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

Percentage Accurate: 100.0% → 100.0%

Time: 14.3s

Alternatives: 14

Speedup: 2.0×

• 99.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \geq 0.5\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \left(t\_0 \cdot t\_0\right) \cdot t\_0\\ t_2 := \left(t\_1 \cdot t\_0\right) \cdot t\_0\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(t\_0 + \frac{1}{2} \cdot t\_1\right) + \frac{3}{4} \cdot t\_2\right) + \frac{15}{8} \cdot \left(\left(t\_2 \cdot t\_0\right) \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \left(t\_0 \cdot t\_0\right) \cdot t\_0\\ t_2 := \left(t\_1 \cdot t\_0\right) \cdot t\_0\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(t\_0 + \frac{1}{2} \cdot t\_1\right) + \frac{3}{4} \cdot t\_2\right) + \frac{15}{8} \cdot \left(\left(t\_2 \cdot t\_0\right) \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[(1.0 / N[(N[(0.75 / N[(N[Abs[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.875 * N[(1.0 / N[(x * N[(N[(x * x), $MachinePrecision] * t$95$0), $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 egg-rr 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\color{blue}{\frac{1}{\frac{1}{\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \frac{0.75}{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}}}} + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
5. Step-by-step derivation

   1. associate-*l* N/A

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

   2. associate-*l* N/A

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

   3. pow3 N/A

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

   4. frac-times N/A

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

   5. metadata-eval N/A

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

   6. sqr-abs N/A

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

   7. cube-div N/A

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

   8. metadata-eval N/A

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

   9. cube-unmult N/A

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

   10. /-lowering-/.f64 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. *-lowering-*.f64 N/A

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

   14. *-lowering-*.f64 N/A

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

   15. *-lowering-*.f64 100.0

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\frac{1}{\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \frac{0.75}{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}}} + \frac{15}{8} \cdot \left(\frac{1}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)} \cdot \frac{1}{\left|x\right|}\right)\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\frac{1}{\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \frac{0.75}{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}}} + \frac{15}{8} \cdot \left(\color{blue}{\frac{1}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}} \cdot \frac{1}{\left|x\right|}\right)\right) \]
7. Step-by-step derivation

   1. un-div-inv N/A

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

   2. associate-/r* N/A

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

   3. /-lowering-/.f64 N/A

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

   4. associate-*r* N/A

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

   5. associate-*l* N/A

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

   6. associate-*l* N/A

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

8. Applied egg-rr 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\frac{1}{\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \frac{0.75}{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}}} + \frac{15}{8} \cdot \color{blue}{\frac{1}{x \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}}\right) \]
9. Step-by-step derivation

   1. metadata-eval 100.0

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

10. Applied egg-rr 100.0%

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

11. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[(N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.75 / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * 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 egg-rr 100.0%

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

5. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 N/A

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

   2. +-lowering-+.f64 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. /-lowering-/.f64 N/A

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

   6. fabs-lowering-fabs.f64 N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. /-lowering-/.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. *-lowering-*.f64 N/A

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

   13. *-lowering-*.f64 N/A

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

   14. fabs-lowering-fabs.f64 N/A

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

   15. metadata-eval N/A

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

   16. pow-sqr N/A

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

   17. unpow2 N/A

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

   18. associate-*l* N/A

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

   19. unpow2 N/A

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

   20. cube-unmult N/A

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

   21. metadata-eval N/A

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

7. Simplified 100.0%

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

9. Applied egg-rr 100.0%

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

Alternative 3: 100.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 N/A

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

   2. +-lowering-+.f64 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. /-lowering-/.f64 N/A

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

   6. fabs-lowering-fabs.f64 N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. /-lowering-/.f64 N/A

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

   10. unpow2 N/A

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

   11. associate-*l* N/A

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

   12. *-lowering-*.f64 N/A

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

   13. *-lowering-*.f64 N/A

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

   14. fabs-lowering-fabs.f64 N/A

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

   15. metadata-eval N/A

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

   16. pow-sqr N/A

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

   17. unpow2 N/A

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

   18. associate-*l* N/A

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

   19. unpow2 N/A

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

   20. cube-unmult N/A

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

   21. metadata-eval N/A

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

7. Simplified 100.0%

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

   1. associate-*l/ N/A

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

   2. associate-*l/ N/A

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

   3. /-lowering-/.f64 N/A

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

9. Applied egg-rr 100.0%

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

Alternative 4: 99.6% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around inf

