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

Percentage Accurate: 100.0% → 100.0%

Time: 18.1s

Alternatives: 14

Speedup: 2.9×

• 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, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ {\left({\left(e^{x}\right)}^{2}\right)}^{\left(x \cdot 0.5\right)} \cdot \frac{\frac{0.5}{{x}^{3}} + \left(t\_0 + {t\_0}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt[3]{{\pi}^{1.5}}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x))))
   (*
    (pow (pow (exp x) 2.0) (* x 0.5))
    (/
     (+
      (/ 0.5 (pow x 3.0))
      (+ t_0 (* (pow t_0 5.0) (+ 0.75 (/ 1.875 (* x x))))))
     (cbrt (pow PI 1.5))))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	return pow(pow(exp(x), 2.0), (x * 0.5)) * (((0.5 / pow(x, 3.0)) + (t_0 + (pow(t_0, 5.0) * (0.75 + (1.875 / (x * x)))))) / cbrt(pow(((double) M_PI), 1.5)));
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	return Math.pow(Math.pow(Math.exp(x), 2.0), (x * 0.5)) * (((0.5 / Math.pow(x, 3.0)) + (t_0 + (Math.pow(t_0, 5.0) * (0.75 + (1.875 / (x * x)))))) / Math.cbrt(Math.pow(Math.PI, 1.5)));
}

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	return Float64(((exp(x) ^ 2.0) ^ Float64(x * 0.5)) * Float64(Float64(Float64(0.5 / (x ^ 3.0)) + Float64(t_0 + Float64((t_0 ^ 5.0) * Float64(0.75 + Float64(1.875 / Float64(x * x)))))) / cbrt((pi ^ 1.5))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[Power[N[Power[N[Exp[x], $MachinePrecision], 2.0], $MachinePrecision], N[(x * 0.5), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(0.5 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 + N[(N[Power[t$95$0, 5.0], $MachinePrecision] * N[(0.75 + N[(1.875 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Power[Pi, 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{e^{x \cdot x} \cdot \frac{\frac{0.5}{{\left(\left|x\right|\right)}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-cube-cbrt 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{\left(\sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}} \cdot \sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}}\right) \cdot \sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   2. associate-*l* 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{\sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}} \cdot \left(\sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}} \cdot \sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}}\right)} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   3. cbrt-div 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{\frac{\sqrt[3]{0.5}}{\sqrt[3]{{\left(\left|x\right|\right)}^{3}}}} \cdot \left(\sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}} \cdot \sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}}\right) + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   4. rem-cbrt-cube 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\sqrt[3]{0.5}}{\color{blue}{\left|x\right|}} \cdot \left(\sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}} \cdot \sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}}\right) + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   5. add-sqr-sqrt 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\sqrt[3]{0.5}}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|} \cdot \left(\sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}} \cdot \sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}}\right) + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   6. fabs-sqr 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\sqrt[3]{0.5}}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \cdot \left(\sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}} \cdot \sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}}\right) + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   7. add-sqr-sqrt 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\sqrt[3]{0.5}}{\color{blue}{x}} \cdot \left(\sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}} \cdot \sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}}\right) + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   8. pow2 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\sqrt[3]{0.5}}{x} \cdot \color{blue}{{\left(\sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}}\right)}^{2}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   9. cbrt-div 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\sqrt[3]{0.5}}{x} \cdot {\color{blue}{\left(\frac{\sqrt[3]{0.5}}{\sqrt[3]{{\left(\left|x\right|\right)}^{3}}}\right)}}^{2} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   10. rem-cbrt-cube 100.0%

       \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\sqrt[3]{0.5}}{x} \cdot {\left(\frac{\sqrt[3]{0.5}}{\color{blue}{\left|x\right|}}\right)}^{2} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   11. add-sqr-sqrt 100.0%

       \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\sqrt[3]{0.5}}{x} \cdot {\left(\frac{\sqrt[3]{0.5}}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}\right)}^{2} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   12. fabs-sqr 100.0%

       \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\sqrt[3]{0.5}}{x} \cdot {\left(\frac{\sqrt[3]{0.5}}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)}^{2} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   13. add-sqr-sqrt 100.0%

       \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\sqrt[3]{0.5}}{x} \cdot {\left(\frac{\sqrt[3]{0.5}}{\color{blue}{x}}\right)}^{2} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

6. Applied egg-rr 100.0%

   \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{\frac{\sqrt[3]{0.5}}{x} \cdot {\left(\frac{\sqrt[3]{0.5}}{x}\right)}^{2}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]
7. Step-by-step derivation

   1. unpow2 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\sqrt[3]{0.5}}{x} \cdot \color{blue}{\left(\frac{\sqrt[3]{0.5}}{x} \cdot \frac{\sqrt[3]{0.5}}{x}\right)} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   2. cube-mult 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{{\left(\frac{\sqrt[3]{0.5}}{x}\right)}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   3. cube-div 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{\frac{{\left(\sqrt[3]{0.5}\right)}^{3}}{{x}^{3}}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   4. rem-cube-cbrt 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\color{blue}{0.5}}{{x}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

8. Simplified 100.0%

   \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{\frac{0.5}{{x}^{3}}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]
9. Step-by-step derivation

   1. exp-prod 100.0%

      \[\leadsto \color{blue}{{\left(e^{x}\right)}^{x}} \cdot \frac{\frac{0.5}{{x}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

10. Applied egg-rr 100.0%

    \[\leadsto \color{blue}{{\left(e^{x}\right)}^{x}} \cdot \frac{\frac{0.5}{{x}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]
11. Step-by-step derivation

    1. add-cbrt-cube 100.0%

       \[\leadsto {\left(e^{x}\right)}^{x} \cdot \frac{\frac{0.5}{{x}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\color{blue}{\sqrt[3]{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}}}} \]

    2. pow1/3 100.0%

       \[\leadsto {\left(e^{x}\right)}^{x} \cdot \frac{\frac{0.5}{{x}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\color{blue}{{\left(\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}\right)}^{0.3333333333333333}}} \]

    3. add-sqr-sqrt 100.0%

       \[\leadsto {\left(e^{x}\right)}^{x} \cdot \frac{\frac{0.5}{{x}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{{\left(\color{blue}{\pi} \cdot \sqrt{\pi}\right)}^{0.3333333333333333}} \]

    4. pow1 100.0%

       \[\leadsto {\left(e^{x}\right)}^{x} \cdot \frac{\frac{0.5}{{x}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{{\left(\color{blue}{{\pi}^{1}} \cdot \sqrt{\pi}\right)}^{0.3333333333333333}} \]

    5. pow1/2 100.0%

       \[\leadsto {\left(e^{x}\right)}^{x} \cdot \frac{\frac{0.5}{{x}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{{\left({\pi}^{1} \cdot \color{blue}{{\pi}^{0.5}}\right)}^{0.3333333333333333}} \]

    6. pow-prod-up 100.0%

       \[\leadsto {\left(e^{x}\right)}^{x} \cdot \frac{\frac{0.5}{{x}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{{\color{blue}{\left({\pi}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}} \]

    7. metadata-eval 100.0%

       \[\leadsto {\left(e^{x}\right)}^{x} \cdot \frac{\frac{0.5}{{x}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{{\left({\pi}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \]

