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

Percentage Accurate: 99.8% → 99.8%

Time: 22.3s

Alternatives: 15

Speedup: 5.7×

• 28.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

6. Applied egg-rr 0

   \[\leadsto expr\]

8. Applied egg-rr 0

   \[\leadsto expr\]
9. Add Preprocessing

Alternative 2: 99.3% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.04:\\ \;\;\;\;\left|\left({\pi}^{-0.5} \cdot \left|x\right|\right) \cdot \left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666666 + \left(x \cdot x\right) \cdot 0.2\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.047619047619047616}{\left|\frac{\sqrt{\pi}}{x}\right|} \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.04)
   (fabs
    (*
     (* (pow PI -0.5) (fabs x))
     (+ 2.0 (* (* x x) (+ 0.6666666666666666 (* (* x x) 0.2))))))
   (*
    (/ (* x 0.047619047619047616) (fabs (/ (sqrt PI) x)))
    (* (* x (* x (* x x))) x))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.04) {
		tmp = fabs(((pow(((double) M_PI), -0.5) * fabs(x)) * (2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * 0.2))))));
	} else {
		tmp = ((x * 0.047619047619047616) / fabs((sqrt(((double) M_PI)) / x))) * ((x * (x * (x * x))) * x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.04) {
		tmp = Math.abs(((Math.pow(Math.PI, -0.5) * Math.abs(x)) * (2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * 0.2))))));
	} else {
		tmp = ((x * 0.047619047619047616) / Math.abs((Math.sqrt(Math.PI) / x))) * ((x * (x * (x * x))) * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.04:
		tmp = math.fabs(((math.pow(math.pi, -0.5) * math.fabs(x)) * (2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * 0.2))))))
	else:
		tmp = ((x * 0.047619047619047616) / math.fabs((math.sqrt(math.pi) / x))) * ((x * (x * (x * x))) * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.04)
		tmp = abs(Float64(Float64((pi ^ -0.5) * abs(x)) * Float64(2.0 + Float64(Float64(x * x) * Float64(0.6666666666666666 + Float64(Float64(x * x) * 0.2))))));
	else
		tmp = Float64(Float64(Float64(x * 0.047619047619047616) / abs(Float64(sqrt(pi) / x))) * Float64(Float64(x * Float64(x * Float64(x * x))) * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.04)
		tmp = abs((((pi ^ -0.5) * abs(x)) * (2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * 0.2))))));
	else
		tmp = ((x * 0.047619047619047616) / abs((sqrt(pi) / x))) * ((x * (x * (x * x))) * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.04], N[Abs[N[(N[(N[Power[Pi, -0.5], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(N[(x * x), $MachinePrecision] * N[(0.6666666666666666 + N[(N[(x * x), $MachinePrecision] * 0.2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[(x * 0.047619047619047616), $MachinePrecision] / N[Abs[N[(N[Sqrt[Pi], $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0400000000000000008

   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 0

      \[\leadsto expr\]
   4. Add Preprocessing

   6. Applied egg-rr 0

      \[\leadsto expr\]

   7. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   9. Simplified 0

      \[\leadsto expr\]

   if 0.0400000000000000008 < (fabs.f64 x)

   1. Initial program 98.8%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   11. Applied egg-rr 0

       \[\leadsto expr\]

   13. Applied egg-rr 0

       \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.2% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.04:\\ \;\;\;\;\left|\left|x\right| \cdot \frac{1}{\sqrt{\pi} \cdot \left(0.5 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.047619047619047616}{\left|\frac{\sqrt{\pi}}{x}\right|} \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.04)
   (fabs
    (*
     (fabs x)
     (/ 1.0 (* (sqrt PI) (+ 0.5 (* x (* x -0.16666666666666666)))))))
   (*
    (/ (* x 0.047619047619047616) (fabs (/ (sqrt PI) x)))
    (* (* x (* x (* x x))) x))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.04) {
		tmp = fabs((fabs(x) * (1.0 / (sqrt(((double) M_PI)) * (0.5 + (x * (x * -0.16666666666666666)))))));
	} else {
		tmp = ((x * 0.047619047619047616) / fabs((sqrt(((double) M_PI)) / x))) * ((x * (x * (x * x))) * x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.04) {
		tmp = Math.abs((Math.abs(x) * (1.0 / (Math.sqrt(Math.PI) * (0.5 + (x * (x * -0.16666666666666666)))))));
	} else {
		tmp = ((x * 0.047619047619047616) / Math.abs((Math.sqrt(Math.PI) / x))) * ((x * (x * (x * x))) * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.04:
		tmp = math.fabs((math.fabs(x) * (1.0 / (math.sqrt(math.pi) * (0.5 + (x * (x * -0.16666666666666666)))))))
	else:
		tmp = ((x * 0.047619047619047616) / math.fabs((math.sqrt(math.pi) / x))) * ((x * (x * (x * x))) * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.04)
		tmp = abs(Float64(abs(x) * Float64(1.0 / Float64(sqrt(pi) * Float64(0.5 + Float64(x * Float64(x * -0.16666666666666666)))))));
	else
		tmp = Float64(Float64(Float64(x * 0.047619047619047616) / abs(Float64(sqrt(pi) / x))) * Float64(Float64(x * Float64(x * Float64(x * x))) * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.04)
		tmp = abs((abs(x) * (1.0 / (sqrt(pi) * (0.5 + (x * (x * -0.16666666666666666)))))));
	else
		tmp = ((x * 0.047619047619047616) / abs((sqrt(pi) / x))) * ((x * (x * (x * x))) * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.04], N[Abs[N[(N[Abs[x], $MachinePrecision] * N[(1.0 / N[(N[Sqrt[Pi], $MachinePrecision] * N[(0.5 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[(x * 0.047619047619047616), $MachinePrecision] / N[Abs[N[(N[Sqrt[Pi], $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0400000000000000008

