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

Percentage Accurate: 100.0% → 99.9%

Time: 28.9s

Alternatives: 5

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: 99.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(3 \cdot \log \left(\sqrt[3]{{\left(e^{0.75}\right)}^{\left({x}^{-4}\right)}}\right) + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp x) x) (* x (sqrt PI)))
  (+
   1.0
   (+
    (* 3.0 (log (cbrt (pow (exp 0.75) (pow x -4.0)))))
    (+ (/ 0.5 (* x x)) (* 3.0 (log (cbrt (exp (/ 1.875 (pow x 6.0)))))))))))

C

double code(double x) {
	return (pow(exp(x), x) / (x * sqrt(((double) M_PI)))) * (1.0 + ((3.0 * log(cbrt(pow(exp(0.75), pow(x, -4.0))))) + ((0.5 / (x * x)) + (3.0 * log(cbrt(exp((1.875 / pow(x, 6.0)))))))));
}

Java

public static double code(double x) {
	return (Math.pow(Math.exp(x), x) / (x * Math.sqrt(Math.PI))) * (1.0 + ((3.0 * Math.log(Math.cbrt(Math.pow(Math.exp(0.75), Math.pow(x, -4.0))))) + ((0.5 / (x * x)) + (3.0 * Math.log(Math.cbrt(Math.exp((1.875 / Math.pow(x, 6.0)))))))));
}

Julia

function code(x)
	return Float64(Float64((exp(x) ^ x) / Float64(x * sqrt(pi))) * Float64(1.0 + Float64(Float64(3.0 * log(cbrt((exp(0.75) ^ (x ^ -4.0))))) + Float64(Float64(0.5 / Float64(x * x)) + Float64(3.0 * log(cbrt(exp(Float64(1.875 / (x ^ 6.0))))))))))
end

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[(x * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(3.0 * N[Log[N[Power[N[Power[N[Exp[0.75], $MachinePrecision], N[Power[x, -4.0], $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[Log[N[Power[N[Exp[N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 100.0%

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

   1. unpow1 100.0%

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

   2. sqr-pow 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi} \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   3. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi} \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   4. sqr-pow 100.0%

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

   5. unpow1 100.0%

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

   6. *-commutative 100.0%

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

6. Simplified 100.0%

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

   1. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \color{blue}{\log \left(e^{\frac{1.875}{{x}^{6}}}\right)}\right)\right)\right) \]

   2. add-cube-cbrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \log \color{blue}{\left(\left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}} \cdot \sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right) \cdot \sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)}\right)\right)\right) \]

   3. log-prod 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \color{blue}{\left(\log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}} \cdot \sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)}\right)\right)\right) \]

   4. div-inv 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{e^{\color{blue}{1.875 \cdot \frac{1}{{x}^{6}}}}} \cdot \sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

   5. exp-prod 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{\color{blue}{{\left(e^{1.875}\right)}^{\left(\frac{1}{{x}^{6}}\right)}}} \cdot \sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

   6. pow-flip 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\color{blue}{\left({x}^{\left(-6\right)}\right)}}} \cdot \sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

   7. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{\color{blue}{-6}}\right)}} \cdot \sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

   8. div-inv 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}} \cdot \sqrt[3]{e^{\color{blue}{1.875 \cdot \frac{1}{{x}^{6}}}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

   9. exp-prod 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}} \cdot \sqrt[3]{\color{blue}{{\left(e^{1.875}\right)}^{\left(\frac{1}{{x}^{6}}\right)}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

   10. pow-flip 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}} \cdot \sqrt[3]{{\left(e^{1.875}\right)}^{\color{blue}{\left({x}^{\left(-6\right)}\right)}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}} \cdot \sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{\color{blue}{-6}}\right)}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

8. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \color{blue}{\left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}} \cdot \sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right) + \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)}\right)\right)\right) \]
9. Step-by-step derivation

   1. log-prod 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\color{blue}{\left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right) + \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)} + \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right)\right) \]

   2. count-2 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)} + \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right)\right) \]

   3. distribute-lft1-in 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \color{blue}{\left(2 + 1\right) \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)}\right)\right)\right) \]

   4. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \color{blue}{3} \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

10. Simplified 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \color{blue}{3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)}\right)\right)\right) \]
11. Step-by-step derivation

    1. add-log-exp 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{\log \left(e^{\frac{0.75}{{\left(\left|x\right|\right)}^{4}}}\right)} + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

    2. add-cube-cbrt 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\log \color{blue}{\left(\left(\sqrt[3]{e^{\frac{0.75}{{\left(\left|x\right|\right)}^{4}}}} \cdot \sqrt[3]{e^{\frac{0.75}{{\left(\left|x\right|\right)}^{4}}}}\right) \cdot \sqrt[3]{e^{\frac{0.75}{{\left(\left|x\right|\right)}^{4}}}}\right)} + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

