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

Percentage Accurate: 100.0% → 100.0%

Time: 11.1s

Alternatives: 9

Speedup: 4.2×

• 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, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate-+l+ 99.9%

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

3. Simplified 99.9%

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

   1. add-cube-cbrt 99.9%

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

   2. pow3 99.9%

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

5. Applied egg-rr 99.9%

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

6. Taylor expanded in x around inf 99.9%

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

   1. unpow2 99.9%

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

   2. exp-prod 100.0%

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

8. Simplified 100.0%

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

9. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate-+l+ 99.9%

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

3. Simplified 99.9%

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

   1. add-cube-cbrt 99.9%

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

   2. pow3 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 3: 100.0% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate-+l+ 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 4: 99.6% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate-+l+ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around inf 99.5%

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

   1. unpow2 99.5%

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

   2. associate-/l/ 99.5%

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

6. Simplified 99.5%

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

7. Final simplification 99.5%

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

Alternative 5: 51.8% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate-+l+ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around inf 99.5%

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

   1. unpow2 99.5%

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

   2. associate-/l/ 99.5%

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

6. Simplified 99.5%

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

7. Taylor expanded in x around 0 47.8%

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

   1. unpow2 47.8%

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

9. Simplified 47.8%

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

    1. add-sqr-sqrt 47.8%

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

    2. fabs-sqr 47.8%

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

    3. add-sqr-sqrt 47.8%

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

    4. div-inv 47.8%

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

    5. +-commutative 47.8%

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

    6. fma-def 47.8%

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

11. Applied egg-rr 47.8%

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

    1. un-div-inv 47.8%

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

    2. fma-udef 47.8%

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

    3. metadata-eval 47.8%

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

    4. *-un-lft-identity 47.8%

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

    5. frac-add 5.2%

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

    6. *-inverses 5.2%

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

    7. +-commutative 5.2%

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

    8. *-inverses 5.2%

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

    9. frac-add 47.8%

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

    10. metadata-eval 47.8%

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

    11. *-rgt-identity 47.8%

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

13. Applied egg-rr 47.8%

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

14. Final simplification 47.8%

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

Alternative 6: 51.8% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(x, x, 1\right) \cdot \frac{1}{x}}{\sqrt{\pi}} \cdot \left(1 + \frac{\frac{0.5}{x}}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (/ (* (fma x x 1.0) (/ 1.0 x)) (sqrt PI)) (+ 1.0 (/ (/ 0.5 x) x))))

C

double code(double x) {
	return ((fma(x, x, 1.0) * (1.0 / x)) / sqrt(((double) M_PI))) * (1.0 + ((0.5 / x) / x));
}

Julia

function code(x)
	return Float64(Float64(Float64(fma(x, x, 1.0) * Float64(1.0 / x)) / sqrt(pi)) * Float64(1.0 + Float64(Float64(0.5 / x) / x)))
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate-+l+ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around inf 99.5%

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

   1. unpow2 99.5%

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

   2. associate-/l/ 99.5%

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

6. Simplified 99.5%

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

7. Taylor expanded in x around 0 47.8%

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

   1. unpow2 47.8%

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

9. Simplified 47.8%

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

    1. add-sqr-sqrt 47.8%

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

    2. fabs-sqr 47.8%

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

    3. add-sqr-sqrt 47.8%

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

    4. div-inv 47.8%

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

    5. +-commutative 47.8%

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

    6. fma-def 47.8%

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

11. Applied egg-rr 47.8%

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

12. Taylor expanded in x around inf 47.8%

    \[\leadsto \frac{\mathsf{fma}\left(x, x, 1\right) \cdot \frac{1}{x}}{\sqrt{\pi}} \cdot \left(1 + \color{blue}{\frac{0.5}{{x}^{2}}}\right) \]
13. Step-by-step derivation

    1. unpow2 2.3%

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

    2. associate-/r* 2.3%

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

14. Simplified 47.8%

    \[\leadsto \frac{\mathsf{fma}\left(x, x, 1\right) \cdot \frac{1}{x}}{\sqrt{\pi}} \cdot \left(1 + \color{blue}{\frac{\frac{0.5}{x}}{x}}\right) \]

