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

Percentage Accurate: 100.0% → 100.0%

Time: 12.5s

Alternatives: 7

Speedup: 4.2×

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

Specification

\[x \geq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 3.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (/ (pow (exp x) x) (fabs x)) (sqrt PI))
  (+ 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)) / sqrt(((double) M_PI))) * (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.sqrt(Math.PI)) * (1.0 + ((1.875 / Math.pow(x, 6.0)) + ((0.5 + (0.75 / (x * x))) / (x * x))));
}

Python

def code(x):
	return ((math.pow(math.exp(x), x) / math.fabs(x)) / math.sqrt(math.pi)) * (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)) / sqrt(pi)) * Float64(1.0 + Float64(Float64(1.875 / (x ^ 6.0)) + Float64(Float64(0.5 + Float64(0.75 / Float64(x * x))) / Float64(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 + (0.75 / (x * x))) / (x * x))));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

   1. add-cbrt-cube 100.0%

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

   2. pow1/3 100.0%

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

   3. add-sqr-sqrt 100.0%

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

   4. pow1 100.0%

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

   5. pow1/2 100.0%

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

   6. pow-prod-up 100.0%

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

   7. metadata-eval 100.0%

      \[\leadsto \frac{\frac{e^{x \cdot x}}{\left|x\right|}}{{\left({\pi}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

5. Applied egg-rr 100.0%

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

   1. unpow1/3 100.0%

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

7. Simplified 100.0%

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

8. Taylor expanded in x around inf 100.0%

   \[\leadsto \frac{\frac{\color{blue}{e^{{x}^{2}}}}{\left|x\right|}}{\sqrt[3]{{\pi}^{1.5}}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]
9. Step-by-step derivation

   1. unpow2 100.0%

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

   2. exp-prod 100.0%

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

10. Simplified 100.0%

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

    1. expm1-log1p-u 100.0%

       \[\leadsto \frac{\frac{{\left(e^{x}\right)}^{x}}{\left|x\right|}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{{\pi}^{1.5}}\right)\right)}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

    2. expm1-udef 100.0%

       \[\leadsto \frac{\frac{{\left(e^{x}\right)}^{x}}{\left|x\right|}}{\color{blue}{e^{\mathsf{log1p}\left(\sqrt[3]{{\pi}^{1.5}}\right)} - 1}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

    3. pow1/3 100.0%

       \[\leadsto \frac{\frac{{\left(e^{x}\right)}^{x}}{\left|x\right|}}{e^{\mathsf{log1p}\left(\color{blue}{{\left({\pi}^{1.5}\right)}^{0.3333333333333333}}\right)} - 1} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

    4. pow-pow 100.0%

       \[\leadsto \frac{\frac{{\left(e^{x}\right)}^{x}}{\left|x\right|}}{e^{\mathsf{log1p}\left(\color{blue}{{\pi}^{\left(1.5 \cdot 0.3333333333333333\right)}}\right)} - 1} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right) \]

    5. metadata-eval 100.0%

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

    6. pow1/2 100.0%

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

12. Applied egg-rr 100.0%

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

    1. expm1-def 100.0%

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

    2. expm1-log1p 100.0%

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

14. Simplified 100.0%

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

15. Final simplification 100.0%

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

Alternative 2: 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 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%

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

4. Final simplification 100.0%

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

Alternative 3: 99.6% accurate, 4.2× speedup

Math

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

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(0.5 / Float64(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[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around inf 99.7%

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

   1. unpow2 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 4: 51.7% accurate, 5.2× speedup

Math

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

FPCore

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

C

double code(double x) {
	return (1.0 + ((1.875 / pow(x, 6.0)) + (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(1.875 / (x ^ 6.0)) + Float64(0.5 / Float64(x * x)))) * Float64(Float64(fma(x, x, 1.0) / x) / sqrt(pi)))
end

Wolfram

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

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

4. Taylor expanded in x around inf 99.7%

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

   1. unpow2 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in x around 0 52.3%

