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

Percentage Accurate: 99.8% → 99.8%

Time: 15.2s

Alternatives: 11

Speedup: 3.0×

• 27.5% 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, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fabs (* x (* x (fabs x))))) (t_1 (* (fabs x) (* (fabs x) t_0))))
   (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) (* (fabs x) (* (fabs x) t_1))))))))

C

double code(double x) {
	double t_0 = fabs((x * (x * fabs(x))));
	double t_1 = fabs(x) * (fabs(x) * t_0);
	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) * (fabs(x) * (fabs(x) * t_1))))));
}

Java

public static double code(double x) {
	double t_0 = Math.abs((x * (x * Math.abs(x))));
	double t_1 = Math.abs(x) * (Math.abs(x) * t_0);
	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) * (Math.abs(x) * (Math.abs(x) * t_1))))));
}

Python

def code(x):
	t_0 = math.fabs((x * (x * math.fabs(x))))
	t_1 = math.fabs(x) * (math.fabs(x) * t_0)
	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) * (math.fabs(x) * (math.fabs(x) * t_1))))))

Julia

function code(x)
	t_0 = abs(Float64(x * Float64(x * abs(x))))
	t_1 = Float64(abs(x) * Float64(abs(x) * t_0))
	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(abs(x) * Float64(abs(x) * t_1))))))
end

MATLAB

function tmp = code(x)
	t_0 = abs((x * (x * abs(x))));
	t_1 = abs(x) * (abs(x) * t_0);
	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) * (abs(x) * (abs(x) * t_1))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[Abs[N[(x * N[(x * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * t$95$0), $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[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 89.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left|x\right|\\ t_1 := \left|x \cdot t\_0\right|\\ t_2 := \left|x\right| \cdot \left(\left|x\right| \cdot t\_1\right)\\ \mathbf{if}\;\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_1\right) + \frac{1}{5} \cdot t\_2\right) + \frac{1}{21} \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot t\_2\right)\right) \leq 0.2:\\ \;\;\;\;\left|x\right| \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|x \cdot \left(0.6666666666666666 \cdot t\_0\right)\right|}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (fabs x)))
        (t_1 (fabs (* x t_0)))
        (t_2 (* (fabs x) (* (fabs x) t_1))))
   (if (<=
        (+
         (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_1)) (* (/ 1.0 5.0) t_2))
         (* (/ 1.0 21.0) (* (fabs x) (* (fabs x) t_2))))
        0.2)
     (* (fabs x) (/ 2.0 (sqrt PI)))
     (/ (fabs (* x (* 0.6666666666666666 t_0))) (sqrt PI)))))

C

double code(double x) {
	double t_0 = x * fabs(x);
	double t_1 = fabs((x * t_0));
	double t_2 = fabs(x) * (fabs(x) * t_1);
	double tmp;
	if (((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_1)) + ((1.0 / 5.0) * t_2)) + ((1.0 / 21.0) * (fabs(x) * (fabs(x) * t_2)))) <= 0.2) {
		tmp = fabs(x) * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = fabs((x * (0.6666666666666666 * t_0))) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = x * Math.abs(x);
	double t_1 = Math.abs((x * t_0));
	double t_2 = Math.abs(x) * (Math.abs(x) * t_1);
	double tmp;
	if (((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_1)) + ((1.0 / 5.0) * t_2)) + ((1.0 / 21.0) * (Math.abs(x) * (Math.abs(x) * t_2)))) <= 0.2) {
		tmp = Math.abs(x) * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = Math.abs((x * (0.6666666666666666 * t_0))) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * math.fabs(x)
	t_1 = math.fabs((x * t_0))
	t_2 = math.fabs(x) * (math.fabs(x) * t_1)
	tmp = 0
	if ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_1)) + ((1.0 / 5.0) * t_2)) + ((1.0 / 21.0) * (math.fabs(x) * (math.fabs(x) * t_2)))) <= 0.2:
		tmp = math.fabs(x) * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.fabs((x * (0.6666666666666666 * t_0))) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	t_0 = Float64(x * abs(x))
	t_1 = abs(Float64(x * t_0))
	t_2 = Float64(abs(x) * Float64(abs(x) * t_1))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_1)) + Float64(Float64(1.0 / 5.0) * t_2)) + Float64(Float64(1.0 / 21.0) * Float64(abs(x) * Float64(abs(x) * t_2)))) <= 0.2)
		tmp = Float64(abs(x) * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64(abs(Float64(x * Float64(0.6666666666666666 * t_0))) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * abs(x);
	t_1 = abs((x * t_0));
	t_2 = abs(x) * (abs(x) * t_1);
	tmp = 0.0;
	if (((((2.0 * abs(x)) + ((2.0 / 3.0) * t_1)) + ((1.0 / 5.0) * t_2)) + ((1.0 / 21.0) * (abs(x) * (abs(x) * t_2)))) <= 0.2)
		tmp = abs(x) * (2.0 / sqrt(pi));
	else
		tmp = abs((x * (0.6666666666666666 * t_0))) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x * t$95$0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.2], N[(N[Abs[x], $MachinePrecision] * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[(x * N[(0.6666666666666666 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) < 0.20000000000000001

