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

Percentage Accurate: 100.0% → 99.6%

Time: 6.6s

Alternatives: 1

Speedup: N/A×

• 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: 99.6% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 100.0%

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

   1. sqr-pow 100.0%

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

   2. associate-/r* 100.0%

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

   3. unpow1 100.0%

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

   4. metadata-eval 100.0%

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

   5. pow-sqr 100.0%

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

   6. unpow1/2 100.0%

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

   7. unpow1/2 100.0%

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

   8. fabs-sqr 100.0%

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

   9. unpow1/2 100.0%

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

   10. unpow1/2 100.0%

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

   11. pow-sqr 100.0%

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

   12. metadata-eval 100.0%

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

   13. unpow1 100.0%

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

   14. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{{x}^{\color{blue}{2.5}}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   15. exp-to-pow 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\frac{1}{\color{blue}{e^{\log x \cdot 2.5}}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   16. exp-neg 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\color{blue}{e^{-\log x \cdot 2.5}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   17. distribute-rgt-neg-in 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{e^{\color{blue}{\log x \cdot \left(-2.5\right)}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   18. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{e^{\log x \cdot \color{blue}{-2.5}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   19. exp-to-pow 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\color{blue}{{x}^{-2.5}}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   20. *-rgt-identity 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \frac{\color{blue}{{x}^{-2.5} \cdot 1}}{{\left(\left|x\right|\right)}^{\left(\frac{5}{2}\right)}} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

6. Simplified 100.0%

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

7. Taylor expanded in x around 0 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\frac{1.875}{{x}^{7}}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

8. Taylor expanded in x around inf 100.0%

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

   1. unpow1 100.0%

      \[\leadsto \left(\frac{1}{\left|\color{blue}{{x}^{1}}\right|} + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   2. metadata-eval 100.0%

      \[\leadsto \left(\frac{1}{\left|{x}^{\color{blue}{\left(2 \cdot 0.5\right)}}\right|} + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   3. pow-sqr 100.0%

      \[\leadsto \left(\frac{1}{\left|\color{blue}{{x}^{0.5} \cdot {x}^{0.5}}\right|} + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   4. unpow1/2 100.0%

      \[\leadsto \left(\frac{1}{\left|\color{blue}{\sqrt{x}} \cdot {x}^{0.5}\right|} + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. unpow1/2 100.0%

      \[\leadsto \left(\frac{1}{\left|\sqrt{x} \cdot \color{blue}{\sqrt{x}}\right|} + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. fabs-sqr 100.0%

      \[\leadsto \left(\frac{1}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. unpow1/2 100.0%

      \[\leadsto \left(\frac{1}{\color{blue}{{x}^{0.5}} \cdot \sqrt{x}} + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. unpow1/2 100.0%

      \[\leadsto \left(\frac{1}{{x}^{0.5} \cdot \color{blue}{{x}^{0.5}}} + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   9. pow-sqr 100.0%

      \[\leadsto \left(\frac{1}{\color{blue}{{x}^{\left(2 \cdot 0.5\right)}}} + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   10. metadata-eval 100.0%

       \[\leadsto \left(\frac{1}{{x}^{\color{blue}{1}}} + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   11. unpow1 100.0%

       \[\leadsto \left(\frac{1}{\color{blue}{x}} + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

10. Simplified 100.0%

    \[\leadsto \left(\color{blue}{\frac{1}{x}} + \frac{1.875}{{x}^{7}}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

11. Taylor expanded in x around inf 100.0%

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

12. Final simplification 100.0%

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

Reproduce

herbie shell --seed 2023274 
(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)))))))