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

Percentage Accurate: 99.8% → 99.9%

Time: 10.7s

Alternatives: 11

Speedup: 3.1×

• 28.9% 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, 3.1× speedup

Math

\[\begin{array}{l} \\ \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7} + \left(0.2 \cdot {x}^{5} + \left(0.6666666666666666 \cdot {x}^{3} + x \cdot 2\right)\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (sqrt (/ 1.0 PI))
   (+
    (* 0.047619047619047616 (pow x 7.0))
    (+
     (* 0.2 (pow x 5.0))
     (+ (* 0.6666666666666666 (pow x 3.0)) (* x 2.0)))))))

C

double code(double x) {
	return fabs((sqrt((1.0 / ((double) M_PI))) * ((0.047619047619047616 * pow(x, 7.0)) + ((0.2 * pow(x, 5.0)) + ((0.6666666666666666 * pow(x, 3.0)) + (x * 2.0))))));
}

Java

public static double code(double x) {
	return Math.abs((Math.sqrt((1.0 / Math.PI)) * ((0.047619047619047616 * Math.pow(x, 7.0)) + ((0.2 * Math.pow(x, 5.0)) + ((0.6666666666666666 * Math.pow(x, 3.0)) + (x * 2.0))))));
}

Python

def code(x):
	return math.fabs((math.sqrt((1.0 / math.pi)) * ((0.047619047619047616 * math.pow(x, 7.0)) + ((0.2 * math.pow(x, 5.0)) + ((0.6666666666666666 * math.pow(x, 3.0)) + (x * 2.0))))))

Julia

function code(x)
	return abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(0.047619047619047616 * (x ^ 7.0)) + Float64(Float64(0.2 * (x ^ 5.0)) + Float64(Float64(0.6666666666666666 * (x ^ 3.0)) + Float64(x * 2.0))))))
end

MATLAB

function tmp = code(x)
	tmp = abs((sqrt((1.0 / pi)) * ((0.047619047619047616 * (x ^ 7.0)) + ((0.2 * (x ^ 5.0)) + ((0.6666666666666666 * (x ^ 3.0)) + (x * 2.0))))));
end

Wolfram

code[x_] := N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.2 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in x around 0 99.9%

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

6. Simplified 99.8%

   \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, x \cdot 2\right)\right) + \left(0.2 \cdot {x}^{4}\right) \cdot x\right)}\right| \]
7. Step-by-step derivation

   1. fma-udef 99.8%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{0.6666666666666666 \cdot {x}^{3} + x \cdot 2}\right) + \left(0.2 \cdot {x}^{4}\right) \cdot x\right)\right| \]

8. Applied egg-rr 99.8%

   \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{0.6666666666666666 \cdot {x}^{3} + x \cdot 2}\right) + \left(0.2 \cdot {x}^{4}\right) \cdot x\right)\right| \]

9. Taylor expanded in x around 0 99.8%

   \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.6666666666666666 \cdot {x}^{3} + x \cdot 2\right) + \color{blue}{0.2 \cdot {x}^{5}}\right)\right| \]

10. Taylor expanded in x around 0 99.8%

    \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(0.047619047619047616 \cdot {x}^{7} + \left(0.2 \cdot {x}^{5} + \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right)\right)\right)}\right| \]

11. Final simplification 99.8%

    \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7} + \left(0.2 \cdot {x}^{5} + \left(0.6666666666666666 \cdot {x}^{3} + x \cdot 2\right)\right)\right)\right| \]

Alternative 2: 99.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.6666666666666666 \cdot {x}^{3}\\ t_1 := \sqrt{\frac{1}{\pi}}\\ t_2 := 0.2 \cdot {x}^{5}\\ \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|t_1 \cdot \left(t_2 + \left(t_0 + x \cdot 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t_1 \cdot \left(0.047619047619047616 \cdot {x}^{7} + \left(t_2 + t_0\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 0.6666666666666666 (pow x 3.0)))
        (t_1 (sqrt (/ 1.0 PI)))
        (t_2 (* 0.2 (pow x 5.0))))
   (if (<= (fabs x) 0.4)
     (fabs (* t_1 (+ t_2 (+ t_0 (* x 2.0)))))
     (fabs (* t_1 (+ (* 0.047619047619047616 (pow x 7.0)) (+ t_2 t_0)))))))

C

double code(double x) {
	double t_0 = 0.6666666666666666 * pow(x, 3.0);
	double t_1 = sqrt((1.0 / ((double) M_PI)));
	double t_2 = 0.2 * pow(x, 5.0);
	double tmp;
	if (fabs(x) <= 0.4) {
		tmp = fabs((t_1 * (t_2 + (t_0 + (x * 2.0)))));
	} else {
		tmp = fabs((t_1 * ((0.047619047619047616 * pow(x, 7.0)) + (t_2 + t_0))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = 0.6666666666666666 * Math.pow(x, 3.0);
	double t_1 = Math.sqrt((1.0 / Math.PI));
	double t_2 = 0.2 * Math.pow(x, 5.0);
	double tmp;
	if (Math.abs(x) <= 0.4) {
		tmp = Math.abs((t_1 * (t_2 + (t_0 + (x * 2.0)))));
	} else {
		tmp = Math.abs((t_1 * ((0.047619047619047616 * Math.pow(x, 7.0)) + (t_2 + t_0))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.6666666666666666 * math.pow(x, 3.0)
	t_1 = math.sqrt((1.0 / math.pi))
	t_2 = 0.2 * math.pow(x, 5.0)
	tmp = 0
	if math.fabs(x) <= 0.4:
		tmp = math.fabs((t_1 * (t_2 + (t_0 + (x * 2.0)))))
	else:
		tmp = math.fabs((t_1 * ((0.047619047619047616 * math.pow(x, 7.0)) + (t_2 + t_0))))
	return tmp

Julia

function code(x)
	t_0 = Float64(0.6666666666666666 * (x ^ 3.0))
	t_1 = sqrt(Float64(1.0 / pi))
	t_2 = Float64(0.2 * (x ^ 5.0))
	tmp = 0.0
	if (abs(x) <= 0.4)
		tmp = abs(Float64(t_1 * Float64(t_2 + Float64(t_0 + Float64(x * 2.0)))));
	else
		tmp = abs(Float64(t_1 * Float64(Float64(0.047619047619047616 * (x ^ 7.0)) + Float64(t_2 + t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.6666666666666666 * (x ^ 3.0);
	t_1 = sqrt((1.0 / pi));
	t_2 = 0.2 * (x ^ 5.0);
	tmp = 0.0;
	if (abs(x) <= 0.4)
		tmp = abs((t_1 * (t_2 + (t_0 + (x * 2.0)))));
	else
		tmp = abs((t_1 * ((0.047619047619047616 * (x ^ 7.0)) + (t_2 + t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(0.2 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.4], N[Abs[N[(t$95$1 * N[(t$95$2 + N[(t$95$0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$1 * N[(N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.40000000000000002