15. Final simplification 47.8%

    \[\leadsto \frac{\mathsf{fma}\left(x, x, 1\right) \cdot \frac{1}{x}}{\sqrt{\pi}} \cdot \left(1 + \frac{\frac{0.5}{x}}{x}\right) \]

Alternative 7: 51.8% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \left(1 + \frac{\frac{0.5}{x}}{x}\right) \cdot \frac{\mathsf{fma}\left(x, x, 1\right)}{x \cdot \sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (+ 1.0 (/ (/ 0.5 x) x)) (/ (fma x x 1.0) (* x (sqrt PI)))))

C

double code(double x) {
	return (1.0 + ((0.5 / x) / x)) * (fma(x, x, 1.0) / (x * sqrt(((double) M_PI))));
}

Julia

function code(x)
	return Float64(Float64(1.0 + Float64(Float64(0.5 / x) / x)) * Float64(fma(x, x, 1.0) / Float64(x * sqrt(pi))))
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate-+l+ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around inf 99.5%

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

   1. unpow2 99.5%

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

   2. associate-/l/ 99.5%

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

6. Simplified 99.5%

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

7. Taylor expanded in x around 0 47.8%

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

   1. unpow2 47.8%

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

9. Simplified 47.8%

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

    1. add-log-exp 99.3%

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

    2. *-un-lft-identity 99.3%

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

    3. log-prod 99.3%

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

    4. metadata-eval 99.3%

       \[\leadsto \left(\color{blue}{0} + \log \left(e^{\frac{\frac{1 + x \cdot x}{\left|x\right|}}{\sqrt{\pi}}}\right)\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{\frac{0.5}{x}}{x}\right)\right) \]

    5. add-log-exp 47.8%

       \[\leadsto \left(0 + \color{blue}{\frac{\frac{1 + x \cdot x}{\left|x\right|}}{\sqrt{\pi}}}\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{\frac{0.5}{x}}{x}\right)\right) \]

    6. add-sqr-sqrt 47.8%

       \[\leadsto \left(0 + \frac{\color{blue}{\sqrt{\frac{1 + x \cdot x}{\left|x\right|}} \cdot \sqrt{\frac{1 + x \cdot x}{\left|x\right|}}}}{\sqrt{\pi}}\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{\frac{0.5}{x}}{x}\right)\right) \]

    7. add-sqr-sqrt 47.8%

       \[\leadsto \left(0 + \frac{\color{blue}{\frac{1 + x \cdot x}{\left|x\right|}}}{\sqrt{\pi}}\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{\frac{0.5}{x}}{x}\right)\right) \]

    8. add-sqr-sqrt 47.8%

       \[\leadsto \left(0 + \frac{\frac{1 + x \cdot x}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}}{\sqrt{\pi}}\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{\frac{0.5}{x}}{x}\right)\right) \]

    9. fabs-sqr 47.8%

       \[\leadsto \left(0 + \frac{\frac{1 + x \cdot x}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}}{\sqrt{\pi}}\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{\frac{0.5}{x}}{x}\right)\right) \]

    10. add-sqr-sqrt 47.8%

        \[\leadsto \left(0 + \frac{\frac{1 + x \cdot x}{\color{blue}{x}}}{\sqrt{\pi}}\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{\frac{0.5}{x}}{x}\right)\right) \]

    11. associate-/l/ 47.4%

        \[\leadsto \left(0 + \color{blue}{\frac{1 + x \cdot x}{\sqrt{\pi} \cdot x}}\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{\frac{0.5}{x}}{x}\right)\right) \]

    12. +-commutative 47.4%

        \[\leadsto \left(0 + \frac{\color{blue}{x \cdot x + 1}}{\sqrt{\pi} \cdot x}\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{\frac{0.5}{x}}{x}\right)\right) \]

    13. fma-def 47.4%

        \[\leadsto \left(0 + \frac{\color{blue}{\mathsf{fma}\left(x, x, 1\right)}}{\sqrt{\pi} \cdot x}\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{\frac{0.5}{x}}{x}\right)\right) \]

