Jmat.Real.erfi, branch x greater than or equal to 5

Percentage Accurate: 100.0% → 100.0%

Time: 13.3s

Alternatives: 9

Speedup: 1.8×

• 99.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \geq 0.5\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \left(t\_0 \cdot t\_0\right) \cdot t\_0\\ t_2 := \left(t\_1 \cdot t\_0\right) \cdot t\_0\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(t\_0 + \frac{1}{2} \cdot t\_1\right) + \frac{3}{4} \cdot t\_2\right) + \frac{15}{8} \cdot \left(\left(t\_2 \cdot t\_0\right) \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x)))
        (t_1 (* (* t_0 t_0) t_0))
        (t_2 (* (* t_1 t_0) t_0)))
   (*
    (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x))))
    (+
     (+ (+ t_0 (* (/ 1.0 2.0) t_1)) (* (/ 3.0 4.0) t_2))
     (* (/ 15.0 8.0) (* (* t_2 t_0) t_0))))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / sqrt(((double) M_PI))) * exp((fabs(x) * fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / Math.sqrt(Math.PI)) * Math.exp((Math.abs(x) * Math.abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	t_1 = (t_0 * t_0) * t_0
	t_2 = (t_1 * t_0) * t_0
	return ((1.0 / math.sqrt(math.pi)) * math.exp((math.fabs(x) * math.fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)))

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	t_1 = Float64(Float64(t_0 * t_0) * t_0)
	t_2 = Float64(Float64(t_1 * t_0) * t_0)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * exp(Float64(abs(x) * abs(x)))) * Float64(Float64(Float64(t_0 + Float64(Float64(1.0 / 2.0) * t_1)) + Float64(Float64(3.0 / 4.0) * t_2)) + Float64(Float64(15.0 / 8.0) * Float64(Float64(t_2 * t_0) * t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / abs(x);
	t_1 = (t_0 * t_0) * t_0;
	t_2 = (t_1 * t_0) * t_0;
	tmp = ((1.0 / sqrt(pi)) * exp((abs(x) * abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$0 + N[(N[(1.0 / 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 / 4.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(15.0 / 8.0), $MachinePrecision] * N[(N[(t$95$2 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \left(t\_0 \cdot t\_0\right) \cdot t\_0\\ t_2 := \left(t\_1 \cdot t\_0\right) \cdot t\_0\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(t\_0 + \frac{1}{2} \cdot t\_1\right) + \frac{3}{4} \cdot t\_2\right) + \frac{15}{8} \cdot \left(\left(t\_2 \cdot t\_0\right) \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x)))
        (t_1 (* (* t_0 t_0) t_0))
        (t_2 (* (* t_1 t_0) t_0)))
   (*
    (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x))))
    (+
     (+ (+ t_0 (* (/ 1.0 2.0) t_1)) (* (/ 3.0 4.0) t_2))
     (* (/ 15.0 8.0) (* (* t_2 t_0) t_0))))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / sqrt(((double) M_PI))) * exp((fabs(x) * fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / Math.sqrt(Math.PI)) * Math.exp((Math.abs(x) * Math.abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	t_1 = (t_0 * t_0) * t_0
	t_2 = (t_1 * t_0) * t_0
	return ((1.0 / math.sqrt(math.pi)) * math.exp((math.fabs(x) * math.fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)))

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	t_1 = Float64(Float64(t_0 * t_0) * t_0)
	t_2 = Float64(Float64(t_1 * t_0) * t_0)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * exp(Float64(abs(x) * abs(x)))) * Float64(Float64(Float64(t_0 + Float64(Float64(1.0 / 2.0) * t_1)) + Float64(Float64(3.0 / 4.0) * t_2)) + Float64(Float64(15.0 / 8.0) * Float64(Float64(t_2 * t_0) * t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / abs(x);
	t_1 = (t_0 * t_0) * t_0;
	t_2 = (t_1 * t_0) * t_0;
	tmp = ((1.0 / sqrt(pi)) * exp((abs(x) * abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$0 + N[(N[(1.0 / 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 / 4.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(15.0 / 8.0), $MachinePrecision] * N[(N[(t$95$2 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Alternative 1: 100.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\frac{0.75}{t\_0} + \frac{1.875}{\left(x \cdot x\right) \cdot t\_0}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) (* (* x x) (* x x)))))
   (*
    (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x))))
    (+
     (/ (+ 1.0 (/ 0.5 (* x x))) (fabs x))
     (+ (/ 0.75 t_0) (/ 1.875 (* (* x x) t_0)))))))

