ 4.0 PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 5e-36) {
		tmp = x * sqrt((4.0 / ((double) M_PI)));
	} else {
		tmp = sqrt(((x * x) * (4.0 / ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 5e-36) {
		tmp = x * Math.sqrt((4.0 / Math.PI));
	} else {
		tmp = Math.sqrt(((x * x) * (4.0 / Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5e-36:
		tmp = x * math.sqrt((4.0 / math.pi))
	else:
		tmp = math.sqrt(((x * x) * (4.0 / math.pi)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5e-36)
		tmp = Float64(x * sqrt(Float64(4.0 / pi)));
	else
		tmp = sqrt(Float64(Float64(x * x) * Float64(4.0 / pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5e-36)
		tmp = x * sqrt((4.0 / pi));
	else
		tmp = sqrt(((x * x) * (4.0 / pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5e-36], N[(x * N[Sqrt[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(x * x), $MachinePrecision] * N[(4.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.00000000000000004e-36

   1. Initial program 99.8%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 69.4%

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

      1. *-commutative 69.4%

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

      2. associate-*r* 69.4%

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

   7. Simplified 69.4%

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

      1. add-sqr-sqrt 68.9%

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

      2. fabs-sqr 68.9%

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

      3. add-sqr-sqrt 69.4%

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

      4. *-commutative 69.4%

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

      5. inv-pow 69.4%

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

      6. sqrt-pow1 69.4%

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

      7. metadata-eval 69.4%

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

      8. *-commutative 69.4%

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

      9. add-sqr-sqrt 35.2%

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

      10. fabs-sqr 35.2%

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

      11. add-sqr-sqrt 37.0%

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

   9. Applied egg-rr 37.0%

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

       1. *-commutative 37.0%

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

       2. *-commutative 37.0%

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

       3. associate-*r* 37.0%

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

       4. add-sqr-sqrt 35.3%

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

       5. sqrt-unprod 50.1%

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

       6. swap-sqr 50.1%

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

       7. metadata-eval 50.1%

          \[\leadsto \sqrt{\color{blue}{4} \cdot \left(\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(x \cdot {\pi}^{-0.5}\right)\right)} \]

       8. swap-sqr 50.0%

          \[\leadsto \sqrt{4 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)\right)}} \]

       9. pow2 50.0%

          \[\leadsto \sqrt{4 \cdot \left(\color{blue}{{x}^{2}} \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)\right)} \]

       10. pow-prod-up 50.1%

           \[\leadsto \sqrt{4 \cdot \left({x}^{2} \cdot \color{blue}{{\pi}^{\left(-0.5 + -0.5\right)}}\right)} \]

       11. metadata-eval 50.1%

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

   11. Applied egg-rr 50.1%

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

       1. associate-*r* 50.1%

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

       2. metadata-eval 50.1%

          \[\leadsto \sqrt{\left(4 \cdot {x}^{2}\right) \cdot {\pi}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       3. pow-sqr 50.0%

          \[\leadsto \sqrt{\left(4 \cdot {x}^{2}\right) \cdot \color{blue}{\left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)}} \]

       4. *-commutative 50.0%

          \[\leadsto \sqrt{\color{blue}{\left({x}^{2} \cdot 4\right)} \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)} \]

       5. associate-*l* 50.0%

          \[\leadsto \sqrt{\color{blue}{{x}^{2} \cdot \left(4 \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)\right)}} \]

       6. pow-sqr 50.1%

          \[\leadsto \sqrt{{x}^{2} \cdot \left(4 \cdot \color{blue}{{\pi}^{\left(2 \cdot -0.5\right)}}\right)} \]

       7. metadata-eval 50.1%

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

       8. unpow-1 50.1%

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

       9. associate-*r/ 50.1%

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

       10. metadata-eval 50.1%

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

   13. Simplified 50.1%

       \[\leadsto \color{blue}{\sqrt{{x}^{2} \cdot \frac{4}{\pi}}} \]
   14. Step-by-step derivation

