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

Percentage Accurate: 100.0% → 100.0%

Time: 18.3s

Alternatives: 11

Speedup: 1.9×

• 99.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \geq 0.5\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \left(t_0 \cdot t_0\right) \cdot t_0\\ t_2 := \left(t_1 \cdot t_0\right) \cdot t_0\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(t_0 + \frac{1}{2} \cdot t_1\right) + \frac{3}{4} \cdot t_2\right) + \frac{15}{8} \cdot \left(\left(t_2 \cdot t_0\right) \cdot t_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x)))
        (t_1 (* (* t_0 t_0) t_0))
        (t_2 (* (* t_1 t_0) t_0)))
   (*
    (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x))))
    (+
     (+ (+ t_0 (* (/ 1.0 2.0) t_1)) (* (/ 3.0 4.0) t_2))
     (* (/ 15.0 8.0) (* (* t_2 t_0) t_0))))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / sqrt(((double) M_PI))) * exp((fabs(x) * fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / Math.sqrt(Math.PI)) * Math.exp((Math.abs(x) * Math.abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	t_1 = (t_0 * t_0) * t_0
	t_2 = (t_1 * t_0) * t_0
	return ((1.0 / math.sqrt(math.pi)) * math.exp((math.fabs(x) * math.fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)))

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	t_1 = Float64(Float64(t_0 * t_0) * t_0)
	t_2 = Float64(Float64(t_1 * t_0) * t_0)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * exp(Float64(abs(x) * abs(x)))) * Float64(Float64(Float64(t_0 + Float64(Float64(1.0 / 2.0) * t_1)) + Float64(Float64(3.0 / 4.0) * t_2)) + Float64(Float64(15.0 / 8.0) * Float64(Float64(t_2 * t_0) * t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / abs(x);
	t_1 = (t_0 * t_0) * t_0;
	t_2 = (t_1 * t_0) * t_0;
	tmp = ((1.0 / sqrt(pi)) * exp((abs(x) * abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$0 + N[(N[(1.0 / 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 / 4.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(15.0 / 8.0), $MachinePrecision] * N[(N[(t$95$2 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \left(t_0 \cdot t_0\right) \cdot t_0\\ t_2 := \left(t_1 \cdot t_0\right) \cdot t_0\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(t_0 + \frac{1}{2} \cdot t_1\right) + \frac{3}{4} \cdot t_2\right) + \frac{15}{8} \cdot \left(\left(t_2 \cdot t_0\right) \cdot t_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x)))
        (t_1 (* (* t_0 t_0) t_0))
        (t_2 (* (* t_1 t_0) t_0)))
   (*
    (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x))))
    (+
     (+ (+ t_0 (* (/ 1.0 2.0) t_1)) (* (/ 3.0 4.0) t_2))
     (* (/ 15.0 8.0) (* (* t_2 t_0) t_0))))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / sqrt(((double) M_PI))) * exp((fabs(x) * fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / Math.sqrt(Math.PI)) * Math.exp((Math.abs(x) * Math.abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	t_1 = (t_0 * t_0) * t_0
	t_2 = (t_1 * t_0) * t_0
	return ((1.0 / math.sqrt(math.pi)) * math.exp((math.fabs(x) * math.fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)))

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	t_1 = Float64(Float64(t_0 * t_0) * t_0)
	t_2 = Float64(Float64(t_1 * t_0) * t_0)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * exp(Float64(abs(x) * abs(x)))) * Float64(Float64(Float64(t_0 + Float64(Float64(1.0 / 2.0) * t_1)) + Float64(Float64(3.0 / 4.0) * t_2)) + Float64(Float64(15.0 / 8.0) * Float64(Float64(t_2 * t_0) * t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / abs(x);
	t_1 = (t_0 * t_0) * t_0;
	t_2 = (t_1 * t_0) * t_0;
	tmp = ((1.0 / sqrt(pi)) * exp((abs(x) * abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$0 + N[(N[(1.0 / 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 / 4.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(15.0 / 8.0), $MachinePrecision] * N[(N[(t$95$2 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Alternative 1: 100.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\sqrt{{\left(e^{0.5}\right)}^{\left({x}^{-2}\right)}}\right)\\ t_1 := \frac{1}{\left|x\right|}\\ \mathsf{fma}\left(0.75, {t_1}^{5}, \mathsf{fma}\left(1.875, {t_1}^{7}, \frac{1 + \left(t_0 + t_0\right)}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (log (sqrt (pow (exp 0.5) (pow x -2.0))))) (t_1 (/ 1.0 (fabs x))))
   (*
    (fma
     0.75
     (pow t_1 5.0)
     (fma 1.875 (pow t_1 7.0) (/ (+ 1.0 (+ t_0 t_0)) (fabs x))))
    (/ (pow (exp x) x) (pow (cbrt (sqrt PI)) 3.0)))))

C

double code(double x) {
	double t_0 = log(sqrt(pow(exp(0.5), pow(x, -2.0))));
	double t_1 = 1.0 / fabs(x);
	return fma(0.75, pow(t_1, 5.0), fma(1.875, pow(t_1, 7.0), ((1.0 + (t_0 + t_0)) / fabs(x)))) * (pow(exp(x), x) / pow(cbrt(sqrt(((double) M_PI))), 3.0));
}