    3. log-prod 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{\left(\log \left(\sqrt[3]{e^{\frac{0.75}{{\left(\left|x\right|\right)}^{4}}}} \cdot \sqrt[3]{e^{\frac{0.75}{{\left(\left|x\right|\right)}^{4}}}}\right) + \log \left(\sqrt[3]{e^{\frac{0.75}{{\left(\left|x\right|\right)}^{4}}}}\right)\right)} + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

12. Applied egg-rr 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{\left(\log \left(\sqrt[3]{{\left(e^{0.75}\right)}^{\left({x}^{-4}\right)}} \cdot \sqrt[3]{{\left(e^{0.75}\right)}^{\left({x}^{-4}\right)}}\right) + \log \left(\sqrt[3]{{\left(e^{0.75}\right)}^{\left({x}^{-4}\right)}}\right)\right)} + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]
13. Step-by-step derivation

    1. log-prod 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\left(\color{blue}{\left(\log \left(\sqrt[3]{{\left(e^{0.75}\right)}^{\left({x}^{-4}\right)}}\right) + \log \left(\sqrt[3]{{\left(e^{0.75}\right)}^{\left({x}^{-4}\right)}}\right)\right)} + \log \left(\sqrt[3]{{\left(e^{0.75}\right)}^{\left({x}^{-4}\right)}}\right)\right) + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

    2. count-2 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\left(\color{blue}{2 \cdot \log \left(\sqrt[3]{{\left(e^{0.75}\right)}^{\left({x}^{-4}\right)}}\right)} + \log \left(\sqrt[3]{{\left(e^{0.75}\right)}^{\left({x}^{-4}\right)}}\right)\right) + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

    3. distribute-lft1-in 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{\left(2 + 1\right) \cdot \log \left(\sqrt[3]{{\left(e^{0.75}\right)}^{\left({x}^{-4}\right)}}\right)} + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

    4. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{3} \cdot \log \left(\sqrt[3]{{\left(e^{0.75}\right)}^{\left({x}^{-4}\right)}}\right) + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

14. Simplified 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{3 \cdot \log \left(\sqrt[3]{{\left(e^{0.75}\right)}^{\left({x}^{-4}\right)}}\right)} + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

15. Taylor expanded in x around 0 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(3 \cdot \log \left(\sqrt[3]{{\left(e^{0.75}\right)}^{\left({x}^{-4}\right)}}\right) + \left(\frac{0.5}{x \cdot x} + 3 \cdot \color{blue}{\log \left({\left(e^{\frac{1.875}{{x}^{6}}}\right)}^{0.3333333333333333}\right)}\right)\right)\right) \]
16. Step-by-step derivation

    1. unpow1/3 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(3 \cdot \log \left(\sqrt[3]{{\left(e^{0.75}\right)}^{\left({x}^{-4}\right)}}\right) + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \color{blue}{\left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)}\right)\right)\right) \]

17. Simplified 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(3 \cdot \log \left(\sqrt[3]{{\left(e^{0.75}\right)}^{\left({x}^{-4}\right)}}\right) + \left(\frac{0.5}{x \cdot x} + 3 \cdot \color{blue}{\log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)}\right)\right)\right) \]

18. Final simplification 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(3 \cdot \log \left(\sqrt[3]{{\left(e^{0.75}\right)}^{\left({x}^{-4}\right)}}\right) + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right) \]

Alternative 2: 99.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{e^{\left(3 \cdot \log \left(x \cdot \sqrt{\pi}\right)\right) \cdot 0.3333333333333333}} \cdot \left(1 + \left(\left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right) + 0.75 \cdot {x}^{-4}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/
   (pow (exp x) x)
   (exp (* (* 3.0 (log (* x (sqrt PI)))) 0.3333333333333333)))
  (+
   1.0
   (+
    (+ (/ 0.5 (* x x)) (* 3.0 (log (cbrt (exp (/ 1.875 (pow x 6.0)))))))
    (* 0.75 (pow x -4.0))))))

C

double code(double x) {
	return (pow(exp(x), x) / exp(((3.0 * log((x * sqrt(((double) M_PI))))) * 0.3333333333333333))) * (1.0 + (((0.5 / (x * x)) + (3.0 * log(cbrt(exp((1.875 / pow(x, 6.0))))))) + (0.75 * pow(x, -4.0))));
}