   1. Initial program 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 98.9%

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

      1. metadata-eval N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. rem-square-sqrt N/A

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

      5. sqrt-prod N/A

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

      6. rem-sqrt-square N/A

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

      7. fabs-div N/A

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

      8. lift-/.f64 N/A

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

      9. fabs-fabs N/A

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

      10. lift-fabs.f64 N/A

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

      11. metadata-eval N/A

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

      12. fabs-mul N/A

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

      13. lift-*.f64 N/A

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

      14. fabs-mul N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

      18. fabs-mul N/A

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

   9. Applied egg-rr 98.9%

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

   if 0.20000000000000001 < (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))))

   1. Initial program 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \mathsf{fma}\left(x \cdot x, \left|x\right| \cdot \frac{2}{3}, \color{blue}{2 \cdot \left|x\right|}\right)\right| \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \mathsf{fma}\left(x \cdot x, \left|x\right| \cdot \frac{2}{3}, \color{blue}{2 \cdot \left|x\right|}\right)\right| \]

      2. lower-fabs.f64 63.2

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

   7. Simplified 63.2%

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

      1. lift-PI.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fabs.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fabs.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-fma.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \left|x\right| \cdot \frac{2}{3}, 2 \cdot \left|x\right|\right)}\right| \]

      10. *-commutative N/A

          \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x \cdot x, \left|x\right| \cdot \frac{2}{3}, 2 \cdot \left|x\right|\right) \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

      11. lift-/.f64 N/A

          \[\leadsto \left|\mathsf{fma}\left(x \cdot x, \left|x\right| \cdot \frac{2}{3}, 2 \cdot \left|x\right|\right) \cdot \color{blue}{\frac{1}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

      12. un-div-inv N/A

          \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \left|x\right| \cdot \frac{2}{3}, 2 \cdot \left|x\right|\right)}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

   9. Applied egg-rr 63.2%

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

   10. Taylor expanded in x around inf

       \[\leadsto \frac{\left|\color{blue}{\frac{2}{3} \cdot \left({x}^{2} \cdot \left|x\right|\right)}\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\left|\frac{2}{3} \cdot \color{blue}{\left(\left|x\right| \cdot {x}^{2}\right)}\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

       2. associate-*r* N/A

          \[\leadsto \frac{\left|\color{blue}{\left(\frac{2}{3} \cdot \left|x\right|\right) \cdot {x}^{2}}\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

       3. unpow2 N/A

          \[\leadsto \frac{\left|\left(\frac{2}{3} \cdot \left|x\right|\right) \cdot \color{blue}{\left(x \cdot x\right)}\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

       4. associate-*r* N/A

          \[\leadsto \frac{\left|\color{blue}{\left(\left(\frac{2}{3} \cdot \left|x\right|\right) \cdot x\right) \cdot x}\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

       5. associate-*r* N/A

          \[\leadsto \frac{\left|\color{blue}{\left(\frac{2}{3} \cdot \left(\left|x\right| \cdot x\right)\right)} \cdot x\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

       6. *-commutative N/A

          \[\leadsto \frac{\left|\left(\frac{2}{3} \cdot \color{blue}{\left(x \cdot \left|x\right|\right)}\right) \cdot x\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

       7. *-commutative N/A

          \[\leadsto \frac{\left|\color{blue}{x \cdot \left(\frac{2}{3} \cdot \left(x \cdot \left|x\right|\right)\right)}\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

       8. lower-*.f64 N/A

          \[\leadsto \frac{\left|\color{blue}{x \cdot \left(\frac{2}{3} \cdot \left(x \cdot \left|x\right|\right)\right)}\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