12. Applied egg-rr 100.0%

    \[\leadsto {\left(e^{x}\right)}^{x} \cdot \frac{\frac{0.5}{{x}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\color{blue}{{\left({\pi}^{1.5}\right)}^{0.3333333333333333}}} \]
13. Step-by-step derivation

    1. unpow1/3 100.0%

       \[\leadsto {\left(e^{x}\right)}^{x} \cdot \frac{\frac{0.5}{{x}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\color{blue}{\sqrt[3]{{\pi}^{1.5}}}} \]

14. Simplified 100.0%

    \[\leadsto {\left(e^{x}\right)}^{x} \cdot \frac{\frac{0.5}{{x}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\color{blue}{\sqrt[3]{{\pi}^{1.5}}}} \]
15. Step-by-step derivation

    1. sqr-pow 100.0%

       \[\leadsto \color{blue}{\left({\left(e^{x}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{x}{2}\right)}\right)} \cdot \frac{\frac{0.5}{{x}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt[3]{{\pi}^{1.5}}} \]

    2. pow-prod-down 100.0%

       \[\leadsto \color{blue}{{\left(e^{x} \cdot e^{x}\right)}^{\left(\frac{x}{2}\right)}} \cdot \frac{\frac{0.5}{{x}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt[3]{{\pi}^{1.5}}} \]

    3. pow2 100.0%

       \[\leadsto {\color{blue}{\left({\left(e^{x}\right)}^{2}\right)}}^{\left(\frac{x}{2}\right)} \cdot \frac{\frac{0.5}{{x}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt[3]{{\pi}^{1.5}}} \]

    4. div-inv 100.0%

       \[\leadsto {\left({\left(e^{x}\right)}^{2}\right)}^{\color{blue}{\left(x \cdot \frac{1}{2}\right)}} \cdot \frac{\frac{0.5}{{x}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt[3]{{\pi}^{1.5}}} \]

    5. metadata-eval 100.0%

       \[\leadsto {\left({\left(e^{x}\right)}^{2}\right)}^{\left(x \cdot \color{blue}{0.5}\right)} \cdot \frac{\frac{0.5}{{x}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt[3]{{\pi}^{1.5}}} \]

16. Applied egg-rr 100.0%

    \[\leadsto \color{blue}{{\left({\left(e^{x}\right)}^{2}\right)}^{\left(x \cdot 0.5\right)}} \cdot \frac{\frac{0.5}{{x}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt[3]{{\pi}^{1.5}}} \]
17. Add Preprocessing

Alternative 2: 100.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ {\left(e^{x}\right)}^{x} \cdot \frac{\frac{0.5}{{x}^{3}} + \left(t\_0 + {t\_0}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x))))
   (*
    (pow (exp x) x)
    (/
     (+
      (/ 0.5 (pow x 3.0))
      (+ t_0 (* (pow t_0 5.0) (+ 0.75 (/ 1.875 (* x x))))))
     (sqrt PI)))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	return pow(exp(x), x) * (((0.5 / pow(x, 3.0)) + (t_0 + (pow(t_0, 5.0) * (0.75 + (1.875 / (x * x)))))) / sqrt(((double) M_PI)));
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	return Math.pow(Math.exp(x), x) * (((0.5 / Math.pow(x, 3.0)) + (t_0 + (Math.pow(t_0, 5.0) * (0.75 + (1.875 / (x * x)))))) / Math.sqrt(Math.PI));
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	return math.pow(math.exp(x), x) * (((0.5 / math.pow(x, 3.0)) + (t_0 + (math.pow(t_0, 5.0) * (0.75 + (1.875 / (x * x)))))) / math.sqrt(math.pi))

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	return Float64((exp(x) ^ x) * Float64(Float64(Float64(0.5 / (x ^ 3.0)) + Float64(t_0 + Float64((t_0 ^ 5.0) * Float64(0.75 + Float64(1.875 / Float64(x * x)))))) / sqrt(pi)))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / abs(x);
	tmp = (exp(x) ^ x) * (((0.5 / (x ^ 3.0)) + (t_0 + ((t_0 ^ 5.0) * (0.75 + (1.875 / (x * x)))))) / sqrt(pi));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] * N[(N[(N[(0.5 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 + N[(N[Power[t$95$0, 5.0], $MachinePrecision] * N[(0.75 + N[(1.875 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{e^{x \cdot x} \cdot \frac{\frac{0.5}{{\left(\left|x\right|\right)}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-cube-cbrt 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{\left(\sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}} \cdot \sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}}\right) \cdot \sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   2. associate-*l* 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{\sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}} \cdot \left(\sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}} \cdot \sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}}\right)} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   3. cbrt-div 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{\frac{\sqrt[3]{0.5}}{\sqrt[3]{{\left(\left|x\right|\right)}^{3}}}} \cdot \left(\sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}} \cdot \sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}}\right) + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   4. rem-cbrt-cube 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\sqrt[3]{0.5}}{\color{blue}{\left|x\right|}} \cdot \left(\sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}} \cdot \sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}}\right) + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   5. add-sqr-sqrt 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\sqrt[3]{0.5}}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|} \cdot \left(\sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}} \cdot \sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}}\right) + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   6. fabs-sqr 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\sqrt[3]{0.5}}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \cdot \left(\sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}} \cdot \sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}}\right) + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   7. add-sqr-sqrt 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\sqrt[3]{0.5}}{\color{blue}{x}} \cdot \left(\sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}} \cdot \sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}}\right) + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   8. pow2 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\sqrt[3]{0.5}}{x} \cdot \color{blue}{{\left(\sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}}\right)}^{2}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   9. cbrt-div 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\sqrt[3]{0.5}}{x} \cdot {\color{blue}{\left(\frac{\sqrt[3]{0.5}}{\sqrt[3]{{\left(\left|x\right|\right)}^{3}}}\right)}}^{2} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   10. rem-cbrt-cube 100.0%

       \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\sqrt[3]{0.5}}{x} \cdot {\left(\frac{\sqrt[3]{0.5}}{\color{blue}{\left|x\right|}}\right)}^{2} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   11. add-sqr-sqrt 100.0%

       \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\sqrt[3]{0.5}}{x} \cdot {\left(\frac{\sqrt[3]{0.5}}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}\right)}^{2} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   12. fabs-sqr 100.0%

       \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\sqrt[3]{0.5}}{x} \cdot {\left(\frac{\sqrt[3]{0.5}}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)}^{2} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   13. add-sqr-sqrt 100.0%

       \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\sqrt[3]{0.5}}{x} \cdot {\left(\frac{\sqrt[3]{0.5}}{\color{blue}{x}}\right)}^{2} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

6. Applied egg-rr 100.0%

   \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{\frac{\sqrt[3]{0.5}}{x} \cdot {\left(\frac{\sqrt[3]{0.5}}{x}\right)}^{2}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]
7. Step-by-step derivation

   1. unpow2 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\sqrt[3]{0.5}}{x} \cdot \color{blue}{\left(\frac{\sqrt[3]{0.5}}{x} \cdot \frac{\sqrt[3]{0.5}}{x}\right)} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   2. cube-mult 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{{\left(\frac{\sqrt[3]{0.5}}{x}\right)}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   3. cube-div 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{\frac{{\left(\sqrt[3]{0.5}\right)}^{3}}{{x}^{3}}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   4. rem-cube-cbrt 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\color{blue}{0.5}}{{x}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