Java

public static double code(double x) {
	return (Math.pow(Math.exp(x), x) / Math.exp(((3.0 * Math.log((x * Math.sqrt(Math.PI)))) * 0.3333333333333333))) * (1.0 + (((0.5 / (x * x)) + (3.0 * Math.log(Math.cbrt(Math.exp((1.875 / Math.pow(x, 6.0))))))) + (0.75 * Math.pow(x, -4.0))));
}

Julia

function code(x)
	return Float64(Float64((exp(x) ^ x) / exp(Float64(Float64(3.0 * log(Float64(x * sqrt(pi)))) * 0.3333333333333333))) * Float64(1.0 + Float64(Float64(Float64(0.5 / Float64(x * x)) + Float64(3.0 * log(cbrt(exp(Float64(1.875 / (x ^ 6.0))))))) + Float64(0.75 * (x ^ -4.0)))))
end

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Exp[N[(N[(3.0 * N[Log[N[(x * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[Log[N[Power[N[Exp[N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.75 * N[Power[x, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

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

   1. add-cbrt-cube 30.1%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\sqrt[3]{\left(\left(\left|x\right| \cdot \sqrt{\pi}\right) \cdot \left(\left|x\right| \cdot \sqrt{\pi}\right)\right) \cdot \left(\left|x\right| \cdot \sqrt{\pi}\right)}}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   2. pow1/3 30.1%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left(\left(\left(\left|x\right| \cdot \sqrt{\pi}\right) \cdot \left(\left|x\right| \cdot \sqrt{\pi}\right)\right) \cdot \left(\left|x\right| \cdot \sqrt{\pi}\right)\right)}^{0.3333333333333333}}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   3. pow-to-exp 30.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{e^{\log \left(\left(\left(\left|x\right| \cdot \sqrt{\pi}\right) \cdot \left(\left|x\right| \cdot \sqrt{\pi}\right)\right) \cdot \left(\left|x\right| \cdot \sqrt{\pi}\right)\right) \cdot 0.3333333333333333}}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   4. pow3 30.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{e^{\log \color{blue}{\left({\left(\left|x\right| \cdot \sqrt{\pi}\right)}^{3}\right)} \cdot 0.3333333333333333}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   5. metadata-eval 30.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{e^{\log \left({\left(\left|x\right| \cdot \sqrt{\pi}\right)}^{\color{blue}{\left(\frac{6}{2}\right)}}\right) \cdot 0.3333333333333333}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   6. log-pow 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{e^{\color{blue}{\left(\frac{6}{2} \cdot \log \left(\left|x\right| \cdot \sqrt{\pi}\right)\right)} \cdot 0.3333333333333333}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   7. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{e^{\left(\color{blue}{3} \cdot \log \left(\left|x\right| \cdot \sqrt{\pi}\right)\right) \cdot 0.3333333333333333}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{e^{\left(3 \cdot \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \sqrt{\pi}\right)\right) \cdot 0.3333333333333333}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   9. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{e^{\left(3 \cdot \log \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \sqrt{\pi}\right)\right) \cdot 0.3333333333333333}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{e^{\left(3 \cdot \log \left(\color{blue}{x} \cdot \sqrt{\pi}\right)\right) \cdot 0.3333333333333333}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

5. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{e^{\left(3 \cdot \log \left(x \cdot \sqrt{\pi}\right)\right) \cdot 0.3333333333333333}}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]
6. Step-by-step derivation

   1. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \color{blue}{\log \left(e^{\frac{1.875}{{x}^{6}}}\right)}\right)\right)\right) \]

   2. add-cube-cbrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \log \color{blue}{\left(\left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}} \cdot \sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right) \cdot \sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)}\right)\right)\right) \]

   3. log-prod 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \color{blue}{\left(\log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}} \cdot \sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)}\right)\right)\right) \]

   4. div-inv 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{e^{\color{blue}{1.875 \cdot \frac{1}{{x}^{6}}}}} \cdot \sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

   5. exp-prod 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{\color{blue}{{\left(e^{1.875}\right)}^{\left(\frac{1}{{x}^{6}}\right)}}} \cdot \sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

   6. pow-flip 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\color{blue}{\left({x}^{\left(-6\right)}\right)}}} \cdot \sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

   7. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{\color{blue}{-6}}\right)}} \cdot \sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

   8. div-inv 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}} \cdot \sqrt[3]{e^{\color{blue}{1.875 \cdot \frac{1}{{x}^{6}}}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

   9. exp-prod 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}} \cdot \sqrt[3]{\color{blue}{{\left(e^{1.875}\right)}^{\left(\frac{1}{{x}^{6}}\right)}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

