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

Percentage Accurate: 100.0% → 100.0%

Time: 14.6s

Alternatives: 8

Speedup: 3.5×

• 99.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \geq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	double t_0 = Math.cbrt(Math.pow(Math.PI, 0.75));
	return (((1.0 + (0.5 / (x * x))) / Math.abs(x)) + (Math.pow((1.0 / Math.abs(x)), 5.0) * (0.75 + ((1.875 / x) / x)))) * (Math.pow(Math.exp(x), x) / (t_0 * t_0));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-cbrt-cube 100.0%

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

   2. pow1/3 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left(\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}\right)}^{0.3333333333333333}}} \]

   3. add-sqr-sqrt 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\color{blue}{\pi} \cdot \sqrt{\pi}\right)}^{0.3333333333333333}} \]

   4. pow1 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\color{blue}{{\pi}^{1}} \cdot \sqrt{\pi}\right)}^{0.3333333333333333}} \]

   5. pow1/2 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{1} \cdot \color{blue}{{\pi}^{0.5}}\right)}^{0.3333333333333333}} \]

   6. pow-prod-up 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{{\color{blue}{\left({\pi}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}} \]

   7. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \]

5. Applied egg-rr 100.0%

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

   1. unpow1/3 100.0%

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

7. Simplified 100.0%

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

   1. pow1/3 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left({\pi}^{1.5}\right)}^{0.3333333333333333}}} \]

   2. add-sqr-sqrt 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{{\color{blue}{\left(\sqrt{{\pi}^{1.5}} \cdot \sqrt{{\pi}^{1.5}}\right)}}^{0.3333333333333333}} \]

   3. unpow-prod-down 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left(\sqrt{{\pi}^{1.5}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{{\pi}^{1.5}}\right)}^{0.3333333333333333}}} \]

   4. sqrt-pow1 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{{\color{blue}{\left({\pi}^{\left(\frac{1.5}{2}\right)}\right)}}^{0.3333333333333333} \cdot {\left(\sqrt{{\pi}^{1.5}}\right)}^{0.3333333333333333}} \]

   5. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{\color{blue}{0.75}}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{{\pi}^{1.5}}\right)}^{0.3333333333333333}} \]

   6. sqrt-pow1 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.75}\right)}^{0.3333333333333333} \cdot {\color{blue}{\left({\pi}^{\left(\frac{1.5}{2}\right)}\right)}}^{0.3333333333333333}} \]

   7. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{0.75}\right)}^{0.3333333333333333} \cdot {\left({\pi}^{\color{blue}{0.75}}\right)}^{0.3333333333333333}} \]

9. Applied egg-rr 100.0%

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

    1. unpow1/3 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{\sqrt[3]{{\pi}^{0.75}}} \cdot {\left({\pi}^{0.75}\right)}^{0.3333333333333333}} \]

    2. unpow1/3 100.0%

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

11. Simplified 100.0%

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

12. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x):
	return (((1.0 + (0.5 / (x * x))) / math.fabs(x)) + ((0.75 + ((1.875 / x) / x)) * ((1.0 + math.pow(x, -5.0)) + -1.0))) * (math.pow(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(0.75 + Float64(Float64(1.875 / x) / x)) * Float64(Float64(1.0 + (x ^ -5.0)) + -1.0))) * Float64((exp(x) ^ x) / sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = (((1.0 + (0.5 / (x * x))) / abs(x)) + ((0.75 + ((1.875 / x) / x)) * ((1.0 + (x ^ -5.0)) + -1.0))) * ((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[(0.75 + N[(N[(1.875 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[x, -5.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-cbrt-cube 100.0%

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

   2. pow1/3 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{{\left(\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}\right)}^{0.3333333333333333}}} \]

   3. add-sqr-sqrt 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\color{blue}{\pi} \cdot \sqrt{\pi}\right)}^{0.3333333333333333}} \]

   4. pow1 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left(\color{blue}{{\pi}^{1}} \cdot \sqrt{\pi}\right)}^{0.3333333333333333}} \]

   5. pow1/2 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{1} \cdot \color{blue}{{\pi}^{0.5}}\right)}^{0.3333333333333333}} \]

   6. pow-prod-up 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{{\color{blue}{\left({\pi}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}} \]

   7. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{{\left({\pi}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \]

