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

Percentage Accurate: 100.0% → 100.0%

Time: 6.4s

Alternatives: 9

Speedup: 4.2×

• 99.3% 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, 4.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ (/ (exp (* x x)) (fabs x)) (sqrt PI))
  (+ 1.0 (+ (/ 1.875 (pow x 6.0)) (/ (+ 0.5 (/ 0.75 (* x x))) (* 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 + (0.75 / (x * x))) / (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 + (0.75 / (x * x))) / (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 + (0.75 / (x * x))) / (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 + 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[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 + 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. Final simplification 100.0%

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

Alternative 2: 99.5% 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 3: 99.5% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

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 inf 99.7%

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

   1. *-commutative 99.7%

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

   2. unpow2 99.7%

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

   3. exp-prod 99.7%

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

   4. unpow1 99.7%

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

   5. sqr-pow 99.7%

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

   6. fabs-sqr 99.7%

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

   7. sqr-pow 99.7%

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

   8. unpow1 99.7%

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

9. Simplified 99.7%

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

    1. div-inv 99.7%

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

11. Applied egg-rr 99.7%

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

12. Final simplification 99.7%

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

Alternative 4: 99.5% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[Exp[N[(x * x), $MachinePrecision]], $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 inf 99.7%

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

   1. *-commutative 99.7%

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

   2. unpow2 99.7%

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

   3. exp-prod 99.7%

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

   4. unpow1 99.7%

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

   5. sqr-pow 99.7%

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

   6. fabs-sqr 99.7%

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

   7. sqr-pow 99.7%

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

   8. unpow1 99.7%

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

9. Simplified 99.7%

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

    1. pow-exp 99.7%

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

11. Applied egg-rr 99.7%

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

12. Final simplification 99.7%

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

Alternative 5: 52.3% accurate, 10.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] + 1.0), $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 inf 99.7%

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

   1. *-commutative 99.7%

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

   2. unpow2 99.7%

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

   3. exp-prod 99.7%

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

   4. unpow1 99.7%

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

   5. sqr-pow 99.7%

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

   6. fabs-sqr 99.7%

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

   7. sqr-pow 99.7%

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

   8. unpow1 99.7%

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

9. Simplified 99.7%

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

10. Taylor expanded in x around 0 44.4%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \frac{\color{blue}{1 + {x}^{2}}}{x} \]
11. Step-by-step derivation

    1. unpow2 44.4%

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

12. Simplified 44.4%

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

13. Final simplification 44.4%

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

Alternative 6: 52.3% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Simplified 100.0%

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

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 inf 99.7%

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

   1. *-commutative 99.7%

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

   2. unpow2 99.7%

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

   3. exp-prod 99.7%

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

   4. unpow1 99.7%

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

   5. sqr-pow 99.7%

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

   6. fabs-sqr 99.7%

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

   7. sqr-pow 99.7%

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

   8. unpow1 99.7%

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

9. Simplified 99.7%

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

10. Taylor expanded in x around 0 44.4%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \frac{\color{blue}{1 + {x}^{2}}}{x} \]
11. Step-by-step derivation

    1. unpow2 44.4%

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

12. Simplified 44.4%

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

13. Taylor expanded in x around inf 44.4%

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

    1. unpow2 44.4%

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

15. Simplified 44.4%

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

16. Final simplification 44.4%

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

Alternative 7: 5.4% accurate, 10.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(x + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $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 inf 99.7%

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

   1. *-commutative 99.7%

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

   2. unpow2 99.7%

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

   3. exp-prod 99.7%

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

   4. unpow1 99.7%

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

   5. sqr-pow 99.7%

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

   6. fabs-sqr 99.7%

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

   7. sqr-pow 99.7%

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

   8. unpow1 99.7%

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

9. Simplified 99.7%

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

10. Taylor expanded in x around 0 5.2%

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

    1. expm1-log1p-u 5.2%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{x} + x\right)\right)\right)} \]

    2. expm1-udef 5.2%

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

    3. sqrt-div 5.2%

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

    4. metadata-eval 5.2%

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

    5. associate-*l/ 5.2%

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

    6. *-un-lft-identity 5.2%

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

    7. +-commutative 5.2%

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

    8. inv-pow 5.2%

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

12. Applied egg-rr 5.2%

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

    1. expm1-def 5.2%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x + {x}^{-1}}{\sqrt{\pi}}\right)\right)} \]

    2. expm1-log1p 5.2%

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

    3. unpow-1 5.2%

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

14. Simplified 5.2%

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

15. Final simplification 5.2%

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

Alternative 8: 5.4% accurate, 10.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(x * N[Sqrt[N[(1.0 / 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 inf 99.7%

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

   1. *-commutative 99.7%

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

   2. unpow2 99.7%

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

   3. exp-prod 99.7%

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

   4. unpow1 99.7%

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

   5. sqr-pow 99.7%

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

   6. fabs-sqr 99.7%

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

   7. sqr-pow 99.7%

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

   8. unpow1 99.7%

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

9. Simplified 99.7%

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

10. Taylor expanded in x around 0 5.2%

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

11. Taylor expanded in x around inf 5.2%

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

12. Final simplification 5.2%

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

Alternative 9: 2.3% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \frac{{\pi}^{-0.5}}{x} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Power[Pi, -0.5], $MachinePrecision] / x), $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 inf 99.7%

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

   1. *-commutative 99.7%

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

   2. unpow2 99.7%

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

   3. exp-prod 99.7%

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

   4. unpow1 99.7%

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

   5. sqr-pow 99.7%

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

   6. fabs-sqr 99.7%

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

   7. sqr-pow 99.7%

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

   8. unpow1 99.7%

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

9. Simplified 99.7%

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

10. Taylor expanded in x around 0 2.4%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \color{blue}{\frac{1}{x}} \]
11. Step-by-step derivation

    1. expm1-log1p-u 2.4%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \frac{1}{x}\right)\right)} \]

    2. expm1-udef 1.7%

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

    3. un-div-inv 1.7%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\sqrt{\frac{1}{\pi}}}{x}}\right)} - 1 \]

    4. inv-pow 1.7%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{{\pi}^{-1}}}}{x}\right)} - 1 \]

    5. sqrt-pow1 1.7%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}}}{x}\right)} - 1 \]

    6. metadata-eval 1.7%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{{\pi}^{\color{blue}{-0.5}}}{x}\right)} - 1 \]

12. Applied egg-rr 1.7%

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

    1. expm1-def 2.4%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\pi}^{-0.5}}{x}\right)\right)} \]

    2. expm1-log1p 2.4%

       \[\leadsto \color{blue}{\frac{{\pi}^{-0.5}}{x}} \]

14. Simplified 2.4%

    \[\leadsto \color{blue}{\frac{{\pi}^{-0.5}}{x}} \]

15. Final simplification 2.4%

    \[\leadsto \frac{{\pi}^{-0.5}}{x} \]

Reproduce

herbie shell --seed 2023193 
(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)))))))