   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%

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

   4. Taylor expanded in x around 0 99.8%

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

   6. Simplified 99.8%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, x \cdot 2\right)\right) + \left(0.2 \cdot {x}^{4}\right) \cdot x\right)}\right| \]
   7. Step-by-step derivation

      1. fma-udef 99.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{0.6666666666666666 \cdot {x}^{3} + x \cdot 2}\right) + \left(0.2 \cdot {x}^{4}\right) \cdot x\right)\right| \]

   8. Applied egg-rr 99.8%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{0.6666666666666666 \cdot {x}^{3} + x \cdot 2}\right) + \left(0.2 \cdot {x}^{4}\right) \cdot x\right)\right| \]

   9. Taylor expanded in x around 0 99.8%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.6666666666666666 \cdot {x}^{3} + x \cdot 2\right) + \color{blue}{0.2 \cdot {x}^{5}}\right)\right| \]

   10. Taylor expanded in x around 0 99.4%

       \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(0.2 \cdot {x}^{5} + \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right)\right)}\right| \]

   if 0.40000000000000002 < (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 99.8%

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

   4. Taylor expanded in x around 0 99.9%

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

   6. Simplified 99.9%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, x \cdot 2\right)\right) + \left(0.2 \cdot {x}^{4}\right) \cdot x\right)}\right| \]
   7. Step-by-step derivation

      1. fma-udef 99.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{0.6666666666666666 \cdot {x}^{3} + x \cdot 2}\right) + \left(0.2 \cdot {x}^{4}\right) \cdot x\right)\right| \]

   8. Applied egg-rr 99.9%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{0.6666666666666666 \cdot {x}^{3} + x \cdot 2}\right) + \left(0.2 \cdot {x}^{4}\right) \cdot x\right)\right| \]

   9. Taylor expanded in x around 0 99.9%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.6666666666666666 \cdot {x}^{3} + x \cdot 2\right) + \color{blue}{0.2 \cdot {x}^{5}}\right)\right| \]

   10. Taylor expanded in x around inf 99.4%

       \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(0.047619047619047616 \cdot {x}^{7} + \left(0.2 \cdot {x}^{5} + 0.6666666666666666 \cdot {x}^{3}\right)\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + \left(0.6666666666666666 \cdot {x}^{3} + x \cdot 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7} + \left(0.2 \cdot {x}^{5} + 0.6666666666666666 \cdot {x}^{3}\right)\right)\right|\\ \end{array} \]

Alternative 3: 99.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + \left(0.6666666666666666 \cdot {x}^{3} + x \cdot 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \left(\frac{21}{{x}^{6}} + \frac{-88.2}{{x}^{8}}\right)\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.4)
   (fabs
    (*
     (sqrt (/ 1.0 PI))
     (+ (* 0.2 (pow x 5.0)) (+ (* 0.6666666666666666 (pow x 3.0)) (* x 2.0)))))
   (/
    (fabs x)
    (fabs (* (sqrt PI) (+ (/ 21.0 (pow x 6.0)) (/ -88.2 (pow x 8.0))))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.4) {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * ((0.2 * pow(x, 5.0)) + ((0.6666666666666666 * pow(x, 3.0)) + (x * 2.0)))));
	} else {
		tmp = fabs(x) / fabs((sqrt(((double) M_PI)) * ((21.0 / pow(x, 6.0)) + (-88.2 / pow(x, 8.0)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.4) {
		tmp = Math.abs((Math.sqrt((1.0 / Math.PI)) * ((0.2 * Math.pow(x, 5.0)) + ((0.6666666666666666 * Math.pow(x, 3.0)) + (x * 2.0)))));
	} else {
		tmp = Math.abs(x) / Math.abs((Math.sqrt(Math.PI) * ((21.0 / Math.pow(x, 6.0)) + (-88.2 / Math.pow(x, 8.0)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.4:
		tmp = math.fabs((math.sqrt((1.0 / math.pi)) * ((0.2 * math.pow(x, 5.0)) + ((0.6666666666666666 * math.pow(x, 3.0)) + (x * 2.0)))))
	else:
		tmp = math.fabs(x) / math.fabs((math.sqrt(math.pi) * ((21.0 / math.pow(x, 6.0)) + (-88.2 / math.pow(x, 8.0)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.4)
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(0.2 * (x ^ 5.0)) + Float64(Float64(0.6666666666666666 * (x ^ 3.0)) + Float64(x * 2.0)))));
	else
		tmp = Float64(abs(x) / abs(Float64(sqrt(pi) * Float64(Float64(21.0 / (x ^ 6.0)) + Float64(-88.2 / (x ^ 8.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.4)
		tmp = abs((sqrt((1.0 / pi)) * ((0.2 * (x ^ 5.0)) + ((0.6666666666666666 * (x ^ 3.0)) + (x * 2.0)))));
	else
		tmp = abs(x) / abs((sqrt(pi) * ((21.0 / (x ^ 6.0)) + (-88.2 / (x ^ 8.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.4], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(0.2 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[x], $MachinePrecision] / N[Abs[N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[(21.0 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(-88.2 / N[Power[x, 8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.40000000000000002

   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%

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

   4. Taylor expanded in x around 0 99.8%

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

   6. Simplified 99.8%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, x \cdot 2\right)\right) + \left(0.2 \cdot {x}^{4}\right) \cdot x\right)}\right| \]
   7. Step-by-step derivation

      1. fma-udef 99.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{0.6666666666666666 \cdot {x}^{3} + x \cdot 2}\right) + \left(0.2 \cdot {x}^{4}\right) \cdot x\right)\right| \]

   8. Applied egg-rr 99.8%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{0.6666666666666666 \cdot {x}^{3} + x \cdot 2}\right) + \left(0.2 \cdot {x}^{4}\right) \cdot x\right)\right| \]

   9. Taylor expanded in x around 0 99.8%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.6666666666666666 \cdot {x}^{3} + x \cdot 2\right) + \color{blue}{0.2 \cdot {x}^{5}}\right)\right| \]

   10. Taylor expanded in x around 0 99.4%

       \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(0.2 \cdot {x}^{5} + \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right)\right)}\right| \]

   if 0.40000000000000002 < (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%

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

   4. Taylor expanded in x around inf 98.8%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{-88.2 \cdot \left(\frac{1}{{x}^{8}} \cdot \sqrt{\pi}\right) + 21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}\right|} \]
   5. Step-by-step derivation

      1. +-commutative 98.8%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right) + -88.2 \cdot \left(\frac{1}{{x}^{8}} \cdot \sqrt{\pi}\right)}\right|} \]

      2. associate-*r* 98.8%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\left(21 \cdot \frac{1}{{x}^{6}}\right) \cdot \sqrt{\pi}} + -88.2 \cdot \left(\frac{1}{{x}^{8}} \cdot \sqrt{\pi}\right)\right|} \]