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

   1. unpow2 52.3%

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

9. Simplified 52.3%

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

    1. add-log-exp 98.5%

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

    2. *-un-lft-identity 98.5%

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

    3. log-prod 98.5%

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

    4. metadata-eval 98.5%

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

    5. add-log-exp 52.3%

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

    6. +-commutative 52.3%

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

    7. fma-def 52.3%

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

    8. add-sqr-sqrt 52.3%

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

    9. fabs-sqr 52.3%

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

    10. add-sqr-sqrt 52.3%

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

11. Applied egg-rr 52.3%

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

    1. +-lft-identity 52.3%

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

13. Simplified 52.3%

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

14. Final simplification 52.3%

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

Alternative 5: 51.6% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

4. Taylor expanded in x around inf 99.7%

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

   1. unpow2 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in x around 0 52.3%

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

   1. unpow2 52.3%

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

9. Simplified 52.3%

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

    1. add-log-exp 98.5%

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

    2. *-un-lft-identity 98.5%

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

    3. log-prod 98.5%

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

    4. metadata-eval 98.5%

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

    5. add-log-exp 52.3%

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

    6. add-sqr-sqrt 52.3%

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

    7. add-sqr-sqrt 52.3%

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

    8. add-sqr-sqrt 52.3%

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

    9. fabs-sqr 52.3%

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

    10. add-sqr-sqrt 52.3%

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

    11. associate-/l/ 51.9%

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

    12. +-commutative 51.9%

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

    13. fma-def 51.9%

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

    14. *-commutative 51.9%

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

11. Applied egg-rr 51.9%

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

    1. +-lft-identity 51.9%

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

13. Simplified 51.9%

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

14. Taylor expanded in x around inf 51.9%

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

    1. unpow2 2.3%

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

16. Simplified 51.9%

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

17. Final simplification 51.9%

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

Alternative 6: 5.4% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around inf 99.7%

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

   1. unpow2 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in x around 0 52.3%

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

   1. unpow2 52.3%

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

9. Simplified 52.3%

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

10. Taylor expanded in x around 0 52.3%

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

    1. unpow2 52.3%

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

    2. associate-/l* 5.4%

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

    3. associate-/r/ 5.4%

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

    4. unpow1 5.4%

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

    5. sqr-pow 5.4%

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

    6. metadata-eval 5.4%

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

    7. unpow1/2 5.4%

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

    8. metadata-eval 5.4%

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

    9. unpow1/2 5.4%

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

    10. fabs-sqr 5.4%

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

    11. unpow1/2 5.4%

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

    12. metadata-eval 5.4%

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

    13. unpow1/2 5.4%

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

    14. metadata-eval 5.4%

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

    15. sqr-pow 5.4%

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

    16. unpow1 5.4%

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

    17. unpow1 5.4%

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

    18. sqr-pow 5.4%

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

    19. metadata-eval 5.4%

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

    20. unpow1/2 5.4%

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

    21. metadata-eval 5.4%

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

    22. unpow1/2 5.4%

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

12. Simplified 5.4%

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

13. Taylor expanded in x around inf 5.4%

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

    1. unpow2 2.3%

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

15. Simplified 5.4%

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

16. Final simplification 5.4%

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

Alternative 7: 2.3% accurate, 10.3× speedup

Math

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

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(0.5 / Float64(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[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[Pi, -0.5], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in x around inf 99.7%

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

   1. unpow2 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in x around 0 2.3%

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

   1. add-log-exp 1.7%

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

   2. *-un-lft-identity 1.7%

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

   3. log-prod 1.7%

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

   4. metadata-eval 1.7%

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

   5. add-log-exp 2.3%

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

   6. div-inv 2.3%

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

   7. clear-num 2.3%

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

   8. associate-*l/ 2.3%

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

   9. *-un-lft-identity 2.3%

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

   10. pow1/2 2.3%

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

   11. pow-flip 2.3%

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

   12. metadata-eval 2.3%

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

   13. add-sqr-sqrt 2.3%

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

   14. fabs-sqr 2.3%

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

   15. add-sqr-sqrt 2.3%

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

   16. /-rgt-identity 2.3%

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

9. Applied egg-rr 2.3%

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

    1. +-lft-identity 2.3%

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

11. Simplified 2.3%

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

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%

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

14. Simplified 2.3%

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

15. Final simplification 2.3%

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

Reproduce

herbie shell --seed 2023188 
(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)))))))