       9. *-rgt-identity N/A

          \[\leadsto \frac{\left|x \cdot \left(\frac{2}{3} \cdot \left(x \cdot \color{blue}{\left(\left|x\right| \cdot 1\right)}\right)\right)\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

       10. *-inverses N/A

           \[\leadsto \frac{\left|x \cdot \left(\frac{2}{3} \cdot \left(x \cdot \left(\left|x\right| \cdot \color{blue}{\frac{{x}^{2}}{{x}^{2}}}\right)\right)\right)\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

       11. associate-/l* N/A

           \[\leadsto \frac{\left|x \cdot \left(\frac{2}{3} \cdot \left(x \cdot \color{blue}{\frac{\left|x\right| \cdot {x}^{2}}{{x}^{2}}}\right)\right)\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

       12. *-commutative N/A

           \[\leadsto \frac{\left|x \cdot \left(\frac{2}{3} \cdot \left(x \cdot \frac{\color{blue}{{x}^{2} \cdot \left|x\right|}}{{x}^{2}}\right)\right)\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

       13. associate-/l* N/A

           \[\leadsto \frac{\left|x \cdot \left(\frac{2}{3} \cdot \color{blue}{\frac{x \cdot \left({x}^{2} \cdot \left|x\right|\right)}{{x}^{2}}}\right)\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

       14. unpow2 N/A

           \[\leadsto \frac{\left|x \cdot \left(\frac{2}{3} \cdot \frac{x \cdot \left({x}^{2} \cdot \left|x\right|\right)}{\color{blue}{x \cdot x}}\right)\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

       15. times-frac N/A

           \[\leadsto \frac{\left|x \cdot \left(\frac{2}{3} \cdot \color{blue}{\left(\frac{x}{x} \cdot \frac{{x}^{2} \cdot \left|x\right|}{x}\right)}\right)\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

       16. *-inverses N/A

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

       17. associate-*r/ N/A

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

       18. *-lft-identity N/A

           \[\leadsto \frac{\left|x \cdot \left(\frac{2}{3} \cdot \color{blue}{\left({x}^{2} \cdot \frac{\left|x\right|}{x}\right)}\right)\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

       19. lower-*.f64 N/A

           \[\leadsto \frac{\left|x \cdot \color{blue}{\left(\frac{2}{3} \cdot \left({x}^{2} \cdot \frac{\left|x\right|}{x}\right)\right)}\right|}{\sqrt{\mathsf{PI}\left(\right)}} \]

       20. *-lft-identity N/A

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

   12. Simplified 63.2%

       \[\leadsto \frac{\left|\color{blue}{x \cdot \left(0.6666666666666666 \cdot \left(x \cdot \left|x\right|\right)\right)}\right|}{\sqrt{\pi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left|x \cdot \left(x \cdot \left|x\right|\right)\right|\right) + \frac{1}{5} \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left|x \cdot \left(x \cdot \left|x\right|\right)\right|\right)\right)\right) + \frac{1}{21} \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left|x \cdot \left(x \cdot \left|x\right|\right)\right|\right)\right)\right)\right) \leq 0.2:\\ \;\;\;\;\left|x\right| \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|x \cdot \left(0.6666666666666666 \cdot \left(x \cdot \left|x\right|\right)\right)\right|}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right| \cdot \left(t\_0 \cdot t\_0\right), 0.047619047619047616, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666 + \left(x \cdot x\right) \cdot 0.2, 2\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (fma
      (* (fabs x) (* t_0 t_0))
      0.047619047619047616
      (*
       (fabs x)
       (fma (* x x) (+ 0.6666666666666666 (* (* x x) 0.2)) 2.0)))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	return fabs(((1.0 / sqrt(((double) M_PI))) * fma((fabs(x) * (t_0 * t_0)), 0.047619047619047616, (fabs(x) * fma((x * x), (0.6666666666666666 + ((x * x) * 0.2)), 2.0)))));
}

