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

Percentage Accurate: 100.0% → 100.0%

Time: 8.2s

Alternatives: 8

Speedup: 2.0×

• 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, 1.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (*
  (* (/ 1.0 (sqrt PI)) (pow (exp x) x))
  (/
   (+
    (/ 3.0 (* (* 4.0 x) (* (* x x) x)))
    (+
     (+ (/ 15.0 (* (* (* x x) 8.0) (* (* x x) (* x x)))) 1.0)
     (/ 1.0 (* (+ x x) x))))
   (fabs x))))

C

double code(double x) {
	return ((1.0 / sqrt(((double) M_PI))) * pow(exp(x), x)) * (((3.0 / ((4.0 * x) * ((x * x) * x))) + (((15.0 / (((x * x) * 8.0) * ((x * x) * (x * x)))) + 1.0) + (1.0 / ((x + x) * x)))) / fabs(x));
}

Java

public static double code(double x) {
	return ((1.0 / Math.sqrt(Math.PI)) * Math.pow(Math.exp(x), x)) * (((3.0 / ((4.0 * x) * ((x * x) * x))) + (((15.0 / (((x * x) * 8.0) * ((x * x) * (x * x)))) + 1.0) + (1.0 / ((x + x) * x)))) / Math.abs(x));
}

Python

def code(x):
	return ((1.0 / math.sqrt(math.pi)) * math.pow(math.exp(x), x)) * (((3.0 / ((4.0 * x) * ((x * x) * x))) + (((15.0 / (((x * x) * 8.0) * ((x * x) * (x * x)))) + 1.0) + (1.0 / ((x + x) * x)))) / math.fabs(x))

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * (exp(x) ^ x)) * Float64(Float64(Float64(3.0 / Float64(Float64(4.0 * x) * Float64(Float64(x * x) * x))) + Float64(Float64(Float64(15.0 / Float64(Float64(Float64(x * x) * 8.0) * Float64(Float64(x * x) * Float64(x * x)))) + 1.0) + Float64(1.0 / Float64(Float64(x + x) * x)))) / abs(x)))
end

MATLAB

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

Wolfram

code[x_] := N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(3.0 / N[(N[(4.0 * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(15.0 / N[(N[(N[(x * x), $MachinePrecision] * 8.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(1.0 / N[(N[(x + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

7. Applied rewrites 100.0%

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

9. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/
   (+
    (+
     (/ 3.0 (* (* 4.0 x) (* (* x x) x)))
     (/ 15.0 (* (* (* x x) 8.0) (* (* x x) (* x x)))))
    (+ (/ 1.0 (* (+ x x) x)) 1.0))
   (fabs x))
  (/ (exp (* x x)) (sqrt PI))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[(N[(N[(3.0 / N[(N[(4.0 * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(15.0 / N[(N[(N[(x * x), $MachinePrecision] * 8.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(N[(x + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

7. Applied rewrites 100.0%

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

9. Applied rewrites 100.0%

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

Alternative 3: 99.9% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/
  (*
   (+
    (+
     (/ 3.0 (* (* 4.0 x) (* (* x x) x)))
     (/ 15.0 (* (* (* x x) 8.0) (* (* x x) (* x x)))))
    (+ (/ 1.0 (* (+ x x) x)) 1.0))
   (exp (* x x)))
  (* (fabs x) (sqrt PI))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[(N[(N[(3.0 / N[(N[(4.0 * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(15.0 / N[(N[(N[(x * x), $MachinePrecision] * 8.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(N[(x + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Abs[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. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

7. Applied rewrites 100.0%

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

9. Applied rewrites 99.9%

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

Alternative 4: 99.5% accurate, 2.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/
  (*
   (fma
    (/ 1.0 (fabs x))
    (/ 0.75 (pow x 4.0))
    (/ (+ (/ 1.0 (* (+ x x) x)) 1.0) (fabs x)))
   (exp (* x x)))
  (sqrt PI)))