      3. associate-*r* 98.8%

         \[\leadsto \frac{\left|x\right|}{\left|\left(21 \cdot \frac{1}{{x}^{6}}\right) \cdot \sqrt{\pi} + \color{blue}{\left(-88.2 \cdot \frac{1}{{x}^{8}}\right) \cdot \sqrt{\pi}}\right|} \]

      4. distribute-rgt-out 98.8%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot \left(21 \cdot \frac{1}{{x}^{6}} + -88.2 \cdot \frac{1}{{x}^{8}}\right)}\right|} \]

      5. associate-*r/ 98.8%

         \[\leadsto \frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \left(\color{blue}{\frac{21 \cdot 1}{{x}^{6}}} + -88.2 \cdot \frac{1}{{x}^{8}}\right)\right|} \]

      6. metadata-eval 98.8%

         \[\leadsto \frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \left(\frac{\color{blue}{21}}{{x}^{6}} + -88.2 \cdot \frac{1}{{x}^{8}}\right)\right|} \]

      7. associate-*r/ 98.8%

         \[\leadsto \frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \left(\frac{21}{{x}^{6}} + \color{blue}{\frac{-88.2 \cdot 1}{{x}^{8}}}\right)\right|} \]

      8. metadata-eval 98.8%

         \[\leadsto \frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \left(\frac{21}{{x}^{6}} + \frac{\color{blue}{-88.2}}{{x}^{8}}\right)\right|} \]

   6. Simplified 98.8%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot \left(\frac{21}{{x}^{6}} + \frac{-88.2}{{x}^{8}}\right)}\right|} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + \left(0.6666666666666666 \cdot {x}^{3} + x \cdot 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \left(\frac{21}{{x}^{6}} + \frac{-88.2}{{x}^{8}}\right)\right|}\\ \end{array} \]

Alternative 4: 99.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.2 \cdot {x}^{5}\\ \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(t_0 + \left(0.6666666666666666 \cdot {x}^{3} + x \cdot 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7} + t_0\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 0.2 (pow x 5.0))))
   (if (<= (fabs x) 0.4)
     (fabs
      (*
       (sqrt (/ 1.0 PI))
       (+ t_0 (+ (* 0.6666666666666666 (pow x 3.0)) (* x 2.0)))))
     (fabs (* (pow PI -0.5) (+ (* 0.047619047619047616 (pow x 7.0)) t_0))))))

C

double code(double x) {
	double t_0 = 0.2 * pow(x, 5.0);
	double tmp;
	if (fabs(x) <= 0.4) {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * (t_0 + ((0.6666666666666666 * pow(x, 3.0)) + (x * 2.0)))));
	} else {
		tmp = fabs((pow(((double) M_PI), -0.5) * ((0.047619047619047616 * pow(x, 7.0)) + t_0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = 0.2 * Math.pow(x, 5.0);
	double tmp;
	if (Math.abs(x) <= 0.4) {
		tmp = Math.abs((Math.sqrt((1.0 / Math.PI)) * (t_0 + ((0.6666666666666666 * Math.pow(x, 3.0)) + (x * 2.0)))));
	} else {
		tmp = Math.abs((Math.pow(Math.PI, -0.5) * ((0.047619047619047616 * Math.pow(x, 7.0)) + t_0)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.2 * math.pow(x, 5.0)
	tmp = 0
	if math.fabs(x) <= 0.4:
		tmp = math.fabs((math.sqrt((1.0 / math.pi)) * (t_0 + ((0.6666666666666666 * math.pow(x, 3.0)) + (x * 2.0)))))
	else:
		tmp = math.fabs((math.pow(math.pi, -0.5) * ((0.047619047619047616 * math.pow(x, 7.0)) + t_0)))
	return tmp

Julia

function code(x)
	t_0 = Float64(0.2 * (x ^ 5.0))
	tmp = 0.0
	if (abs(x) <= 0.4)
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(t_0 + Float64(Float64(0.6666666666666666 * (x ^ 3.0)) + Float64(x * 2.0)))));
	else
		tmp = abs(Float64((pi ^ -0.5) * Float64(Float64(0.047619047619047616 * (x ^ 7.0)) + t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.2 * (x ^ 5.0);
	tmp = 0.0;
	if (abs(x) <= 0.4)
		tmp = abs((sqrt((1.0 / pi)) * (t_0 + ((0.6666666666666666 * (x ^ 3.0)) + (x * 2.0)))));
	else
		tmp = abs(((pi ^ -0.5) * ((0.047619047619047616 * (x ^ 7.0)) + t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.2 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.4], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 + N[(N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.40000000000000002

   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%

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

   4. Taylor expanded in x around 0 99.8%

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

   6. Simplified 99.8%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, x \cdot 2\right)\right) + \left(0.2 \cdot {x}^{4}\right) \cdot x\right)}\right| \]
   7. Step-by-step derivation

      1. fma-udef 99.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{0.6666666666666666 \cdot {x}^{3} + x \cdot 2}\right) + \left(0.2 \cdot {x}^{4}\right) \cdot x\right)\right| \]

   8. Applied egg-rr 99.8%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{0.6666666666666666 \cdot {x}^{3} + x \cdot 2}\right) + \left(0.2 \cdot {x}^{4}\right) \cdot x\right)\right| \]

   9. Taylor expanded in x around 0 99.8%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.6666666666666666 \cdot {x}^{3} + x \cdot 2\right) + \color{blue}{0.2 \cdot {x}^{5}}\right)\right| \]

   10. Taylor expanded in x around 0 99.4%

       \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(0.2 \cdot {x}^{5} + \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right)\right)}\right| \]

   if 0.40000000000000002 < (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 99.8%

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

   4. Taylor expanded in x around inf 98.7%

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

      1. +-commutative 98.7%

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

      2. associate-*r* 98.7%

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

      3. associate-*r* 98.7%

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

      4. distribute-rgt-out 98.7%

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

      5. associate-*r* 98.7%

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

      6. unpow1 98.7%

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

      7. sqr-pow 0.0%

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

      8. fabs-sqr 0.0%

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

      9. sqr-pow 22.3%

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

      10. unpow1 22.3%

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

      11. unpow1 22.3%

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

      12. sqr-pow 0.0%

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

      13. fabs-sqr 0.0%

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

      14. sqr-pow 98.7%

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

      15. unpow1 98.7%

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

      16. pow-plus 98.7%

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

      17. metadata-eval 98.7%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\left(0.2 \cdot {x}^{4}\right) \cdot x + 0.047619047619047616 \cdot {x}^{\color{blue}{7}}\right)\right| \]

   6. Simplified 98.7%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left(0.2 \cdot {x}^{4}\right) \cdot x + 0.047619047619047616 \cdot {x}^{7}\right)}\right| \]
   7. Step-by-step derivation

      1. distribute-lft-in 98.7%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left(0.2 \cdot {x}^{4}\right) \cdot x\right) + \sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)}\right| \]