   10. pow-flip 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}} \cdot \sqrt[3]{{\left(e^{1.875}\right)}^{\color{blue}{\left({x}^{\left(-6\right)}\right)}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}} \cdot \sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{\color{blue}{-6}}\right)}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

7. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{e^{\left(3 \cdot \log \left(x \cdot \sqrt{\pi}\right)\right) \cdot 0.3333333333333333}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \color{blue}{\left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}} \cdot \sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right) + \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)}\right)\right)\right) \]
8. Step-by-step derivation

   1. log-prod 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\color{blue}{\left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right) + \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)} + \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right)\right) \]

   2. count-2 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)} + \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right)\right) \]

   3. distribute-lft1-in 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \color{blue}{\left(2 + 1\right) \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)}\right)\right)\right) \]

   4. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \color{blue}{3} \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

9. Simplified 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{e^{\left(3 \cdot \log \left(x \cdot \sqrt{\pi}\right)\right) \cdot 0.3333333333333333}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \color{blue}{3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)}\right)\right)\right) \]
10. Step-by-step derivation

    1. clear-num 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{\frac{1}{\frac{{\left(\left|x\right|\right)}^{4}}{0.75}}} + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

    2. associate-/r/ 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{\frac{1}{{\left(\left|x\right|\right)}^{4}} \cdot 0.75} + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

    3. pow-flip 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{{\left(\left|x\right|\right)}^{\left(-4\right)}} \cdot 0.75 + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

    4. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left({\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-4\right)} \cdot 0.75 + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

    5. fabs-sqr 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left({\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-4\right)} \cdot 0.75 + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

    6. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left({\color{blue}{x}}^{\left(-4\right)} \cdot 0.75 + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

    7. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left({x}^{\color{blue}{-4}} \cdot 0.75 + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

11. Applied egg-rr 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{e^{\left(3 \cdot \log \left(x \cdot \sqrt{\pi}\right)\right) \cdot 0.3333333333333333}} \cdot \left(1 + \left(\color{blue}{{x}^{-4} \cdot 0.75} + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

12. Taylor expanded in x around 0 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{e^{\left(3 \cdot \log \left(x \cdot \sqrt{\pi}\right)\right) \cdot 0.3333333333333333}} \cdot \left(1 + \left({x}^{-4} \cdot 0.75 + \left(\frac{0.5}{x \cdot x} + 3 \cdot \color{blue}{\log \left({\left(e^{\frac{1.875}{{x}^{6}}}\right)}^{0.3333333333333333}\right)}\right)\right)\right) \]
13. Step-by-step derivation

    1. unpow1/3 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(3 \cdot \log \left(\sqrt[3]{{\left(e^{0.75}\right)}^{\left({x}^{-4}\right)}}\right) + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \color{blue}{\left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)}\right)\right)\right) \]

14. Simplified 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{e^{\left(3 \cdot \log \left(x \cdot \sqrt{\pi}\right)\right) \cdot 0.3333333333333333}} \cdot \left(1 + \left({x}^{-4} \cdot 0.75 + \left(\frac{0.5}{x \cdot x} + 3 \cdot \color{blue}{\log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)}\right)\right)\right) \]

15. Final simplification 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{e^{\left(3 \cdot \log \left(x \cdot \sqrt{\pi}\right)\right) \cdot 0.3333333333333333}} \cdot \left(1 + \left(\left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right) + 0.75 \cdot {x}^{-4}\right)\right) \]

Alternative 3: 99.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \sqrt{\pi}\right)\right)} \cdot \left(1 + \left(0.75 \cdot {x}^{-4} + \left(\frac{0.5}{x \cdot x} + 3 \cdot \frac{0.625}{{x}^{6}}\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp x) x) (expm1 (log1p (* x (sqrt PI)))))
  (+
   1.0
   (+
    (* 0.75 (pow x -4.0))
    (+ (/ 0.5 (* x x)) (* 3.0 (/ 0.625 (pow x 6.0))))))))

C

double code(double x) {
	return (pow(exp(x), x) / expm1(log1p((x * sqrt(((double) M_PI)))))) * (1.0 + ((0.75 * pow(x, -4.0)) + ((0.5 / (x * x)) + (3.0 * (0.625 / pow(x, 6.0))))));
}

Java

public static double code(double x) {
	return (Math.pow(Math.exp(x), x) / Math.expm1(Math.log1p((x * Math.sqrt(Math.PI))))) * (1.0 + ((0.75 * Math.pow(x, -4.0)) + ((0.5 / (x * x)) + (3.0 * (0.625 / Math.pow(x, 6.0))))));
}

Python

def code(x):
	return (math.pow(math.exp(x), x) / math.expm1(math.log1p((x * math.sqrt(math.pi))))) * (1.0 + ((0.75 * math.pow(x, -4.0)) + ((0.5 / (x * x)) + (3.0 * (0.625 / math.pow(x, 6.0))))))