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

   1. /-lowering-/.f64 N/A

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

   2. *-lowering-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. pow-sqr N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. unpow2 N/A

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

   8. cube-unmult N/A

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

   9. metadata-eval N/A

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

   10. pow-plus N/A

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

   11. *-lowering-*.f64 N/A

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

   12. pow-plus N/A

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

   13. metadata-eval N/A

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

   14. cube-unmult N/A

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

   15. unpow2 N/A

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

   16. *-lowering-*.f64 N/A

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

   17. unpow2 N/A

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

   18. *-lowering-*.f64 N/A

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

   19. fabs-lowering-fabs.f64 99.8

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

7. Simplified 99.8%

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

9. Applied egg-rr 99.8%

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

10. Final simplification 99.8%

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

Alternative 5: 99.6% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\color{blue}{\frac{1}{\frac{1}{\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \frac{0.75}{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}}}} + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
5. Step-by-step derivation

   1. associate-*l* N/A

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

   2. associate-*l* N/A

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

   3. pow3 N/A

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

   4. frac-times N/A

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

   5. metadata-eval N/A

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

   6. sqr-abs N/A

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

   7. cube-div N/A

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

   8. metadata-eval N/A

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

   9. cube-unmult N/A

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

   10. /-lowering-/.f64 N/A

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

   11. *-lowering-*.f64 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. *-lowering-*.f64 N/A

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

   14. *-lowering-*.f64 N/A

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

   15. *-lowering-*.f64 100.0

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\frac{1}{\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \frac{0.75}{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}}} + \frac{15}{8} \cdot \left(\frac{1}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right)} \cdot \frac{1}{\left|x\right|}\right)\right) \]

6. Applied egg-rr 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{\frac{1}{\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \frac{0.75}{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}}} + \frac{15}{8} \cdot \left(\color{blue}{\frac{1}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}} \cdot \frac{1}{\left|x\right|}\right)\right) \]

7. Taylor expanded in x around inf

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

   1. associate-/l/ N/A

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

   2. associate-*r/ N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-inverses N/A

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

   6. associate-/r* N/A

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

   7. sqr-abs N/A

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

   8. unpow2 N/A

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

   9. times-frac N/A

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

   10. pow-sqr N/A

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

   11. metadata-eval N/A

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

   12. *-rgt-identity N/A

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

   13. times-frac N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. distribute-rgt-out N/A

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

   17. +-lowering-+.f64 N/A

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

9. Simplified 99.8%

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

    1. associate-*l/ N/A

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

    2. associate-*l/ N/A

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

    3. /-lowering-/.f64 N/A

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

11. Applied egg-rr 99.8%

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

12. Final simplification 99.8%

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

Alternative 6: 99.4% 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 egg-rr 100.0%

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

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. associate-/l* N/A

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

   4. *-lowering-*.f64 N/A

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

   5. unpow2 N/A

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

   6. sqr-abs N/A

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

   7. unpow2 N/A

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

   8. exp-lowering-exp.f64 N/A

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. /-lowering-/.f64 N/A

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

   12. sqrt-lowering-sqrt.f64 N/A

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

   13. *-inverses N/A

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

   14. /-lowering-/.f64 N/A

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

   15. *-inverses N/A

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

   16. PI-lowering-PI.f64 N/A

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

   17. fabs-lowering-fabs.f64 99.7

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

7. Simplified 99.7%

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

   1. sqr-abs N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. *-lft-identity N/A

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

   5. /-lowering-/.f64 N/A

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

   6. *-lft-identity N/A

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

   7. sqr-abs N/A

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

   8. exp-lowering-exp.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. sqrt-div N/A