Julia

function code(x)
	t_0 = log(sqrt((exp(0.5) ^ (x ^ -2.0))))
	t_1 = Float64(1.0 / abs(x))
	return Float64(fma(0.75, (t_1 ^ 5.0), fma(1.875, (t_1 ^ 7.0), Float64(Float64(1.0 + Float64(t_0 + t_0)) / abs(x)))) * Float64((exp(x) ^ x) / (cbrt(sqrt(pi)) ^ 3.0)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Log[N[Sqrt[N[Power[N[Exp[0.5], $MachinePrecision], N[Power[x, -2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(0.75 * N[Power[t$95$1, 5.0], $MachinePrecision] + N[(1.875 * N[Power[t$95$1, 7.0], $MachinePrecision] + N[(N[(1.0 + N[(t$95$0 + t$95$0), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Power[N[Power[N[Sqrt[Pi], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $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%

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

   1. add-cube-cbrt 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \sqrt[3]{\sqrt{\pi}}}} \]

   2. pow3 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}}} \]

5. Applied egg-rr 100.0%

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

   1. add-log-exp 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \color{blue}{\log \left(e^{\frac{0.5}{x \cdot x}}\right)}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}} \]

   2. add-sqr-sqrt 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \log \color{blue}{\left(\sqrt{e^{\frac{0.5}{x \cdot x}}} \cdot \sqrt{e^{\frac{0.5}{x \cdot x}}}\right)}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}} \]

   3. log-prod 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \color{blue}{\left(\log \left(\sqrt{e^{\frac{0.5}{x \cdot x}}}\right) + \log \left(\sqrt{e^{\frac{0.5}{x \cdot x}}}\right)\right)}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}} \]

   4. div-inv 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \left(\log \left(\sqrt{e^{\color{blue}{0.5 \cdot \frac{1}{x \cdot x}}}}\right) + \log \left(\sqrt{e^{\frac{0.5}{x \cdot x}}}\right)\right)}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}} \]

   5. exp-prod 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \left(\log \left(\sqrt{\color{blue}{{\left(e^{0.5}\right)}^{\left(\frac{1}{x \cdot x}\right)}}}\right) + \log \left(\sqrt{e^{\frac{0.5}{x \cdot x}}}\right)\right)}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}} \]

   6. pow2 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \left(\log \left(\sqrt{{\left(e^{0.5}\right)}^{\left(\frac{1}{\color{blue}{{x}^{2}}}\right)}}\right) + \log \left(\sqrt{e^{\frac{0.5}{x \cdot x}}}\right)\right)}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}} \]

   7. pow-flip 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \left(\log \left(\sqrt{{\left(e^{0.5}\right)}^{\color{blue}{\left({x}^{\left(-2\right)}\right)}}}\right) + \log \left(\sqrt{e^{\frac{0.5}{x \cdot x}}}\right)\right)}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}} \]

   8. metadata-eval 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \left(\log \left(\sqrt{{\left(e^{0.5}\right)}^{\left({x}^{\color{blue}{-2}}\right)}}\right) + \log \left(\sqrt{e^{\frac{0.5}{x \cdot x}}}\right)\right)}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}} \]

   9. div-inv 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \left(\log \left(\sqrt{{\left(e^{0.5}\right)}^{\left({x}^{-2}\right)}}\right) + \log \left(\sqrt{e^{\color{blue}{0.5 \cdot \frac{1}{x \cdot x}}}}\right)\right)}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}} \]

   10. exp-prod 100.0%

       \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \left(\log \left(\sqrt{{\left(e^{0.5}\right)}^{\left({x}^{-2}\right)}}\right) + \log \left(\sqrt{\color{blue}{{\left(e^{0.5}\right)}^{\left(\frac{1}{x \cdot x}\right)}}}\right)\right)}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}} \]

   11. pow2 100.0%

       \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \left(\log \left(\sqrt{{\left(e^{0.5}\right)}^{\left({x}^{-2}\right)}}\right) + \log \left(\sqrt{{\left(e^{0.5}\right)}^{\left(\frac{1}{\color{blue}{{x}^{2}}}\right)}}\right)\right)}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}} \]

   12. pow-flip 100.0%

       \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \left(\log \left(\sqrt{{\left(e^{0.5}\right)}^{\left({x}^{-2}\right)}}\right) + \log \left(\sqrt{{\left(e^{0.5}\right)}^{\color{blue}{\left({x}^{\left(-2\right)}\right)}}}\right)\right)}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}} \]

   13. metadata-eval 100.0%

       \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \left(\log \left(\sqrt{{\left(e^{0.5}\right)}^{\left({x}^{-2}\right)}}\right) + \log \left(\sqrt{{\left(e^{0.5}\right)}^{\left({x}^{\color{blue}{-2}}\right)}}\right)\right)}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}} \]

