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

Percentage Accurate: 100.0% → 99.8%

Time: 5.0s

Alternatives: 3

Speedup: N/A×

• 99.4% 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: 99.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt[3]{{\pi}^{1.5}}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (exp (* x x)) (* (fabs x) (cbrt (pow PI 1.5))))
  (+ (/ (+ 0.5 (/ 0.75 (* x x))) (* x x)) (+ 1.0 (/ 1.875 (pow x 6.0))))))

C

double code(double x) {
	return (exp((x * x)) / (fabs(x) * cbrt(pow(((double) M_PI), 1.5)))) * (((0.5 + (0.75 / (x * x))) / (x * x)) + (1.0 + (1.875 / pow(x, 6.0))));
}

Java

public static double code(double x) {
	return (Math.exp((x * x)) / (Math.abs(x) * Math.cbrt(Math.pow(Math.PI, 1.5)))) * (((0.5 + (0.75 / (x * x))) / (x * x)) + (1.0 + (1.875 / Math.pow(x, 6.0))));
}

Julia

function code(x)
	return Float64(Float64(exp(Float64(x * x)) / Float64(abs(x) * cbrt((pi ^ 1.5)))) * Float64(Float64(Float64(0.5 + Float64(0.75 / Float64(x * x))) / Float64(x * x)) + Float64(1.0 + Float64(1.875 / (x ^ 6.0)))))
end

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[(N[Abs[x], $MachinePrecision] * N[Power[N[Power[Pi, 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.5 + N[(0.75 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(1.875 / N[Power[x, 6.0], $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%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)} \]
4. Step-by-step derivation

   1. add-cbrt-cube 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\left|x\right| \cdot \color{blue}{\sqrt[3]{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}}}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

   2. pow1/3 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\left|x\right| \cdot \color{blue}{{\left(\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}\right)}^{0.3333333333333333}}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

   3. add-sqr-sqrt 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\left|x\right| \cdot {\left(\color{blue}{\pi} \cdot \sqrt{\pi}\right)}^{0.3333333333333333}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

   4. pow1 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\left|x\right| \cdot {\left(\color{blue}{{\pi}^{1}} \cdot \sqrt{\pi}\right)}^{0.3333333333333333}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

   5. pow1/2 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\left|x\right| \cdot {\left({\pi}^{1} \cdot \color{blue}{{\pi}^{0.5}}\right)}^{0.3333333333333333}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

   6. pow-prod-up 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\left|x\right| \cdot {\color{blue}{\left({\pi}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

   7. metadata-eval 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\left|x\right| \cdot {\left({\pi}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

5. Applied egg-rr 100.0%

   \[\leadsto \frac{e^{x \cdot x}}{\left|x\right| \cdot \color{blue}{{\left({\pi}^{1.5}\right)}^{0.3333333333333333}}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]
6. Step-by-step derivation

   1. unpow1/3 100.0%

      \[\leadsto \frac{e^{x \cdot x}}{\left|x\right| \cdot \color{blue}{\sqrt[3]{{\pi}^{1.5}}}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

7. Simplified 100.0%

   \[\leadsto \frac{e^{x \cdot x}}{\left|x\right| \cdot \color{blue}{\sqrt[3]{{\pi}^{1.5}}}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

8. Final simplification 100.0%

   \[\leadsto \frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt[3]{{\pi}^{1.5}}} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

Alternative 2: 99.8% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \cdot \frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (+ (/ (+ 0.5 (/ 0.75 (* x x))) (* x x)) (+ 1.0 (/ 1.875 (pow x 6.0))))
  (/ (exp (* x x)) (* (fabs x) (sqrt PI)))))

C

double code(double x) {
	return (((0.5 + (0.75 / (x * x))) / (x * x)) + (1.0 + (1.875 / pow(x, 6.0)))) * (exp((x * x)) / (fabs(x) * sqrt(((double) M_PI))));
}

Java

public static double code(double x) {
	return (((0.5 + (0.75 / (x * x))) / (x * x)) + (1.0 + (1.875 / Math.pow(x, 6.0)))) * (Math.exp((x * x)) / (Math.abs(x) * Math.sqrt(Math.PI)));
}

Python

def code(x):
	return (((0.5 + (0.75 / (x * x))) / (x * x)) + (1.0 + (1.875 / math.pow(x, 6.0)))) * (math.exp((x * x)) / (math.fabs(x) * math.sqrt(math.pi)))

Julia

function code(x)
	return Float64(Float64(Float64(Float64(0.5 + Float64(0.75 / Float64(x * x))) / Float64(x * x)) + Float64(1.0 + Float64(1.875 / (x ^ 6.0)))) * Float64(exp(Float64(x * x)) / Float64(abs(x) * sqrt(pi))))
end