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * fma(Float64(abs(x) * Float64(t_0 * t_0)), 0.047619047619047616, Float64(abs(x) * fma(Float64(x * x), Float64(0.6666666666666666 + Float64(Float64(x * x) * 0.2)), 2.0)))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Abs[x], $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] * 0.047619047619047616 + N[(N[Abs[x], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(0.6666666666666666 + N[(N[(x * x), $MachinePrecision] * 0.2), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 4: 99.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.1:\\ \;\;\;\;\left|\left(\left|x\right| \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right), 2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{x \cdot \left(x \cdot \left(\mathsf{fma}\left(x, x \cdot 0.047619047619047616, 0.2\right) \cdot \left(x \cdot \left|x\right|\right)\right)\right)}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.1)
   (fabs
    (*
     (* (fabs x) (sqrt (/ 1.0 PI)))
     (fma x (* x (fma (* x x) 0.2 0.6666666666666666)) 2.0)))
   (fabs
    (*
     x
     (/
      (* x (* x (* (fma x (* x 0.047619047619047616) 0.2) (* x (fabs x)))))
      (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.1) {
		tmp = fabs(((fabs(x) * sqrt((1.0 / ((double) M_PI)))) * fma(x, (x * fma((x * x), 0.2, 0.6666666666666666)), 2.0)));
	} else {
		tmp = fabs((x * ((x * (x * (fma(x, (x * 0.047619047619047616), 0.2) * (x * fabs(x))))) / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.1)
		tmp = abs(Float64(Float64(abs(x) * sqrt(Float64(1.0 / pi))) * fma(x, Float64(x * fma(Float64(x * x), 0.2, 0.6666666666666666)), 2.0)));
	else
		tmp = abs(Float64(x * Float64(Float64(x * Float64(x * Float64(fma(x, Float64(x * 0.047619047619047616), 0.2) * Float64(x * abs(x))))) / sqrt(pi))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.1], N[Abs[N[(N[(N[Abs[x], $MachinePrecision] * N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.2 + 0.6666666666666666), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(x * N[(N[(x * N[(x * N[(N[(x * N[(x * 0.047619047619047616), $MachinePrecision] + 0.2), $MachinePrecision] * N[(x * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.10000000000000001

   1. Initial program 99.9%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(2 \cdot \left|x\right| + {x}^{2} \cdot \left(\frac{2}{3} \cdot \left|x\right| + {x}^{2} \cdot \left(\frac{1}{21} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{1}{5} \cdot \left|x\right|\right)\right)\right)}\right| \]

   7. Simplified 99.9%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 99.6%

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

   if 0.10000000000000001 < (fabs.f64 x)

   1. Initial program 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left({x}^{6} \cdot \left(\frac{1}{21} \cdot \left|x\right| + \frac{1}{5} \cdot \frac{\left|x\right|}{{x}^{2}}\right)\right)}\right| \]

   7. Simplified 98.3%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left|x\right|\right)\right) \cdot \mathsf{fma}\left(x, x \cdot 0.047619047619047616, 0.2\right)\right)\right)}\right| \]
   8. Step-by-step derivation

      1. lift-PI.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\color{blue}{\mathsf{PI}\left(\right)}}} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left|x\right|\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{21}\right) + \frac{1}{5}\right)\right)\right)\right| \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left|\frac{1}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left|x\right|\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{21}\right) + \frac{1}{5}\right)\right)\right)\right| \]

      3. lift-/.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{1}{\sqrt{\mathsf{PI}\left(\right)}}} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left|x\right|\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{21}\right) + \frac{1}{5}\right)\right)\right)\right| \]

      4. lift-*.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\left(x \cdot \left(x \cdot \left|x\right|\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{21}\right) + \frac{1}{5}\right)\right)\right)\right| \]

      5. lift-fabs.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \color{blue}{\left|x\right|}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{21}\right) + \frac{1}{5}\right)\right)\right)\right| \]

      6. lift-*.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \color{blue}{\left(x \cdot \left|x\right|\right)}\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{21}\right) + \frac{1}{5}\right)\right)\right)\right| \]

      7. lift-*.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot \left(x \cdot \left|x\right|\right)\right)} \cdot \left(x \cdot \left(x \cdot \frac{1}{21}\right) + \frac{1}{5}\right)\right)\right)\right| \]

      8. lift-*.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left|x\right|\right)\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{21}\right)} + \frac{1}{5}\right)\right)\right)\right| \]

      9. lift-fma.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left|x\right|\right)\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{21}, \frac{1}{5}\right)}\right)\right)\right| \]

      10. lift-*.f64 N/A

          \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(\left(x \cdot \left(x \cdot \left|x\right|\right)\right) \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{21}, \frac{1}{5}\right)\right)}\right)\right| \]

      11. lift-*.f64 N/A

          \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left|x\right|\right)\right) \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{21}, \frac{1}{5}\right)\right)\right)}\right| \]

      12. *-commutative N/A

          \[\leadsto \left|\color{blue}{\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left|x\right|\right)\right) \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{21}, \frac{1}{5}\right)\right)\right) \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