8. Simplified 100.0%

   \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{\frac{0.5}{{x}^{3}}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]
9. Step-by-step derivation

   1. exp-prod 100.0%

      \[\leadsto \color{blue}{{\left(e^{x}\right)}^{x}} \cdot \frac{\frac{0.5}{{x}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

10. Applied egg-rr 100.0%

    \[\leadsto \color{blue}{{\left(e^{x}\right)}^{x}} \cdot \frac{\frac{0.5}{{x}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]
11. Add Preprocessing

Alternative 3: 100.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ e^{x \cdot x} \cdot \frac{\left(t\_0 + {t\_0}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) + \left(\left(1 + 0.5 \cdot {x}^{-3}\right) + -1\right)}{\sqrt{\pi}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x))))
   (*
    (exp (* x x))
    (/
     (+
      (+ t_0 (* (pow t_0 5.0) (+ 0.75 (/ 1.875 (* x x)))))
      (+ (+ 1.0 (* 0.5 (pow x -3.0))) -1.0))
     (sqrt PI)))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	return exp((x * x)) * (((t_0 + (pow(t_0, 5.0) * (0.75 + (1.875 / (x * x))))) + ((1.0 + (0.5 * pow(x, -3.0))) + -1.0)) / sqrt(((double) M_PI)));
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	return Math.exp((x * x)) * (((t_0 + (Math.pow(t_0, 5.0) * (0.75 + (1.875 / (x * x))))) + ((1.0 + (0.5 * Math.pow(x, -3.0))) + -1.0)) / Math.sqrt(Math.PI));
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	return math.exp((x * x)) * (((t_0 + (math.pow(t_0, 5.0) * (0.75 + (1.875 / (x * x))))) + ((1.0 + (0.5 * math.pow(x, -3.0))) + -1.0)) / math.sqrt(math.pi))

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	return Float64(exp(Float64(x * x)) * Float64(Float64(Float64(t_0 + Float64((t_0 ^ 5.0) * Float64(0.75 + Float64(1.875 / Float64(x * x))))) + Float64(Float64(1.0 + Float64(0.5 * (x ^ -3.0))) + -1.0)) / sqrt(pi)))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / abs(x);
	tmp = exp((x * x)) * (((t_0 + ((t_0 ^ 5.0) * (0.75 + (1.875 / (x * x))))) + ((1.0 + (0.5 * (x ^ -3.0))) + -1.0)) / sqrt(pi));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(t$95$0 + N[(N[Power[t$95$0, 5.0], $MachinePrecision] * N[(0.75 + N[(1.875 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(0.5 * N[Power[x, -3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{e^{x \cdot x} \cdot \frac{\frac{0.5}{{\left(\left|x\right|\right)}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. expm1-log1p-u 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.5}{{\left(\left|x\right|\right)}^{3}}\right)\right)} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   2. expm1-undefine 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0.5}{{\left(\left|x\right|\right)}^{3}}\right)} - 1\right)} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   3. div-inv 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\left(e^{\mathsf{log1p}\left(\color{blue}{0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}}\right)} - 1\right) + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   4. pow-flip 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\left(e^{\mathsf{log1p}\left(0.5 \cdot \color{blue}{{\left(\left|x\right|\right)}^{\left(-3\right)}}\right)} - 1\right) + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   5. add-sqr-sqrt 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\left(e^{\mathsf{log1p}\left(0.5 \cdot {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-3\right)}\right)} - 1\right) + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   6. fabs-sqr 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\left(e^{\mathsf{log1p}\left(0.5 \cdot {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-3\right)}\right)} - 1\right) + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   7. add-sqr-sqrt 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\left(e^{\mathsf{log1p}\left(0.5 \cdot {\color{blue}{x}}^{\left(-3\right)}\right)} - 1\right) + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   8. metadata-eval 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\left(e^{\mathsf{log1p}\left(0.5 \cdot {x}^{\color{blue}{-3}}\right)} - 1\right) + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

6. Applied egg-rr 100.0%

   \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{\left(e^{\mathsf{log1p}\left(0.5 \cdot {x}^{-3}\right)} - 1\right)} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]
7. Step-by-step derivation

   1. log1p-undefine 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\left(e^{\color{blue}{\log \left(1 + 0.5 \cdot {x}^{-3}\right)}} - 1\right) + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   2. rem-exp-log 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\left(\color{blue}{\left(1 + 0.5 \cdot {x}^{-3}\right)} - 1\right) + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

8. Applied egg-rr 100.0%

   \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{\left(\left(1 + 0.5 \cdot {x}^{-3}\right) - 1\right)} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

9. Final simplification 100.0%

   \[\leadsto e^{x \cdot x} \cdot \frac{\left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) + \left(\left(1 + 0.5 \cdot {x}^{-3}\right) + -1\right)}{\sqrt{\pi}} \]
10. Add Preprocessing

Alternative 4: 100.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ e^{x \cdot x} \cdot \frac{\left(t\_0 + {t\_0}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) + \frac{\frac{0.5}{{x}^{2}}}{x}}{\sqrt{\pi}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x))))
   (*
    (exp (* x x))
    (/
     (+
      (+ t_0 (* (pow t_0 5.0) (+ 0.75 (/ 1.875 (* x x)))))
      (/ (/ 0.5 (pow x 2.0)) x))
     (sqrt PI)))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	return exp((x * x)) * (((t_0 + (pow(t_0, 5.0) * (0.75 + (1.875 / (x * x))))) + ((0.5 / pow(x, 2.0)) / x)) / sqrt(((double) M_PI)));
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	return Math.exp((x * x)) * (((t_0 + (Math.pow(t_0, 5.0) * (0.75 + (1.875 / (x * x))))) + ((0.5 / Math.pow(x, 2.0)) / x)) / Math.sqrt(Math.PI));
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	return math.exp((x * x)) * (((t_0 + (math.pow(t_0, 5.0) * (0.75 + (1.875 / (x * x))))) + ((0.5 / math.pow(x, 2.0)) / x)) / math.sqrt(math.pi))

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	return Float64(exp(Float64(x * x)) * Float64(Float64(Float64(t_0 + Float64((t_0 ^ 5.0) * Float64(0.75 + Float64(1.875 / Float64(x * x))))) + Float64(Float64(0.5 / (x ^ 2.0)) / x)) / sqrt(pi)))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / abs(x);
	tmp = exp((x * x)) * (((t_0 + ((t_0 ^ 5.0) * (0.75 + (1.875 / (x * x))))) + ((0.5 / (x ^ 2.0)) / x)) / sqrt(pi));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(t$95$0 + N[(N[Power[t$95$0, 5.0], $MachinePrecision] * N[(0.75 + N[(1.875 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{e^{x \cdot x} \cdot \frac{\frac{0.5}{{\left(\left|x\right|\right)}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\frac{0.5}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   2. fabs-sqr 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\frac{0.5}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   3. add-sqr-sqrt 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\frac{0.5}{{\color{blue}{x}}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   4. pow3 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\frac{0.5}{\color{blue}{\left(x \cdot x\right) \cdot x}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   5. add-sqr-sqrt 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\frac{0.5}{\left(x \cdot x\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   6. fabs-sqr 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\frac{0.5}{\left(x \cdot x\right) \cdot \color{blue}{\left|\sqrt{x} \cdot \sqrt{x}\right|}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   7. add-sqr-sqrt 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\frac{0.5}{\left(x \cdot x\right) \cdot \left|\color{blue}{x}\right|} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   8. associate-/r* 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{\frac{\frac{0.5}{x \cdot x}}{\left|x\right|}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   9. pow2 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\frac{0.5}{\color{blue}{{x}^{2}}}}{\left|x\right|} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\frac{0.5}{{x}^{2}}}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   11. fabs-sqr 100.0%