7. Applied egg-rr 100.0%

   \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \color{blue}{\left(\log \left(\sqrt{{\left(e^{0.5}\right)}^{\left({x}^{-2}\right)}}\right) + \log \left(\sqrt{{\left(e^{0.5}\right)}^{\left({x}^{-2}\right)}}\right)\right)}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}} \]

8. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \left(\log \left(\sqrt{{\left(e^{0.5}\right)}^{\left({x}^{-2}\right)}}\right) + \log \left(\sqrt{{\left(e^{0.5}\right)}^{\left({x}^{-2}\right)}}\right)\right)}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}} \]

Alternative 2: 100.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ \frac{{\left(e^{x}\right)}^{x}}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}} \cdot \mathsf{fma}\left(0.75, {t_0}^{5}, \mathsf{fma}\left(1.875, {t_0}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x))))
   (*
    (/ (pow (exp x) x) (pow (cbrt (sqrt PI)) 3.0))
    (fma
     0.75
     (pow t_0 5.0)
     (fma 1.875 (pow t_0 7.0) (/ (+ 1.0 (/ 0.5 (* x x))) (fabs x)))))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	return (pow(exp(x), x) / pow(cbrt(sqrt(((double) M_PI))), 3.0)) * fma(0.75, pow(t_0, 5.0), fma(1.875, pow(t_0, 7.0), ((1.0 + (0.5 / (x * x))) / fabs(x))));
}

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	return Float64(Float64((exp(x) ^ x) / (cbrt(sqrt(pi)) ^ 3.0)) * fma(0.75, (t_0 ^ 5.0), fma(1.875, (t_0 ^ 7.0), Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) / abs(x)))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Power[N[Power[N[Sqrt[Pi], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(0.75 * N[Power[t$95$0, 5.0], $MachinePrecision] + N[(1.875 * N[Power[t$95$0, 7.0], $MachinePrecision] + N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-cube-cbrt 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \sqrt[3]{\sqrt{\pi}}}} \]

   2. pow3 100.0%

      \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}}} \]

5. Applied egg-rr 100.0%

   \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}}} \]

6. Final simplification 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}} \cdot \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \]

Alternative 3: 100.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ \mathsf{fma}\left(0.75, {t_0}^{5}, \mathsf{fma}\left(1.875, {t_0}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x))))
   (*
    (fma
     0.75
     (pow t_0 5.0)
     (fma 1.875 (pow t_0 7.0) (/ (+ 1.0 (/ 0.5 (* x x))) (fabs x))))
    (/ (pow (exp x) x) (sqrt PI)))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	return fma(0.75, pow(t_0, 5.0), fma(1.875, pow(t_0, 7.0), ((1.0 + (0.5 / (x * x))) / fabs(x)))) * (pow(exp(x), x) / sqrt(((double) M_PI)));
}

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	return Float64(fma(0.75, (t_0 ^ 5.0), fma(1.875, (t_0 ^ 7.0), Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) / abs(x)))) * Float64((exp(x) ^ x) / sqrt(pi)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(0.75 * N[Power[t$95$0, 5.0], $MachinePrecision] + N[(1.875 * N[Power[t$95$0, 7.0], $MachinePrecision] + N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}}} \]

4. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(0.75, {\left(\frac{1}{\left|x\right|}\right)}^{5}, \mathsf{fma}\left(1.875, {\left(\frac{1}{\left|x\right|}\right)}^{7}, \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

Alternative 4: 99.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{{\left(\sqrt[3]{x}\right)}^{2} \cdot \frac{-1}{\frac{-{\pi}^{-0.5}}{\sqrt[3]{x}}}} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/
   (pow (exp x) x)
   (* (pow (cbrt x) 2.0) (/ -1.0 (/ (- (pow PI -0.5)) (cbrt x)))))
  (+
   1.0
   (+ (/ 0.75 (pow (fabs x) 4.0)) (+ (/ 0.5 (* x x)) (/ 1.875 (pow x 6.0)))))))

C

double code(double x) {
	return (pow(exp(x), x) / (pow(cbrt(x), 2.0) * (-1.0 / (-pow(((double) M_PI), -0.5) / cbrt(x))))) * (1.0 + ((0.75 / pow(fabs(x), 4.0)) + ((0.5 / (x * x)) + (1.875 / pow(x, 6.0)))));
}

Java

public static double code(double x) {
	return (Math.pow(Math.exp(x), x) / (Math.pow(Math.cbrt(x), 2.0) * (-1.0 / (-Math.pow(Math.PI, -0.5) / Math.cbrt(x))))) * (1.0 + ((0.75 / Math.pow(Math.abs(x), 4.0)) + ((0.5 / (x * x)) + (1.875 / Math.pow(x, 6.0)))));
}

Julia

function code(x)
	return Float64(Float64((exp(x) ^ x) / Float64((cbrt(x) ^ 2.0) * Float64(-1.0 / Float64(Float64(-(pi ^ -0.5)) / cbrt(x))))) * Float64(1.0 + Float64(Float64(0.75 / (abs(x) ^ 4.0)) + Float64(Float64(0.5 / Float64(x * x)) + Float64(1.875 / (x ^ 6.0))))))
end

