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

Percentage Accurate: 100.0% → 100.0%

Time: 12.9s

Alternatives: 15

Speedup: 1.8×

• 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, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, 1 + \frac{0.5}{x \cdot x}, \frac{0.75}{t\_0} + \frac{1.875}{\left(x \cdot x\right) \cdot t\_0}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) (* (* x x) (* x x)))))
   (*
    (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x))))
    (fma
     (/ 1.0 (fabs x))
     (+ 1.0 (/ 0.5 (* x x)))
     (+ (/ 0.75 t_0) (/ 1.875 (* (* x x) t_0)))))))

C

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

Julia

function code(x)
	t_0 = Float64(abs(x) * Float64(Float64(x * x) * Float64(x * x)))
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * exp(Float64(abs(x) * abs(x)))) * fma(Float64(1.0 / abs(x)), Float64(1.0 + Float64(0.5 / Float64(x * x))), Float64(Float64(0.75 / t_0) + Float64(1.875 / Float64(Float64(x * x) * t_0)))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $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[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.75 / t$95$0), $MachinePrecision] + N[(1.875 / N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\frac{0.75}{t\_0} + \frac{1.875}{\left(x \cdot x\right) \cdot t\_0}\right) + \frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}\right) \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

def code(x):
	t_0 = math.fabs(x) * ((x * x) * (x * x))
	return ((1.0 / math.sqrt(math.pi)) * math.exp((math.fabs(x) * math.fabs(x)))) * (((0.75 / t_0) + (1.875 / ((x * x) * t_0))) + ((1.0 + (0.5 / (x * x))) / math.fabs(x)))

Julia

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

MATLAB

function tmp = code(x)
	t_0 = abs(x) * ((x * x) * (x * x));
	tmp = ((1.0 / sqrt(pi)) * exp((abs(x) * abs(x)))) * (((0.75 / t_0) + (1.875 / ((x * x) * t_0))) + ((1.0 + (0.5 / (x * x))) / abs(x)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $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[(0.75 / t$95$0), $MachinePrecision] + N[(1.875 / N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 3: 99.7% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\frac{e^{{\left(\left|x\right|\right)}^{2}}}{{x}^{2} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + \left(\frac{3}{4} \cdot \left(\frac{e^{{\left(\left|x\right|\right)}^{2}}}{{x}^{4} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \frac{e^{{\left(\left|x\right|\right)}^{2}}}{\left|x\right|}\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

7. Applied rewrites 99.7%

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

9. Applied rewrites 99.7%

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

Alternative 4: 99.6% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\frac{e^{{\left(\left|x\right|\right)}^{2}}}{{x}^{2} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + \left(\frac{3}{4} \cdot \left(\frac{e^{{\left(\left|x\right|\right)}^{2}}}{{x}^{4} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \frac{e^{{\left(\left|x\right|\right)}^{2}}}{\left|x\right|}\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

7. Applied rewrites 99.7%

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

9. Applied rewrites 99.7%

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

10. Final simplification 99.7%

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

Alternative 5: 99.6% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-PI.f64 N/A

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

   9. associate-*r/ N/A

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

7. Applied rewrites 99.7%

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

Alternative 6: 99.5% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. sqr-abs N/A

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

   7. unpow2 N/A

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

   8. lower-exp.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-PI.f64 N/A

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

   14. lower-fabs.f64 99.7

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

7. Applied rewrites 99.7%

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

9. Applied rewrites 99.7%

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

Alternative 7: 88.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5, x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{+76}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\pi}} \cdot \frac{\mathsf{fma}\left(x \cdot x, t\_0 \cdot t\_0, -1\right)}{\mathsf{fma}\left(x, t\_0, -1\right)}}{\left|x\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\pi} \cdot \left|x\right|} \cdot \mathsf{fma}\left(x, t\_0, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma 0.5 (* x (* x x)) x)))
   (if (<= (fabs x) 2e+76)
     (/
      (* (sqrt (/ 1.0 PI)) (/ (fma (* x x) (* t_0 t_0) -1.0) (fma x t_0 -1.0)))
      (fabs x))
     (* (/ 1.0 (* (sqrt PI) (fabs x))) (fma x t_0 1.0)))))