       \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\frac{0.5}{{x}^{2}}}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   12. add-sqr-sqrt 100.0%

       \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\frac{0.5}{{x}^{2}}}{\color{blue}{x}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

6. Applied egg-rr 100.0%

   \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{\frac{\frac{0.5}{{x}^{2}}}{x}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

7. Final simplification 100.0%

   \[\leadsto e^{x \cdot x} \cdot \frac{\left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) + \frac{\frac{0.5}{{x}^{2}}}{x}}{\sqrt{\pi}} \]
8. Add Preprocessing

Alternative 5: 100.0% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ \frac{\frac{0.5}{{x}^{3}} + \left(t\_0 + {t\_0}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \cdot e^{x \cdot x} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x))))
   (*
    (/
     (+
      (/ 0.5 (pow x 3.0))
      (+ t_0 (* (pow t_0 5.0) (+ 0.75 (/ 1.875 (* x x))))))
     (sqrt PI))
    (exp (* x x)))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	return (((0.5 / pow(x, 3.0)) + (t_0 + (pow(t_0, 5.0) * (0.75 + (1.875 / (x * x)))))) / sqrt(((double) M_PI))) * exp((x * x));
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	return (((0.5 / Math.pow(x, 3.0)) + (t_0 + (Math.pow(t_0, 5.0) * (0.75 + (1.875 / (x * x)))))) / Math.sqrt(Math.PI)) * Math.exp((x * x));
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	return (((0.5 / math.pow(x, 3.0)) + (t_0 + (math.pow(t_0, 5.0) * (0.75 + (1.875 / (x * x)))))) / math.sqrt(math.pi)) * math.exp((x * x))

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	return Float64(Float64(Float64(Float64(0.5 / (x ^ 3.0)) + Float64(t_0 + Float64((t_0 ^ 5.0) * Float64(0.75 + Float64(1.875 / Float64(x * x)))))) / sqrt(pi)) * exp(Float64(x * x)))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / abs(x);
	tmp = (((0.5 / (x ^ 3.0)) + (t_0 + ((t_0 ^ 5.0) * (0.75 + (1.875 / (x * x)))))) / sqrt(pi)) * exp((x * x));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(0.5 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 + N[(N[Power[t$95$0, 5.0], $MachinePrecision] * N[(0.75 + N[(1.875 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{e^{x \cdot x} \cdot \frac{\frac{0.5}{{\left(\left|x\right|\right)}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-cube-cbrt 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{\left(\sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}} \cdot \sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}}\right) \cdot \sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   2. associate-*l* 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{\sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}} \cdot \left(\sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}} \cdot \sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}}\right)} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   3. cbrt-div 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{\frac{\sqrt[3]{0.5}}{\sqrt[3]{{\left(\left|x\right|\right)}^{3}}}} \cdot \left(\sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}} \cdot \sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}}\right) + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   4. rem-cbrt-cube 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\sqrt[3]{0.5}}{\color{blue}{\left|x\right|}} \cdot \left(\sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}} \cdot \sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}}\right) + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   5. add-sqr-sqrt 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\sqrt[3]{0.5}}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|} \cdot \left(\sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}} \cdot \sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}}\right) + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   6. fabs-sqr 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\sqrt[3]{0.5}}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \cdot \left(\sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}} \cdot \sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}}\right) + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   7. add-sqr-sqrt 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\sqrt[3]{0.5}}{\color{blue}{x}} \cdot \left(\sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}} \cdot \sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}}\right) + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   8. pow2 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\sqrt[3]{0.5}}{x} \cdot \color{blue}{{\left(\sqrt[3]{\frac{0.5}{{\left(\left|x\right|\right)}^{3}}}\right)}^{2}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   9. cbrt-div 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\sqrt[3]{0.5}}{x} \cdot {\color{blue}{\left(\frac{\sqrt[3]{0.5}}{\sqrt[3]{{\left(\left|x\right|\right)}^{3}}}\right)}}^{2} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   10. rem-cbrt-cube 100.0%

       \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\sqrt[3]{0.5}}{x} \cdot {\left(\frac{\sqrt[3]{0.5}}{\color{blue}{\left|x\right|}}\right)}^{2} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   11. add-sqr-sqrt 100.0%

       \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\sqrt[3]{0.5}}{x} \cdot {\left(\frac{\sqrt[3]{0.5}}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}\right)}^{2} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   12. fabs-sqr 100.0%

       \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\sqrt[3]{0.5}}{x} \cdot {\left(\frac{\sqrt[3]{0.5}}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)}^{2} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   13. add-sqr-sqrt 100.0%

       \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\sqrt[3]{0.5}}{x} \cdot {\left(\frac{\sqrt[3]{0.5}}{\color{blue}{x}}\right)}^{2} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

6. Applied egg-rr 100.0%

   \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{\frac{\sqrt[3]{0.5}}{x} \cdot {\left(\frac{\sqrt[3]{0.5}}{x}\right)}^{2}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]
7. Step-by-step derivation

   1. unpow2 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\sqrt[3]{0.5}}{x} \cdot \color{blue}{\left(\frac{\sqrt[3]{0.5}}{x} \cdot \frac{\sqrt[3]{0.5}}{x}\right)} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   2. cube-mult 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{{\left(\frac{\sqrt[3]{0.5}}{x}\right)}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   3. cube-div 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{\frac{{\left(\sqrt[3]{0.5}\right)}^{3}}{{x}^{3}}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

   4. rem-cube-cbrt 100.0%

      \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\color{blue}{0.5}}{{x}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

8. Simplified 100.0%

   \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{\frac{0.5}{{x}^{3}}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

9. Final simplification 100.0%

   \[\leadsto \frac{\frac{0.5}{{x}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \cdot e^{x \cdot x} \]
10. Add Preprocessing

Alternative 6: 99.7% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{x}\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (pow (exp x) x)
  (*
   (sqrt (/ 1.0 PI))
   (+ (/ 0.75 (pow x 5.0)) (+ (/ 0.5 (pow x 3.0)) (/ 1.0 x))))))

C

double code(double x) {
	return pow(exp(x), x) * (sqrt((1.0 / ((double) M_PI))) * ((0.75 / pow(x, 5.0)) + ((0.5 / pow(x, 3.0)) + (1.0 / x))));
}

