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

Percentage Accurate: 100.0% → 100.0%

Time: 19.1s

Alternatives: 8

Speedup: 4.2×

• 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, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{{\pi}^{0.75}}\\ \frac{{\left(e^{x}\right)}^{x}}{t_0 \cdot t_0} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{0.75}{{x}^{5}} + \frac{\frac{1.875}{{x}^{6}}}{x}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (pow PI 0.75))))
   (*
    (/ (pow (exp x) x) (* t_0 t_0))
    (fma
     (+ 1.0 (/ 0.5 (* x x)))
     (/ 1.0 (fabs x))
     (+ (/ 0.75 (pow x 5.0)) (/ (/ 1.875 (pow x 6.0)) x))))))

C

double code(double x) {
	double t_0 = cbrt(pow(((double) M_PI), 0.75));
	return (pow(exp(x), x) / (t_0 * t_0)) * fma((1.0 + (0.5 / (x * x))), (1.0 / fabs(x)), ((0.75 / pow(x, 5.0)) + ((1.875 / pow(x, 6.0)) / x)));
}

Julia

function code(x)
	t_0 = cbrt((pi ^ 0.75))
	return Float64(Float64((exp(x) ^ x) / Float64(t_0 * t_0)) * fma(Float64(1.0 + Float64(0.5 / Float64(x * x))), Float64(1.0 / abs(x)), Float64(Float64(0.75 / (x ^ 5.0)) + Float64(Float64(1.875 / (x ^ 6.0)) / x))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Power[Pi, 0.75], $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(0.75 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

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

   1. add-cbrt-cube 100.0%

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

   2. pow1/3 100.0%

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

   3. add-sqr-sqrt 100.0%

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

   4. pow1 100.0%

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

   5. pow1/2 100.0%

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

   6. pow-prod-up 100.0%

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

   7. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{{\left(\frac{1}{\left|x\right|}\right)}^{4}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

5. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left({\pi}^{1.5}\right)}^{0.3333333333333333}}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{{\left(\frac{1}{\left|x\right|}\right)}^{4}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]
6. Step-by-step derivation

   1. unpow1/3 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\sqrt[3]{{\pi}^{1.5}}}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{{\left(\frac{1}{\left|x\right|}\right)}^{4}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

7. Simplified 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\sqrt[3]{{\pi}^{1.5}}}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{{\left(\frac{1}{\left|x\right|}\right)}^{4}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]
8. Step-by-step derivation

   1. pow1/3 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left({\pi}^{1.5}\right)}^{0.3333333333333333}}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{{\left(\frac{1}{\left|x\right|}\right)}^{4}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   2. add-sqr-sqrt 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\color{blue}{\left(\sqrt{{\pi}^{1.5}} \cdot \sqrt{{\pi}^{1.5}}\right)}}^{0.3333333333333333}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{{\left(\frac{1}{\left|x\right|}\right)}^{4}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   3. unpow-prod-down 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left(\sqrt{{\pi}^{1.5}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{{\pi}^{1.5}}\right)}^{0.3333333333333333}}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{{\left(\frac{1}{\left|x\right|}\right)}^{4}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   4. sqrt-pow1 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\color{blue}{\left({\pi}^{\left(\frac{1.5}{2}\right)}\right)}}^{0.3333333333333333} \cdot {\left(\sqrt{{\pi}^{1.5}}\right)}^{0.3333333333333333}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{{\left(\frac{1}{\left|x\right|}\right)}^{4}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   5. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{\color{blue}{0.75}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{{\pi}^{1.5}}\right)}^{0.3333333333333333}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{{\left(\frac{1}{\left|x\right|}\right)}^{4}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   6. sqrt-pow1 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.75}\right)}^{0.3333333333333333} \cdot {\color{blue}{\left({\pi}^{\left(\frac{1.5}{2}\right)}\right)}}^{0.3333333333333333}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{{\left(\frac{1}{\left|x\right|}\right)}^{4}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