      2. inv-pow 98.7%

         \[\leadsto \left|\sqrt{\color{blue}{{\pi}^{-1}}} \cdot \left(\left(0.2 \cdot {x}^{4}\right) \cdot x\right) + \sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right| \]

      3. sqrt-pow1 98.7%

         \[\leadsto \left|\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \cdot \left(\left(0.2 \cdot {x}^{4}\right) \cdot x\right) + \sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right| \]

      4. metadata-eval 98.7%

         \[\leadsto \left|{\pi}^{\color{blue}{-0.5}} \cdot \left(\left(0.2 \cdot {x}^{4}\right) \cdot x\right) + \sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right| \]

      5. associate-*l* 98.7%

         \[\leadsto \left|{\pi}^{-0.5} \cdot \color{blue}{\left(0.2 \cdot \left({x}^{4} \cdot x\right)\right)} + \sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right| \]

      6. pow-plus 98.7%

         \[\leadsto \left|{\pi}^{-0.5} \cdot \left(0.2 \cdot \color{blue}{{x}^{\left(4 + 1\right)}}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right| \]

      7. metadata-eval 98.7%

         \[\leadsto \left|{\pi}^{-0.5} \cdot \left(0.2 \cdot {x}^{\color{blue}{5}}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right| \]

      8. inv-pow 98.7%

         \[\leadsto \left|{\pi}^{-0.5} \cdot \left(0.2 \cdot {x}^{5}\right) + \sqrt{\color{blue}{{\pi}^{-1}}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right| \]

      9. sqrt-pow1 98.7%

         \[\leadsto \left|{\pi}^{-0.5} \cdot \left(0.2 \cdot {x}^{5}\right) + \color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right| \]

      10. metadata-eval 98.7%

          \[\leadsto \left|{\pi}^{-0.5} \cdot \left(0.2 \cdot {x}^{5}\right) + {\pi}^{\color{blue}{-0.5}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right| \]

   8. Applied egg-rr 98.7%

      \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \left(0.2 \cdot {x}^{5}\right) + {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)}\right| \]
   9. Step-by-step derivation

      1. distribute-lft-out 98.7%

         \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)}\right| \]

   10. Simplified 98.7%

       \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + \left(0.6666666666666666 \cdot {x}^{3} + x \cdot 2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7} + 0.2 \cdot {x}^{5}\right)\right|\\ \end{array} \]

Alternative 5: 99.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3} + x \cdot 2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7} + 0.2 \cdot {x}^{5}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.4)
   (fabs
    (* (sqrt (/ 1.0 PI)) (+ (* 0.6666666666666666 (pow x 3.0)) (* x 2.0))))
   (fabs
    (*
     (pow PI -0.5)
     (+ (* 0.047619047619047616 (pow x 7.0)) (* 0.2 (pow x 5.0)))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.4) {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * ((0.6666666666666666 * pow(x, 3.0)) + (x * 2.0))));
	} else {
		tmp = fabs((pow(((double) M_PI), -0.5) * ((0.047619047619047616 * pow(x, 7.0)) + (0.2 * pow(x, 5.0)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.4) {
		tmp = Math.abs((Math.sqrt((1.0 / Math.PI)) * ((0.6666666666666666 * Math.pow(x, 3.0)) + (x * 2.0))));
	} else {
		tmp = Math.abs((Math.pow(Math.PI, -0.5) * ((0.047619047619047616 * Math.pow(x, 7.0)) + (0.2 * Math.pow(x, 5.0)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.4:
		tmp = math.fabs((math.sqrt((1.0 / math.pi)) * ((0.6666666666666666 * math.pow(x, 3.0)) + (x * 2.0))))
	else:
		tmp = math.fabs((math.pow(math.pi, -0.5) * ((0.047619047619047616 * math.pow(x, 7.0)) + (0.2 * math.pow(x, 5.0)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.4)
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(0.6666666666666666 * (x ^ 3.0)) + Float64(x * 2.0))));
	else
		tmp = abs(Float64((pi ^ -0.5) * Float64(Float64(0.047619047619047616 * (x ^ 7.0)) + Float64(0.2 * (x ^ 5.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.4)
		tmp = abs((sqrt((1.0 / pi)) * ((0.6666666666666666 * (x ^ 3.0)) + (x * 2.0))));
	else
		tmp = abs(((pi ^ -0.5) * ((0.047619047619047616 * (x ^ 7.0)) + (0.2 * (x ^ 5.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.4], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] + N[(0.2 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.40000000000000002

   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%

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

   4. Taylor expanded in x around 0 99.8%

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

   6. Simplified 99.8%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, x \cdot 2\right)\right) + \left(0.2 \cdot {x}^{4}\right) \cdot x\right)}\right| \]
   7. Step-by-step derivation

      1. fma-udef 99.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{0.6666666666666666 \cdot {x}^{3} + x \cdot 2}\right) + \left(0.2 \cdot {x}^{4}\right) \cdot x\right)\right| \]

   8. Applied egg-rr 99.8%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{0.6666666666666666 \cdot {x}^{3} + x \cdot 2}\right) + \left(0.2 \cdot {x}^{4}\right) \cdot x\right)\right| \]

   9. Taylor expanded in x around 0 99.8%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.6666666666666666 \cdot {x}^{3} + x \cdot 2\right) + \color{blue}{0.2 \cdot {x}^{5}}\right)\right| \]

   10. Taylor expanded in x around 0 99.1%

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

   if 0.40000000000000002 < (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 99.8%

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

   4. Taylor expanded in x around inf 98.7%

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

      1. +-commutative 98.7%

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

      2. associate-*r* 98.7%

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

      3. associate-*r* 98.7%

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

      4. distribute-rgt-out 98.7%

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

      5. associate-*r* 98.7%

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

      6. unpow1 98.7%

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

      7. sqr-pow 0.0%

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

      8. fabs-sqr 0.0%

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

      9. sqr-pow 22.3%

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

      10. unpow1 22.3%

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

      11. unpow1 22.3%

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

      12. sqr-pow 0.0%

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

      13. fabs-sqr 0.0%

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

      14. sqr-pow 98.7%

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

      15. unpow1 98.7%

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

      16. pow-plus 98.7%

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

      17. metadata-eval 98.7%

          \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\left(0.2 \cdot {x}^{4}\right) \cdot x + 0.047619047619047616 \cdot {x}^{\color{blue}{7}}\right)\right| \]

   6. Simplified 98.7%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left(0.2 \cdot {x}^{4}\right) \cdot x + 0.047619047619047616 \cdot {x}^{7}\right)}\right| \]
   7. Step-by-step derivation

      1. distribute-lft-in 98.7%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left(0.2 \cdot {x}^{4}\right) \cdot x\right) + \sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)}\right| \]

      2. inv-pow 98.7%

         \[\leadsto \left|\sqrt{\color{blue}{{\pi}^{-1}}} \cdot \left(\left(0.2 \cdot {x}^{4}\right) \cdot x\right) + \sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right| \]