       1. *-commutative 50.1%

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

       2. sqrt-prod 50.1%

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

       3. sqrt-pow1 37.0%

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

       4. metadata-eval 37.0%

          \[\leadsto \sqrt{\frac{4}{\pi}} \cdot {x}^{\color{blue}{1}} \]

       5. pow1 37.0%

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

   15. Applied egg-rr 37.0%

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

   if 5.00000000000000004e-36 < x

   1. Initial program 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in x around 0 97.2%

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

      1. *-commutative 97.2%

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

      2. associate-*r* 97.2%

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

   7. Simplified 97.2%

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

      1. add-sqr-sqrt 96.4%

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

      2. fabs-sqr 96.4%

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

      3. add-sqr-sqrt 97.2%

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

      4. *-commutative 97.2%

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

      5. inv-pow 97.2%

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

      6. sqrt-pow1 97.2%

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

      7. metadata-eval 97.2%

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

      8. *-commutative 97.2%

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

      9. add-sqr-sqrt 97.1%

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

      10. fabs-sqr 97.1%

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

      11. add-sqr-sqrt 97.2%

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

   9. Applied egg-rr 97.2%

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

       1. *-commutative 97.2%

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

       2. *-commutative 97.2%

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

       3. associate-*r* 97.2%

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

       4. add-sqr-sqrt 96.4%

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

       5. sqrt-unprod 97.2%

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

       6. swap-sqr 97.2%

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

       7. metadata-eval 97.2%

          \[\leadsto \sqrt{\color{blue}{4} \cdot \left(\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(x \cdot {\pi}^{-0.5}\right)\right)} \]

       8. swap-sqr 97.2%

          \[\leadsto \sqrt{4 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)\right)}} \]

       9. pow2 97.2%

          \[\leadsto \sqrt{4 \cdot \left(\color{blue}{{x}^{2}} \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)\right)} \]

       10. pow-prod-up 97.7%

           \[\leadsto \sqrt{4 \cdot \left({x}^{2} \cdot \color{blue}{{\pi}^{\left(-0.5 + -0.5\right)}}\right)} \]

       11. metadata-eval 97.7%

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

   11. Applied egg-rr 97.7%

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

       1. associate-*r* 97.7%

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

       2. metadata-eval 97.7%

          \[\leadsto \sqrt{\left(4 \cdot {x}^{2}\right) \cdot {\pi}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

       3. pow-sqr 97.2%

          \[\leadsto \sqrt{\left(4 \cdot {x}^{2}\right) \cdot \color{blue}{\left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)}} \]

       4. *-commutative 97.2%

          \[\leadsto \sqrt{\color{blue}{\left({x}^{2} \cdot 4\right)} \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)} \]

       5. associate-*l* 97.2%

          \[\leadsto \sqrt{\color{blue}{{x}^{2} \cdot \left(4 \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)\right)}} \]

       6. pow-sqr 97.7%

          \[\leadsto \sqrt{{x}^{2} \cdot \left(4 \cdot \color{blue}{{\pi}^{\left(2 \cdot -0.5\right)}}\right)} \]

       7. metadata-eval 97.7%

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

       8. unpow-1 97.7%

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

       9. associate-*r/ 97.7%

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

       10. metadata-eval 97.7%

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

   13. Simplified 97.7%

       \[\leadsto \color{blue}{\sqrt{{x}^{2} \cdot \frac{4}{\pi}}} \]
   14. Step-by-step derivation

       1. pow2 99.5%

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

   15. Applied egg-rr 97.7%

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

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-36}:\\ \;\;\;\;x \cdot \sqrt{\frac{4}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(x \cdot x\right) \cdot \frac{4}{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 34.6% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ x \cdot \sqrt{\frac{4}{\pi}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (sqrt (/ 4.0 PI))))

C

double code(double x) {
	return x * sqrt((4.0 / ((double) M_PI)));
}

Java

public static double code(double x) {
	return x * Math.sqrt((4.0 / Math.PI));
}

Python

def code(x):
	return x * math.sqrt((4.0 / math.pi))