Julia

function code(x)
	return Float64(Float64((exp(x) ^ x) / expm1(log1p(Float64(x * sqrt(pi))))) * Float64(1.0 + Float64(Float64(0.75 * (x ^ -4.0)) + Float64(Float64(0.5 / Float64(x * x)) + Float64(3.0 * Float64(0.625 / (x ^ 6.0)))))))
end

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[(Exp[N[Log[1 + N[(x * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(0.75 * N[Power[x, -4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(0.625 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

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

   1. add-sqr-sqrt 100.0%

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

   2. fabs-sqr 100.0%

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

   3. add-sqr-sqrt 100.0%

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

   4. expm1-log1p-u 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \sqrt{\pi}\right)\right)}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

5. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \sqrt{\pi}\right)\right)}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]
6. Step-by-step derivation

   1. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \color{blue}{\log \left(e^{\frac{1.875}{{x}^{6}}}\right)}\right)\right)\right) \]

   2. add-cube-cbrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \log \color{blue}{\left(\left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}} \cdot \sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right) \cdot \sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)}\right)\right)\right) \]

   3. log-prod 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \color{blue}{\left(\log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}} \cdot \sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)}\right)\right)\right) \]

   4. div-inv 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{e^{\color{blue}{1.875 \cdot \frac{1}{{x}^{6}}}}} \cdot \sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

   5. exp-prod 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{\color{blue}{{\left(e^{1.875}\right)}^{\left(\frac{1}{{x}^{6}}\right)}}} \cdot \sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

   6. pow-flip 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\color{blue}{\left({x}^{\left(-6\right)}\right)}}} \cdot \sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

   7. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{\color{blue}{-6}}\right)}} \cdot \sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

   8. div-inv 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}} \cdot \sqrt[3]{e^{\color{blue}{1.875 \cdot \frac{1}{{x}^{6}}}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

   9. exp-prod 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}} \cdot \sqrt[3]{\color{blue}{{\left(e^{1.875}\right)}^{\left(\frac{1}{{x}^{6}}\right)}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

   10. pow-flip 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}} \cdot \sqrt[3]{{\left(e^{1.875}\right)}^{\color{blue}{\left({x}^{\left(-6\right)}\right)}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}} \cdot \sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{\color{blue}{-6}}\right)}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

7. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \sqrt{\pi}\right)\right)} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \color{blue}{\left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}} \cdot \sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right) + \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)}\right)\right)\right) \]
8. Step-by-step derivation

   1. log-prod 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\color{blue}{\left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right) + \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)} + \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right)\right) \]

   2. count-2 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)} + \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right)\right) \]

   3. distribute-lft1-in 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \color{blue}{\left(2 + 1\right) \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)}\right)\right)\right) \]

   4. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \color{blue}{3} \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

9. Simplified 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \sqrt{\pi}\right)\right)} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \color{blue}{3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)}\right)\right)\right) \]
10. Step-by-step derivation

    1. clear-num 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{\frac{1}{\frac{{\left(\left|x\right|\right)}^{4}}{0.75}}} + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

    2. associate-/r/ 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{\frac{1}{{\left(\left|x\right|\right)}^{4}} \cdot 0.75} + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

    3. pow-flip 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{{\left(\left|x\right|\right)}^{\left(-4\right)}} \cdot 0.75 + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

    4. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left({\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-4\right)} \cdot 0.75 + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

    5. fabs-sqr 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left({\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-4\right)} \cdot 0.75 + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

    6. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left({\color{blue}{x}}^{\left(-4\right)} \cdot 0.75 + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

    7. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left({x}^{\color{blue}{-4}} \cdot 0.75 + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

11. Applied egg-rr 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \sqrt{\pi}\right)\right)} \cdot \left(1 + \left(\color{blue}{{x}^{-4} \cdot 0.75} + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

12. Taylor expanded in x around 0 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \sqrt{\pi}\right)\right)} \cdot \left(1 + \left({x}^{-4} \cdot 0.75 + \left(\frac{0.5}{x \cdot x} + 3 \cdot \color{blue}{\log \left({\left(e^{\frac{1.875}{{x}^{6}}}\right)}^{0.3333333333333333}\right)}\right)\right)\right) \]
13. Step-by-step derivation

    1. log-pow 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left({x}^{-4} \cdot 0.75 + \left(\frac{0.5}{x \cdot x} + 3 \cdot \color{blue}{\left(0.3333333333333333 \cdot \log \left(e^{\frac{1.875}{{x}^{6}}}\right)\right)}\right)\right)\right) \]