    14. *-commutative 47.4%

        \[\leadsto \left(0 + \frac{\mathsf{fma}\left(x, x, 1\right)}{\color{blue}{x \cdot \sqrt{\pi}}}\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{\frac{0.5}{x}}{x}\right)\right) \]

11. Applied egg-rr 47.4%

    \[\leadsto \color{blue}{\left(0 + \frac{\mathsf{fma}\left(x, x, 1\right)}{x \cdot \sqrt{\pi}}\right)} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{\frac{0.5}{x}}{x}\right)\right) \]
12. Step-by-step derivation

    1. +-lft-identity 47.4%

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

13. Simplified 47.4%

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

14. Taylor expanded in x around inf 47.4%

    \[\leadsto \frac{\mathsf{fma}\left(x, x, 1\right)}{x \cdot \sqrt{\pi}} \cdot \left(1 + \color{blue}{\frac{0.5}{{x}^{2}}}\right) \]
15. Step-by-step derivation

    1. unpow2 2.3%

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

    2. associate-/r* 2.3%

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

16. Simplified 47.4%

    \[\leadsto \frac{\mathsf{fma}\left(x, x, 1\right)}{x \cdot \sqrt{\pi}} \cdot \left(1 + \color{blue}{\frac{\frac{0.5}{x}}{x}}\right) \]

17. Final simplification 47.4%

    \[\leadsto \left(1 + \frac{\frac{0.5}{x}}{x}\right) \cdot \frac{\mathsf{fma}\left(x, x, 1\right)}{x \cdot \sqrt{\pi}} \]

Alternative 8: 5.4% accurate, 10.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (* (+ 1.0 (/ (/ 0.5 x) x)) (/ (+ (/ 1.0 x) (/ x (/ x x))) (sqrt PI))))

C

double code(double x) {
	return (1.0 + ((0.5 / x) / x)) * (((1.0 / x) + (x / (x / x))) / sqrt(((double) M_PI)));
}

Java

public static double code(double x) {
	return (1.0 + ((0.5 / x) / x)) * (((1.0 / x) + (x / (x / x))) / Math.sqrt(Math.PI));
}

Python

def code(x):
	return (1.0 + ((0.5 / x) / x)) * (((1.0 / x) + (x / (x / x))) / math.sqrt(math.pi))

Julia

function code(x)
	return Float64(Float64(1.0 + Float64(Float64(0.5 / x) / x)) * Float64(Float64(Float64(1.0 / x) + Float64(x / Float64(x / x))) / sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 + ((0.5 / x) / x)) * (((1.0 / x) + (x / (x / x))) / sqrt(pi));
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate-+l+ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around inf 99.5%

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

   1. unpow2 99.5%

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

   2. associate-/l/ 99.5%

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

6. Simplified 99.5%

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

7. Taylor expanded in x around 0 47.8%

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

   1. unpow2 47.8%

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

9. Simplified 47.8%

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

10. Taylor expanded in x around 0 47.8%

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

    1. unpow2 47.8%

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

    2. associate-/l* 5.2%

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

    3. unpow1 5.2%

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

    4. sqr-pow 5.2%

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

    5. metadata-eval 5.2%

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

    6. unpow1/2 5.2%

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

    7. metadata-eval 5.2%

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

    8. unpow1/2 5.2%

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

    9. fabs-sqr 5.2%

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

    10. unpow1/2 5.2%

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

    11. metadata-eval 5.2%

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

    12. unpow1/2 5.2%

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

    13. metadata-eval 5.2%

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

    14. sqr-pow 5.2%

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

    15. unpow1 5.2%

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

    16. unpow1 5.2%

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

    17. sqr-pow 5.2%

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

    18. metadata-eval 5.2%

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

    19. unpow1/2 5.2%

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

    20. metadata-eval 5.2%

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

    21. unpow1/2 5.2%

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

12. Simplified 5.2%

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

13. Taylor expanded in x around inf 5.2%

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

    1. unpow2 2.3%

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

    2. associate-/r* 2.3%

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

15. Simplified 5.2%

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

16. Final simplification 5.2%

    \[\leadsto \left(1 + \frac{\frac{0.5}{x}}{x}\right) \cdot \frac{\frac{1}{x} + \frac{x}{\frac{x}{x}}}{\sqrt{\pi}} \]