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[(N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[(-1.0 / N[((-N[Power[Pi, -0.5], $MachinePrecision]) / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(0.75 / N[Power[N[Abs[x], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 99.6%

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

4. Taylor expanded in x around 0 99.6%

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

   1. *-commutative 99.6%

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

   2. unpow1 99.6%

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

   3. sqr-pow 99.6%

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

   4. fabs-sqr 99.6%

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

   5. sqr-pow 99.6%

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

   6. unpow1 99.6%

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

6. Simplified 99.6%

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

   1. *-un-lft-identity 99.6%

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

   2. associate-*r* 99.6%

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

   3. metadata-eval 99.6%

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

   4. div-inv 99.6%

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

   5. associate-/r/ 99.6%

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

   6. add-cube-cbrt 99.6%

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

   7. metadata-eval 99.6%

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

   8. sqrt-div 99.6%

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

   9. associate-/l* 99.6%

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

   10. pow2 99.6%

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

   11. inv-pow 99.6%

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

   12. sqrt-pow1 99.6%

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

   13. metadata-eval 99.6%

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

8. Applied egg-rr 99.6%

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

   1. frac-2neg 99.6%

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

   2. div-inv 99.6%

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

   3. distribute-neg-frac 99.6%

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

10. Applied egg-rr 99.6%

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

11. Final simplification 99.6%

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

Alternative 5: 99.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{\frac{{\left(\sqrt[3]{x}\right)}^{2}}{\frac{{\pi}^{-0.5}}{\sqrt[3]{x}}}} \cdot \left(1 + \left(\left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right) + \frac{0.75}{{x}^{4}}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp x) x) (/ (pow (cbrt x) 2.0) (/ (pow PI -0.5) (cbrt x))))
  (+ 1.0 (+ (+ (/ 0.5 (* x x)) (/ 1.875 (pow x 6.0))) (/ 0.75 (pow x 4.0))))))

C

double code(double x) {
	return (pow(exp(x), x) / (pow(cbrt(x), 2.0) / (pow(((double) M_PI), -0.5) / cbrt(x)))) * (1.0 + (((0.5 / (x * x)) + (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.pow(Math.cbrt(x), 2.0) / (Math.pow(Math.PI, -0.5) / Math.cbrt(x)))) * (1.0 + (((0.5 / (x * x)) + (1.875 / Math.pow(x, 6.0))) + (0.75 / Math.pow(x, 4.0))));
}

Julia

function code(x)
	return Float64(Float64((exp(x) ^ x) / Float64((cbrt(x) ^ 2.0) / Float64((pi ^ -0.5) / cbrt(x)))) * Float64(1.0 + Float64(Float64(Float64(0.5 / Float64(x * x)) + 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[(N[Power[N[Power[x, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Power[Pi, -0.5], $MachinePrecision] / N[Power[x, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.75 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 99.6%

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

4. Taylor expanded in x around 0 99.6%

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

   1. *-commutative 99.6%

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

   2. unpow1 99.6%

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

   3. sqr-pow 99.6%

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

   4. fabs-sqr 99.6%

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

   5. sqr-pow 99.6%

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

   6. unpow1 99.6%

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

6. Simplified 99.6%

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

   1. *-un-lft-identity 99.6%

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

   2. associate-*r* 99.6%

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

   3. metadata-eval 99.6%

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

   4. div-inv 99.6%

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

   5. associate-/r/ 99.6%

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

   6. add-cube-cbrt 99.6%

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

   7. metadata-eval 99.6%

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

   8. sqrt-div 99.6%

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

   9. associate-/l* 99.6%

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

   10. pow2 99.6%

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

   11. inv-pow 99.6%

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

   12. sqrt-pow1 99.6%

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

   13. metadata-eval 99.6%

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

8. Applied egg-rr 99.6%

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

   1. frac-2neg 2.2%

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

   2. div-inv 2.2%

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

   3. metadata-eval 2.2%

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

   4. metadata-eval 2.2%

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

   5. add-sqr-sqrt 2.2%

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

   6. fabs-sqr 2.2%

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

   7. add-sqr-sqrt 2.2%

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

   8. pow-prod-up 2.2%

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

   9. pow-prod-up 2.2%

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

   10. metadata-eval 2.2%

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

10. Applied egg-rr 99.6%

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

    1. associate-*r/ 2.2%

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

    2. metadata-eval 2.2%

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

    3. neg-mul-1 2.2%

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

    4. associate-/r* 2.2%

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

    5. metadata-eval 2.2%

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

12. Simplified 99.6%

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

13. Final simplification 99.6%

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

Alternative 6: 99.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \left(1 + \left(\left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right) + \frac{0.75}{{x}^{4}}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot \sqrt{\pi}\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (+ 1.0 (+ (+ (/ 0.5 (* x x)) (/ 1.875 (pow x 6.0))) (/ 0.75 (pow x 4.0))))
  (/ (pow (exp x) x) (expm1 (log1p (* x (sqrt PI)))))))

C

double code(double x) {
	return (1.0 + (((0.5 / (x * x)) + (1.875 / pow(x, 6.0))) + (0.75 / pow(x, 4.0)))) * (pow(exp(x), x) / expm1(log1p((x * sqrt(((double) M_PI))))));
}