C

double code(double x) {
	double t_0 = fabs(x) * ((x * x) * (x * x));
	return ((1.0 / sqrt(((double) M_PI))) * exp((fabs(x) * fabs(x)))) * (((1.0 + (0.5 / (x * x))) / fabs(x)) + ((0.75 / t_0) + (1.875 / ((x * x) * t_0))));
}

Java

public static double code(double x) {
	double t_0 = Math.abs(x) * ((x * x) * (x * x));
	return ((1.0 / Math.sqrt(Math.PI)) * Math.exp((Math.abs(x) * Math.abs(x)))) * (((1.0 + (0.5 / (x * x))) / Math.abs(x)) + ((0.75 / t_0) + (1.875 / ((x * x) * t_0))));
}

Python

def code(x):
	t_0 = math.fabs(x) * ((x * x) * (x * x))
	return ((1.0 / math.sqrt(math.pi)) * math.exp((math.fabs(x) * math.fabs(x)))) * (((1.0 + (0.5 / (x * x))) / math.fabs(x)) + ((0.75 / t_0) + (1.875 / ((x * x) * t_0))))

Julia

function code(x)
	t_0 = Float64(abs(x) * Float64(Float64(x * x) * Float64(x * x)))
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * exp(Float64(abs(x) * abs(x)))) * Float64(Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) / abs(x)) + Float64(Float64(0.75 / t_0) + Float64(1.875 / Float64(Float64(x * x) * t_0)))))
end

MATLAB

function tmp = code(x)
	t_0 = abs(x) * ((x * x) * (x * x));
	tmp = ((1.0 / sqrt(pi)) * exp((abs(x) * abs(x)))) * (((1.0 + (0.5 / (x * x))) / abs(x)) + ((0.75 / t_0) + (1.875 / ((x * x) * t_0))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(0.75 / t$95$0), $MachinePrecision] + N[(1.875 / N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.9%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Add Preprocessing

4. Applied egg-rr 99.9%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{0.75}{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} + \frac{1.875}{\left(x \cdot x\right) \cdot \left(\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}\right)\right)} \]

5. Final simplification 99.9%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\frac{0.75}{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} + \frac{1.875}{\left(x \cdot x\right) \cdot \left(\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]
6. Add Preprocessing

Alternative 2: 88.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \mathsf{fma}\left(x \cdot x, 0.16666666666666666, 0.5\right)\\ t_1 := x \cdot t\_0\\ \mathbf{if}\;\left|x\right| \leq 10^{+76}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{x \cdot \mathsf{fma}\left(t\_1, t\_1, -1\right)}{\mathsf{fma}\left(x, t\_0, -1\right)}, 1\right) \cdot \frac{\sqrt{\frac{1}{\pi}}}{\left|x\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5 \cdot \left(x \cdot x\right), x\right), 1\right)}{\sqrt{\pi} \cdot \left|x\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (fma (* x x) 0.16666666666666666 0.5))) (t_1 (* x t_0)))
   (if (<= (fabs x) 1e+76)
     (*
      (fma x (/ (* x (fma t_1 t_1 -1.0)) (fma x t_0 -1.0)) 1.0)
      (/ (sqrt (/ 1.0 PI)) (fabs x)))
     (/ (fma x (fma x (* 0.5 (* x x)) x) 1.0) (* (sqrt PI) (fabs x))))))