5. Applied egg-rr 100.0%

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

   1. unpow1/3 100.0%

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

7. Simplified 100.0%

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

   1. expm1-log1p-u 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{1}{\left|x\right|}\right)}^{5}\right)\right)} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \]

   2. expm1-udef 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{1}{\left|x\right|}\right)}^{5}\right)} - 1\right)} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \]

   3. inv-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(e^{\mathsf{log1p}\left({\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}\right)} - 1\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \]

   4. pow-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}\right)} - 1\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \]

   5. add-sqr-sqrt 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(e^{\mathsf{log1p}\left({\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}\right)} - 1\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \]

   6. fabs-sqr 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(e^{\mathsf{log1p}\left({\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}\right)} - 1\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \]

   7. add-sqr-sqrt 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(e^{\mathsf{log1p}\left({\color{blue}{x}}^{\left(-1 \cdot 5\right)}\right)} - 1\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \]

   8. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(e^{\mathsf{log1p}\left({x}^{\color{blue}{-5}}\right)} - 1\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \]

9. Applied egg-rr 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-5}\right)} - 1\right)} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \]
10. Step-by-step derivation

    1. sub-neg 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-5}\right)} + \left(-1\right)\right)} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt[3]{{\pi}^{1.5}}} \]

    2. log1p-udef 100.0%

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

    3. add-exp-log 100.0%

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

    4. metadata-eval 100.0%

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

11. Applied egg-rr 100.0%

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

    1. add-log-exp 100.0%

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

    2. pow1/3 100.0%

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

    3. pow-pow 100.0%

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

    4. metadata-eval 100.0%

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

    5. pow1/2 100.0%

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

    6. *-un-lft-identity 100.0%

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

    7. log-prod 100.0%

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

    8. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\left(1 + {x}^{-5}\right) + -1\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{0} + \log \left(e^{\sqrt{\pi}}\right)} \]

    9. add-log-exp 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\left(1 + {x}^{-5}\right) + -1\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{0 + \color{blue}{\sqrt{\pi}}} \]

13. Applied egg-rr 100.0%

    \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\left(1 + {x}^{-5}\right) + -1\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\color{blue}{0 + \sqrt{\pi}}} \]
14. Step-by-step derivation

    1. +-lft-identity 100.0%

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

15. Simplified 100.0%

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

16. Final simplification 100.0%

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

Alternative 3: 100.0% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right)\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. add-log-exp 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. inv-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. pow-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   9. fabs-sqr 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{x}}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   11. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {x}^{\color{blue}{-5}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

5. Applied egg-rr 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\left(0 + {x}^{-5}\right)} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
6. Step-by-step derivation

   1. +-lft-identity 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 4: 99.6% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $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[(0.75 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right)\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. add-log-exp 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. inv-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. pow-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   9. fabs-sqr 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{x}}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   11. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {x}^{\color{blue}{-5}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

5. Applied egg-rr 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\left(0 + {x}^{-5}\right)} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
6. Step-by-step derivation

   1. +-lft-identity 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 99.5%

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

9. Final simplification 99.5%

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

Alternative 5: 99.6% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x):
	return (((1.0 + (0.5 / (x * x))) / math.fabs(x)) + (0.75 / math.pow(x, 5.0))) * (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(0.75 / (x ^ 5.0))) * Float64(exp(Float64(x * x)) / sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = (((1.0 + (0.5 / (x * x))) / abs(x)) + (0.75 / (x ^ 5.0))) * (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[(0.75 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right)\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. add-log-exp 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. inv-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. pow-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   9. fabs-sqr 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{x}}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   11. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {x}^{\color{blue}{-5}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

5. Applied egg-rr 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\left(0 + {x}^{-5}\right)} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
6. Step-by-step derivation

   1. +-lft-identity 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 99.5%

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

9. Taylor expanded in x around inf 99.5%

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

    1. unpow2 99.5%

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

11. Simplified 99.5%

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

12. Final simplification 99.5%

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

Alternative 6: 99.6% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right)\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. add-log-exp 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. inv-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. pow-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   9. fabs-sqr 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{x}}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   11. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {x}^{\color{blue}{-5}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

5. Applied egg-rr 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\left(0 + {x}^{-5}\right)} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
6. Step-by-step derivation