   9. Applied egg-rr 98.3%

      \[\leadsto \left|\color{blue}{x \cdot \frac{x \cdot \left(x \cdot \left(\mathsf{fma}\left(x, x \cdot 0.047619047619047616, 0.2\right) \cdot \left(x \cdot \left|x\right|\right)\right)\right)}{\sqrt{\pi}}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.1:\\ \;\;\;\;\left|\left(\left|x\right| \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right), 2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{x \cdot \left(x \cdot \left(\mathsf{fma}\left(x, x \cdot 0.047619047619047616, 0.2\right) \cdot \left(x \cdot \left|x\right|\right)\right)\right)}{\sqrt{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.1:\\ \;\;\;\;\left|\left(\left|x\right| \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right), 2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot 0.047619047619047616\right)\right)\right) \cdot \left|\frac{x}{\sqrt{\pi}}\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.1)
   (fabs
    (*
     (* (fabs x) (sqrt (/ 1.0 PI)))
     (fma x (* x (fma (* x x) 0.2 0.6666666666666666)) 2.0)))
   (*
    (* x (* x x))
    (* (* x (* x (* x 0.047619047619047616))) (fabs (/ x (sqrt PI)))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.1) {
		tmp = fabs(((fabs(x) * sqrt((1.0 / ((double) M_PI)))) * fma(x, (x * fma((x * x), 0.2, 0.6666666666666666)), 2.0)));
	} else {
		tmp = (x * (x * x)) * ((x * (x * (x * 0.047619047619047616))) * fabs((x / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.1)
		tmp = abs(Float64(Float64(abs(x) * sqrt(Float64(1.0 / pi))) * fma(x, Float64(x * fma(Float64(x * x), 0.2, 0.6666666666666666)), 2.0)));
	else
		tmp = Float64(Float64(x * Float64(x * x)) * Float64(Float64(x * Float64(x * Float64(x * 0.047619047619047616))) * abs(Float64(x / sqrt(pi)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.1], N[Abs[N[(N[(N[Abs[x], $MachinePrecision] * N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.2 + 0.6666666666666666), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(x * N[(x * N[(x * 0.047619047619047616), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[N[(x / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.10000000000000001

   1. Initial program 99.9%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(2 \cdot \left|x\right| + {x}^{2} \cdot \left(\frac{2}{3} \cdot \left|x\right| + {x}^{2} \cdot \left(\frac{1}{21} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{1}{5} \cdot \left|x\right|\right)\right)\right)}\right| \]

   7. Simplified 99.9%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 99.6%

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

   if 0.10000000000000001 < (fabs.f64 x)

   1. Initial program 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 97.6%

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

   9. Applied egg-rr 97.6%

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

       1. associate-*r* N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{\color{blue}{\left(x \cdot \left(x \cdot \frac{1}{21}\right)\right)} \cdot \left(x \cdot \left|x\right|\right)}{\sqrt{\mathsf{PI}\left(\right)}} \]

       2. lift-*.f64 N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{\left(x \cdot \color{blue}{\left(x \cdot \frac{1}{21}\right)}\right) \cdot \left(x \cdot \left|x\right|\right)}{\sqrt{\mathsf{PI}\left(\right)}} \]

       3. lift-fabs.f64 N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{\left(x \cdot \left(x \cdot \frac{1}{21}\right)\right) \cdot \left(x \cdot \color{blue}{\left|x\right|}\right)}{\sqrt{\mathsf{PI}\left(\right)}} \]

       4. associate-*r* N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{\color{blue}{\left(\left(x \cdot \left(x \cdot \frac{1}{21}\right)\right) \cdot x\right) \cdot \left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}} \]

       5. lift-PI.f64 N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{\left(\left(x \cdot \left(x \cdot \frac{1}{21}\right)\right) \cdot x\right) \cdot \left|x\right|}{\sqrt{\color{blue}{\mathsf{PI}\left(\right)}}} \]

       6. lift-sqrt.f64 N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{\left(\left(x \cdot \left(x \cdot \frac{1}{21}\right)\right) \cdot x\right) \cdot \left|x\right|}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

       7. associate-/l* N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\left(\left(x \cdot \left(x \cdot \frac{1}{21}\right)\right) \cdot x\right) \cdot \frac{\left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\left(\left(x \cdot \left(x \cdot \frac{1}{21}\right)\right) \cdot x\right) \cdot \frac{\left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\right)} \]

