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

Percentage Accurate: 100.0% → 100.0%

Time: 14.1s

Alternatives: 15

Speedup: 1.9×

• 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, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 rewrites 100.0%

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

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{15}{8} \cdot \left(\frac{e^{{\left(\left|x\right|\right)}^{2}}}{{x}^{6} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \frac{e^{{\left(\left|x\right|\right)}^{2}}}{\left|x\right|} + \frac{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{1}{\left|x\right|} + \frac{3}{4} \cdot \frac{1}{\left|{x}^{3}\right|}\right)}{{x}^{2}} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)} \]

7. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \frac{e^{x \cdot x}}{\left|x\right|}\right) \cdot \left(\frac{\frac{0.5}{\left|x\right|} + \frac{0.75}{\left|x \cdot \left(x \cdot x\right)\right|}}{\left|x\right|} + \left(\frac{1.875}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} + 1\right)\right)} \]
8. Step-by-step derivation

9. Applied rewrites 100.0%

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

10. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 rewrites 100.0%

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

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{15}{8} \cdot \left(\frac{e^{{\left(\left|x\right|\right)}^{2}}}{{x}^{6} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \frac{e^{{\left(\left|x\right|\right)}^{2}}}{\left|x\right|} + \frac{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{1}{\left|x\right|} + \frac{3}{4} \cdot \frac{1}{\left|{x}^{3}\right|}\right)}{{x}^{2}} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)} \]

7. Applied rewrites 100.0%

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

8. Final simplification 100.0%

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

Alternative 3: 99.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 rewrites 100.0%

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

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{15}{8} \cdot \left(\frac{e^{{\left(\left|x\right|\right)}^{2}}}{{x}^{6} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \frac{e^{{\left(\left|x\right|\right)}^{2}}}{\left|x\right|} + \frac{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{1}{2} \cdot \frac{1}{\left|x\right|} + \frac{3}{4} \cdot \frac{1}{\left|{x}^{3}\right|}\right)}{{x}^{2}} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)} \]

7. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \frac{e^{x \cdot x}}{\left|x\right|}\right) \cdot \left(\frac{\frac{0.5}{\left|x\right|} + \frac{0.75}{\left|x \cdot \left(x \cdot x\right)\right|}}{\left|x\right|} + \left(\frac{1.875}{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} + 1\right)\right)} \]
8. Step-by-step derivation

9. Applied rewrites 100.0%

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

10. Final simplification 100.0%

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

Alternative 4: 99.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x \cdot x\right|}\right) \cdot \mathsf{fma}\left(\frac{1}{\left|x \cdot \left(x \cdot x\right)\right|}, 0.5 + \frac{0.75}{x \cdot x}, \frac{1}{\left|x\right|}\right) \end{array} \]

FPCore

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

C

double code(double x) {
	return ((1.0 / sqrt(((double) M_PI))) * exp(fabs((x * x)))) * fma((1.0 / fabs((x * (x * x)))), (0.5 + (0.75 / (x * x))), (1.0 / fabs(x)));
}

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * exp(abs(Float64(x * x)))) * fma(Float64(1.0 / abs(Float64(x * Float64(x * x)))), Float64(0.5 + Float64(0.75 / Float64(x * x))), Float64(1.0 / abs(x))))
end

Wolfram

code[x_] := N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Exp[N[Abs[N[(x * x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[Abs[N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(0.75 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 rewrites 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

   \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{\frac{1}{2}}{{x}^{2} \cdot \left|x\right|} + \left(\frac{\frac{3}{4}}{{x}^{4} \cdot \left|x\right|} + \frac{1}{\left|x\right|}\right)\right)} \]

7. Applied rewrites 98.6%

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

8. Final simplification 98.6%

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

Alternative 5: 99.6% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[(N[Abs[x], $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(0.75 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 rewrites 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

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\frac{e^{{\left(\left|x\right|\right)}^{2}}}{{x}^{2} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + \left(\frac{3}{4} \cdot \left(\frac{e^{{\left(\left|x\right|\right)}^{2}}}{{x}^{4} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \frac{e^{{\left(\left|x\right|\right)}^{2}}}{\left|x\right|}\right)} \]

7. Applied rewrites 98.6%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \frac{e^{x \cdot x}}{\left|x\right|}\right) \cdot \left(\frac{0.75}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \left(1 + \frac{0.5}{x \cdot x}\right)\right)} \]
8. Step-by-step derivation

9. Applied rewrites 98.6%

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

10. Final simplification 98.6%

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

Alternative 6: 99.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{1}{\left|x\right|} \cdot \left(1 + \frac{0.5}{x \cdot x}\right)\right) \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 rewrites 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

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

   1. associate-/r* N/A

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

   2. associate-*r/ N/A

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

   3. *-rgt-identity N/A

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

   4. associate-*r/ N/A

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

   5. distribute-rgt1-in N/A

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

   6. +-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-fabs.f64 98.5

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

7. Applied rewrites 98.5%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\left(1 + \frac{0.5}{x \cdot x}\right) \cdot \frac{1}{\left|x\right|}\right)} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-lft-identity 98.5