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(0.75 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 / N[(N[(x + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $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. Applied rewrites 100.0%

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

4. Taylor expanded in x around inf

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

6. Applied rewrites 99.5%

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

8. Applied rewrites 99.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{\left|x\right|}, \frac{0.75}{{x}^{4}}, \frac{\frac{1}{\left(x + x\right) \cdot x} + 1}{\left|x\right|}\right) \cdot e^{x \cdot x}}}{\sqrt{\pi}} \]
9. Add Preprocessing

Alternative 5: 99.5% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(1.875 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 / N[(N[(x + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $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. Applied rewrites 100.0%

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

4. Taylor expanded in x around 0

   \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{\left|x\right|}, \color{blue}{\frac{\frac{15}{8}}{{x}^{6}}}, \left(\frac{1}{2 \cdot \left(x \cdot x\right)} + 1\right) \cdot \frac{1}{\left|x\right|}\right) \cdot e^{x \cdot x}}{\sqrt{\pi}} \]

6. Applied rewrites 99.5%

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

8. Applied rewrites 99.5%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{\left|x\right|}, \frac{1.875}{{x}^{6}}, \frac{\frac{1}{\left(x + x\right) \cdot x} + 1}{\left|x\right|}\right) \cdot e^{x \cdot x}}}{\sqrt{\pi}} \]
9. Add Preprocessing

Alternative 6: 99.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}\right) \cdot e^{x \cdot x}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (*
   (+ (/ 1.0 (fabs x)) (* 0.5 (/ 1.0 (* (pow x 2.0) (fabs x)))))
   (exp (* x x)))
  (sqrt PI)))

C

double code(double x) {
	return (((1.0 / fabs(x)) + (0.5 * (1.0 / (pow(x, 2.0) * fabs(x))))) * exp((x * x))) / sqrt(((double) M_PI));
}

Java

public static double code(double x) {
	return (((1.0 / Math.abs(x)) + (0.5 * (1.0 / (Math.pow(x, 2.0) * Math.abs(x))))) * Math.exp((x * x))) / Math.sqrt(Math.PI);
}

Python

def code(x):
	return (((1.0 / math.fabs(x)) + (0.5 * (1.0 / (math.pow(x, 2.0) * math.fabs(x))))) * math.exp((x * x))) / math.sqrt(math.pi)

Julia

function code(x)
	return Float64(Float64(Float64(Float64(1.0 / abs(x)) + Float64(0.5 * Float64(1.0 / Float64((x ^ 2.0) * abs(x))))) * exp(Float64(x * x))) / sqrt(pi))
end

MATLAB

function tmp = code(x)
	tmp = (((1.0 / abs(x)) + (0.5 * (1.0 / ((x ^ 2.0) * abs(x))))) * exp((x * x))) / sqrt(pi);
end

Wolfram

code[x_] := N[(N[(N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[(N[Power[x, 2.0], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $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. Applied rewrites 100.0%

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

4. Taylor expanded in x around inf

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

6. Applied rewrites 99.5%

   \[\leadsto \frac{\color{blue}{\left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}\right)} \cdot e^{x \cdot x}}{\sqrt{\pi}} \]
7. Add Preprocessing

Alternative 7: 99.4% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[(N[Abs[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. Applied rewrites 100.0%

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

4. Taylor expanded in x around inf

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

6. Applied rewrites 99.4%

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

8. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x}}{\left|x\right| \cdot \sqrt{\pi}}} \]
9. Add Preprocessing

Alternative 8: 2.3% accurate, 15.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(1.0 / N[(N[Abs[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. Applied rewrites 100.0%

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

4. Taylor expanded in x around inf

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

6. Applied rewrites 99.4%

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

7. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{\left|x\right| \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

9. Applied rewrites 2.3%

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

Reproduce

herbie shell --seed 2025153 
(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)))))))