Java

public static double code(double x) {
	return Math.pow(Math.exp(x), x) * (Math.sqrt((1.0 / Math.PI)) * ((0.75 / Math.pow(x, 5.0)) + ((0.5 / Math.pow(x, 3.0)) + (1.0 / x))));
}

Python

def code(x):
	return math.pow(math.exp(x), x) * (math.sqrt((1.0 / math.pi)) * ((0.75 / math.pow(x, 5.0)) + ((0.5 / math.pow(x, 3.0)) + (1.0 / x))))

Julia

function code(x)
	return Float64((exp(x) ^ x) * Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(0.75 / (x ^ 5.0)) + Float64(Float64(0.5 / (x ^ 3.0)) + Float64(1.0 / x)))))
end

MATLAB

function tmp = code(x)
	tmp = (exp(x) ^ x) * (sqrt((1.0 / pi)) * ((0.75 / (x ^ 5.0)) + ((0.5 / (x ^ 3.0)) + (1.0 / x))));
end

Wolfram

code[x_] := N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(0.75 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{e^{x \cdot x} \cdot \frac{\frac{0.5}{{\left(\left|x\right|\right)}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}}} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 99.5%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r/ 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\frac{0.75 \cdot 1}{{\left(\left|x\right|\right)}^{5}}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   2. metadata-eval 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{\color{blue}{0.75}}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   3. rem-square-sqrt 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   4. fabs-sqr 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   5. rem-square-sqrt 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{\color{blue}{x}}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   6. +-commutative 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \color{blue}{\left(0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)}\right)\right) \]

   7. associate-*r/ 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\color{blue}{\frac{0.5 \cdot 1}{{\left(\left|x\right|\right)}^{3}}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   8. metadata-eval 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{\color{blue}{0.5}}{{\left(\left|x\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   9. rem-square-sqrt 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   10. fabs-sqr 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   11. rem-square-sqrt 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{\color{blue}{x}}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   12. rem-square-sqrt 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}\right)\right)\right) \]

   13. fabs-sqr 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)\right)\right) \]

   14. rem-square-sqrt 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{\color{blue}{x}}\right)\right)\right) \]

7. Simplified 99.5%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{x}\right)\right)\right)} \]
8. Step-by-step derivation

   1. exp-prod 100.0%

      \[\leadsto \color{blue}{{\left(e^{x}\right)}^{x}} \cdot \frac{\frac{0.5}{{x}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

9. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{{\left(e^{x}\right)}^{x}} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{x}\right)\right)\right) \]
10. Add Preprocessing

Alternative 7: 99.7% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{x}\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (exp (* x x))
  (*
   (sqrt (/ 1.0 PI))
   (+ (/ 0.75 (pow x 5.0)) (+ (/ 0.5 (pow x 3.0)) (/ 1.0 x))))))

C

double code(double x) {
	return exp((x * x)) * (sqrt((1.0 / ((double) M_PI))) * ((0.75 / pow(x, 5.0)) + ((0.5 / pow(x, 3.0)) + (1.0 / x))));
}

Java

public static double code(double x) {
	return Math.exp((x * x)) * (Math.sqrt((1.0 / Math.PI)) * ((0.75 / Math.pow(x, 5.0)) + ((0.5 / Math.pow(x, 3.0)) + (1.0 / x))));
}

Python

def code(x):
	return math.exp((x * x)) * (math.sqrt((1.0 / math.pi)) * ((0.75 / math.pow(x, 5.0)) + ((0.5 / math.pow(x, 3.0)) + (1.0 / x))))

Julia

function code(x)
	return Float64(exp(Float64(x * x)) * Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(0.75 / (x ^ 5.0)) + Float64(Float64(0.5 / (x ^ 3.0)) + Float64(1.0 / x)))))
end

MATLAB

function tmp = code(x)
	tmp = exp((x * x)) * (sqrt((1.0 / pi)) * ((0.75 / (x ^ 5.0)) + ((0.5 / (x ^ 3.0)) + (1.0 / x))));
end

Wolfram

code[x_] := N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(0.75 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{e^{x \cdot x} \cdot \frac{\frac{0.5}{{\left(\left|x\right|\right)}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}}} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 99.5%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r/ 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\frac{0.75 \cdot 1}{{\left(\left|x\right|\right)}^{5}}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   2. metadata-eval 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{\color{blue}{0.75}}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   3. rem-square-sqrt 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   4. fabs-sqr 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   5. rem-square-sqrt 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{\color{blue}{x}}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   6. +-commutative 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \color{blue}{\left(0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)}\right)\right) \]

   7. associate-*r/ 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\color{blue}{\frac{0.5 \cdot 1}{{\left(\left|x\right|\right)}^{3}}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   8. metadata-eval 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{\color{blue}{0.5}}{{\left(\left|x\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   9. rem-square-sqrt 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   10. fabs-sqr 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   11. rem-square-sqrt 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{\color{blue}{x}}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   12. rem-square-sqrt 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}\right)\right)\right) \]

   13. fabs-sqr 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)\right)\right) \]

   14. rem-square-sqrt 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{\color{blue}{x}}\right)\right)\right) \]

7. Simplified 99.5%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{x}\right)\right)\right)} \]
8. Add Preprocessing

Alternative 8: 99.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ {\left(e^{x}\right)}^{x} \cdot \frac{\sqrt{\frac{1}{\pi}} \cdot \left(1 + \frac{0.5}{{x}^{2}}\right)}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (pow (exp x) x) (/ (* (sqrt (/ 1.0 PI)) (+ 1.0 (/ 0.5 (pow x 2.0)))) x)))

C

double code(double x) {
	return pow(exp(x), x) * ((sqrt((1.0 / ((double) M_PI))) * (1.0 + (0.5 / pow(x, 2.0)))) / x);
}

Java

public static double code(double x) {
	return Math.pow(Math.exp(x), x) * ((Math.sqrt((1.0 / Math.PI)) * (1.0 + (0.5 / Math.pow(x, 2.0)))) / x);
}

Python

def code(x):
	return math.pow(math.exp(x), x) * ((math.sqrt((1.0 / math.pi)) * (1.0 + (0.5 / math.pow(x, 2.0)))) / x)

Julia

function code(x)
	return Float64((exp(x) ^ x) * Float64(Float64(sqrt(Float64(1.0 / pi)) * Float64(1.0 + Float64(0.5 / (x ^ 2.0)))) / x))
end

MATLAB

function tmp = code(x)
	tmp = (exp(x) ^ x) * ((sqrt((1.0 / pi)) * (1.0 + (0.5 / (x ^ 2.0)))) / x);
end

Wolfram

code[x_] := N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] * N[(N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{e^{x \cdot x} \cdot \frac{\frac{0.5}{{\left(\left|x\right|\right)}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}}} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 99.5%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r/ 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\frac{0.75 \cdot 1}{{\left(\left|x\right|\right)}^{5}}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   2. metadata-eval 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{\color{blue}{0.75}}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   3. rem-square-sqrt 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   4. fabs-sqr 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   5. rem-square-sqrt 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{\color{blue}{x}}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   6. +-commutative 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \color{blue}{\left(0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)}\right)\right) \]