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

   6. lift-*.f64 N/A

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

   7. lift-fabs.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. sqr-abs N/A

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

   10. lift-*.f64 98.5

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

9. Applied rewrites 98.5%

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

10. Final simplification 98.5%

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

Alternative 7: 99.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x \cdot x} \cdot \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

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 rewrites 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

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

   1. associate-/r* N/A

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

   2. associate-*r/ N/A

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

   3. *-rgt-identity N/A

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

   4. associate-*r/ N/A

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

   5. distribute-rgt1-in N/A

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

   6. +-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-fabs.f64 98.5

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

7. Applied rewrites 98.5%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\left(1 + \frac{0.5}{x \cdot x}\right) \cdot \frac{1}{\left|x\right|}\right)} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

9. Applied rewrites 98.5%

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

10. Final simplification 98.5%

    \[\leadsto \frac{e^{x \cdot x} \cdot \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \]
11. Add Preprocessing

Alternative 8: 99.6% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 rewrites 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. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. sqr-abs N/A

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

   8. unpow2 N/A

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

   9. lower-exp.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-fabs.f64 98.5

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

7. Applied rewrites 98.5%

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

9. Applied rewrites 98.5%

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

10. Final simplification 98.5%

    \[\leadsto \frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\pi}} \]
11. Add Preprocessing

Alternative 9: 84.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{0.75}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot 0.16666666666666666, 0.5\right), x\right), 1\right)}{\left|x\right| \cdot \sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (+ 1.0 (+ (/ 0.5 (* x x)) (/ 0.75 (* x (* x (* x x))))))
  (/
   (fma x (fma x (* (* x x) (fma x (* x 0.16666666666666666) 0.5)) x) 1.0)
   (* (fabs x) (sqrt PI)))))

C

double code(double x) {
	return (1.0 + ((0.5 / (x * x)) + (0.75 / (x * (x * (x * x)))))) * (fma(x, fma(x, ((x * x) * fma(x, (x * 0.16666666666666666), 0.5)), x), 1.0) / (fabs(x) * sqrt(((double) M_PI))));
}

Julia

function code(x)
	return Float64(Float64(1.0 + Float64(Float64(0.5 / Float64(x * x)) + Float64(0.75 / Float64(x * Float64(x * Float64(x * x)))))) * Float64(fma(x, fma(x, Float64(Float64(x * x) * fma(x, Float64(x * 0.16666666666666666), 0.5)), x), 1.0) / Float64(abs(x) * sqrt(pi))))
end

Wolfram

code[x_] := N[(N[(1.0 + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(0.75 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Abs[x], $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 rewrites 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

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\frac{e^{{\left(\left|x\right|\right)}^{2}}}{{x}^{2} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + \left(\frac{3}{4} \cdot \left(\frac{e^{{\left(\left|x\right|\right)}^{2}}}{{x}^{4} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \frac{e^{{\left(\left|x\right|\right)}^{2}}}{\left|x\right|}\right)} \]

7. Applied rewrites 98.6%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \frac{e^{x \cdot x}}{\left|x\right|}\right) \cdot \left(\frac{0.75}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \left(1 + \frac{0.5}{x \cdot x}\right)\right)} \]
8. Step-by-step derivation

9. Applied rewrites 98.6%

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

10. Taylor expanded in x around 0

    \[\leadsto \frac{1 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)}{\sqrt{\mathsf{PI}\left(\right)} \cdot \left|x\right|} \cdot \left(1 + \left(\frac{\frac{1}{2}}{x \cdot x} + \frac{\frac{3}{4}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right) \]
11. Step-by-step derivation

12. Applied rewrites 83.8%

    \[\leadsto \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot 0.16666666666666666, 0.5\right), x\right), 1\right)}{\sqrt{\pi} \cdot \left|x\right|} \cdot \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{0.75}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right) \]

13. Final simplification 83.8%

    \[\leadsto \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{0.75}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot 0.16666666666666666, 0.5\right), x\right), 1\right)}{\left|x\right| \cdot \sqrt{\pi}} \]
14. Add Preprocessing

Alternative 10: 84.4% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{\pi}} \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot 0.16666666666666666, 0.5\right), x\right), 1\right)}{\left|x\right|} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (sqrt (/ 1.0 PI))
  (/
   (fma x (fma x (* (* x x) (fma x (* x 0.16666666666666666) 0.5)) x) 1.0)
   (fabs x))))

C

double code(double x) {
	return sqrt((1.0 / ((double) M_PI))) * (fma(x, fma(x, ((x * x) * fma(x, (x * 0.16666666666666666), 0.5)), x), 1.0) / fabs(x));
}