      3. sqrt-pow1 98.7%

         \[\leadsto \left|\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \cdot \left(\left(0.2 \cdot {x}^{4}\right) \cdot x\right) + \sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right| \]

      4. metadata-eval 98.7%

         \[\leadsto \left|{\pi}^{\color{blue}{-0.5}} \cdot \left(\left(0.2 \cdot {x}^{4}\right) \cdot x\right) + \sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right| \]

      5. associate-*l* 98.7%

         \[\leadsto \left|{\pi}^{-0.5} \cdot \color{blue}{\left(0.2 \cdot \left({x}^{4} \cdot x\right)\right)} + \sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right| \]

      6. pow-plus 98.7%

         \[\leadsto \left|{\pi}^{-0.5} \cdot \left(0.2 \cdot \color{blue}{{x}^{\left(4 + 1\right)}}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right| \]

      7. metadata-eval 98.7%

         \[\leadsto \left|{\pi}^{-0.5} \cdot \left(0.2 \cdot {x}^{\color{blue}{5}}\right) + \sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right| \]

      8. inv-pow 98.7%

         \[\leadsto \left|{\pi}^{-0.5} \cdot \left(0.2 \cdot {x}^{5}\right) + \sqrt{\color{blue}{{\pi}^{-1}}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right| \]

      9. sqrt-pow1 98.7%

         \[\leadsto \left|{\pi}^{-0.5} \cdot \left(0.2 \cdot {x}^{5}\right) + \color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right| \]

      10. metadata-eval 98.7%

          \[\leadsto \left|{\pi}^{-0.5} \cdot \left(0.2 \cdot {x}^{5}\right) + {\pi}^{\color{blue}{-0.5}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right| \]

   8. Applied egg-rr 98.7%

      \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \left(0.2 \cdot {x}^{5}\right) + {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)}\right| \]
   9. Step-by-step derivation

      1. distribute-lft-out 98.7%

         \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)}\right| \]

   10. Simplified 98.7%

       \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3} + x \cdot 2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot {x}^{7} + 0.2 \cdot {x}^{5}\right)\right|\\ \end{array} \]

Alternative 6: 66.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3} + x \cdot 2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{0.047619047619047616}{{x}^{-6}}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.4)
   (fabs
    (* (sqrt (/ 1.0 PI)) (+ (* 0.6666666666666666 (pow x 3.0)) (* x 2.0))))
   (/ (* x (/ 0.047619047619047616 (pow x -6.0))) (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.4) {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * ((0.6666666666666666 * pow(x, 3.0)) + (x * 2.0))));
	} else {
		tmp = (x * (0.047619047619047616 / pow(x, -6.0))) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.4) {
		tmp = Math.abs((Math.sqrt((1.0 / Math.PI)) * ((0.6666666666666666 * Math.pow(x, 3.0)) + (x * 2.0))));
	} else {
		tmp = (x * (0.047619047619047616 / Math.pow(x, -6.0))) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.4:
		tmp = math.fabs((math.sqrt((1.0 / math.pi)) * ((0.6666666666666666 * math.pow(x, 3.0)) + (x * 2.0))))
	else:
		tmp = (x * (0.047619047619047616 / math.pow(x, -6.0))) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.4)
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(0.6666666666666666 * (x ^ 3.0)) + Float64(x * 2.0))));
	else
		tmp = Float64(Float64(x * Float64(0.047619047619047616 / (x ^ -6.0))) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.4)
		tmp = abs((sqrt((1.0 / pi)) * ((0.6666666666666666 * (x ^ 3.0)) + (x * 2.0))));
	else
		tmp = (x * (0.047619047619047616 / (x ^ -6.0))) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.4], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(x * N[(0.047619047619047616 / N[Power[x, -6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.40000000000000002

   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%

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

   4. Taylor expanded in x around 0 99.8%

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

   6. Simplified 99.8%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, x \cdot 2\right)\right) + \left(0.2 \cdot {x}^{4}\right) \cdot x\right)}\right| \]
   7. Step-by-step derivation

      1. fma-udef 99.8%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{0.6666666666666666 \cdot {x}^{3} + x \cdot 2}\right) + \left(0.2 \cdot {x}^{4}\right) \cdot x\right)\right| \]

   8. Applied egg-rr 99.8%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{7}, \color{blue}{0.6666666666666666 \cdot {x}^{3} + x \cdot 2}\right) + \left(0.2 \cdot {x}^{4}\right) \cdot x\right)\right| \]

   9. Taylor expanded in x around 0 99.8%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.6666666666666666 \cdot {x}^{3} + x \cdot 2\right) + \color{blue}{0.2 \cdot {x}^{5}}\right)\right| \]

   10. Taylor expanded in x around 0 99.1%

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

   if 0.40000000000000002 < (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%

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

   4. Taylor expanded in x around inf 97.7%

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

      1. associate-*r* 97.6%

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

      2. *-commutative 97.6%

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

      3. associate-*r/ 97.6%

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

      4. metadata-eval 97.6%

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

   6. Simplified 97.6%

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

      1. add-sqr-sqrt 0.0%

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

      2. fabs-sqr 0.0%

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

      3. add-sqr-sqrt 0.1%

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

      4. add-sqr-sqrt 0.1%

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

      5. fabs-sqr 0.1%

         \[\leadsto \frac{x}{\color{blue}{\sqrt{\sqrt{\pi} \cdot \frac{21}{{x}^{6}}} \cdot \sqrt{\sqrt{\pi} \cdot \frac{21}{{x}^{6}}}}} \]

      6. add-sqr-sqrt 0.1%

         \[\leadsto \frac{x}{\color{blue}{\sqrt{\pi} \cdot \frac{21}{{x}^{6}}}} \]

      7. expm1-log1p-u 0.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\sqrt{\pi} \cdot \frac{21}{{x}^{6}}}\right)\right)} \]

      8. *-commutative 0.0%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\color{blue}{\frac{21}{{x}^{6}} \cdot \sqrt{\pi}}}\right)\right) \]

      9. div-inv 0.0%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\color{blue}{\left(21 \cdot \frac{1}{{x}^{6}}\right)} \cdot \sqrt{\pi}}\right)\right) \]

      10. pow-flip 0.0%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\left(21 \cdot \color{blue}{{x}^{\left(-6\right)}}\right) \cdot \sqrt{\pi}}\right)\right) \]

      11. metadata-eval 0.0%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\left(21 \cdot {x}^{\color{blue}{-6}}\right) \cdot \sqrt{\pi}}\right)\right) \]

   8. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\left(21 \cdot {x}^{-6}\right) \cdot \sqrt{\pi}}\right)\right)} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 0.1%

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

      2. div-inv 0.1%

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

   10. Applied egg-rr 0.1%

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

       1. associate-/r* 0.1%

          \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{21 \cdot {x}^{-6}}}{\sqrt{\pi}}} \]