   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 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   10. Taylor expanded in x around 0 0

       \[\leadsto expr\]

   12. Simplified 0

       \[\leadsto expr\]

   if 0.0400000000000000008 < (fabs.f64 x)

   1. Initial program 98.8%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   11. Applied egg-rr 0

       \[\leadsto expr\]

   13. Applied egg-rr 0

       \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.8% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

6. Applied egg-rr 0

   \[\leadsto expr\]
7. Add Preprocessing

Alternative 5: 99.8% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return Math.abs(x) * Math.abs(((2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * (0.2 + (0.047619047619047616 * (x * x))))))) / Math.sqrt(Math.PI)));
}

Python

def code(x):
	return math.fabs(x) * math.fabs(((2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * (0.2 + (0.047619047619047616 * (x * x))))))) / math.sqrt(math.pi)))

Julia

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

MATLAB

function tmp = code(x)
	tmp = abs(x) * abs(((2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * (0.2 + (0.047619047619047616 * (x * x))))))) / sqrt(pi)));
end

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

6. Applied egg-rr 0

   \[\leadsto expr\]
7. Add Preprocessing

Alternative 6: 99.4% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return Math.abs(((Math.abs(x) / Math.sqrt(Math.PI)) * (2.0 + ((x * x) * (0.6666666666666666 + ((x * (0.2 + (x * (x * 0.047619047619047616)))) * x))))));
}

Python

def code(x):
	return math.fabs(((math.fabs(x) / math.sqrt(math.pi)) * (2.0 + ((x * x) * (0.6666666666666666 + ((x * (0.2 + (x * (x * 0.047619047619047616)))) * x))))))

Julia

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

MATLAB

function tmp = code(x)
	tmp = abs(((abs(x) / sqrt(pi)) * (2.0 + ((x * x) * (0.6666666666666666 + ((x * (0.2 + (x * (x * 0.047619047619047616)))) * x))))));
end

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

6. Applied egg-rr 0

   \[\leadsto expr\]

8. Applied egg-rr 0

   \[\leadsto expr\]

10. Applied egg-rr 0

    \[\leadsto expr\]
11. Add Preprocessing

Alternative 7: 99.4% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return Math.abs(((Math.abs(x) / Math.sqrt(Math.PI)) * (2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * (0.2 + (x * (x * 0.047619047619047616)))))))));
}

Python

def code(x):
	return math.fabs(((math.fabs(x) / math.sqrt(math.pi)) * (2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * (0.2 + (x * (x * 0.047619047619047616)))))))))

Julia

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

MATLAB

function tmp = code(x)
	tmp = abs(((abs(x) / sqrt(pi)) * (2.0 + ((x * x) * (0.6666666666666666 + ((x * x) * (0.2 + (x * (x * 0.047619047619047616)))))))));
end