       9. *-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{21}\right)\right)\right)} \cdot \frac{\left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{21}\right)\right)\right)} \cdot \frac{\left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \color{blue}{\left(x \cdot \left(x \cdot \frac{1}{21}\right)\right)}\right) \cdot \frac{\left|x\right|}{\sqrt{\mathsf{PI}\left(\right)}}\right) \]

       12. lift-fabs.f64 N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{21}\right)\right)\right) \cdot \frac{\color{blue}{\left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}}\right) \]

       13. rem-square-sqrt N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{21}\right)\right)\right) \cdot \frac{\left|x\right|}{\color{blue}{\sqrt{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{\sqrt{\mathsf{PI}\left(\right)}}}}\right) \]

       14. sqrt-prod N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{21}\right)\right)\right) \cdot \frac{\left|x\right|}{\color{blue}{\sqrt{\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}}}}\right) \]

       15. rem-sqrt-square N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{21}\right)\right)\right) \cdot \frac{\left|x\right|}{\color{blue}{\left|\sqrt{\mathsf{PI}\left(\right)}\right|}}\right) \]

       16. div-fabs N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{21}\right)\right)\right) \cdot \color{blue}{\left|\frac{x}{\sqrt{\mathsf{PI}\left(\right)}}\right|}\right) \]

       17. lower-fabs.f64 N/A

           \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot \frac{1}{21}\right)\right)\right) \cdot \color{blue}{\left|\frac{x}{\sqrt{\mathsf{PI}\left(\right)}}\right|}\right) \]

       18. lower-/.f64 97.7

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

   11. Applied egg-rr 97.7%

       \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\left(x \cdot \left(x \cdot \left(x \cdot 0.047619047619047616\right)\right)\right) \cdot \left|\frac{x}{\sqrt{\pi}}\right|\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.1:\\ \;\;\;\;\left|\left(\left|x\right| \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right), 2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot \left(x \cdot \left(x \cdot 0.047619047619047616\right)\right)\right) \cdot \left|\frac{x}{\sqrt{\pi}}\right|\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(2 \cdot \left|x\right| + {x}^{2} \cdot \left(\frac{2}{3} \cdot \left|x\right| + {x}^{2} \cdot \left(\frac{1}{21} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{1}{5} \cdot \left|x\right|\right)\right)\right)}\right| \]

7. Simplified 99.8%

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

9. Applied egg-rr 99.9%

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

    1. lift-*.f64 N/A

       \[\leadsto \left|x\right| \cdot \left|\frac{x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \frac{1}{21}\right) + \frac{1}{5}\right) + \frac{2}{3}\right)\right) + 2}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

    2. lift-*.f64 N/A

       \[\leadsto \left|x\right| \cdot \left|\frac{x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{21}\right)} + \frac{1}{5}\right) + \frac{2}{3}\right)\right) + 2}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

    3. lift-fma.f64 N/A

       \[\leadsto \left|x\right| \cdot \left|\frac{x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{21}, \frac{1}{5}\right)} + \frac{2}{3}\right)\right) + 2}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

    4. lift-fma.f64 N/A

       \[\leadsto \left|x\right| \cdot \left|\frac{x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{21}, \frac{1}{5}\right), \frac{2}{3}\right)}\right) + 2}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

    5. lift-*.f64 N/A

       \[\leadsto \left|x\right| \cdot \left|\frac{x \cdot \color{blue}{\left(x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{21}, \frac{1}{5}\right), \frac{2}{3}\right)\right)} + 2}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

    6. lift-fma.f64 N/A

       \[\leadsto \left|x\right| \cdot \left|\frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{21}, \frac{1}{5}\right), \frac{2}{3}\right), 2\right)}}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

    7. lift-PI.f64 N/A

       \[\leadsto \left|x\right| \cdot \left|\frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{21}, \frac{1}{5}\right), \frac{2}{3}\right), 2\right)}{\sqrt{\color{blue}{\mathsf{PI}\left(\right)}}}\right| \]

    8. lift-sqrt.f64 N/A

       \[\leadsto \left|x\right| \cdot \left|\frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{21}, \frac{1}{5}\right), \frac{2}{3}\right), 2\right)}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

    9. lift-/.f64 N/A

       \[\leadsto \left|x\right| \cdot \left|\color{blue}{\frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{21}, \frac{1}{5}\right), \frac{2}{3}\right), 2\right)}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