C

double code(double x) {
	double t_0 = fma(0.5, (x * (x * x)), x);
	double tmp;
	if (fabs(x) <= 2e+76) {
		tmp = (sqrt((1.0 / ((double) M_PI))) * (fma((x * x), (t_0 * t_0), -1.0) / fma(x, t_0, -1.0))) / fabs(x);
	} else {
		tmp = (1.0 / (sqrt(((double) M_PI)) * fabs(x))) * fma(x, t_0, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(0.5, Float64(x * Float64(x * x)), x)
	tmp = 0.0
	if (abs(x) <= 2e+76)
		tmp = Float64(Float64(sqrt(Float64(1.0 / pi)) * Float64(fma(Float64(x * x), Float64(t_0 * t_0), -1.0) / fma(x, t_0, -1.0))) / abs(x));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(pi) * abs(x))) * fma(x, t_0, 1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 2e+76], N[(N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision] + -1.0), $MachinePrecision] / N[(x * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[Pi], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 2.0000000000000001e76

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. sqr-abs N/A

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

      7. unpow2 N/A

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

      8. lower-exp.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-PI.f64 N/A

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

      14. lower-fabs.f64 98.6

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + {x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \]
   9. Step-by-step derivation

   10. Applied rewrites 4.5%

       \[\leadsto \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.5, x\right), 1\right) \cdot \sqrt{\frac{1}{\pi}}}{\left|x\right|} \]
   11. Step-by-step derivation

   12. Applied rewrites 52.0%

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

   if 2.0000000000000001e76 < (fabs.f64 x)

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. sqr-abs N/A

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

      7. unpow2 N/A

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

      8. lower-exp.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-PI.f64 N/A

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

      14. lower-fabs.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + {x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \]
   9. Step-by-step derivation

   10. Applied rewrites 100.0%

       \[\leadsto \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.5, x\right), 1\right) \cdot \sqrt{\frac{1}{\pi}}}{\left|x\right|} \]
   11. Step-by-step derivation

   12. Applied rewrites 100.0%

       \[\leadsto \frac{1}{\sqrt{\pi} \cdot \left|x\right|} \cdot \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.5, x \cdot \left(x \cdot x\right), x\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{+76}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\pi}} \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.5, x \cdot \left(x \cdot x\right), x\right) \cdot \mathsf{fma}\left(0.5, x \cdot \left(x \cdot x\right), x\right), -1\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.5, x \cdot \left(x \cdot x\right), x\right), -1\right)}}{\left|x\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\pi} \cdot \left|x\right|} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(0.5, x \cdot \left(x \cdot x\right), x\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \left(\frac{1}{\sqrt{\pi} \cdot \left|x\right|} \cdot \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{0.75}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fma (* x x) (fma (* x x) (fma x (* x 0.16666666666666666) 0.5) 1.0) 1.0)
  (*
   (/ 1.0 (* (sqrt PI) (fabs x)))
   (+ 1.0 (+ (/ 0.5 (* x x)) (/ 0.75 (* x (* x (* x x)))))))))

C

double code(double x) {
	return fma((x * x), fma((x * x), fma(x, (x * 0.16666666666666666), 0.5), 1.0), 1.0) * ((1.0 / (sqrt(((double) M_PI)) * fabs(x))) * (1.0 + ((0.5 / (x * x)) + (0.75 / (x * (x * (x * x)))))));
}