   1. +-lft-identity 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 99.5%

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

9. Taylor expanded in x around inf 99.4%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \frac{e^{{x}^{2}}}{\left|x\right|}} \]
10. Step-by-step derivation

    1. unpow2 99.4%

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

    2. exp-prod 99.4%

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

    3. unpow1 99.4%

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

    4. sqr-pow 99.4%

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

    5. fabs-sqr 99.4%

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

    6. sqr-pow 99.4%

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

    7. unpow1 99.4%

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

11. Simplified 99.4%

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

12. Final simplification 99.4%

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

Alternative 7: 52.0% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{\pi}} \cdot \frac{x \cdot x}{\left|x\right|} \end{array} \]

FPCore

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

C

double code(double x) {
	return sqrt((1.0 / ((double) M_PI))) * ((x * x) / fabs(x));
}

Java

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

Python

def code(x):
	return math.sqrt((1.0 / math.pi)) * ((x * x) / math.fabs(x))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right)\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. add-log-exp 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. inv-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. pow-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   9. fabs-sqr 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{x}}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   11. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {x}^{\color{blue}{-5}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

5. Applied egg-rr 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\left(0 + {x}^{-5}\right)} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
6. Step-by-step derivation

   1. +-lft-identity 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around 0 99.5%

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

9. Taylor expanded in x around 0 48.3%

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

    1. unpow2 48.3%

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

11. Simplified 48.3%

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

12. Taylor expanded in x around inf 48.2%

    \[\leadsto \color{blue}{\frac{{x}^{2}}{\left|x\right|} \cdot \sqrt{\frac{1}{\pi}}} \]
13. Step-by-step derivation

    1. *-commutative 48.2%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \frac{{x}^{2}}{\left|x\right|}} \]

    2. unpow2 48.2%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \frac{\color{blue}{x \cdot x}}{\left|x\right|} \]

14. Simplified 48.2%

    \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \frac{x \cdot x}{\left|x\right|}} \]

15. Final simplification 48.2%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \frac{x \cdot x}{\left|x\right|} \]

Alternative 8: 5.4% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{\pi}} \cdot \frac{x}{\frac{x}{x}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-log-exp 100.0%

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

   2. *-un-lft-identity 100.0%

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

   3. log-prod 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\color{blue}{0} + \log \left(e^{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right)\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   5. add-log-exp 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + \color{blue}{{\left(\frac{1}{\left|x\right|}\right)}^{5}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   6. inv-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{5}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   7. pow-pow 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 5\right)}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   8. add-sqr-sqrt 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   9. fabs-sqr 100.0%

      \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   10. add-sqr-sqrt 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {\color{blue}{x}}^{\left(-1 \cdot 5\right)}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

   11. metadata-eval 100.0%

       \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(0 + {x}^{\color{blue}{-5}}\right) \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]

5. Applied egg-rr 100.0%

   \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\left(0 + {x}^{-5}\right)} \cdot \left(0.75 + \frac{\frac{1.875}{x}}{x}\right)\right) \cdot \frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \]
6. Step-by-step derivation

   1. +-lft-identity 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around 0 99.5%

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

9. Taylor expanded in x around 0 48.3%

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

    1. unpow2 48.3%

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

11. Simplified 48.3%

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

12. Taylor expanded in x around inf 48.2%

    \[\leadsto \color{blue}{\frac{{x}^{2}}{\left|x\right|} \cdot \sqrt{\frac{1}{\pi}}} \]
13. Step-by-step derivation

    1. *-commutative 48.2%

       \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \frac{{x}^{2}}{\left|x\right|}} \]

    2. unpow2 48.2%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \frac{\color{blue}{x \cdot x}}{\left|x\right|} \]

    3. associate-/l* 5.3%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \color{blue}{\frac{x}{\frac{\left|x\right|}{x}}} \]

    4. unpow1 5.3%

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

    5. sqr-pow 5.3%

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

    6. fabs-sqr 5.3%

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

    7. sqr-pow 5.3%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \frac{x}{\frac{\color{blue}{{x}^{1}}}{x}} \]

    8. unpow1 5.3%

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \frac{x}{\frac{\color{blue}{x}}{x}} \]

14. Simplified 5.3%

    \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \frac{x}{\frac{x}{x}}} \]

15. Final simplification 5.3%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \frac{x}{\frac{x}{x}} \]

Reproduce

herbie shell --seed 2023271 
(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)))))))