Alternative 9: 2.3% accurate, 10.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (* (+ 1.0 (/ (/ 0.5 x) x)) (/ (pow PI -0.5) x)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate-+l+ 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around inf 99.5%

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

   1. unpow2 99.5%

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

   2. associate-/l/ 99.5%

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

6. Simplified 99.5%

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

7. Taylor expanded in x around 0 2.3%

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

   1. add-log-exp 1.7%

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

   2. *-un-lft-identity 1.7%

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

   3. log-prod 1.7%

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

   4. metadata-eval 1.7%

      \[\leadsto \left(\color{blue}{0} + \log \left(e^{\frac{\frac{1}{\left|x\right|}}{\sqrt{\pi}}}\right)\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{\frac{0.5}{x}}{x}\right)\right) \]

   5. add-log-exp 2.3%

      \[\leadsto \left(0 + \color{blue}{\frac{\frac{1}{\left|x\right|}}{\sqrt{\pi}}}\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{\frac{0.5}{x}}{x}\right)\right) \]

   6. div-inv 2.3%

      \[\leadsto \left(0 + \color{blue}{\frac{1}{\left|x\right|} \cdot \frac{1}{\sqrt{\pi}}}\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{\frac{0.5}{x}}{x}\right)\right) \]

   7. clear-num 2.3%

      \[\leadsto \left(0 + \color{blue}{\frac{1}{\frac{\left|x\right|}{1}}} \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{\frac{0.5}{x}}{x}\right)\right) \]

   8. associate-*l/ 2.3%

      \[\leadsto \left(0 + \color{blue}{\frac{1 \cdot \frac{1}{\sqrt{\pi}}}{\frac{\left|x\right|}{1}}}\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{\frac{0.5}{x}}{x}\right)\right) \]

   9. *-un-lft-identity 2.3%

      \[\leadsto \left(0 + \frac{\color{blue}{\frac{1}{\sqrt{\pi}}}}{\frac{\left|x\right|}{1}}\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{\frac{0.5}{x}}{x}\right)\right) \]

   10. pow1/2 2.3%

       \[\leadsto \left(0 + \frac{\frac{1}{\color{blue}{{\pi}^{0.5}}}}{\frac{\left|x\right|}{1}}\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{\frac{0.5}{x}}{x}\right)\right) \]

   11. pow-flip 2.3%

       \[\leadsto \left(0 + \frac{\color{blue}{{\pi}^{\left(-0.5\right)}}}{\frac{\left|x\right|}{1}}\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{\frac{0.5}{x}}{x}\right)\right) \]

   12. metadata-eval 2.3%

       \[\leadsto \left(0 + \frac{{\pi}^{\color{blue}{-0.5}}}{\frac{\left|x\right|}{1}}\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{\frac{0.5}{x}}{x}\right)\right) \]

   13. add-sqr-sqrt 2.3%

       \[\leadsto \left(0 + \frac{{\pi}^{-0.5}}{\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{1}}\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{\frac{0.5}{x}}{x}\right)\right) \]

   14. fabs-sqr 2.3%

       \[\leadsto \left(0 + \frac{{\pi}^{-0.5}}{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{1}}\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{\frac{0.5}{x}}{x}\right)\right) \]

   15. add-sqr-sqrt 2.3%

       \[\leadsto \left(0 + \frac{{\pi}^{-0.5}}{\frac{\color{blue}{x}}{1}}\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{\frac{0.5}{x}}{x}\right)\right) \]

   16. /-rgt-identity 2.3%

       \[\leadsto \left(0 + \frac{{\pi}^{-0.5}}{\color{blue}{x}}\right) \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{\frac{0.5}{x}}{x}\right)\right) \]

9. Applied egg-rr 2.3%

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

    1. +-lft-identity 2.3%

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

11. Simplified 2.3%

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

12. Taylor expanded in x around inf 2.3%

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

    1. unpow2 2.3%

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

    2. associate-/r* 2.3%

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

14. Simplified 2.3%

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

15. Final simplification 2.3%

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

Reproduce

herbie shell --seed 2023178 
(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)))))))