       2. associate-*r/ 0.1%

          \[\leadsto \color{blue}{\frac{x \cdot \frac{1}{21 \cdot {x}^{-6}}}{\sqrt{\pi}}} \]

       3. associate-/r* 0.1%

          \[\leadsto \frac{x \cdot \color{blue}{\frac{\frac{1}{21}}{{x}^{-6}}}}{\sqrt{\pi}} \]

       4. metadata-eval 0.1%

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

   12. Simplified 0.1%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3} + x \cdot 2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{0.047619047619047616}{{x}^{-6}}}{\sqrt{\pi}}\\ \end{array} \]

Alternative 7: 65.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|\left(x \cdot 2\right) \cdot {\pi}^{-0.5}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{0.047619047619047616}{{x}^{-6}}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.4)
   (fabs (* (* x 2.0) (pow PI -0.5)))
   (/ (* x (/ 0.047619047619047616 (pow x -6.0))) (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.4) {
		tmp = fabs(((x * 2.0) * pow(((double) M_PI), -0.5)));
	} else {
		tmp = (x * (0.047619047619047616 / pow(x, -6.0))) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.4) {
		tmp = Math.abs(((x * 2.0) * Math.pow(Math.PI, -0.5)));
	} else {
		tmp = (x * (0.047619047619047616 / Math.pow(x, -6.0))) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.4:
		tmp = math.fabs(((x * 2.0) * math.pow(math.pi, -0.5)))
	else:
		tmp = (x * (0.047619047619047616 / math.pow(x, -6.0))) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.4)
		tmp = abs(Float64(Float64(x * 2.0) * (pi ^ -0.5)));
	else
		tmp = Float64(Float64(x * Float64(0.047619047619047616 / (x ^ -6.0))) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.4)
		tmp = abs(((x * 2.0) * (pi ^ -0.5)));
	else
		tmp = (x * (0.047619047619047616 / (x ^ -6.0))) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.4], N[Abs[N[(N[(x * 2.0), $MachinePrecision] * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(x * N[(0.047619047619047616 / N[Power[x, -6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.40000000000000002

   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%

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

   4. Taylor expanded in x around 0 98.0%

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

      1. *-commutative 98.0%

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

      2. associate-*l* 98.0%

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

      3. unpow1 98.0%

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

      4. sqr-pow 43.3%

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

      5. fabs-sqr 43.3%

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

      6. sqr-pow 98.0%

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

      7. unpow1 98.0%

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

   6. Simplified 98.0%

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

      1. expm1-log1p-u 98.0%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2\right)\right)\right)}\right| \]

      2. expm1-udef 8.7%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2\right)\right)} - 1}\right| \]

      3. inv-pow 8.7%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\sqrt{\color{blue}{{\pi}^{-1}}} \cdot \left(x \cdot 2\right)\right)} - 1\right| \]

      4. sqrt-pow1 8.7%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \cdot \left(x \cdot 2\right)\right)} - 1\right| \]

      5. metadata-eval 8.7%

         \[\leadsto \left|e^{\mathsf{log1p}\left({\pi}^{\color{blue}{-0.5}} \cdot \left(x \cdot 2\right)\right)} - 1\right| \]

   8. Applied egg-rr 8.7%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(x \cdot 2\right)\right)} - 1}\right| \]
   9. Step-by-step derivation

      1. expm1-def 98.0%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(x \cdot 2\right)\right)\right)}\right| \]

      2. expm1-log1p 98.0%

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

      3. *-commutative 98.0%

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

   10. Simplified 98.0%

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

   if 0.40000000000000002 < (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%

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

   4. Taylor expanded in x around inf 97.7%

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

      1. associate-*r* 97.6%

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

      2. *-commutative 97.6%

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

      3. associate-*r/ 97.6%

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

      4. metadata-eval 97.6%

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

   6. Simplified 97.6%

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

      1. add-sqr-sqrt 0.0%

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

      2. fabs-sqr 0.0%

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

      3. add-sqr-sqrt 0.1%

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

      4. add-sqr-sqrt 0.1%

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

      5. fabs-sqr 0.1%

         \[\leadsto \frac{x}{\color{blue}{\sqrt{\sqrt{\pi} \cdot \frac{21}{{x}^{6}}} \cdot \sqrt{\sqrt{\pi} \cdot \frac{21}{{x}^{6}}}}} \]

      6. add-sqr-sqrt 0.1%

         \[\leadsto \frac{x}{\color{blue}{\sqrt{\pi} \cdot \frac{21}{{x}^{6}}}} \]

      7. expm1-log1p-u 0.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\sqrt{\pi} \cdot \frac{21}{{x}^{6}}}\right)\right)} \]

      8. *-commutative 0.0%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\color{blue}{\frac{21}{{x}^{6}} \cdot \sqrt{\pi}}}\right)\right) \]

      9. div-inv 0.0%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\color{blue}{\left(21 \cdot \frac{1}{{x}^{6}}\right)} \cdot \sqrt{\pi}}\right)\right) \]

      10. pow-flip 0.0%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\left(21 \cdot \color{blue}{{x}^{\left(-6\right)}}\right) \cdot \sqrt{\pi}}\right)\right) \]

      11. metadata-eval 0.0%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\left(21 \cdot {x}^{\color{blue}{-6}}\right) \cdot \sqrt{\pi}}\right)\right) \]

   8. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\left(21 \cdot {x}^{-6}\right) \cdot \sqrt{\pi}}\right)\right)} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 0.1%

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

      2. div-inv 0.1%

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

   10. Applied egg-rr 0.1%

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

       1. associate-/r* 0.1%

          \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{21 \cdot {x}^{-6}}}{\sqrt{\pi}}} \]

       2. associate-*r/ 0.1%

          \[\leadsto \color{blue}{\frac{x \cdot \frac{1}{21 \cdot {x}^{-6}}}{\sqrt{\pi}}} \]

       3. associate-/r* 0.1%

          \[\leadsto \frac{x \cdot \color{blue}{\frac{\frac{1}{21}}{{x}^{-6}}}}{\sqrt{\pi}} \]

       4. metadata-eval 0.1%

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

   12. Simplified 0.1%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.4:\\ \;\;\;\;\left|\left(x \cdot 2\right) \cdot {\pi}^{-0.5}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{0.047619047619047616}{{x}^{-6}}}{\sqrt{\pi}}\\ \end{array} \]