Julia

function code(x)
	return Float64(fma(Float64(x * x), fma(Float64(x * x), fma(x, Float64(x * 0.16666666666666666), 0.5), 1.0), 1.0) * Float64(Float64(1.0 / Float64(sqrt(pi) * abs(x))) * Float64(1.0 + Float64(Float64(0.5 / Float64(x * x)) + Float64(0.75 / Float64(x * Float64(x * Float64(x * x))))))))
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(1.0 / N[(N[Sqrt[Pi], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(0.75 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\frac{e^{{\left(\left|x\right|\right)}^{2}}}{{x}^{2} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + \left(\frac{3}{4} \cdot \left(\frac{e^{{\left(\left|x\right|\right)}^{2}}}{{x}^{4} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \frac{e^{{\left(\left|x\right|\right)}^{2}}}{\left|x\right|}\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{\left(1 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \cdot \left(\left(1 + \frac{\frac{1}{2}}{x \cdot x}\right) + \frac{\frac{3}{4}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right) \]
9. Step-by-step derivation

10. Applied rewrites 85.9%

    \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \sqrt{\frac{1}{\pi}}}{\left|x\right|} \cdot \left(\left(1 + \frac{0.5}{x \cdot x}\right) + \frac{0.75}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right) \]
11. Step-by-step derivation

12. Applied rewrites 85.9%

    \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \color{blue}{\left(\frac{1}{\sqrt{\pi} \cdot \left|x\right|} \cdot \left(1 + \left(\frac{0.5}{x \cdot x} + \frac{0.75}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right)} \]
13. Add Preprocessing

Alternative 9: 84.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{\pi}} \cdot \left(\left(1 + \frac{0.5}{x \cdot x}\right) \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.16666666666666666, 0.5\right), 1\right), 1\right)}{\left|x\right|}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (sqrt (/ 1.0 PI))
  (*
   (+ 1.0 (/ 0.5 (* x x)))
   (/
    (fma (* x x) (fma (* x x) (fma x (* x 0.16666666666666666) 0.5) 1.0) 1.0)
    (fabs x)))))

C

double code(double x) {
	return sqrt((1.0 / ((double) M_PI))) * ((1.0 + (0.5 / (x * x))) * (fma((x * x), fma((x * x), fma(x, (x * 0.16666666666666666), 0.5), 1.0), 1.0) / fabs(x)));
}

Julia

function code(x)
	return Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) * Float64(fma(Float64(x * x), fma(Float64(x * x), fma(x, Float64(x * 0.16666666666666666), 0.5), 1.0), 1.0) / abs(x))))
end

Wolfram

code[x_] := N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-PI.f64 N/A

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

   9. associate-*r/ N/A

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 85.9%

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\left(1 + \frac{0.5}{x \cdot x}\right) \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.16666666666666666, 0.5\right), 1\right), 1\right)}{\left|\color{blue}{x}\right|}\right) \]
11. Add Preprocessing

Alternative 10: 84.4% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.16666666666666666, 0.5\right), 1\right), 1\right)}{\left|x\right|} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (*
   (sqrt (/ 1.0 PI))
   (fma (* x x) (fma (* x x) (fma x (* x 0.16666666666666666) 0.5) 1.0) 1.0))
  (fabs x)))

C

double code(double x) {
	return (sqrt((1.0 / ((double) M_PI))) * fma((x * x), fma((x * x), fma(x, (x * 0.16666666666666666), 0.5), 1.0), 1.0)) / fabs(x);
}

Julia

function code(x)
	return Float64(Float64(sqrt(Float64(1.0 / pi)) * fma(Float64(x * x), fma(Float64(x * x), fma(x, Float64(x * 0.16666666666666666), 0.5), 1.0), 1.0)) / abs(x))
end

Wolfram

code[x_] := N[(N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. sqr-abs N/A

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

   7. unpow2 N/A

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

   8. lower-exp.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-PI.f64 N/A

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

   14. lower-fabs.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{\left(1 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right)\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \]
9. Step-by-step derivation

10. Applied rewrites 85.9%

    \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.16666666666666666, 0.5\right), 1\right), 1\right) \cdot \sqrt{\frac{1}{\pi}}}{\left|x\right|} \]

11. Final simplification 85.9%

    \[\leadsto \frac{\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.16666666666666666, 0.5\right), 1\right), 1\right)}{\left|x\right|} \]
12. Add Preprocessing