    2. rem-log-exp 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left({x}^{-4} \cdot 0.75 + \left(\frac{0.5}{x \cdot x} + 3 \cdot \left(0.3333333333333333 \cdot \color{blue}{\frac{1.875}{{x}^{6}}}\right)\right)\right)\right) \]

    3. associate-*r/ 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left({x}^{-4} \cdot 0.75 + \left(\frac{0.5}{x \cdot x} + 3 \cdot \color{blue}{\frac{0.3333333333333333 \cdot 1.875}{{x}^{6}}}\right)\right)\right) \]

    4. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left({x}^{-4} \cdot 0.75 + \left(\frac{0.5}{x \cdot x} + 3 \cdot \frac{\color{blue}{0.625}}{{x}^{6}}\right)\right)\right) \]

14. Simplified 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \sqrt{\pi}\right)\right)} \cdot \left(1 + \left({x}^{-4} \cdot 0.75 + \left(\frac{0.5}{x \cdot x} + 3 \cdot \color{blue}{\frac{0.625}{{x}^{6}}}\right)\right)\right) \]

15. Final simplification 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \sqrt{\pi}\right)\right)} \cdot \left(1 + \left(0.75 \cdot {x}^{-4} + \left(\frac{0.5}{x \cdot x} + 3 \cdot \frac{0.625}{{x}^{6}}\right)\right)\right) \]

Alternative 4: 99.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(0.75 \cdot {x}^{-4} + \left(\frac{0.5}{x \cdot x} + 3 \cdot \frac{0.625}{{x}^{6}}\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp x) x) (* x (sqrt PI)))
  (+
   1.0
   (+
    (* 0.75 (pow x -4.0))
    (+ (/ 0.5 (* x x)) (* 3.0 (/ 0.625 (pow x 6.0))))))))

C

double code(double x) {
	return (pow(exp(x), x) / (x * sqrt(((double) M_PI)))) * (1.0 + ((0.75 * pow(x, -4.0)) + ((0.5 / (x * x)) + (3.0 * (0.625 / pow(x, 6.0))))));
}

Java

public static double code(double x) {
	return (Math.pow(Math.exp(x), x) / (x * Math.sqrt(Math.PI))) * (1.0 + ((0.75 * Math.pow(x, -4.0)) + ((0.5 / (x * x)) + (3.0 * (0.625 / Math.pow(x, 6.0))))));
}

Python

def code(x):
	return (math.pow(math.exp(x), x) / (x * math.sqrt(math.pi))) * (1.0 + ((0.75 * math.pow(x, -4.0)) + ((0.5 / (x * x)) + (3.0 * (0.625 / math.pow(x, 6.0))))))

Julia

function code(x)
	return Float64(Float64((exp(x) ^ x) / Float64(x * sqrt(pi))) * Float64(1.0 + Float64(Float64(0.75 * (x ^ -4.0)) + Float64(Float64(0.5 / Float64(x * x)) + Float64(3.0 * Float64(0.625 / (x ^ 6.0)))))))
end

MATLAB

function tmp = code(x)
	tmp = ((exp(x) ^ x) / (x * sqrt(pi))) * (1.0 + ((0.75 * (x ^ -4.0)) + ((0.5 / (x * x)) + (3.0 * (0.625 / (x ^ 6.0))))));
end

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[(x * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(0.75 * N[Power[x, -4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(3.0 * N[(0.625 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 100.0%

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

   1. unpow1 100.0%

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

   2. sqr-pow 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi} \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   3. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi} \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   4. sqr-pow 100.0%

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

   5. unpow1 100.0%

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

   6. *-commutative 100.0%

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

6. Simplified 100.0%

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

   1. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \color{blue}{\log \left(e^{\frac{1.875}{{x}^{6}}}\right)}\right)\right)\right) \]

   2. add-cube-cbrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \log \color{blue}{\left(\left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}} \cdot \sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right) \cdot \sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)}\right)\right)\right) \]

   3. log-prod 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \color{blue}{\left(\log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}} \cdot \sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)}\right)\right)\right) \]

   4. div-inv 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{e^{\color{blue}{1.875 \cdot \frac{1}{{x}^{6}}}}} \cdot \sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

   5. exp-prod 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{\color{blue}{{\left(e^{1.875}\right)}^{\left(\frac{1}{{x}^{6}}\right)}}} \cdot \sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

   6. pow-flip 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\color{blue}{\left({x}^{\left(-6\right)}\right)}}} \cdot \sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

   7. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{\color{blue}{-6}}\right)}} \cdot \sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