11. Applied egg-rr 99.9%

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

Alternative 7: 93.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.1:\\ \;\;\;\;\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right)}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{x \cdot \left(0.2 \cdot \left(x \cdot \left(x \cdot \left|x\right|\right)\right)\right)}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.1)
   (* (fabs x) (fabs (/ (fma x (* x 0.6666666666666666) 2.0) (sqrt PI))))
   (fabs (* x (/ (* x (* 0.2 (* x (* x (fabs x))))) (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.1) {
		tmp = fabs(x) * fabs((fma(x, (x * 0.6666666666666666), 2.0) / sqrt(((double) M_PI))));
	} else {
		tmp = fabs((x * ((x * (0.2 * (x * (x * fabs(x))))) / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.1)
		tmp = Float64(abs(x) * abs(Float64(fma(x, Float64(x * 0.6666666666666666), 2.0) / sqrt(pi))));
	else
		tmp = abs(Float64(x * Float64(Float64(x * Float64(0.2 * Float64(x * Float64(x * abs(x))))) / sqrt(pi))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.1], N[(N[Abs[x], $MachinePrecision] * N[Abs[N[(N[(x * N[(x * 0.6666666666666666), $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(x * N[(N[(x * N[(0.2 * N[(x * N[(x * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.10000000000000001

   1. Initial program 99.9%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(2 \cdot \left|x\right| + {x}^{2} \cdot \left(\frac{2}{3} \cdot \left|x\right| + {x}^{2} \cdot \left(\frac{1}{21} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{1}{5} \cdot \left|x\right|\right)\right)\right)}\right| \]

   7. Simplified 99.9%

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in x around 0

       \[\leadsto \left|x\right| \cdot \left|\frac{\mathsf{fma}\left(x, \color{blue}{\frac{2}{3} \cdot x}, 2\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left|x\right| \cdot \left|\frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{2}{3}}, 2\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

       2. lower-*.f64 99.4

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

   12. Simplified 99.4%

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

   if 0.10000000000000001 < (fabs.f64 x)

   1. Initial program 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left({x}^{6} \cdot \left(\frac{1}{21} \cdot \left|x\right| + \frac{1}{5} \cdot \frac{\left|x\right|}{{x}^{2}}\right)\right)}\right| \]

   7. Simplified 98.3%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left|x\right|\right)\right) \cdot \mathsf{fma}\left(x, x \cdot 0.047619047619047616, 0.2\right)\right)\right)}\right| \]
   8. Step-by-step derivation

      1. lift-PI.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\color{blue}{\mathsf{PI}\left(\right)}}} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left|x\right|\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{21}\right) + \frac{1}{5}\right)\right)\right)\right| \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \left|\frac{1}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left|x\right|\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{21}\right) + \frac{1}{5}\right)\right)\right)\right| \]

      3. lift-/.f64 N/A

         \[\leadsto \left|\color{blue}{\frac{1}{\sqrt{\mathsf{PI}\left(\right)}}} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left|x\right|\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{21}\right) + \frac{1}{5}\right)\right)\right)\right| \]

      4. lift-*.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\left(x \cdot \left(x \cdot \left|x\right|\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{21}\right) + \frac{1}{5}\right)\right)\right)\right| \]

      5. lift-fabs.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \color{blue}{\left|x\right|}\right)\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{21}\right) + \frac{1}{5}\right)\right)\right)\right| \]

      6. lift-*.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \color{blue}{\left(x \cdot \left|x\right|\right)}\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{21}\right) + \frac{1}{5}\right)\right)\right)\right| \]

      7. lift-*.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\left(x \cdot \left(x \cdot \left|x\right|\right)\right)} \cdot \left(x \cdot \left(x \cdot \frac{1}{21}\right) + \frac{1}{5}\right)\right)\right)\right| \]

      8. lift-*.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left|x\right|\right)\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{21}\right)} + \frac{1}{5}\right)\right)\right)\right| \]

      9. lift-fma.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left|x\right|\right)\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{21}, \frac{1}{5}\right)}\right)\right)\right| \]

      10. lift-*.f64 N/A

          \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(\left(x \cdot \left(x \cdot \left|x\right|\right)\right) \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{21}, \frac{1}{5}\right)\right)}\right)\right| \]

      11. lift-*.f64 N/A

          \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left|x\right|\right)\right) \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{21}, \frac{1}{5}\right)\right)\right)}\right| \]

      12. *-commutative N/A

          \[\leadsto \left|\color{blue}{\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot \left(x \cdot \left|x\right|\right)\right) \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{21}, \frac{1}{5}\right)\right)\right) \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}}}\right| \]