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

   11. metadata-eval N/A

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

   12. associate-/r/ N/A

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

   13. /-rgt-identity N/A

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

   14. *-lowering-*.f64 N/A

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

   15. fabs-lowering-fabs.f64 N/A

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

   16. sqrt-lowering-sqrt.f64 N/A

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

   17. PI-lowering-PI.f64 99.7

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

9. Applied egg-rr 99.7%

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

10. Final simplification 99.7%

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

Alternative 7: 87.7% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5, x \cdot x, 1\right)\\ \mathbf{if}\;\left|x\right| \leq 10^{+61}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, t\_0 \cdot \left(\left(x \cdot x\right) \cdot t\_0\right), -1\right)}{\left(\sqrt{\pi} \cdot \left|x\right|\right) \cdot \mathsf{fma}\left(x \cdot x, t\_0, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma 0.5 (* x x) 1.0)))
   (if (<= (fabs x) 1e+61)
     (/
      (fma (* x x) (* t_0 (* (* x x) t_0)) -1.0)
      (* (* (sqrt PI) (fabs x)) (fma (* x x) t_0 -1.0)))
     (*
      (* (* x x) (* x x))
      (* (sqrt (/ 1.0 PI)) (* (fabs x) 0.16666666666666666))))))

C

double code(double x) {
	double t_0 = fma(0.5, (x * x), 1.0);
	double tmp;
	if (fabs(x) <= 1e+61) {
		tmp = fma((x * x), (t_0 * ((x * x) * t_0)), -1.0) / ((sqrt(((double) M_PI)) * fabs(x)) * fma((x * x), t_0, -1.0));
	} else {
		tmp = ((x * x) * (x * x)) * (sqrt((1.0 / ((double) M_PI))) * (fabs(x) * 0.16666666666666666));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(0.5, Float64(x * x), 1.0)
	tmp = 0.0
	if (abs(x) <= 1e+61)
		tmp = Float64(fma(Float64(x * x), Float64(t_0 * Float64(Float64(x * x) * t_0)), -1.0) / Float64(Float64(sqrt(pi) * abs(x)) * fma(Float64(x * x), t_0, -1.0)));
	else
		tmp = Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(sqrt(Float64(1.0 / pi)) * Float64(abs(x) * 0.16666666666666666)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 1e+61], N[(N[(N[(x * x), $MachinePrecision] * N[(t$95$0 * N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 9.99999999999999949e60

   1. Initial program 99.9%

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

   4. Applied egg-rr 99.9%

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

   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. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. sqr-abs N/A

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

      7. unpow2 N/A

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

      8. exp-lowering-exp.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. sqrt-lowering-sqrt.f64 N/A

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

      13. *-inverses N/A

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

      14. /-lowering-/.f64 N/A

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

      15. *-inverses N/A

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

      16. PI-lowering-PI.f64 N/A

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

      17. fabs-lowering-fabs.f64 98.7

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

   7. Simplified 98.7%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 4.1

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

   10. Simplified 4.1%

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

       1. flip-+ N/A

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

       2. clear-num N/A

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

       3. frac-times N/A

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

       4. /-lowering-/.f64 N/A

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

   12. Applied egg-rr 43.7%

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

   if 9.99999999999999949e60 < (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 egg-rr 100.0%

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

   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. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. sqr-abs N/A

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

      7. unpow2 N/A

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

      8. exp-lowering-exp.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. sqrt-lowering-sqrt.f64 N/A

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

      13. *-inverses N/A

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

      14. /-lowering-/.f64 N/A

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

      15. *-inverses N/A

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

      16. PI-lowering-PI.f64 N/A

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

      17. fabs-lowering-fabs.f64 100.0

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 100.0%

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

   11. Taylor expanded in x around inf

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

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(\left({x}^{4} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{6}} \]

       2. associate-*l* N/A

          \[\leadsto \color{blue}{\left({x}^{4} \cdot \left(\left|x\right| \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)} \cdot \frac{1}{6} \]

       3. *-commutative N/A

          \[\leadsto \left({x}^{4} \cdot \color{blue}{\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)}\right) \cdot \frac{1}{6} \]

       4. associate-*r* N/A

          \[\leadsto \color{blue}{{x}^{4} \cdot \left(\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right) \cdot \frac{1}{6}\right)} \]