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing
5. Add Preprocessing

Alternative 8: 99.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.04:\\ \;\;\;\;\left|\frac{2 + x \cdot \left(x \cdot 0.6666666666666666\right)}{\sqrt{\pi}} \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.047619047619047616}{\left|\frac{\sqrt{\pi}}{x}\right|} \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.04)
   (fabs (* (/ (+ 2.0 (* x (* x 0.6666666666666666))) (sqrt PI)) x))
   (*
    (/ (* x 0.047619047619047616) (fabs (/ (sqrt PI) x)))
    (* (* x (* x (* x x))) x))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.04) {
		tmp = fabs((((2.0 + (x * (x * 0.6666666666666666))) / sqrt(((double) M_PI))) * x));
	} else {
		tmp = ((x * 0.047619047619047616) / fabs((sqrt(((double) M_PI)) / x))) * ((x * (x * (x * x))) * x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.04) {
		tmp = Math.abs((((2.0 + (x * (x * 0.6666666666666666))) / Math.sqrt(Math.PI)) * x));
	} else {
		tmp = ((x * 0.047619047619047616) / Math.abs((Math.sqrt(Math.PI) / x))) * ((x * (x * (x * x))) * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.04:
		tmp = math.fabs((((2.0 + (x * (x * 0.6666666666666666))) / math.sqrt(math.pi)) * x))
	else:
		tmp = ((x * 0.047619047619047616) / math.fabs((math.sqrt(math.pi) / x))) * ((x * (x * (x * x))) * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.04)
		tmp = abs(Float64(Float64(Float64(2.0 + Float64(x * Float64(x * 0.6666666666666666))) / sqrt(pi)) * x));
	else
		tmp = Float64(Float64(Float64(x * 0.047619047619047616) / abs(Float64(sqrt(pi) / x))) * Float64(Float64(x * Float64(x * Float64(x * x))) * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.04)
		tmp = abs((((2.0 + (x * (x * 0.6666666666666666))) / sqrt(pi)) * x));
	else
		tmp = ((x * 0.047619047619047616) / abs((sqrt(pi) / x))) * ((x * (x * (x * x))) * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.04], N[Abs[N[(N[(N[(2.0 + N[(x * N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision], N[(N[(N[(x * 0.047619047619047616), $MachinePrecision] / N[Abs[N[(N[Sqrt[Pi], $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0400000000000000008

   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 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   if 0.0400000000000000008 < (fabs.f64 x)

   1. Initial program 98.8%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   11. Applied egg-rr 0

       \[\leadsto expr\]

   13. Applied egg-rr 0

       \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 99.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.04:\\ \;\;\;\;\left|\frac{2 + x \cdot \left(x \cdot 0.6666666666666666\right)}{\sqrt{\pi}} \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot \left(0.047619047619047616 \cdot \left(x \cdot \left|x\right|\right)\right)\right) \cdot x\right) \cdot \frac{x \cdot \left(x \cdot x\right)}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.04)
   (fabs (* (/ (+ 2.0 (* x (* x 0.6666666666666666))) (sqrt PI)) x))
   (*
    (* (* x (* 0.047619047619047616 (* x (fabs x)))) x)
    (/ (* x (* x x)) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.04) {
		tmp = fabs((((2.0 + (x * (x * 0.6666666666666666))) / sqrt(((double) M_PI))) * x));
	} else {
		tmp = ((x * (0.047619047619047616 * (x * fabs(x)))) * x) * ((x * (x * x)) / sqrt(((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.04) {
		tmp = Math.abs((((2.0 + (x * (x * 0.6666666666666666))) / Math.sqrt(Math.PI)) * x));
	} else {
		tmp = ((x * (0.047619047619047616 * (x * Math.abs(x)))) * x) * ((x * (x * x)) / Math.sqrt(Math.PI));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.04:
		tmp = math.fabs((((2.0 + (x * (x * 0.6666666666666666))) / math.sqrt(math.pi)) * x))
	else:
		tmp = ((x * (0.047619047619047616 * (x * math.fabs(x)))) * x) * ((x * (x * x)) / math.sqrt(math.pi))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.04)
		tmp = abs(Float64(Float64(Float64(2.0 + Float64(x * Float64(x * 0.6666666666666666))) / sqrt(pi)) * x));
	else
		tmp = Float64(Float64(Float64(x * Float64(0.047619047619047616 * Float64(x * abs(x)))) * x) * Float64(Float64(x * Float64(x * x)) / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.04)
		tmp = abs((((2.0 + (x * (x * 0.6666666666666666))) / sqrt(pi)) * x));
	else
		tmp = ((x * (0.047619047619047616 * (x * abs(x)))) * x) * ((x * (x * x)) / sqrt(pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.04], N[Abs[N[(N[(N[(2.0 + N[(x * N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision], N[(N[(N[(x * N[(0.047619047619047616 * N[(x * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0400000000000000008

   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 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   if 0.0400000000000000008 < (fabs.f64 x)