MATLAB

function tmp = code(x)
	tmp = (((0.5 + (0.75 / (x * x))) / (x * x)) + (1.0 + (1.875 / (x ^ 6.0)))) * (exp((x * x)) / (abs(x) * sqrt(pi)));
end

Wolfram

code[x_] := N[(N[(N[(N[(0.5 + N[(0.75 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[(N[Abs[x], $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 3: 99.4% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x \cdot x}}{x \cdot \sqrt{\pi}} \cdot \left(\left(1 + \frac{1.875}{{x}^{6}}\right) + \frac{0.5}{x \cdot x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (exp (* x x)) (* x (sqrt PI)))
  (+ (+ 1.0 (/ 1.875 (pow x 6.0))) (/ 0.5 (* x x)))))

C

double code(double x) {
	return (exp((x * x)) / (x * sqrt(((double) M_PI)))) * ((1.0 + (1.875 / pow(x, 6.0))) + (0.5 / (x * x)));
}

Java

public static double code(double x) {
	return (Math.exp((x * x)) / (x * Math.sqrt(Math.PI))) * ((1.0 + (1.875 / Math.pow(x, 6.0))) + (0.5 / (x * x)));
}

Python

def code(x):
	return (math.exp((x * x)) / (x * math.sqrt(math.pi))) * ((1.0 + (1.875 / math.pow(x, 6.0))) + (0.5 / (x * x)))

Julia

function code(x)
	return Float64(Float64(exp(Float64(x * x)) / Float64(x * sqrt(pi))) * Float64(Float64(1.0 + Float64(1.875 / (x ^ 6.0))) + Float64(0.5 / Float64(x * x))))
end

MATLAB

function tmp = code(x)
	tmp = (exp((x * x)) / (x * sqrt(pi))) * ((1.0 + (1.875 / (x ^ 6.0))) + (0.5 / (x * x)));
end

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[(x * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around inf 99.7%

   \[\leadsto \frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(\color{blue}{\frac{0.5}{{x}^{2}}} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]
5. Step-by-step derivation

   1. unpow2 99.7%

      \[\leadsto \frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(\frac{0.5}{\color{blue}{x \cdot x}} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

6. Simplified 99.7%

   \[\leadsto \frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\pi}} \cdot \left(\color{blue}{\frac{0.5}{x \cdot x}} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]
7. Step-by-step derivation

   1. expm1-log1p-u 99.7%

      \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left|x\right| \cdot \sqrt{\pi}\right)\right)}} \cdot \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

   2. expm1-udef 99.7%

      \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{e^{\mathsf{log1p}\left(\left|x\right| \cdot \sqrt{\pi}\right)} - 1}} \cdot \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

   3. log1p-udef 99.7%

      \[\leadsto \frac{e^{x \cdot x}}{e^{\color{blue}{\log \left(1 + \left|x\right| \cdot \sqrt{\pi}\right)}} - 1} \cdot \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

   4. add-exp-log 99.7%

      \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{\left(1 + \left|x\right| \cdot \sqrt{\pi}\right)} - 1} \cdot \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

   5. add-sqr-sqrt 99.7%

      \[\leadsto \frac{e^{x \cdot x}}{\left(1 + \left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \sqrt{\pi}\right) - 1} \cdot \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

   6. fabs-sqr 99.7%

      \[\leadsto \frac{e^{x \cdot x}}{\left(1 + \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \sqrt{\pi}\right) - 1} \cdot \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

   7. add-sqr-sqrt 99.7%

      \[\leadsto \frac{e^{x \cdot x}}{\left(1 + \color{blue}{x} \cdot \sqrt{\pi}\right) - 1} \cdot \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

8. Applied egg-rr 99.7%

   \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{\left(1 + x \cdot \sqrt{\pi}\right) - 1}} \cdot \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]
9. Step-by-step derivation

   1. +-commutative 99.7%

      \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{\left(x \cdot \sqrt{\pi} + 1\right)} - 1} \cdot \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

   2. associate--l+ 99.7%

      \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{x \cdot \sqrt{\pi} + \left(1 - 1\right)}} \cdot \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

   3. metadata-eval 99.7%

      \[\leadsto \frac{e^{x \cdot x}}{x \cdot \sqrt{\pi} + \color{blue}{0}} \cdot \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

   4. +-rgt-identity 99.7%

      \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{x \cdot \sqrt{\pi}}} \cdot \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

   5. *-commutative 99.7%

      \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{\sqrt{\pi} \cdot x}} \cdot \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

10. Simplified 99.7%

    \[\leadsto \frac{e^{x \cdot x}}{\color{blue}{\sqrt{\pi} \cdot x}} \cdot \left(\frac{0.5}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \]

11. Final simplification 99.7%

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

Reproduce

herbie shell --seed 2023182 
(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)))))))