Alternative 8: 67.9% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left|\left(x \cdot 2\right) \cdot {\pi}^{-0.5}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{21 \cdot \left(\sqrt{\pi} \cdot {x}^{-6}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (fabs (* (* x 2.0) (pow PI -0.5)))
   (/ x (* 21.0 (* (sqrt PI) (pow x -6.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = fabs(((x * 2.0) * pow(((double) M_PI), -0.5)));
	} else {
		tmp = x / (21.0 * (sqrt(((double) M_PI)) * pow(x, -6.0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = Math.abs(((x * 2.0) * Math.pow(Math.PI, -0.5)));
	} else {
		tmp = x / (21.0 * (Math.sqrt(Math.PI) * Math.pow(x, -6.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = math.fabs(((x * 2.0) * math.pow(math.pi, -0.5)))
	else:
		tmp = x / (21.0 * (math.sqrt(math.pi) * math.pow(x, -6.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = abs(Float64(Float64(x * 2.0) * (pi ^ -0.5)));
	else
		tmp = Float64(x / Float64(21.0 * Float64(sqrt(pi) * (x ^ -6.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = abs(((x * 2.0) * (pi ^ -0.5)));
	else
		tmp = x / (21.0 * (sqrt(pi) * (x ^ -6.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[Abs[N[(N[(x * 2.0), $MachinePrecision] * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(x / N[(21.0 * N[(N[Sqrt[Pi], $MachinePrecision] * N[Power[x, -6.0], $MachinePrecision]), $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%

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

   4. Taylor expanded in x around 0 64.5%

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

      1. *-commutative 64.5%

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

      2. associate-*l* 64.5%

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

      3. unpow1 64.5%

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

      4. sqr-pow 27.5%

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

      5. fabs-sqr 27.5%

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

      6. sqr-pow 64.5%

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

      7. unpow1 64.5%

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

   6. Simplified 64.5%

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

      1. expm1-log1p-u 62.4%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2\right)\right)\right)}\right| \]

      2. expm1-udef 5.6%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2\right)\right)} - 1}\right| \]

      3. inv-pow 5.6%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\sqrt{\color{blue}{{\pi}^{-1}}} \cdot \left(x \cdot 2\right)\right)} - 1\right| \]

      4. sqrt-pow1 5.6%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \cdot \left(x \cdot 2\right)\right)} - 1\right| \]

      5. metadata-eval 5.6%

         \[\leadsto \left|e^{\mathsf{log1p}\left({\pi}^{\color{blue}{-0.5}} \cdot \left(x \cdot 2\right)\right)} - 1\right| \]

   8. Applied egg-rr 5.6%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(x \cdot 2\right)\right)} - 1}\right| \]
   9. Step-by-step derivation

      1. expm1-def 62.4%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(x \cdot 2\right)\right)\right)}\right| \]

      2. expm1-log1p 64.5%

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

      3. *-commutative 64.5%

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

   10. Simplified 64.5%

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

   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 99.4%

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

   4. Taylor expanded in x around inf 39.1%

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

      1. associate-*r* 39.1%

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

      2. *-commutative 39.1%

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

      3. associate-*r/ 39.1%

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

      4. metadata-eval 39.1%

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

   6. Simplified 39.1%

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

      1. expm1-log1p-u 38.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \frac{21}{{x}^{6}}\right|}\right)\right)} \]

      2. expm1-udef 38.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \frac{21}{{x}^{6}}\right|}\right)} - 1} \]

   8. Applied egg-rr 3.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{\left(21 \cdot {x}^{-6}\right) \cdot \sqrt{\pi}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 3.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\left(21 \cdot {x}^{-6}\right) \cdot \sqrt{\pi}}\right)\right)} \]

      2. expm1-log1p 3.5%

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

      3. associate-*l* 3.5%

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

   10. Simplified 3.5%

       \[\leadsto \color{blue}{\frac{x}{21 \cdot \left({x}^{-6} \cdot \sqrt{\pi}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left|\left(x \cdot 2\right) \cdot {\pi}^{-0.5}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{21 \cdot \left(\sqrt{\pi} \cdot {x}^{-6}\right)}\\ \end{array} \]

Alternative 9: 67.9% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left|\left(x \cdot 2\right) \cdot {\pi}^{-0.5}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{\pi} \cdot \left(21 \cdot {x}^{-6}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (fabs (* (* x 2.0) (pow PI -0.5)))
   (/ x (* (sqrt PI) (* 21.0 (pow x -6.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = fabs(((x * 2.0) * pow(((double) M_PI), -0.5)));
	} else {
		tmp = x / (sqrt(((double) M_PI)) * (21.0 * pow(x, -6.0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = Math.abs(((x * 2.0) * Math.pow(Math.PI, -0.5)));
	} else {
		tmp = x / (Math.sqrt(Math.PI) * (21.0 * Math.pow(x, -6.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = math.fabs(((x * 2.0) * math.pow(math.pi, -0.5)))
	else:
		tmp = x / (math.sqrt(math.pi) * (21.0 * math.pow(x, -6.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = abs(Float64(Float64(x * 2.0) * (pi ^ -0.5)));
	else
		tmp = Float64(x / Float64(sqrt(pi) * Float64(21.0 * (x ^ -6.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = abs(((x * 2.0) * (pi ^ -0.5)));
	else
		tmp = x / (sqrt(pi) * (21.0 * (x ^ -6.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[Abs[N[(N[(x * 2.0), $MachinePrecision] * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(x / N[(N[Sqrt[Pi], $MachinePrecision] * N[(21.0 * N[Power[x, -6.0], $MachinePrecision]), $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%

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

   4. Taylor expanded in x around 0 64.5%

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

      1. *-commutative 64.5%

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

      2. associate-*l* 64.5%

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

      3. unpow1 64.5%

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

      4. sqr-pow 27.5%

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

      5. fabs-sqr 27.5%

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

      6. sqr-pow 64.5%

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

      7. unpow1 64.5%

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

   6. Simplified 64.5%

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

      1. expm1-log1p-u 62.4%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2\right)\right)\right)}\right| \]

      2. expm1-udef 5.6%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2\right)\right)} - 1}\right| \]

      3. inv-pow 5.6%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\sqrt{\color{blue}{{\pi}^{-1}}} \cdot \left(x \cdot 2\right)\right)} - 1\right| \]

      4. sqrt-pow1 5.6%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \cdot \left(x \cdot 2\right)\right)} - 1\right| \]

      5. metadata-eval 5.6%

         \[\leadsto \left|e^{\mathsf{log1p}\left({\pi}^{\color{blue}{-0.5}} \cdot \left(x \cdot 2\right)\right)} - 1\right| \]

   8. Applied egg-rr 5.6%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(x \cdot 2\right)\right)} - 1}\right| \]
   9. Step-by-step derivation

      1. expm1-def 62.4%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(x \cdot 2\right)\right)\right)}\right| \]

      2. expm1-log1p 64.5%

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

      3. *-commutative 64.5%

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

   10. Simplified 64.5%

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

   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 99.4%

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

   4. Taylor expanded in x around inf 39.1%

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

      1. associate-*r* 39.1%

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

      2. *-commutative 39.1%

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

      3. associate-*r/ 39.1%

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

      4. metadata-eval 39.1%

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

   6. Simplified 39.1%

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

      1. add-sqr-sqrt 1.6%

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

      2. fabs-sqr 1.6%

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

      3. add-sqr-sqrt 3.5%

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

      4. add-sqr-sqrt 3.5%

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

      5. fabs-sqr 3.5%

         \[\leadsto \frac{x}{\color{blue}{\sqrt{\sqrt{\pi} \cdot \frac{21}{{x}^{6}}} \cdot \sqrt{\sqrt{\pi} \cdot \frac{21}{{x}^{6}}}}} \]