Julia

function code(x)
	return Float64(sqrt(Float64(1.0 / pi)) * Float64(fma(x, fma(x, Float64(Float64(x * x) * fma(x, Float64(x * 0.16666666666666666), 0.5)), x), 1.0) / abs(x)))
end

Wolfram

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

Derivation

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 rewrites 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. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. sqr-abs N/A

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

   8. unpow2 N/A

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

   9. lower-exp.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-fabs.f64 98.5

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

7. Applied rewrites 98.5%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 83.8%

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

Alternative 11: 76.6% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{\pi}} \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5 \cdot \left(x \cdot x\right), x\right), 1\right)}{\left|x\right|} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (sqrt (/ 1.0 PI)) (/ (fma x (fma x (* 0.5 (* x x)) x) 1.0) (fabs x))))

C

double code(double x) {
	return sqrt((1.0 / ((double) M_PI))) * (fma(x, fma(x, (0.5 * (x * x)), x), 1.0) / fabs(x));
}

Julia

function code(x)
	return Float64(sqrt(Float64(1.0 / pi)) * Float64(fma(x, fma(x, Float64(0.5 * Float64(x * x)), x), 1.0) / abs(x)))
end

Wolfram

code[x_] := N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(x * N[(x * N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] + 1.0), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 rewrites 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. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. sqr-abs N/A

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

   8. unpow2 N/A

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

   9. lower-exp.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-fabs.f64 98.5

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

7. Applied rewrites 98.5%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 77.0%

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

11. Final simplification 77.0%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5 \cdot \left(x \cdot x\right), x\right), 1\right)}{\left|x\right|} \]
12. Add Preprocessing

Alternative 12: 69.2% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, 0.5, 1\right), \frac{1}{\left|x\right|}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (sqrt (/ 1.0 PI)) (fma (fabs x) (fma (* x x) 0.5 1.0) (/ 1.0 (fabs x)))))

C

double code(double x) {
	return sqrt((1.0 / ((double) M_PI))) * fma(fabs(x), fma((x * x), 0.5, 1.0), (1.0 / fabs(x)));
}

Julia

function code(x)
	return Float64(sqrt(Float64(1.0 / pi)) * fma(abs(x), fma(Float64(x * x), 0.5, 1.0), Float64(1.0 / abs(x))))
end

Wolfram

code[x_] := N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] + N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 rewrites 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. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. sqr-abs N/A

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

   8. unpow2 N/A

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

   9. lower-exp.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-fabs.f64 98.5

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

7. Applied rewrites 98.5%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 70.9%

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

Alternative 13: 52.1% accurate, 10.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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 rewrites 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. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. sqr-abs N/A

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

   8. unpow2 N/A

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

   9. lower-exp.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-fabs.f64 98.5

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

7. Applied rewrites 98.5%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 54.4%

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

Alternative 14: 5.4% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| + \frac{1}{\left|x\right|}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (sqrt (/ 1.0 PI)) (+ (fabs x) (/ 1.0 (fabs x)))))

C

double code(double x) {
	return sqrt((1.0 / ((double) M_PI))) * (fabs(x) + (1.0 / fabs(x)));
}

Java

public static double code(double x) {
	return Math.sqrt((1.0 / Math.PI)) * (Math.abs(x) + (1.0 / Math.abs(x)));
}

Python

def code(x):
	return math.sqrt((1.0 / math.pi)) * (math.fabs(x) + (1.0 / math.fabs(x)))

Julia

function code(x)
	return Float64(sqrt(Float64(1.0 / pi)) * Float64(abs(x) + Float64(1.0 / abs(x))))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((1.0 / pi)) * (abs(x) + (1.0 / abs(x)));
end

Wolfram

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

Derivation

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 rewrites 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. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. sqr-abs N/A

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

   8. unpow2 N/A

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

   9. lower-exp.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-fabs.f64 98.5

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

7. Applied rewrites 98.5%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 5.6%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| + \color{blue}{\frac{1}{\left|x\right|}}\right) \]
11. Add Preprocessing

Alternative 15: 2.3% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left|x\right| \cdot \sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (* (fabs x) (sqrt PI))))

C

double code(double x) {
	return 1.0 / (fabs(x) * sqrt(((double) M_PI)));
}

Java

public static double code(double x) {
	return 1.0 / (Math.abs(x) * Math.sqrt(Math.PI));
}

Python

def code(x):
	return 1.0 / (math.fabs(x) * math.sqrt(math.pi))

Julia

function code(x)
	return Float64(1.0 / Float64(abs(x) * sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (abs(x) * sqrt(pi));
end

Wolfram

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

Derivation

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 rewrites 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. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. sqr-abs N/A

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

   8. unpow2 N/A

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

   9. lower-exp.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-fabs.f64 98.5

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

7. Applied rewrites 98.5%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 2.4%

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

12. Applied rewrites 2.4%

    \[\leadsto \frac{1}{\color{blue}{\left|x\right| \cdot \sqrt{\pi}}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(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)))))))