   8. div-inv 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}} \cdot \sqrt[3]{e^{\color{blue}{1.875 \cdot \frac{1}{{x}^{6}}}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

   9. exp-prod 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}} \cdot \sqrt[3]{\color{blue}{{\left(e^{1.875}\right)}^{\left(\frac{1}{{x}^{6}}\right)}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

   10. pow-flip 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}} \cdot \sqrt[3]{{\left(e^{1.875}\right)}^{\color{blue}{\left({x}^{\left(-6\right)}\right)}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}} \cdot \sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{\color{blue}{-6}}\right)}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

8. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \color{blue}{\left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}} \cdot \sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right) + \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)}\right)\right)\right) \]
9. Step-by-step derivation

   1. log-prod 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\color{blue}{\left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right) + \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)} + \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right)\right) \]

   2. count-2 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)} + \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right)\right) \]

   3. distribute-lft1-in 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \color{blue}{\left(2 + 1\right) \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)}\right)\right)\right) \]

   4. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \color{blue}{3} \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

10. Simplified 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \color{blue}{3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)}\right)\right)\right) \]
11. Step-by-step derivation

    1. clear-num 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{\frac{1}{\frac{{\left(\left|x\right|\right)}^{4}}{0.75}}} + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

    2. associate-/r/ 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{\frac{1}{{\left(\left|x\right|\right)}^{4}} \cdot 0.75} + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

    3. pow-flip 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{{\left(\left|x\right|\right)}^{\left(-4\right)}} \cdot 0.75 + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

    4. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left({\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-4\right)} \cdot 0.75 + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

    5. fabs-sqr 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left({\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-4\right)} \cdot 0.75 + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

    6. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left({\color{blue}{x}}^{\left(-4\right)} \cdot 0.75 + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

    7. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left({x}^{\color{blue}{-4}} \cdot 0.75 + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

12. Applied egg-rr 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{{x}^{-4} \cdot 0.75} + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

13. Taylor expanded in x around 0 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left({x}^{-4} \cdot 0.75 + \left(\frac{0.5}{x \cdot x} + 3 \cdot \color{blue}{\log \left({\left(e^{\frac{1.875}{{x}^{6}}}\right)}^{0.3333333333333333}\right)}\right)\right)\right) \]
14. Step-by-step derivation

    1. log-pow 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left({x}^{-4} \cdot 0.75 + \left(\frac{0.5}{x \cdot x} + 3 \cdot \color{blue}{\left(0.3333333333333333 \cdot \log \left(e^{\frac{1.875}{{x}^{6}}}\right)\right)}\right)\right)\right) \]

    2. rem-log-exp 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left({x}^{-4} \cdot 0.75 + \left(\frac{0.5}{x \cdot x} + 3 \cdot \left(0.3333333333333333 \cdot \color{blue}{\frac{1.875}{{x}^{6}}}\right)\right)\right)\right) \]

    3. associate-*r/ 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left({x}^{-4} \cdot 0.75 + \left(\frac{0.5}{x \cdot x} + 3 \cdot \color{blue}{\frac{0.3333333333333333 \cdot 1.875}{{x}^{6}}}\right)\right)\right) \]

    4. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left({x}^{-4} \cdot 0.75 + \left(\frac{0.5}{x \cdot x} + 3 \cdot \frac{\color{blue}{0.625}}{{x}^{6}}\right)\right)\right) \]

15. Simplified 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left({x}^{-4} \cdot 0.75 + \left(\frac{0.5}{x \cdot x} + 3 \cdot \color{blue}{\frac{0.625}{{x}^{6}}}\right)\right)\right) \]

16. Final simplification 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(0.75 \cdot {x}^{-4} + \left(\frac{0.5}{x \cdot x} + 3 \cdot \frac{0.625}{{x}^{6}}\right)\right)\right) \]

Alternative 5: 99.6% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.5}{x \cdot x} + 0.75 \cdot {x}^{-4}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp x) x) (* x (sqrt PI)))
  (+ 1.0 (+ (/ 0.5 (* x x)) (* 0.75 (pow x -4.0))))))

C

double code(double x) {
	return (pow(exp(x), x) / (x * sqrt(((double) M_PI)))) * (1.0 + ((0.5 / (x * x)) + (0.75 * pow(x, -4.0))));
}

Java

public static double code(double x) {
	return (Math.pow(Math.exp(x), x) / (x * Math.sqrt(Math.PI))) * (1.0 + ((0.5 / (x * x)) + (0.75 * Math.pow(x, -4.0))));
}

Python

def code(x):
	return (math.pow(math.exp(x), x) / (x * math.sqrt(math.pi))) * (1.0 + ((0.5 / (x * x)) + (0.75 * math.pow(x, -4.0))))