Julia

function code(x)
	return Float64(x * sqrt(Float64(4.0 / pi)))
end

MATLAB

function tmp = code(x)
	tmp = x * sqrt((4.0 / pi));
end

Wolfram

code[x_] := N[(x * N[Sqrt[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 69.8%

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

   1. *-commutative 69.8%

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

   2. associate-*r* 69.8%

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

7. Simplified 69.8%

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

   1. add-sqr-sqrt 69.2%

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

   2. fabs-sqr 69.2%

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

   3. add-sqr-sqrt 69.8%

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

   4. *-commutative 69.8%

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

   5. inv-pow 69.8%

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

   6. sqrt-pow1 69.8%

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

   7. metadata-eval 69.8%

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

   8. *-commutative 69.8%

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

   9. add-sqr-sqrt 35.9%

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

   10. fabs-sqr 35.9%

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

   11. add-sqr-sqrt 37.7%

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

9. Applied egg-rr 37.7%

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

    1. *-commutative 37.7%

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

    2. *-commutative 37.7%

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

    3. associate-*r* 37.7%

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

    4. add-sqr-sqrt 36.0%

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

    5. sqrt-unprod 50.7%

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

    6. swap-sqr 50.7%

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

    7. metadata-eval 50.7%

       \[\leadsto \sqrt{\color{blue}{4} \cdot \left(\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(x \cdot {\pi}^{-0.5}\right)\right)} \]

    8. swap-sqr 50.5%

       \[\leadsto \sqrt{4 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)\right)}} \]

    9. pow2 50.5%

       \[\leadsto \sqrt{4 \cdot \left(\color{blue}{{x}^{2}} \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)\right)} \]

    10. pow-prod-up 50.6%

        \[\leadsto \sqrt{4 \cdot \left({x}^{2} \cdot \color{blue}{{\pi}^{\left(-0.5 + -0.5\right)}}\right)} \]

    11. metadata-eval 50.6%

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

11. Applied egg-rr 50.6%

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

    1. associate-*r* 50.7%

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

    2. metadata-eval 50.7%

       \[\leadsto \sqrt{\left(4 \cdot {x}^{2}\right) \cdot {\pi}^{\color{blue}{\left(2 \cdot -0.5\right)}}} \]

    3. pow-sqr 50.5%

       \[\leadsto \sqrt{\left(4 \cdot {x}^{2}\right) \cdot \color{blue}{\left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)}} \]

    4. *-commutative 50.5%

       \[\leadsto \sqrt{\color{blue}{\left({x}^{2} \cdot 4\right)} \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)} \]

    5. associate-*l* 50.5%

       \[\leadsto \sqrt{\color{blue}{{x}^{2} \cdot \left(4 \cdot \left({\pi}^{-0.5} \cdot {\pi}^{-0.5}\right)\right)}} \]

    6. pow-sqr 50.7%

       \[\leadsto \sqrt{{x}^{2} \cdot \left(4 \cdot \color{blue}{{\pi}^{\left(2 \cdot -0.5\right)}}\right)} \]

    7. metadata-eval 50.7%

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

    8. unpow-1 50.7%

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

    9. associate-*r/ 50.7%

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

    10. metadata-eval 50.7%

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

13. Simplified 50.7%

    \[\leadsto \color{blue}{\sqrt{{x}^{2} \cdot \frac{4}{\pi}}} \]
14. Step-by-step derivation

    1. *-commutative 50.7%

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

    2. sqrt-prod 50.7%

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

    3. sqrt-pow1 37.7%

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

    4. metadata-eval 37.7%

       \[\leadsto \sqrt{\frac{4}{\pi}} \cdot {x}^{\color{blue}{1}} \]

    5. pow1 37.7%

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

15. Applied egg-rr 37.7%

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

16. Final simplification 37.7%

    \[\leadsto x \cdot \sqrt{\frac{4}{\pi}} \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2024158 
(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)))))))