   7. associate-*r/ 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\color{blue}{\frac{0.5 \cdot 1}{{\left(\left|x\right|\right)}^{3}}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   8. metadata-eval 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{\color{blue}{0.5}}{{\left(\left|x\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   9. rem-square-sqrt 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   10. fabs-sqr 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   11. rem-square-sqrt 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{\color{blue}{x}}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   12. rem-square-sqrt 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}\right)\right)\right) \]

   13. fabs-sqr 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)\right)\right) \]

   14. rem-square-sqrt 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{\color{blue}{x}}\right)\right)\right) \]

7. Simplified 99.5%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{x}\right)\right)\right)} \]

8. Taylor expanded in x around inf 99.5%

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

   1. +-commutative 99.5%

      \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{0.5 \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right) + \sqrt{\frac{1}{\pi}}}}{x} \]

   2. associate-*r* 99.5%

      \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{\left(0.5 \cdot \frac{1}{{x}^{2}}\right) \cdot \sqrt{\frac{1}{\pi}}} + \sqrt{\frac{1}{\pi}}}{x} \]

   3. *-lft-identity 99.5%

      \[\leadsto e^{x \cdot x} \cdot \frac{\left(0.5 \cdot \frac{1}{{x}^{2}}\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{1 \cdot \sqrt{\frac{1}{\pi}}}}{x} \]

   4. distribute-rgt-out 99.5%

      \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.5 \cdot \frac{1}{{x}^{2}} + 1\right)}}{x} \]

   5. associate-*r/ 99.5%

      \[\leadsto e^{x \cdot x} \cdot \frac{\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} + 1\right)}{x} \]

   6. metadata-eval 99.5%

      \[\leadsto e^{x \cdot x} \cdot \frac{\sqrt{\frac{1}{\pi}} \cdot \left(\frac{\color{blue}{0.5}}{{x}^{2}} + 1\right)}{x} \]

10. Simplified 99.5%

    \[\leadsto e^{x \cdot x} \cdot \color{blue}{\frac{\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{2}} + 1\right)}{x}} \]
11. Step-by-step derivation

    1. exp-prod 100.0%

       \[\leadsto \color{blue}{{\left(e^{x}\right)}^{x}} \cdot \frac{\frac{0.5}{{x}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

12. Applied egg-rr 99.5%

    \[\leadsto \color{blue}{{\left(e^{x}\right)}^{x}} \cdot \frac{\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{2}} + 1\right)}{x} \]

13. Final simplification 99.5%

    \[\leadsto {\left(e^{x}\right)}^{x} \cdot \frac{\sqrt{\frac{1}{\pi}} \cdot \left(1 + \frac{0.5}{{x}^{2}}\right)}{x} \]
14. Add Preprocessing

Alternative 9: 99.6% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ e^{x \cdot x} \cdot \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x \cdot \sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (exp (* x x)) (/ (fma 0.5 (pow x -2.0) 1.0) (* x (sqrt PI)))))

C

double code(double x) {
	return exp((x * x)) * (fma(0.5, pow(x, -2.0), 1.0) / (x * sqrt(((double) M_PI))));
}

Julia

function code(x)
	return Float64(exp(Float64(x * x)) * Float64(fma(0.5, (x ^ -2.0), 1.0) / Float64(x * sqrt(pi))))
end

Wolfram

code[x_] := N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] * N[(N[(0.5 * N[Power[x, -2.0], $MachinePrecision] + 1.0), $MachinePrecision] / N[(x * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{e^{x \cdot x} \cdot \frac{\frac{0.5}{{\left(\left|x\right|\right)}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}}} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 99.5%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r/ 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\frac{0.75 \cdot 1}{{\left(\left|x\right|\right)}^{5}}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   2. metadata-eval 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{\color{blue}{0.75}}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   3. rem-square-sqrt 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   4. fabs-sqr 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   5. rem-square-sqrt 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{\color{blue}{x}}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   6. +-commutative 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \color{blue}{\left(0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)}\right)\right) \]

   7. associate-*r/ 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\color{blue}{\frac{0.5 \cdot 1}{{\left(\left|x\right|\right)}^{3}}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   8. metadata-eval 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{\color{blue}{0.5}}{{\left(\left|x\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   9. rem-square-sqrt 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   10. fabs-sqr 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   11. rem-square-sqrt 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{\color{blue}{x}}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   12. rem-square-sqrt 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}\right)\right)\right) \]

   13. fabs-sqr 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)\right)\right) \]

   14. rem-square-sqrt 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{\color{blue}{x}}\right)\right)\right) \]

7. Simplified 99.5%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{x}\right)\right)\right)} \]

8. Taylor expanded in x around inf 99.5%

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

   1. +-commutative 99.5%

      \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{0.5 \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right) + \sqrt{\frac{1}{\pi}}}}{x} \]

   2. associate-*r* 99.5%

      \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{\left(0.5 \cdot \frac{1}{{x}^{2}}\right) \cdot \sqrt{\frac{1}{\pi}}} + \sqrt{\frac{1}{\pi}}}{x} \]

   3. *-lft-identity 99.5%

      \[\leadsto e^{x \cdot x} \cdot \frac{\left(0.5 \cdot \frac{1}{{x}^{2}}\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{1 \cdot \sqrt{\frac{1}{\pi}}}}{x} \]

   4. distribute-rgt-out 99.5%

      \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.5 \cdot \frac{1}{{x}^{2}} + 1\right)}}{x} \]

   5. associate-*r/ 99.5%

      \[\leadsto e^{x \cdot x} \cdot \frac{\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} + 1\right)}{x} \]

   6. metadata-eval 99.5%

      \[\leadsto e^{x \cdot x} \cdot \frac{\sqrt{\frac{1}{\pi}} \cdot \left(\frac{\color{blue}{0.5}}{{x}^{2}} + 1\right)}{x} \]

10. Simplified 99.5%

    \[\leadsto e^{x \cdot x} \cdot \color{blue}{\frac{\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{2}} + 1\right)}{x}} \]
11. Step-by-step derivation

    1. associate-/l* 99.5%

       \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \frac{\frac{0.5}{{x}^{2}} + 1}{x}\right)} \]

    2. sqrt-div 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}} \cdot \frac{\frac{0.5}{{x}^{2}} + 1}{x}\right) \]

    3. metadata-eval 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\frac{\color{blue}{1}}{\sqrt{\pi}} \cdot \frac{\frac{0.5}{{x}^{2}} + 1}{x}\right) \]

    4. frac-times 99.5%

       \[\leadsto e^{x \cdot x} \cdot \color{blue}{\frac{1 \cdot \left(\frac{0.5}{{x}^{2}} + 1\right)}{\sqrt{\pi} \cdot x}} \]

    5. *-un-lft-identity 99.5%

       \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{\frac{0.5}{{x}^{2}} + 1}}{\sqrt{\pi} \cdot x} \]

    6. pow2 99.5%

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

    7. div-inv 99.5%

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

    8. fma-define 99.5%

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

    9. pow2 99.5%

       \[\leadsto e^{x \cdot x} \cdot \frac{\mathsf{fma}\left(0.5, \frac{1}{\color{blue}{{x}^{2}}}, 1\right)}{\sqrt{\pi} \cdot x} \]