Java

public static double code(double x) {
	return (1.0 + (((0.5 / (x * x)) + (1.875 / Math.pow(x, 6.0))) + (0.75 / Math.pow(x, 4.0)))) * (Math.pow(Math.exp(x), x) / Math.expm1(Math.log1p((x * Math.sqrt(Math.PI)))));
}

Python

def code(x):
	return (1.0 + (((0.5 / (x * x)) + (1.875 / math.pow(x, 6.0))) + (0.75 / math.pow(x, 4.0)))) * (math.pow(math.exp(x), x) / math.expm1(math.log1p((x * math.sqrt(math.pi)))))

Julia

function code(x)
	return Float64(Float64(1.0 + Float64(Float64(Float64(0.5 / Float64(x * x)) + Float64(1.875 / (x ^ 6.0))) + Float64(0.75 / (x ^ 4.0)))) * Float64((exp(x) ^ x) / expm1(log1p(Float64(x * sqrt(pi))))))
end

Wolfram

code[x_] := N[(N[(1.0 + N[(N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.75 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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]), $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 99.6%

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

4. Taylor expanded in x around 0 99.6%

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

   1. *-commutative 99.6%

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

   2. unpow1 99.6%

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

   3. sqr-pow 99.6%

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

   4. fabs-sqr 99.6%

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

   5. sqr-pow 99.6%

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

   6. unpow1 99.6%

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

6. Simplified 99.6%

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

   1. frac-2neg 2.2%

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

   2. div-inv 2.2%

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

   3. metadata-eval 2.2%

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

   4. metadata-eval 2.2%

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

   5. add-sqr-sqrt 2.2%

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

   6. fabs-sqr 2.2%

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

   7. add-sqr-sqrt 2.2%

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

   8. pow-prod-up 2.2%

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

   9. pow-prod-up 2.2%

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

   10. metadata-eval 2.2%

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

8. Applied egg-rr 99.6%

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

   1. associate-*r/ 2.2%

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

   2. metadata-eval 2.2%

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

   3. neg-mul-1 2.2%

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

   4. associate-/r* 2.2%

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

   5. metadata-eval 2.2%

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

10. Simplified 99.6%

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

    1. expm1-log1p-u 99.6%

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

12. Applied egg-rr 99.6%

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

13. Final simplification 99.6%

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

Alternative 7: 99.9% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x):
	return (1.0 + (((0.5 / (x * x)) + (1.875 / math.pow(x, 6.0))) + (0.75 / math.pow(x, 4.0)))) * (math.pow(math.exp(x), x) / (x * math.sqrt(math.pi)))

Julia

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

MATLAB

function tmp = code(x)
	tmp = (1.0 + (((0.5 / (x * x)) + (1.875 / (x ^ 6.0))) + (0.75 / (x ^ 4.0)))) * ((exp(x) ^ x) / (x * sqrt(pi)));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 99.6%

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

4. Taylor expanded in x around 0 99.6%

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

   1. *-commutative 99.6%

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

   2. unpow1 99.6%

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

   3. sqr-pow 99.6%

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

   4. fabs-sqr 99.6%

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

   5. sqr-pow 99.6%

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

   6. unpow1 99.6%

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

6. Simplified 99.6%

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

   1. frac-2neg 2.2%

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

   2. div-inv 2.2%

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

   3. metadata-eval 2.2%

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

   4. metadata-eval 2.2%

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

   5. add-sqr-sqrt 2.2%

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

   6. fabs-sqr 2.2%

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

   7. add-sqr-sqrt 2.2%

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

   8. pow-prod-up 2.2%

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

   9. pow-prod-up 2.2%

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

   10. metadata-eval 2.2%

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

8. Applied egg-rr 99.6%

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

   1. associate-*r/ 2.2%

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

   2. metadata-eval 2.2%

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

   3. neg-mul-1 2.2%

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

   4. associate-/r* 2.2%

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

   5. metadata-eval 2.2%

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

10. Simplified 99.6%

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

11. Final simplification 99.6%

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

Alternative 8: 51.7% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \left(1 + \left(\left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right) + \frac{0.75}{{x}^{4}}\right)\right) \cdot \sqrt{\frac{{\left(x + \frac{1}{x}\right)}^{2}}{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (+ 1.0 (+ (+ (/ 0.5 (* x x)) (/ 1.875 (pow x 6.0))) (/ 0.75 (pow x 4.0))))
  (sqrt (/ (pow (+ x (/ 1.0 x)) 2.0) PI))))

C

double code(double x) {
	return (1.0 + (((0.5 / (x * x)) + (1.875 / pow(x, 6.0))) + (0.75 / pow(x, 4.0)))) * sqrt((pow((x + (1.0 / x)), 2.0) / ((double) M_PI)));
}

Java

public static double code(double x) {
	return (1.0 + (((0.5 / (x * x)) + (1.875 / Math.pow(x, 6.0))) + (0.75 / Math.pow(x, 4.0)))) * Math.sqrt((Math.pow((x + (1.0 / x)), 2.0) / Math.PI));
}