Alternative 11: 84.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{\frac{1}{\pi}} \cdot \left(0.16666666666666666 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)}{\left|x\right|} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (* (sqrt (/ 1.0 PI)) (* 0.16666666666666666 (* (* x x) (* (* x x) (* x x)))))
  (fabs x)))

C

double code(double x) {
	return (sqrt((1.0 / ((double) M_PI))) * (0.16666666666666666 * ((x * x) * ((x * x) * (x * x))))) / fabs(x);
}

Java

public static double code(double x) {
	return (Math.sqrt((1.0 / Math.PI)) * (0.16666666666666666 * ((x * x) * ((x * x) * (x * x))))) / Math.abs(x);
}

Python

def code(x):
	return (math.sqrt((1.0 / math.pi)) * (0.16666666666666666 * ((x * x) * ((x * x) * (x * x))))) / math.fabs(x)

Julia

function code(x)
	return Float64(Float64(sqrt(Float64(1.0 / pi)) * Float64(0.16666666666666666 * Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(x * x))))) / abs(x))
end

MATLAB

function tmp = code(x)
	tmp = (sqrt((1.0 / pi)) * (0.16666666666666666 * ((x * x) * ((x * x) * (x * x))))) / abs(x);
end

Wolfram

code[x_] := N[(N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(0.16666666666666666 * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. sqr-abs N/A

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

   7. unpow2 N/A

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

   8. lower-exp.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-PI.f64 N/A

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

   14. lower-fabs.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 85.9%

    \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, \sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.16666666666666666, 0.5\right), 1\right), \sqrt{\frac{1}{\pi}}\right)}{\left|\color{blue}{x}\right|} \]

11. Taylor expanded in x around inf

    \[\leadsto \frac{\frac{1}{6} \cdot \left({x}^{6} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)}{\left|x\right|} \]
12. Step-by-step derivation

13. Applied rewrites 85.9%

    \[\leadsto \frac{\sqrt{\frac{1}{\pi}} \cdot \left(0.16666666666666666 \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\right)}{\left|x\right|} \]
14. Add Preprocessing

Alternative 12: 76.4% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. sqr-abs N/A

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

   7. unpow2 N/A

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

   8. lower-exp.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-PI.f64 N/A

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

   14. lower-fabs.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{\left(1 + {x}^{2} \cdot \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \]
9. Step-by-step derivation

10. Applied rewrites 78.4%

    \[\leadsto \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.5, x\right), 1\right) \cdot \sqrt{\frac{1}{\pi}}}{\left|x\right|} \]
11. Step-by-step derivation

12. Applied rewrites 78.4%

    \[\leadsto \frac{1}{\sqrt{\pi} \cdot \left|x\right|} \cdot \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.5, x \cdot \left(x \cdot x\right), x\right), 1\right)} \]
13. Add Preprocessing

Alternative 13: 52.1% accurate, 10.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. sqr-abs N/A

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

   7. unpow2 N/A

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

   8. lower-exp.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-PI.f64 N/A

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

   14. lower-fabs.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{\left(1 + {x}^{2}\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \]
9. Step-by-step derivation

10. Applied rewrites 53.5%

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

11. Final simplification 53.5%

    \[\leadsto \frac{\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(x, x, 1\right)}{\left|x\right|} \]
12. Add Preprocessing

Alternative 14: 5.4% accurate, 10.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. sqr-abs N/A

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

   7. unpow2 N/A

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

   8. lower-exp.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-PI.f64 N/A

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

   14. lower-fabs.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in x around 0

   \[\leadsto \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \frac{1}{\left|x\right|} + \color{blue}{\frac{{x}^{2}}{\left|x\right|} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}} \]
9. Step-by-step derivation

10. Applied rewrites 5.4%

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

Alternative 15: 2.3% accurate, 13.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. sqr-abs N/A

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

   7. unpow2 N/A

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

   8. lower-exp.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-PI.f64 N/A

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

   14. lower-fabs.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|\color{blue}{x}\right|} \]
9. Step-by-step derivation

10. Applied rewrites 2.3%

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

Reproduce

herbie shell --seed 2024233 
(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)))))))