    10. pow-flip 99.5%

        \[\leadsto e^{x \cdot x} \cdot \frac{\mathsf{fma}\left(0.5, \color{blue}{{x}^{\left(-2\right)}}, 1\right)}{\sqrt{\pi} \cdot x} \]

    11. metadata-eval 99.5%

        \[\leadsto e^{x \cdot x} \cdot \frac{\mathsf{fma}\left(0.5, {x}^{\color{blue}{-2}}, 1\right)}{\sqrt{\pi} \cdot x} \]

12. Applied egg-rr 99.5%

    \[\leadsto e^{x \cdot x} \cdot \color{blue}{\frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{\sqrt{\pi} \cdot x}} \]

13. Final simplification 99.5%

    \[\leadsto e^{x \cdot x} \cdot \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x \cdot \sqrt{\pi}} \]
14. Add Preprocessing

Alternative 10: 99.7% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ e^{x \cdot x} \cdot \frac{\sqrt{\frac{1}{\pi}} \cdot \left(1 + \frac{0.5}{{x}^{2}}\right)}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (exp (* x x)) (/ (* (sqrt (/ 1.0 PI)) (+ 1.0 (/ 0.5 (pow x 2.0)))) x)))

C

double code(double x) {
	return exp((x * x)) * ((sqrt((1.0 / ((double) M_PI))) * (1.0 + (0.5 / pow(x, 2.0)))) / x);
}

Java

public static double code(double x) {
	return Math.exp((x * x)) * ((Math.sqrt((1.0 / Math.PI)) * (1.0 + (0.5 / Math.pow(x, 2.0)))) / x);
}

Python

def code(x):
	return math.exp((x * x)) * ((math.sqrt((1.0 / math.pi)) * (1.0 + (0.5 / math.pow(x, 2.0)))) / x)

Julia

function code(x)
	return Float64(exp(Float64(x * x)) * Float64(Float64(sqrt(Float64(1.0 / pi)) * Float64(1.0 + Float64(0.5 / (x ^ 2.0)))) / x))
end

MATLAB

function tmp = code(x)
	tmp = exp((x * x)) * ((sqrt((1.0 / pi)) * (1.0 + (0.5 / (x ^ 2.0)))) / x);
end

Wolfram

code[x_] := N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(0.5 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{e^{x \cdot x} \cdot \frac{\frac{0.5}{{\left(\left|x\right|\right)}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}}} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 99.5%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r/ 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\frac{0.75 \cdot 1}{{\left(\left|x\right|\right)}^{5}}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   2. metadata-eval 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{\color{blue}{0.75}}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   3. rem-square-sqrt 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   4. fabs-sqr 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   5. rem-square-sqrt 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{\color{blue}{x}}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   6. +-commutative 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \color{blue}{\left(0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)}\right)\right) \]

   7. associate-*r/ 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\color{blue}{\frac{0.5 \cdot 1}{{\left(\left|x\right|\right)}^{3}}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   8. metadata-eval 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{\color{blue}{0.5}}{{\left(\left|x\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   9. rem-square-sqrt 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   10. fabs-sqr 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   11. rem-square-sqrt 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{\color{blue}{x}}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   12. rem-square-sqrt 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}\right)\right)\right) \]

   13. fabs-sqr 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)\right)\right) \]

   14. rem-square-sqrt 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{\color{blue}{x}}\right)\right)\right) \]

7. Simplified 99.5%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{x}\right)\right)\right)} \]

8. Taylor expanded in x around inf 99.5%

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

   1. +-commutative 99.5%

      \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{0.5 \cdot \left(\frac{1}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right) + \sqrt{\frac{1}{\pi}}}}{x} \]

   2. associate-*r* 99.5%

      \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{\left(0.5 \cdot \frac{1}{{x}^{2}}\right) \cdot \sqrt{\frac{1}{\pi}}} + \sqrt{\frac{1}{\pi}}}{x} \]

   3. *-lft-identity 99.5%

      \[\leadsto e^{x \cdot x} \cdot \frac{\left(0.5 \cdot \frac{1}{{x}^{2}}\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{1 \cdot \sqrt{\frac{1}{\pi}}}}{x} \]

   4. distribute-rgt-out 99.5%

      \[\leadsto e^{x \cdot x} \cdot \frac{\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.5 \cdot \frac{1}{{x}^{2}} + 1\right)}}{x} \]

   5. associate-*r/ 99.5%

      \[\leadsto e^{x \cdot x} \cdot \frac{\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}} + 1\right)}{x} \]

   6. metadata-eval 99.5%

      \[\leadsto e^{x \cdot x} \cdot \frac{\sqrt{\frac{1}{\pi}} \cdot \left(\frac{\color{blue}{0.5}}{{x}^{2}} + 1\right)}{x} \]

10. Simplified 99.5%

    \[\leadsto e^{x \cdot x} \cdot \color{blue}{\frac{\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.5}{{x}^{2}} + 1\right)}{x}} \]

11. Final simplification 99.5%

    \[\leadsto e^{x \cdot x} \cdot \frac{\sqrt{\frac{1}{\pi}} \cdot \left(1 + \frac{0.5}{{x}^{2}}\right)}{x} \]
12. Add Preprocessing

Alternative 11: 99.6% accurate, 6.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	return pow(exp(x), x) * (1.0 / (x * sqrt(((double) M_PI))));
}

Java

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

Python

def code(x):
	return math.pow(math.exp(x), x) * (1.0 / (x * math.sqrt(math.pi)))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{e^{x \cdot x} \cdot \frac{\frac{0.5}{{\left(\left|x\right|\right)}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}}} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 99.5%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r/ 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\frac{0.75 \cdot 1}{{\left(\left|x\right|\right)}^{5}}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   2. metadata-eval 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{\color{blue}{0.75}}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   3. rem-square-sqrt 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   4. fabs-sqr 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   5. rem-square-sqrt 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{\color{blue}{x}}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   6. +-commutative 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \color{blue}{\left(0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)}\right)\right) \]

   7. associate-*r/ 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\color{blue}{\frac{0.5 \cdot 1}{{\left(\left|x\right|\right)}^{3}}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   8. metadata-eval 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{\color{blue}{0.5}}{{\left(\left|x\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   9. rem-square-sqrt 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   10. fabs-sqr 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   11. rem-square-sqrt 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{\color{blue}{x}}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   12. rem-square-sqrt 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}\right)\right)\right) \]

   13. fabs-sqr 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)\right)\right) \]

   14. rem-square-sqrt 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{\color{blue}{x}}\right)\right)\right) \]

7. Simplified 99.5%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{x}\right)\right)\right)} \]

8. Taylor expanded in x around inf 99.4%

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

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

10. Simplified 99.4%

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

    1. div-inv 99.4%

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

    2. sqrt-div 99.4%

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

    3. metadata-eval 99.4%

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

    4. frac-times 99.4%

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

    5. metadata-eval 99.4%

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

12. Applied egg-rr 99.4%

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

    1. exp-prod 100.0%

       \[\leadsto \color{blue}{{\left(e^{x}\right)}^{x}} \cdot \frac{\frac{0.5}{{x}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}} \]