Python

def code(x):
	return (1.0 + (((0.5 / (x * x)) + (1.875 / math.pow(x, 6.0))) + (0.75 / math.pow(x, 4.0)))) * math.sqrt((math.pow((x + (1.0 / x)), 2.0) / math.pi))

Julia

function code(x)
	return Float64(Float64(1.0 + Float64(Float64(Float64(0.5 / Float64(x * x)) + Float64(1.875 / (x ^ 6.0))) + Float64(0.75 / (x ^ 4.0)))) * sqrt(Float64((Float64(x + Float64(1.0 / x)) ^ 2.0) / pi)))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 + (((0.5 / (x * x)) + (1.875 / (x ^ 6.0))) + (0.75 / (x ^ 4.0)))) * sqrt((((x + (1.0 / x)) ^ 2.0) / pi));
end

Wolfram

code[x_] := N[(N[(1.0 + N[(N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.75 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Power[N[(x + N[(1.0 / x), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / 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 99.6%

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

4. Taylor expanded in x around 0 56.1%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \frac{1}{\left|x\right|} + \frac{{x}^{2}}{\left|x\right|} \cdot \sqrt{\frac{1}{\pi}}\right)} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]
5. Step-by-step derivation

   1. *-commutative 56.1%

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

   2. distribute-lft-out 56.1%

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

   3. unpow1 56.1%

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

   4. sqr-pow 56.1%

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

   5. fabs-sqr 56.1%

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

   6. sqr-pow 56.1%

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

   7. unpow1 56.1%

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

   8. unpow1 56.1%

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

   9. sqr-pow 56.1%

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

   10. fabs-sqr 56.1%

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

   11. sqr-pow 56.1%

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

   12. unpow1 56.1%

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

6. Simplified 56.1%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{x} + \frac{{x}^{2}}{x}\right)\right)} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]
7. Step-by-step derivation

   1. expm1-log1p-u 56.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{x} + \frac{{x}^{2}}{x}\right)\right)\right)} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   2. expm1-udef 56.1%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{x} + \frac{{x}^{2}}{x}\right)\right)} - 1\right)} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   3. *-commutative 56.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(\frac{1}{x} + \frac{{x}^{2}}{x}\right) \cdot \sqrt{\frac{1}{\pi}}}\right)} - 1\right) \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   4. sqrt-div 56.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\left(\frac{1}{x} + \frac{{x}^{2}}{x}\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right)} - 1\right) \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   5. metadata-eval 56.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\left(\frac{1}{x} + \frac{{x}^{2}}{x}\right) \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   6. un-div-inv 56.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{x} + \frac{{x}^{2}}{x}}{\sqrt{\pi}}}\right)} - 1\right) \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   7. +-commutative 56.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{{x}^{2}}{x} + \frac{1}{x}}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   8. pow1 56.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\frac{{x}^{2}}{\color{blue}{{x}^{1}}} + \frac{1}{x}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   9. pow-div 5.7%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{x}^{\left(2 - 1\right)}} + \frac{1}{x}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   10. metadata-eval 5.7%

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

   11. pow1 5.7%

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

8. Applied egg-rr 5.7%

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

   1. expm1-def 5.7%

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

   2. expm1-log1p 5.7%

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

10. Simplified 5.7%

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

    1. frac-2neg 2.2%

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

    2. div-inv 2.2%

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

    3. metadata-eval 2.2%

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

    4. metadata-eval 2.2%

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

    5. add-sqr-sqrt 2.2%

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

    6. fabs-sqr 2.2%

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

    7. add-sqr-sqrt 2.2%

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

    8. pow-prod-up 2.2%

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

    9. pow-prod-up 2.2%

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

    10. metadata-eval 2.2%

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

12. Applied egg-rr 5.7%

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

    1. associate-*r/ 2.2%

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

    2. metadata-eval 2.2%

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

    3. neg-mul-1 2.2%

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

    4. associate-/r* 2.2%

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

    5. metadata-eval 2.2%

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

14. Simplified 5.7%

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

    1. add-sqr-sqrt 5.7%

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

    2. sqrt-unprod 56.1%

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

    3. frac-times 56.1%

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

    4. pow2 56.1%

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

    5. add-sqr-sqrt 56.1%

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

16. Applied egg-rr 56.1%

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

17. Final simplification 56.1%

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

Alternative 9: 5.4% accurate, 5.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (*
  (+ 1.0 (+ (+ (/ 0.5 (* x x)) (/ 1.875 (pow x 6.0))) (/ 0.75 (pow x 4.0))))
  (/ (+ x (/ 1.0 x)) (sqrt PI))))

C

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

Java

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

Python

def code(x):
	return (1.0 + (((0.5 / (x * x)) + (1.875 / math.pow(x, 6.0))) + (0.75 / math.pow(x, 4.0)))) * ((x + (1.0 / x)) / math.sqrt(math.pi))

Julia

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

MATLAB

function tmp = code(x)
	tmp = (1.0 + (((0.5 / (x * x)) + (1.875 / (x ^ 6.0))) + (0.75 / (x ^ 4.0)))) * ((x + (1.0 / x)) / sqrt(pi));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 99.6%