Julia

function code(x)
	return Float64(Float64((exp(x) ^ x) / Float64(x * sqrt(pi))) * Float64(1.0 + Float64(Float64(0.5 / Float64(x * x)) + Float64(0.75 * (x ^ -4.0)))))
end

MATLAB

function tmp = code(x)
	tmp = ((exp(x) ^ x) / (x * sqrt(pi))) * (1.0 + ((0.5 / (x * x)) + (0.75 * (x ^ -4.0))));
end

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[(x * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(0.75 * N[Power[x, -4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around 0 100.0%

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

   1. unpow1 100.0%

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

   2. sqr-pow 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi} \cdot \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   3. fabs-sqr 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi} \cdot \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   4. sqr-pow 100.0%

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

   5. unpow1 100.0%

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

   6. *-commutative 100.0%

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

6. Simplified 100.0%

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

   1. add-log-exp 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \color{blue}{\log \left(e^{\frac{1.875}{{x}^{6}}}\right)}\right)\right)\right) \]

   2. add-cube-cbrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \log \color{blue}{\left(\left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}} \cdot \sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right) \cdot \sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)}\right)\right)\right) \]

   3. log-prod 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \color{blue}{\left(\log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}} \cdot \sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)}\right)\right)\right) \]

   4. div-inv 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{e^{\color{blue}{1.875 \cdot \frac{1}{{x}^{6}}}}} \cdot \sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

   5. exp-prod 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{\color{blue}{{\left(e^{1.875}\right)}^{\left(\frac{1}{{x}^{6}}\right)}}} \cdot \sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

   6. pow-flip 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\color{blue}{\left({x}^{\left(-6\right)}\right)}}} \cdot \sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

   7. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{\color{blue}{-6}}\right)}} \cdot \sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

   8. div-inv 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}} \cdot \sqrt[3]{e^{\color{blue}{1.875 \cdot \frac{1}{{x}^{6}}}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

   9. exp-prod 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}} \cdot \sqrt[3]{\color{blue}{{\left(e^{1.875}\right)}^{\left(\frac{1}{{x}^{6}}\right)}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

   10. pow-flip 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}} \cdot \sqrt[3]{{\left(e^{1.875}\right)}^{\color{blue}{\left({x}^{\left(-6\right)}\right)}}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}} \cdot \sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{\color{blue}{-6}}\right)}}\right) + \log \left(\sqrt[3]{e^{\frac{1.875}{{x}^{6}}}}\right)\right)\right)\right)\right) \]

8. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \color{blue}{\left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}} \cdot \sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right) + \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)}\right)\right)\right) \]
9. Step-by-step derivation

   1. log-prod 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\color{blue}{\left(\log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right) + \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)} + \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right)\right) \]

   2. count-2 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)} + \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right)\right) \]

   3. distribute-lft1-in 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \color{blue}{\left(2 + 1\right) \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)}\right)\right)\right) \]

   4. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \color{blue}{3} \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

10. Simplified 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \color{blue}{3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)}\right)\right)\right) \]
11. Step-by-step derivation

    1. clear-num 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{\frac{1}{\frac{{\left(\left|x\right|\right)}^{4}}{0.75}}} + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

    2. associate-/r/ 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{\frac{1}{{\left(\left|x\right|\right)}^{4}} \cdot 0.75} + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

    3. pow-flip 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{{\left(\left|x\right|\right)}^{\left(-4\right)}} \cdot 0.75 + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

    4. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left({\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-4\right)} \cdot 0.75 + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

    5. fabs-sqr 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left({\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-4\right)} \cdot 0.75 + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

    6. add-sqr-sqrt 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left({\color{blue}{x}}^{\left(-4\right)} \cdot 0.75 + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

    7. metadata-eval 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left({x}^{\color{blue}{-4}} \cdot 0.75 + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

12. Applied egg-rr 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\color{blue}{{x}^{-4} \cdot 0.75} + \left(\frac{0.5}{x \cdot x} + 3 \cdot \log \left(\sqrt[3]{{\left(e^{1.875}\right)}^{\left({x}^{-6}\right)}}\right)\right)\right)\right) \]

13. Taylor expanded in x around inf 99.7%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left({x}^{-4} \cdot 0.75 + \left(\frac{0.5}{x \cdot x} + 3 \cdot \color{blue}{0}\right)\right)\right) \]

14. Final simplification 99.7%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{x \cdot \sqrt{\pi}} \cdot \left(1 + \left(\frac{0.5}{x \cdot x} + 0.75 \cdot {x}^{-4}\right)\right) \]

Reproduce

herbie shell --seed 2023313 
(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)))))))