   7. metadata-eval 100.0%

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

9. Applied egg-rr 100.0%

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

    1. unpow1/3 100.0%

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

    2. unpow1/3 100.0%

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

11. Simplified 100.0%

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

    1. expm1-log1p-u 100.0%

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

    2. expm1-udef 100.0%

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

    3. inv-pow 100.0%

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

    4. pow-pow 100.0%

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

    5. add-sqr-sqrt 100.0%

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

    6. fabs-sqr 100.0%

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

    7. add-sqr-sqrt 100.0%

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

    8. metadata-eval 100.0%

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

13. Applied egg-rr 100.0%

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

    1. expm1-def 100.0%

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

    2. expm1-log1p 100.0%

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

15. Simplified 100.0%

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

16. Taylor expanded in x around 0 100.0%

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

    1. associate-*r/ 100.0%

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

    2. metadata-eval 100.0%

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

    3. *-commutative 100.0%

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

    4. rem-square-sqrt 100.0%

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

    5. fabs-sqr 100.0%

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

    6. rem-square-sqrt 100.0%

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

    7. pow-plus 100.0%

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

    8. metadata-eval 100.0%

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

    9. associate-*r/ 100.0%

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

    10. metadata-eval 100.0%

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

    11. metadata-eval 100.0%

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

    12. pow-sqr 100.0%

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

    13. cube-prod 100.0%

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

    14. *-commutative 100.0%

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

    15. associate-/r* 100.0%

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

    16. cube-prod 100.0%

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

    17. pow-sqr 100.0%

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

    18. metadata-eval 100.0%

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

    19. rem-square-sqrt 100.0%

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

    20. fabs-sqr 100.0%

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

    21. rem-square-sqrt 100.0%

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

18. Simplified 100.0%

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

19. Final simplification 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{0.75}} \cdot \sqrt[3]{{\pi}^{0.75}}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{0.75}{{x}^{5}} + \frac{\frac{1.875}{{x}^{6}}}{x}\right) \]

Alternative 2: 100.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, e^{\mathsf{log1p}\left(\frac{{x}^{-4}}{x} \cdot \mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right)\right)} + -1\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (pow (exp x) x) (sqrt PI))
  (fma
   (+ 1.0 (/ 0.5 (* x x)))
   (/ 1.0 (fabs x))
   (+
    (exp (log1p (* (/ (pow x -4.0) x) (fma 1.875 (pow x -2.0) 0.75))))
    -1.0))))

C

double code(double x) {
	return (pow(exp(x), x) / sqrt(((double) M_PI))) * fma((1.0 + (0.5 / (x * x))), (1.0 / fabs(x)), (exp(log1p(((pow(x, -4.0) / x) * fma(1.875, pow(x, -2.0), 0.75)))) + -1.0));
}

Julia

function code(x)
	return Float64(Float64((exp(x) ^ x) / sqrt(pi)) * fma(Float64(1.0 + Float64(0.5 / Float64(x * x))), Float64(1.0 / abs(x)), Float64(exp(log1p(Float64(Float64((x ^ -4.0) / x) * fma(1.875, (x ^ -2.0), 0.75)))) + -1.0)))
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 100.0%

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

   3. inv-pow 100.0%

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

   4. pow-pow 100.0%

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

   5. add-sqr-sqrt 100.0%

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

   6. fabs-sqr 100.0%

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

   7. add-sqr-sqrt 100.0%

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

   8. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

7. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 100.0%

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

   3. associate-*l/ 100.0%

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

   4. add-sqr-sqrt 100.0%

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

   5. fabs-sqr 100.0%

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

   6. add-sqr-sqrt 100.0%

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

   7. associate-*l/ 100.0%

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

   8. +-commutative 100.0%

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

   9. div-inv 100.0%

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

   10. fma-def 100.0%

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

   11. pow2 100.0%

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

   12. pow-flip 100.0%

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

   13. metadata-eval 100.0%

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

9. Applied egg-rr 100.0%

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

10. Final simplification 100.0%

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

Alternative 3: 100.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{0.75}{{x}^{5}} + \frac{\frac{1.875}{{x}^{6}}}{x}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fma
   (+ 1.0 (/ 0.5 (* x x)))
   (/ 1.0 (fabs x))
   (+ (/ 0.75 (pow x 5.0)) (/ (/ 1.875 (pow x 6.0)) x)))
  (/ (pow (exp x) x) (cbrt (pow PI 1.5)))))

C

double code(double x) {
	return fma((1.0 + (0.5 / (x * x))), (1.0 / fabs(x)), ((0.75 / pow(x, 5.0)) + ((1.875 / pow(x, 6.0)) / x))) * (pow(exp(x), x) / cbrt(pow(((double) M_PI), 1.5)));
}

Julia

function code(x)
	return Float64(fma(Float64(1.0 + Float64(0.5 / Float64(x * x))), Float64(1.0 / abs(x)), Float64(Float64(0.75 / (x ^ 5.0)) + Float64(Float64(1.875 / (x ^ 6.0)) / x))) * Float64((exp(x) ^ x) / cbrt((pi ^ 1.5))))
end

Wolfram

code[x_] := N[(N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(0.75 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Power[N[Power[Pi, 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

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

   1. add-cbrt-cube 100.0%

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

   2. pow1/3 100.0%

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

   3. add-sqr-sqrt 100.0%

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

   4. pow1 100.0%

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

   5. pow1/2 100.0%

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

   6. pow-prod-up 100.0%

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

   7. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{{\left(\frac{1}{\left|x\right|}\right)}^{4}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

5. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left({\pi}^{1.5}\right)}^{0.3333333333333333}}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{{\left(\frac{1}{\left|x\right|}\right)}^{4}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]
6. Step-by-step derivation

   1. unpow1/3 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\sqrt[3]{{\pi}^{1.5}}}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{{\left(\frac{1}{\left|x\right|}\right)}^{4}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

7. Simplified 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\sqrt[3]{{\pi}^{1.5}}}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{{\left(\frac{1}{\left|x\right|}\right)}^{4}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]
8. Step-by-step derivation

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 100.0%

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

   3. inv-pow 100.0%

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

   4. pow-pow 100.0%

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

   5. add-sqr-sqrt 100.0%

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

   6. fabs-sqr 100.0%

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

   7. add-sqr-sqrt 100.0%

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

   8. metadata-eval 100.0%

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

9. Applied egg-rr 100.0%

   \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{\color{blue}{e^{\mathsf{log1p}\left({x}^{-4}\right)} - 1}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]
10. Step-by-step derivation

    1. expm1-def 100.0%

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

    2. expm1-log1p 100.0%

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

11. Simplified 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{\color{blue}{{x}^{-4}}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

12. Taylor expanded in x around 0 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \color{blue}{0.75 \cdot \frac{1}{\left|x\right| \cdot {x}^{4}} + 1.875 \cdot \frac{1}{\left|x\right| \cdot {x}^{6}}}\right) \]
13. Step-by-step derivation

    1. associate-*r/ 100.0%

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

    2. metadata-eval 100.0%

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

    3. *-commutative 100.0%

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

    4. rem-square-sqrt 100.0%

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

    5. fabs-sqr 100.0%

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

    6. rem-square-sqrt 100.0%

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

    7. pow-plus 100.0%

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

    8. metadata-eval 100.0%

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

    9. associate-*r/ 100.0%

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

    10. metadata-eval 100.0%

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

    11. metadata-eval 100.0%

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

    12. pow-sqr 100.0%

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

    13. cube-prod 100.0%

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

    14. *-commutative 100.0%

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

    15. associate-/r* 100.0%

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

    16. cube-prod 100.0%

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

    17. pow-sqr 100.0%

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

    18. metadata-eval 100.0%

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

    19. rem-square-sqrt 100.0%

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

    20. fabs-sqr 100.0%

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

    21. rem-square-sqrt 100.0%

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

14. Simplified 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \color{blue}{\frac{0.75}{{x}^{5}} + \frac{\frac{1.875}{{x}^{6}}}{x}}\right) \]

15. Final simplification 100.0%

    \[\leadsto \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{0.75}{{x}^{5}} + \frac{\frac{1.875}{{x}^{6}}}{x}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \]

Alternative 4: 100.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{0.75}{{x}^{5}} + \frac{\frac{1.875}{{x}^{6}}}{x}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fma
   (+ 1.0 (/ 0.5 (* x x)))
   (/ 1.0 (fabs x))
   (+ (/ 0.75 (pow x 5.0)) (/ (/ 1.875 (pow x 6.0)) x)))
  (/ (pow (exp x) x) (sqrt PI))))

C

double code(double x) {
	return fma((1.0 + (0.5 / (x * x))), (1.0 / fabs(x)), ((0.75 / pow(x, 5.0)) + ((1.875 / pow(x, 6.0)) / x))) * (pow(exp(x), x) / sqrt(((double) M_PI)));
}

Julia

function code(x)
	return Float64(fma(Float64(1.0 + Float64(0.5 / Float64(x * x))), Float64(1.0 / abs(x)), Float64(Float64(0.75 / (x ^ 5.0)) + Float64(Float64(1.875 / (x ^ 6.0)) / x))) * Float64((exp(x) ^ x) / sqrt(pi)))
end

Wolfram

code[x_] := N[(N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(0.75 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

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

   2. expm1-udef 100.0%

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

   3. inv-pow 100.0%

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

   4. pow-pow 100.0%

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

   5. add-sqr-sqrt 100.0%

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

   6. fabs-sqr 100.0%

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

   7. add-sqr-sqrt 100.0%

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

   8. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

   1. expm1-def 100.0%

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

   2. expm1-log1p 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around 0 100.0%