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

4. Taylor expanded in x around 0 56.1%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \frac{1}{\left|x\right|} + \frac{{x}^{2}}{\left|x\right|} \cdot \sqrt{\frac{1}{\pi}}\right)} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]
5. Step-by-step derivation

   1. *-commutative 56.1%

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

   2. distribute-lft-out 56.1%

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

   3. unpow1 56.1%

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

   4. sqr-pow 56.1%

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

   5. fabs-sqr 56.1%

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

   6. sqr-pow 56.1%

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

   7. unpow1 56.1%

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

   8. unpow1 56.1%

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

   9. sqr-pow 56.1%

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

   10. fabs-sqr 56.1%

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

   11. sqr-pow 56.1%

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

   12. unpow1 56.1%

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

6. Simplified 56.1%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{x} + \frac{{x}^{2}}{x}\right)\right)} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]
7. Step-by-step derivation

   1. expm1-log1p-u 56.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{x} + \frac{{x}^{2}}{x}\right)\right)\right)} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   2. expm1-udef 56.1%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{x} + \frac{{x}^{2}}{x}\right)\right)} - 1\right)} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   3. *-commutative 56.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(\frac{1}{x} + \frac{{x}^{2}}{x}\right) \cdot \sqrt{\frac{1}{\pi}}}\right)} - 1\right) \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   4. sqrt-div 56.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\left(\frac{1}{x} + \frac{{x}^{2}}{x}\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right)} - 1\right) \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   5. metadata-eval 56.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\left(\frac{1}{x} + \frac{{x}^{2}}{x}\right) \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   6. un-div-inv 56.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{x} + \frac{{x}^{2}}{x}}{\sqrt{\pi}}}\right)} - 1\right) \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   7. +-commutative 56.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{{x}^{2}}{x} + \frac{1}{x}}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   8. pow1 56.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\frac{{x}^{2}}{\color{blue}{{x}^{1}}} + \frac{1}{x}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   9. pow-div 5.7%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{x}^{\left(2 - 1\right)}} + \frac{1}{x}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   10. metadata-eval 5.7%

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

   11. pow1 5.7%

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

8. Applied egg-rr 5.7%

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

   1. expm1-def 5.7%

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

   2. expm1-log1p 5.7%

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

10. Simplified 5.7%

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

    1. frac-2neg 2.2%

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

    2. div-inv 2.2%

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

    3. metadata-eval 2.2%

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

    4. metadata-eval 2.2%

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

    5. add-sqr-sqrt 2.2%

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

    6. fabs-sqr 2.2%

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

    7. add-sqr-sqrt 2.2%

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

    8. pow-prod-up 2.2%

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

    9. pow-prod-up 2.2%

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

    10. metadata-eval 2.2%

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

12. Applied egg-rr 5.7%

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

    1. associate-*r/ 2.2%

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

    2. metadata-eval 2.2%

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

    3. neg-mul-1 2.2%

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

    4. associate-/r* 2.2%

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

    5. metadata-eval 2.2%

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

14. Simplified 5.7%

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

15. Final simplification 5.7%

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

Alternative 10: 5.4% accurate, 5.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (*
  (+ 1.0 (+ (+ (/ 0.5 (* x x)) (/ 1.875 (pow x 6.0))) (/ 0.75 (pow x 4.0))))
  (* x (sqrt (/ 1.0 PI)))))

C

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

Java

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

Python

def code(x):
	return (1.0 + (((0.5 / (x * x)) + (1.875 / math.pow(x, 6.0))) + (0.75 / math.pow(x, 4.0)))) * (x * math.sqrt((1.0 / math.pi)))

Julia

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

MATLAB

function tmp = code(x)
	tmp = (1.0 + (((0.5 / (x * x)) + (1.875 / (x ^ 6.0))) + (0.75 / (x ^ 4.0)))) * (x * sqrt((1.0 / pi)));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 99.6%

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

4. Taylor expanded in x around 0 56.1%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \frac{1}{\left|x\right|} + \frac{{x}^{2}}{\left|x\right|} \cdot \sqrt{\frac{1}{\pi}}\right)} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]
5. Step-by-step derivation

   1. *-commutative 56.1%

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

   2. distribute-lft-out 56.1%

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

   3. unpow1 56.1%

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

   4. sqr-pow 56.1%

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

   5. fabs-sqr 56.1%

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

   6. sqr-pow 56.1%

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

   7. unpow1 56.1%

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

   8. unpow1 56.1%

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

   9. sqr-pow 56.1%

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

   10. fabs-sqr 56.1%

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

   11. sqr-pow 56.1%

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

   12. unpow1 56.1%

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

6. Simplified 56.1%

   \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{x} + \frac{{x}^{2}}{x}\right)\right)} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]
7. Step-by-step derivation

