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

Percentage Accurate: 99.8% → 99.9%

Time: 9.8s

Alternatives: 7

Speedup: 3.0×

• 28.3% 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, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.9%

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

   1. fma-undefine 99.9%

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

6. Applied egg-rr 99.9%

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

Alternative 2: 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 99.9%

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

5. Taylor expanded in x around inf 99.2%

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

Alternative 3: 98.9% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = abs(x) * abs((((0.047619047619047616 * (x ^ 6.0)) + 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] + 2.0), $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 99.1%

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

6. Taylor expanded in x around inf 99.0%

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

Alternative 4: 33.9% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[Pi], $MachinePrecision] / x), $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 99.8%

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

   5. Taylor expanded in x around 0 72.4%

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

      1. add-sqr-sqrt 71.9%

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

      2. fabs-sqr 71.9%

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

      3. add-sqr-sqrt 72.4%

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

      4. associate-*r* 72.4%

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

      5. sqrt-div 72.4%

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

      6. metadata-eval 72.4%

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

      7. un-div-inv 72.4%

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

      8. add-sqr-sqrt 36.4%

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

      9. fabs-sqr 36.4%

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

      10. add-sqr-sqrt 38.1%

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

   7. Applied egg-rr 38.1%

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

   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 99.8%

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

   5. Taylor expanded in x around inf 32.4%

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

      1. associate-*l* 32.4%

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

   7. Simplified 32.4%

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

      1. add-sqr-sqrt 32.4%

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

      2. fabs-sqr 32.4%

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

      3. add-sqr-sqrt 32.4%

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

      4. associate-*r* 32.4%

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

      5. add-sqr-sqrt 2.3%

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

      6. fabs-sqr 2.3%

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

      7. add-sqr-sqrt 4.1%

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

      8. sqrt-div 4.1%

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

      9. metadata-eval 4.1%

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

      10. un-div-inv 4.1%

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

   9. Applied egg-rr 4.1%

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

       1. clear-num 4.1%

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

       2. un-div-inv 4.1%

          \[\leadsto \color{blue}{\frac{0.047619047619047616 \cdot {x}^{6}}{\frac{\sqrt{\pi}}{x}}} \]

   11. Applied egg-rr 4.1%

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

4. Final simplification 38.1%

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

Alternative 5: 33.9% accurate, 8.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = 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 (x <= 1.85) {
		tmp = 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 x <= 1.85:
		tmp = 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 (x <= 1.85)
		tmp = Float64(x * Float64(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 (x <= 1.85)
		tmp = x * (2.0 / sqrt(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[(2.0 / 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 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 99.8%

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

   5. Taylor expanded in x around 0 72.4%

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

      1. add-sqr-sqrt 71.9%

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

      2. fabs-sqr 71.9%

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

      3. add-sqr-sqrt 72.4%

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

      4. associate-*r* 72.4%

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

      5. sqrt-div 72.4%

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

      6. metadata-eval 72.4%

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

      7. un-div-inv 72.4%

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

      8. add-sqr-sqrt 36.4%

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

      9. fabs-sqr 36.4%

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

      10. add-sqr-sqrt 38.1%

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

   7. Applied egg-rr 38.1%

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

   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 99.8%

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

   5. Taylor expanded in x around inf 32.4%

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

      1. associate-*l* 32.4%

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

   7. Simplified 32.4%

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

      1. *-un-lft-identity 32.4%

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

      2. add-sqr-sqrt 32.4%

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

      3. fabs-sqr 32.4%

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

      4. add-sqr-sqrt 32.4%

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

      5. associate-*r* 32.4%

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

      6. add-sqr-sqrt 2.3%

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

      7. fabs-sqr 2.3%

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

      8. add-sqr-sqrt 4.1%

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

      9. sqrt-div 4.1%

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

      10. metadata-eval 4.1%

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

      11. un-div-inv 4.1%

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

   9. Applied egg-rr 4.1%

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

       1. *-lft-identity 4.1%

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

       2. associate-*r* 4.1%

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

       3. associate-*r/ 4.1%

          \[\leadsto 0.047619047619047616 \cdot \color{blue}{\frac{{x}^{6} \cdot x}{\sqrt{\pi}}} \]

       4. pow-plus 4.1%

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

       5. metadata-eval 4.1%

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

   11. Simplified 4.1%

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

4. Final simplification 38.1%

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

Alternative 6: 33.5% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\frac{\sqrt{\pi}}{0.047619047619047616 \cdot {x}^{6} + 2}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(x / N[(N[Sqrt[Pi], $MachinePrecision] / N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + 2.0), $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 99.1%

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

   1. add-sqr-sqrt 36.4%

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

   2. fabs-sqr 36.4%

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

   3. add-sqr-sqrt 36.1%

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

   4. fabs-sqr 36.1%

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

   5. add-sqr-sqrt 37.4%

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

   6. add-sqr-sqrt 38.0%

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

   7. clear-num 38.0%

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

   8. un-div-inv 37.7%

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

   9. +-commutative 37.7%

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

7. Applied egg-rr 37.7%

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

8. Taylor expanded in x around inf 37.7%

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

9. Final simplification 37.7%

   \[\leadsto \frac{x}{\frac{\sqrt{\pi}}{0.047619047619047616 \cdot {x}^{6} + 2}} \]
10. Add Preprocessing

Alternative 7: 33.9% 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 72.4%

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

   1. add-sqr-sqrt 71.9%

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

   2. fabs-sqr 71.9%

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

   3. add-sqr-sqrt 72.4%

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

   4. associate-*r* 72.4%

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

   5. sqrt-div 72.4%

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

   6. metadata-eval 72.4%

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

   7. un-div-inv 72.4%

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

   8. add-sqr-sqrt 36.4%

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

   9. fabs-sqr 36.4%

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

   10. add-sqr-sqrt 38.1%

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

7. Applied egg-rr 38.1%

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

8. Final simplification 38.1%

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

Reproduce

herbie shell --seed 2024180 
(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)))))))