      6. add-sqr-sqrt 3.5%

         \[\leadsto \frac{x}{\color{blue}{\sqrt{\pi} \cdot \frac{21}{{x}^{6}}}} \]

      7. expm1-log1p-u 3.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\sqrt{\pi} \cdot \frac{21}{{x}^{6}}}\right)\right)} \]

      8. *-commutative 3.4%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\color{blue}{\frac{21}{{x}^{6}} \cdot \sqrt{\pi}}}\right)\right) \]

      9. div-inv 3.4%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\color{blue}{\left(21 \cdot \frac{1}{{x}^{6}}\right)} \cdot \sqrt{\pi}}\right)\right) \]

      10. pow-flip 3.4%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\left(21 \cdot \color{blue}{{x}^{\left(-6\right)}}\right) \cdot \sqrt{\pi}}\right)\right) \]

      11. metadata-eval 3.4%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\left(21 \cdot {x}^{\color{blue}{-6}}\right) \cdot \sqrt{\pi}}\right)\right) \]

   8. Applied egg-rr 3.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\left(21 \cdot {x}^{-6}\right) \cdot \sqrt{\pi}}\right)\right)} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 3.5%

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

   10. Applied egg-rr 3.5%

       \[\leadsto \color{blue}{\frac{x}{\left(21 \cdot {x}^{-6}\right) \cdot \sqrt{\pi}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left|\left(x \cdot 2\right) \cdot {\pi}^{-0.5}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{\pi} \cdot \left(21 \cdot {x}^{-6}\right)}\\ \end{array} \]

Alternative 10: 67.9% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \left|\left(x \cdot 2\right) \cdot {\pi}^{-0.5}\right| \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fabs (* (* x 2.0) (pow PI -0.5))))

C

double code(double x) {
	return fabs(((x * 2.0) * pow(((double) M_PI), -0.5)));
}

Java

public static double code(double x) {
	return Math.abs(((x * 2.0) * Math.pow(Math.PI, -0.5)));
}

Python

def code(x):
	return math.fabs(((x * 2.0) * math.pow(math.pi, -0.5)))

Julia

function code(x)
	return abs(Float64(Float64(x * 2.0) * (pi ^ -0.5)))
end

MATLAB

function tmp = code(x)
	tmp = abs(((x * 2.0) * (pi ^ -0.5)));
end

Wolfram

code[x_] := N[Abs[N[(N[(x * 2.0), $MachinePrecision] * N[Power[Pi, -0.5], $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%

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

4. Taylor expanded in x around 0 64.5%

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

   1. *-commutative 64.5%

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

   2. associate-*l* 64.5%

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

   3. unpow1 64.5%

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

   4. sqr-pow 27.5%

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

   5. fabs-sqr 27.5%

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

   6. sqr-pow 64.5%

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

   7. unpow1 64.5%

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

6. Simplified 64.5%

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

   1. expm1-log1p-u 62.4%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2\right)\right)\right)}\right| \]

   2. expm1-udef 5.6%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2\right)\right)} - 1}\right| \]

   3. inv-pow 5.6%

      \[\leadsto \left|e^{\mathsf{log1p}\left(\sqrt{\color{blue}{{\pi}^{-1}}} \cdot \left(x \cdot 2\right)\right)} - 1\right| \]

   4. sqrt-pow1 5.6%

      \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}} \cdot \left(x \cdot 2\right)\right)} - 1\right| \]

   5. metadata-eval 5.6%

      \[\leadsto \left|e^{\mathsf{log1p}\left({\pi}^{\color{blue}{-0.5}} \cdot \left(x \cdot 2\right)\right)} - 1\right| \]

8. Applied egg-rr 5.6%

   \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(x \cdot 2\right)\right)} - 1}\right| \]
9. Step-by-step derivation

   1. expm1-def 62.4%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(x \cdot 2\right)\right)\right)}\right| \]

   2. expm1-log1p 64.5%

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

   3. *-commutative 64.5%

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

10. Simplified 64.5%

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

11. Final simplification 64.5%

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

Alternative 11: 4.1% accurate, 19.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{expm1}\left(0\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (expm1 0.0))

C

double code(double x) {
	return expm1(0.0);
}

Java

public static double code(double x) {
	return Math.expm1(0.0);
}

Python

def code(x):
	return math.expm1(0.0)

Julia

function code(x)
	return expm1(0.0)
end

Wolfram

code[x_] := N[(Exp[0.0] - 1), $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 99.4%

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

4. Taylor expanded in x around inf 39.1%

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

   1. associate-*r* 39.1%

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

   2. *-commutative 39.1%

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

   3. associate-*r/ 39.1%

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

   4. metadata-eval 39.1%

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

6. Simplified 39.1%

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

   1. add-sqr-sqrt 1.6%

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

   2. fabs-sqr 1.6%

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

   3. add-sqr-sqrt 3.5%

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

   4. add-sqr-sqrt 3.5%

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

   5. fabs-sqr 3.5%

      \[\leadsto \frac{x}{\color{blue}{\sqrt{\sqrt{\pi} \cdot \frac{21}{{x}^{6}}} \cdot \sqrt{\sqrt{\pi} \cdot \frac{21}{{x}^{6}}}}} \]

   6. add-sqr-sqrt 3.5%

      \[\leadsto \frac{x}{\color{blue}{\sqrt{\pi} \cdot \frac{21}{{x}^{6}}}} \]

   7. expm1-log1p-u 3.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\sqrt{\pi} \cdot \frac{21}{{x}^{6}}}\right)\right)} \]

   8. *-commutative 3.4%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\color{blue}{\frac{21}{{x}^{6}} \cdot \sqrt{\pi}}}\right)\right) \]

   9. div-inv 3.4%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\color{blue}{\left(21 \cdot \frac{1}{{x}^{6}}\right)} \cdot \sqrt{\pi}}\right)\right) \]

   10. pow-flip 3.4%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\left(21 \cdot \color{blue}{{x}^{\left(-6\right)}}\right) \cdot \sqrt{\pi}}\right)\right) \]

   11. metadata-eval 3.4%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\left(21 \cdot {x}^{\color{blue}{-6}}\right) \cdot \sqrt{\pi}}\right)\right) \]

8. Applied egg-rr 3.4%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\left(21 \cdot {x}^{-6}\right) \cdot \sqrt{\pi}}\right)\right)} \]

9. Taylor expanded in x around 0 4.0%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{0}\right) \]

10. Final simplification 4.0%

    \[\leadsto \mathsf{expm1}\left(0\right) \]

Reproduce

herbie shell --seed 2023337 
(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)))))))