   1. expm1-log1p-u 56.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{x} + \frac{{x}^{2}}{x}\right)\right)\right)} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   2. expm1-udef 56.1%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{x} + \frac{{x}^{2}}{x}\right)\right)} - 1\right)} \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   3. *-commutative 56.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\left(\frac{1}{x} + \frac{{x}^{2}}{x}\right) \cdot \sqrt{\frac{1}{\pi}}}\right)} - 1\right) \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   4. sqrt-div 56.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\left(\frac{1}{x} + \frac{{x}^{2}}{x}\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right)} - 1\right) \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   5. metadata-eval 56.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\left(\frac{1}{x} + \frac{{x}^{2}}{x}\right) \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   6. un-div-inv 56.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{1}{x} + \frac{{x}^{2}}{x}}{\sqrt{\pi}}}\right)} - 1\right) \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   7. +-commutative 56.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{{x}^{2}}{x} + \frac{1}{x}}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   8. pow1 56.1%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\frac{{x}^{2}}{\color{blue}{{x}^{1}}} + \frac{1}{x}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   9. pow-div 5.7%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{x}^{\left(2 - 1\right)}} + \frac{1}{x}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(1 + \left(\frac{0.75}{{\left(\left|x\right|\right)}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

   10. metadata-eval 5.7%

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

   11. pow1 5.7%

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

8. Applied egg-rr 5.7%

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

   1. expm1-def 5.7%

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

   2. expm1-log1p 5.7%

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

10. Simplified 5.7%

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

    1. frac-2neg 2.2%

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

    2. div-inv 2.2%

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

    3. metadata-eval 2.2%

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

    4. metadata-eval 2.2%

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

    5. add-sqr-sqrt 2.2%

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

    6. fabs-sqr 2.2%

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

    7. add-sqr-sqrt 2.2%

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

    8. pow-prod-up 2.2%

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

    9. pow-prod-up 2.2%

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

    10. metadata-eval 2.2%

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

12. Applied egg-rr 5.7%

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

    1. associate-*r/ 2.2%

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

    2. metadata-eval 2.2%

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

    3. neg-mul-1 2.2%

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

    4. associate-/r* 2.2%

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

    5. metadata-eval 2.2%

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

14. Simplified 5.7%

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

15. Taylor expanded in x around inf 5.7%

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

16. Final simplification 5.7%

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

Alternative 11: 2.3% accurate, 5.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (*
  (+ 1.0 (+ (+ (/ 0.5 (* x x)) (/ 1.875 (pow x 6.0))) (/ 0.75 (pow x 4.0))))
  (/ (pow PI -0.5) x)))

C

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

Java

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

Python

def code(x):
	return (1.0 + (((0.5 / (x * x)) + (1.875 / math.pow(x, 6.0))) + (0.75 / math.pow(x, 4.0)))) * (math.pow(math.pi, -0.5) / x)

Julia

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

MATLAB

function tmp = code(x)
	tmp = (1.0 + (((0.5 / (x * x)) + (1.875 / (x ^ 6.0))) + (0.75 / (x ^ 4.0)))) * ((pi ^ -0.5) / x);
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 99.6%

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

4. Taylor expanded in x around 0 2.2%

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

   1. associate-*r/ 2.2%

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

   2. *-rgt-identity 2.2%

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

   3. unpow1 2.2%

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

   4. sqr-pow 2.2%

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

   5. fabs-sqr 2.2%

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

   6. sqr-pow 2.2%

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

   7. unpow1 2.2%

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

6. Simplified 2.2%

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

   1. frac-2neg 2.2%

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

   2. div-inv 2.2%

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

   3. metadata-eval 2.2%

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

   4. metadata-eval 2.2%

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

   5. add-sqr-sqrt 2.2%

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

   6. fabs-sqr 2.2%

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

   7. add-sqr-sqrt 2.2%

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

   8. pow-prod-up 2.2%

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

   9. pow-prod-up 2.2%

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

   10. metadata-eval 2.2%

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

8. Applied egg-rr 2.2%

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

   1. associate-*r/ 2.2%

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

   2. metadata-eval 2.2%

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

   3. neg-mul-1 2.2%

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

   4. associate-/r* 2.2%

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

   5. metadata-eval 2.2%

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

10. Simplified 2.2%

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

    1. add-sqr-sqrt 2.2%

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

    2. fabs-sqr 2.2%

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

    3. add-sqr-sqrt 2.2%

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

    4. expm1-log1p-u 2.2%

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

    5. un-div-inv 2.2%

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

    6. expm1-udef 1.6%

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

    7. un-div-inv 1.6%

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

    8. inv-pow 1.6%

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

    9. sqrt-pow1 1.6%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}}}{\left|x\right|}\right)} - 1\right) \cdot \left(1 + \left(\frac{0.75}{{x}^{4}} + \left(\frac{0.5}{x \cdot x} + \frac{1.875}{{x}^{6}}\right)\right)\right) \]

    10. metadata-eval 1.6%

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

    11. add-sqr-sqrt 1.6%

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

    12. fabs-sqr 1.6%

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

    13. add-sqr-sqrt 1.6%

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

12. Applied egg-rr 1.6%

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

    1. expm1-def 2.2%

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

    2. expm1-log1p 2.2%

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

14. Simplified 2.2%

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

15. Final simplification 2.2%

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

Reproduce

herbie shell --seed 2023300 
(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)))))))