   1. Initial program 98.8%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   11. Applied egg-rr 0

       \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 99.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.04:\\ \;\;\;\;\left|\frac{2 + x \cdot \left(x \cdot 0.6666666666666666\right)}{\sqrt{\pi}} \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{0.047619047619047616 \cdot \left(x \cdot \left|x\right|\right)}{\frac{\sqrt{\pi}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.04)
   (fabs (* (/ (+ 2.0 (* x (* x 0.6666666666666666))) (sqrt PI)) x))
   (*
    x
    (/
     (* 0.047619047619047616 (* x (fabs x)))
     (/ (sqrt PI) (* x (* x (* x x))))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.04) {
		tmp = fabs((((2.0 + (x * (x * 0.6666666666666666))) / sqrt(((double) M_PI))) * x));
	} else {
		tmp = x * ((0.047619047619047616 * (x * fabs(x))) / (sqrt(((double) M_PI)) / (x * (x * (x * x)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.04) {
		tmp = Math.abs((((2.0 + (x * (x * 0.6666666666666666))) / Math.sqrt(Math.PI)) * x));
	} else {
		tmp = x * ((0.047619047619047616 * (x * Math.abs(x))) / (Math.sqrt(Math.PI) / (x * (x * (x * x)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.04:
		tmp = math.fabs((((2.0 + (x * (x * 0.6666666666666666))) / math.sqrt(math.pi)) * x))
	else:
		tmp = x * ((0.047619047619047616 * (x * math.fabs(x))) / (math.sqrt(math.pi) / (x * (x * (x * x)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.04)
		tmp = abs(Float64(Float64(Float64(2.0 + Float64(x * Float64(x * 0.6666666666666666))) / sqrt(pi)) * x));
	else
		tmp = Float64(x * Float64(Float64(0.047619047619047616 * Float64(x * abs(x))) / Float64(sqrt(pi) / Float64(x * Float64(x * Float64(x * x))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.04)
		tmp = abs((((2.0 + (x * (x * 0.6666666666666666))) / sqrt(pi)) * x));
	else
		tmp = x * ((0.047619047619047616 * (x * abs(x))) / (sqrt(pi) / (x * (x * (x * x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.04], N[Abs[N[(N[(N[(2.0 + N[(x * N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision], N[(x * N[(N[(0.047619047619047616 * N[(x * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[Pi], $MachinePrecision] / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0400000000000000008

   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 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   if 0.0400000000000000008 < (fabs.f64 x)

   1. Initial program 98.8%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]

   11. Applied egg-rr 0

       \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 99.3% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

6. Applied egg-rr 0

   \[\leadsto expr\]

7. Taylor expanded in x around inf 0

   \[\leadsto expr\]

9. Simplified 0

   \[\leadsto expr\]
10. Add Preprocessing

Alternative 12: 98.9% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

5. Taylor expanded in x around inf 0

   \[\leadsto expr\]

7. Simplified 0

   \[\leadsto expr\]

9. Applied egg-rr 0

   \[\leadsto expr\]
10. Add Preprocessing

Alternative 13: 98.4% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(2 + 0.047619047619047616 \cdot \left(t\_0 \cdot t\_0\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (fabs
    (* (/ (fabs x) (sqrt PI)) (+ 2.0 (* 0.047619047619047616 (* t_0 t_0)))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	return fabs(((fabs(x) / sqrt(((double) M_PI))) * (2.0 + (0.047619047619047616 * (t_0 * t_0)))));
}

Java

public static double code(double x) {
	double t_0 = x * (x * x);
	return Math.abs(((Math.abs(x) / Math.sqrt(Math.PI)) * (2.0 + (0.047619047619047616 * (t_0 * t_0)))));
}

Python

def code(x):
	t_0 = x * (x * x)
	return math.fabs(((math.fabs(x) / math.sqrt(math.pi)) * (2.0 + (0.047619047619047616 * (t_0 * t_0)))))

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	return abs(Float64(Float64(abs(x) / sqrt(pi)) * Float64(2.0 + Float64(0.047619047619047616 * Float64(t_0 * t_0)))))
end

MATLAB

function tmp = code(x)
	t_0 = x * (x * x);
	tmp = abs(((abs(x) / sqrt(pi)) * (2.0 + (0.047619047619047616 * (t_0 * t_0)))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(N[Abs[x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(0.047619047619047616 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.5%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

5. Taylor expanded in x around inf 0

   \[\leadsto expr\]

7. Simplified 0

   \[\leadsto expr\]
8. Add Preprocessing

Alternative 14: 89.3% accurate, 8.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x):
	return math.fabs((((2.0 + (x * (x * 0.6666666666666666))) / math.sqrt(math.pi)) * x))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

5. Taylor expanded in x around 0 0

   \[\leadsto expr\]

7. Simplified 0

   \[\leadsto expr\]

9. Applied egg-rr 0

   \[\leadsto expr\]
10. Add Preprocessing

Alternative 15: 67.9% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

5. Taylor expanded in x around 0 0

   \[\leadsto expr\]

7. Simplified 0

   \[\leadsto expr\]

9. Applied egg-rr 0

   \[\leadsto expr\]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024110 
(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)))))))