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

   1. associate-*r/ 100.0%

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

   2. metadata-eval 100.0%

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

   3. *-commutative 100.0%

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

   4. rem-square-sqrt 100.0%

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

   5. fabs-sqr 100.0%

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

   6. rem-square-sqrt 100.0%

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

   7. pow-plus 100.0%

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

   8. metadata-eval 100.0%

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

   9. associate-*r/ 100.0%

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

   10. metadata-eval 100.0%

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

   11. metadata-eval 100.0%

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

   12. pow-sqr 100.0%

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

   13. cube-prod 100.0%

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

   14. *-commutative 100.0%

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

   15. associate-/r* 100.0%

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

   16. cube-prod 100.0%

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

   17. pow-sqr 100.0%

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

   18. metadata-eval 100.0%

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

   19. rem-square-sqrt 100.0%

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

   20. fabs-sqr 100.0%

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

   21. rem-square-sqrt 100.0%

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

10. Simplified 100.0%

    \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \color{blue}{\frac{0.75}{{x}^{5}} + \frac{\frac{1.875}{{x}^{6}}}{x}}\right) \]

11. Final simplification 100.0%

    \[\leadsto \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{0.75}{{x}^{5}} + \frac{\frac{1.875}{{x}^{6}}}{x}\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

Alternative 5: 100.0% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Power[N[Power[Pi, 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 + N[Power[x, -5.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] * N[(0.75 + N[(1.875 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

   1. add-log-exp 99.9%

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

   2. inv-pow 99.9%

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

   3. pow-pow 99.9%

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

   4. add-sqr-sqrt 99.9%

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

   5. fabs-sqr 99.9%

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

   6. add-sqr-sqrt 99.9%

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

   7. metadata-eval 99.9%

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

5. Applied egg-rr 99.9%

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

   1. add-log-exp 99.9%

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

   2. expm1-log1p-u 99.9%

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

   3. expm1-def 100.0%

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

   4. log1p-udef 100.0%

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

   5. add-exp-log 100.0%

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

   6. +-commutative 100.0%

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

7. Applied egg-rr 100.0%

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

   1. add-cbrt-cube 100.0%

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

   2. pow1/3 100.0%

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

   3. add-sqr-sqrt 100.0%

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

   4. pow1 100.0%

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

   5. pow1/2 100.0%

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

   6. pow-prod-up 100.0%

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

   7. metadata-eval 100.0%

      \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{{\left(\frac{1}{\left|x\right|}\right)}^{4}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

9. Applied egg-rr 100.0%

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

    1. unpow1/3 100.0%

       \[\leadsto \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\sqrt[3]{{\pi}^{1.5}}}} \cdot \mathsf{fma}\left(1 + \frac{0.5}{x \cdot x}, \frac{1}{\left|x\right|}, \frac{{\left(\frac{1}{\left|x\right|}\right)}^{4}}{\left|x\right|} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right) \]

11. Simplified 100.0%

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

12. Final simplification 100.0%

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

Alternative 6: 100.0% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 + N[Power[x, -5.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] * N[(0.75 + N[(1.875 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

   1. add-log-exp 99.9%

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

   2. inv-pow 99.9%

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

   3. pow-pow 99.9%

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

   4. add-sqr-sqrt 99.9%

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

   5. fabs-sqr 99.9%

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

   6. add-sqr-sqrt 99.9%

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

   7. metadata-eval 99.9%

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

5. Applied egg-rr 99.9%

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

   1. add-log-exp 99.9%

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

   2. expm1-log1p-u 99.9%

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

   3. expm1-def 100.0%

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

   4. log1p-udef 100.0%

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

   5. add-exp-log 100.0%

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

   6. +-commutative 100.0%

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

7. Applied egg-rr 100.0%

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

8. Final simplification 100.0%

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

Alternative 7: 100.0% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[Power[x, -5.0], $MachinePrecision] * N[(0.75 + N[(1.875 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

   1. expm1-log1p-u 99.9%

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

   2. expm1-udef 100.0%

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

   3. inv-pow 100.0%

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

   4. pow-pow 100.0%

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

   5. add-sqr-sqrt 100.0%

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

   6. fabs-sqr 100.0%

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

   7. add-sqr-sqrt 100.0%

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

   8. metadata-eval 100.0%

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

5. Applied egg-rr 100.0%

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

   1. expm1-def 99.9%

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

   2. expm1-log1p 99.9%

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Alternative 8: 99.9% accurate, 5.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.9%

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

   1. expm1-log1p-u 99.9%

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

   2. expm1-udef 99.9%

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

   3. *-commutative 99.9%

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

   4. add-sqr-sqrt 99.9%

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

   5. fabs-sqr 99.9%

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

   6. add-sqr-sqrt 99.9%

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

5. Applied egg-rr 99.9%

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

   1. expm1-def 99.9%

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

   2. expm1-log1p 99.9%

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

   3. *-commutative 99.9%

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Reproduce

herbie shell --seed 2023199 
(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)))))))