14. Applied egg-rr 99.4%

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

15. Final simplification 99.4%

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

Alternative 12: 99.6% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ e^{x \cdot x} \cdot \frac{1}{x \cdot \sqrt{\pi}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{e^{x \cdot x} \cdot \frac{\frac{0.5}{{\left(\left|x\right|\right)}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}}} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 99.5%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r/ 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\frac{0.75 \cdot 1}{{\left(\left|x\right|\right)}^{5}}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   2. metadata-eval 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{\color{blue}{0.75}}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   3. rem-square-sqrt 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   4. fabs-sqr 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   5. rem-square-sqrt 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{\color{blue}{x}}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   6. +-commutative 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \color{blue}{\left(0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)}\right)\right) \]

   7. associate-*r/ 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\color{blue}{\frac{0.5 \cdot 1}{{\left(\left|x\right|\right)}^{3}}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   8. metadata-eval 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{\color{blue}{0.5}}{{\left(\left|x\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   9. rem-square-sqrt 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   10. fabs-sqr 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   11. rem-square-sqrt 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{\color{blue}{x}}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   12. rem-square-sqrt 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}\right)\right)\right) \]

   13. fabs-sqr 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)\right)\right) \]

   14. rem-square-sqrt 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{\color{blue}{x}}\right)\right)\right) \]

7. Simplified 99.5%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{x}\right)\right)\right)} \]

8. Taylor expanded in x around inf 99.4%

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

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

10. Simplified 99.4%

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

    1. div-inv 99.4%

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

    2. sqrt-div 99.4%

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

    3. metadata-eval 99.4%

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

    4. frac-times 99.4%

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

    5. metadata-eval 99.4%

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

12. Applied egg-rr 99.4%

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

13. Final simplification 99.4%

    \[\leadsto e^{x \cdot x} \cdot \frac{1}{x \cdot \sqrt{\pi}} \]
14. Add Preprocessing

Alternative 13: 99.7% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ e^{x \cdot x} \cdot \frac{{\pi}^{-0.5}}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (exp (* x x)) (/ (pow PI -0.5) x)))

C

double code(double x) {
	return exp((x * x)) * (pow(((double) M_PI), -0.5) / x);
}

Java

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

Python

def code(x):
	return math.exp((x * x)) * (math.pow(math.pi, -0.5) / x)

Julia

function code(x)
	return Float64(exp(Float64(x * x)) * Float64((pi ^ -0.5) / x))
end

MATLAB

function tmp = code(x)
	tmp = exp((x * x)) * ((pi ^ -0.5) / x);
end

Wolfram

code[x_] := N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] * N[(N[Power[Pi, -0.5], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{e^{x \cdot x} \cdot \frac{\frac{0.5}{{\left(\left|x\right|\right)}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}}} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 99.5%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r/ 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\frac{0.75 \cdot 1}{{\left(\left|x\right|\right)}^{5}}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   2. metadata-eval 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{\color{blue}{0.75}}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   3. rem-square-sqrt 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   4. fabs-sqr 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   5. rem-square-sqrt 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{\color{blue}{x}}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   6. +-commutative 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \color{blue}{\left(0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)}\right)\right) \]

   7. associate-*r/ 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\color{blue}{\frac{0.5 \cdot 1}{{\left(\left|x\right|\right)}^{3}}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   8. metadata-eval 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{\color{blue}{0.5}}{{\left(\left|x\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   9. rem-square-sqrt 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   10. fabs-sqr 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   11. rem-square-sqrt 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{\color{blue}{x}}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   12. rem-square-sqrt 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}\right)\right)\right) \]

   13. fabs-sqr 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)\right)\right) \]

   14. rem-square-sqrt 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{\color{blue}{x}}\right)\right)\right) \]

7. Simplified 99.5%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{x}\right)\right)\right)} \]

8. Taylor expanded in x around inf 99.4%

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

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

10. Simplified 99.4%

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

    1. *-un-lft-identity 99.4%

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

    2. inv-pow 99.4%

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

    3. sqrt-pow1 99.4%

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

    4. metadata-eval 99.4%

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

12. Applied egg-rr 99.4%

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

    1. *-lft-identity 99.4%

       \[\leadsto e^{x \cdot x} \cdot \color{blue}{\frac{{\pi}^{-0.5}}{x}} \]

14. Simplified 99.4%

    \[\leadsto e^{x \cdot x} \cdot \color{blue}{\frac{{\pi}^{-0.5}}{x}} \]
15. Add Preprocessing

Alternative 14: 2.3% accurate, 20.0× speedup

Math

\[\begin{array}{l} \\ \frac{{\pi}^{-0.5}}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (pow PI -0.5) x))

C

double code(double x) {
	return pow(((double) M_PI), -0.5) / x;
}

Java

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

Python

def code(x):
	return math.pow(math.pi, -0.5) / x

Julia

function code(x)
	return Float64((pi ^ -0.5) / x)
end

MATLAB

function tmp = code(x)
	tmp = (pi ^ -0.5) / x;
end

Wolfram

code[x_] := N[(N[Power[Pi, -0.5], $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{e^{x \cdot x} \cdot \frac{\frac{0.5}{{\left(\left|x\right|\right)}^{3}} + \left(\frac{1}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}}} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 99.5%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r/ 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\frac{0.75 \cdot 1}{{\left(\left|x\right|\right)}^{5}}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   2. metadata-eval 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{\color{blue}{0.75}}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   3. rem-square-sqrt 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   4. fabs-sqr 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   5. rem-square-sqrt 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{\color{blue}{x}}^{5}} + \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right) \]

   6. +-commutative 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \color{blue}{\left(0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)}\right)\right) \]

   7. associate-*r/ 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\color{blue}{\frac{0.5 \cdot 1}{{\left(\left|x\right|\right)}^{3}}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   8. metadata-eval 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{\color{blue}{0.5}}{{\left(\left|x\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   9. rem-square-sqrt 99.5%

      \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   10. fabs-sqr 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   11. rem-square-sqrt 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{\color{blue}{x}}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \]

   12. rem-square-sqrt 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}\right)\right)\right) \]

   13. fabs-sqr 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)\right)\right) \]

   14. rem-square-sqrt 99.5%

       \[\leadsto e^{x \cdot x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{\color{blue}{x}}\right)\right)\right) \]

7. Simplified 99.5%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \left(\frac{0.5}{{x}^{3}} + \frac{1}{x}\right)\right)\right)} \]

8. Taylor expanded in x around inf 99.4%

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

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

10. Simplified 99.4%

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

11. Taylor expanded in x around 0 2.3%

    \[\leadsto \color{blue}{1} \cdot \frac{\sqrt{\frac{1}{\pi}}}{x} \]
12. Step-by-step derivation

    1. *-un-lft-identity 99.4%

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

    2. inv-pow 99.4%

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

    3. sqrt-pow1 99.4%

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

    4. metadata-eval 99.4%

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

13. Applied egg-rr 2.3%

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

    1. *-lft-identity 99.4%

       \[\leadsto e^{x \cdot x} \cdot \color{blue}{\frac{{\pi}^{-0.5}}{x}} \]

15. Simplified 2.3%

    \[\leadsto 1 \cdot \color{blue}{\frac{{\pi}^{-0.5}}{x}} \]

16. Final simplification 2.3%

    \[\leadsto \frac{{\pi}^{-0.5}}{x} \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(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)))))))