C

double code(double x) {
	double t_0 = x * fma((x * x), 0.16666666666666666, 0.5);
	double t_1 = x * t_0;
	double tmp;
	if (fabs(x) <= 1e+76) {
		tmp = fma(x, ((x * fma(t_1, t_1, -1.0)) / fma(x, t_0, -1.0)), 1.0) * (sqrt((1.0 / ((double) M_PI))) / fabs(x));
	} else {
		tmp = fma(x, fma(x, (0.5 * (x * x)), x), 1.0) / (sqrt(((double) M_PI)) * fabs(x));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x * fma(Float64(x * x), 0.16666666666666666, 0.5))
	t_1 = Float64(x * t_0)
	tmp = 0.0
	if (abs(x) <= 1e+76)
		tmp = Float64(fma(x, Float64(Float64(x * fma(t_1, t_1, -1.0)) / fma(x, t_0, -1.0)), 1.0) * Float64(sqrt(Float64(1.0 / pi)) / abs(x)));
	else
		tmp = Float64(fma(x, fma(x, Float64(0.5 * Float64(x * x)), x), 1.0) / Float64(sqrt(pi) * abs(x)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(N[(x * x), $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * t$95$0), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 1e+76], N[(N[(x * N[(N[(x * N[(t$95$1 * t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision] / N[(x * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x * N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[Pi], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1e76

   1. Initial program 99.9%

      \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 99.9%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{0.75}{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} + \frac{1.875}{\left(x \cdot x\right) \cdot \left(\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}\right)\right)} \]

   5. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. sqr-abs N/A

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

      7. unpow2 N/A

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

      8. lower-exp.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. lower-fabs.f64 98.5

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

   7. Simplified 98.5%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \]

      14. lower-*.f64 35.2

          \[\leadsto \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \frac{\sqrt{\frac{1}{\pi}}}{\left|x\right|} \]

   10. Simplified 35.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.16666666666666666, 0.5\right), 1\right), 1\right)} \cdot \frac{\sqrt{\frac{1}{\pi}}}{\left|x\right|} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-fma.f64 N/A

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

       4. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{6}, \frac{1}{2}\right), 1\right)}, 1\right) \cdot \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{6}, \frac{1}{2}\right), 1\right) \cdot x}, 1\right) \cdot \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \]

       6. lift-fma.f64 N/A

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

       7. flip-+ N/A

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

       8. associate-*l/ N/A

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

       9. lower-/.f64 N/A

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

   12. Applied egg-rr 56.6%

       \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{\mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.16666666666666666, 0.5\right)\right), x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.16666666666666666, 0.5\right)\right), -1\right) \cdot x}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.16666666666666666, 0.5\right), -1\right)}}, 1\right) \cdot \frac{\sqrt{\frac{1}{\pi}}}{\left|x\right|} \]

   if 1e76 < (fabs.f64 x)

   1. Initial program 100.0%

      \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
   2. Add Preprocessing

   4. Applied egg-rr 100.0%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{0.75}{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} + \frac{1.875}{\left(x \cdot x\right) \cdot \left(\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}\right)\right)} \]

   5. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. sqr-abs N/A

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

      7. unpow2 N/A

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

      8. lower-exp.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. lower-fabs.f64 100.0

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

   7. Simplified 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. *-lft-identity N/A

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

      10. lift-*.f64 N/A

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

      11. sqr-abs N/A

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

      12. lift-fabs.f64 N/A

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

      13. lift-fabs.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. lower-/.f64 N/A

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

   9. Applied egg-rr 99.9%

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

   10. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. lower-fma.f64 N/A

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

       5. +-commutative N/A

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

       6. distribute-lft-in N/A

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

       7. *-rgt-identity N/A

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

       8. lower-fma.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 99.5

           \[\leadsto \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot 0.5, x\right), 1\right)}{\left|x\right| \cdot \sqrt{\pi}} \]

   12. Simplified 99.5%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.5, x\right), 1\right)}}{\left|x\right| \cdot \sqrt{\pi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{+76}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{x \cdot \mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.16666666666666666, 0.5\right)\right), x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.16666666666666666, 0.5\right)\right), -1\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.16666666666666666, 0.5\right), -1\right)}, 1\right) \cdot \frac{\sqrt{\frac{1}{\pi}}}{\left|x\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5 \cdot \left(x \cdot x\right), x\right), 1\right)}{\sqrt{\pi} \cdot \left|x\right|}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024218 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x greater than or equal to 5"
  :precision binary64
  :pre (>= x 0.5)
  (* (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x)))) (+ (+ (+ (/ 1.0 (fabs x)) (* (/ 1.0 2.0) (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 3.0 4.0) (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 15.0 8.0) (* (* (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))))