   9. Applied egg-rr 98.3%

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

   10. Taylor expanded in x around 0

       \[\leadsto \left|x \cdot \frac{\color{blue}{\frac{1}{5} \cdot \left({x}^{3} \cdot \left|x\right|\right)}}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

   12. Simplified 79.6%

       \[\leadsto \left|x \cdot \frac{\color{blue}{x \cdot \left(0.2 \cdot \left(x \cdot \left(x \cdot \left|x\right|\right)\right)\right)}}{\sqrt{\pi}}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 93.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \left|\left(\left|x\right| \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.2, 0.6666666666666666\right), 2\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (* (fabs x) (sqrt (/ 1.0 PI)))
   (fma x (* x (fma (* x x) 0.2 0.6666666666666666)) 2.0))))

C

double code(double x) {
	return fabs(((fabs(x) * sqrt((1.0 / ((double) M_PI)))) * fma(x, (x * fma((x * x), 0.2, 0.6666666666666666)), 2.0)));
}

Julia

function code(x)
	return abs(Float64(Float64(abs(x) * sqrt(Float64(1.0 / pi))) * fma(x, Float64(x * fma(Float64(x * x), 0.2, 0.6666666666666666)), 2.0)))
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(2 \cdot \left|x\right| + {x}^{2} \cdot \left(\frac{2}{3} \cdot \left|x\right| + {x}^{2} \cdot \left(\frac{1}{21} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{1}{5} \cdot \left|x\right|\right)\right)\right)}\right| \]

7. Simplified 99.8%

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

8. Taylor expanded in x around 0

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

10. Simplified 93.7%

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

11. Final simplification 93.7%

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

Alternative 9: 93.7% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(2 \cdot \left|x\right| + {x}^{2} \cdot \left(\frac{2}{3} \cdot \left|x\right| + {x}^{2} \cdot \left(\frac{1}{21} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{1}{5} \cdot \left|x\right|\right)\right)\right)}\right| \]

7. Simplified 99.8%

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

9. Applied egg-rr 99.9%

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

10. Taylor expanded in x around 0

    \[\leadsto \left|x\right| \cdot \left|\frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{2}{3} + \frac{1}{5} \cdot {x}^{2}\right)}, 2\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]
11. Step-by-step derivation

    1. lower-*.f64 N/A

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

    2. +-commutative N/A

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

    3. *-commutative N/A

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

    4. lower-fma.f64 N/A

       \[\leadsto \left|x\right| \cdot \left|\frac{\mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{5}, \frac{2}{3}\right)}, 2\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

    5. unpow2 N/A

       \[\leadsto \left|x\right| \cdot \left|\frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{5}, \frac{2}{3}\right), 2\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

    6. lower-*.f64 93.7

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

12. Simplified 93.7%

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

Alternative 10: 89.8% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(2 \cdot \left|x\right| + {x}^{2} \cdot \left(\frac{2}{3} \cdot \left|x\right| + {x}^{2} \cdot \left(\frac{1}{21} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{1}{5} \cdot \left|x\right|\right)\right)\right)}\right| \]

7. Simplified 99.8%

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

9. Applied egg-rr 99.9%

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

10. Taylor expanded in x around 0

    \[\leadsto \left|x\right| \cdot \left|\frac{\mathsf{fma}\left(x, \color{blue}{\frac{2}{3} \cdot x}, 2\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \left|x\right| \cdot \left|\frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{2}{3}}, 2\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

    2. lower-*.f64 88.6

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

12. Simplified 88.6%

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

Alternative 11: 68.9% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around 0

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

7. Simplified 71.1%

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

   1. metadata-eval N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. rem-square-sqrt N/A

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

   5. sqrt-prod N/A

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

   6. rem-sqrt-square N/A

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

   7. fabs-div N/A

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

   8. lift-/.f64 N/A

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

   9. fabs-fabs N/A

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

   10. lift-fabs.f64 N/A

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

   11. metadata-eval N/A

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

   12. fabs-mul N/A

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

   13. lift-*.f64 N/A

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

   14. fabs-mul N/A

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

   15. lift-*.f64 N/A

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

   16. associate-*r* N/A

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

   17. *-commutative N/A

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

   18. fabs-mul N/A

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

9. Applied egg-rr 71.1%

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

Reproduce

herbie shell --seed 2024207 
(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)))))))