       5. *-commutative N/A

          \[\leadsto {x}^{4} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{{x}^{4} \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right)} \]

       7. metadata-eval N/A

          \[\leadsto {x}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right) \]

       8. pow-sqr N/A

          \[\leadsto \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)} \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)} \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \left(\color{blue}{\left(x \cdot x\right)} \cdot {x}^{2}\right) \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \left(\color{blue}{\left(x \cdot x\right)} \cdot {x}^{2}\right) \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right) \cdot \frac{1}{6}\right)} \]

       15. associate-*l* N/A

           \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \frac{1}{6}\right)\right)} \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \frac{1}{6}\right)\right)} \]

       17. sqrt-lowering-sqrt.f64 N/A

           \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}} \cdot \left(\left|x\right| \cdot \frac{1}{6}\right)\right) \]

       18. /-lowering-/.f64 N/A

           \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{\color{blue}{\frac{1}{\mathsf{PI}\left(\right)}}} \cdot \left(\left|x\right| \cdot \frac{1}{6}\right)\right) \]

       19. PI-lowering-PI.f64 N/A

           \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{\frac{1}{\color{blue}{\mathsf{PI}\left(\right)}}} \cdot \left(\left|x\right| \cdot \frac{1}{6}\right)\right) \]

       20. *-lowering-*.f64 N/A

           \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(\left|x\right| \cdot \frac{1}{6}\right)}\right) \]

       21. fabs-lowering-fabs.f64 100.0

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

   13. Simplified 100.0%

       \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot 0.16666666666666666\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{+61}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.5, x \cdot x, 1\right) \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(0.5, x \cdot x, 1\right)\right), -1\right)}{\left(\sqrt{\pi} \cdot \left|x\right|\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.5, x \cdot x, 1\right), -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot 0.16666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.6% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 100.0%

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

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. associate-/l* N/A

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

   4. *-lowering-*.f64 N/A

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

   5. unpow2 N/A

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

   6. sqr-abs N/A

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

   7. unpow2 N/A

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

   8. exp-lowering-exp.f64 N/A

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. /-lowering-/.f64 N/A

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

   12. sqrt-lowering-sqrt.f64 N/A

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

   13. *-inverses N/A

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

   14. /-lowering-/.f64 N/A

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

   15. *-inverses N/A

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

   16. PI-lowering-PI.f64 N/A

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

   17. fabs-lowering-fabs.f64 99.7

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

7. Simplified 99.7%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. accelerator-lowering-fma.f64 N/A

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

   13. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6}, \frac{1}{2}\right), 1\right), 1\right) \cdot \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}}{\left|x\right|} \]

   14. *-lowering-*.f64 84.3

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

10. Simplified 84.3%

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

Alternative 9: 80.4% accurate, 6.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 100.0%

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

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. associate-/l* N/A

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

   4. *-lowering-*.f64 N/A

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

   5. unpow2 N/A

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

   6. sqr-abs N/A

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

   7. unpow2 N/A

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

   8. exp-lowering-exp.f64 N/A

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. /-lowering-/.f64 N/A

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

   12. sqrt-lowering-sqrt.f64 N/A

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

   13. *-inverses N/A

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

   14. /-lowering-/.f64 N/A

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

   15. *-inverses N/A

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

   16. PI-lowering-PI.f64 N/A

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

   17. fabs-lowering-fabs.f64 99.7

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

7. Simplified 99.7%

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

8. Taylor expanded in x around 0

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

10. Simplified 80.5%

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

11. Taylor expanded in x around inf

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

    1. *-lowering-*.f64 N/A

       \[\leadsto \color{blue}{{x}^{4} \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right) + \frac{1}{2} \cdot \left(\frac{\left|x\right|}{{x}^{2}} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)} \]

    2. metadata-eval N/A

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

    3. pow-sqr N/A

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

    4. *-lowering-*.f64 N/A

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

    5. unpow2 N/A

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

    6. *-lowering-*.f64 N/A

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

    7. unpow2 N/A

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

    8. *-lowering-*.f64 N/A

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

    9. *-commutative N/A

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

    10. associate-*r* N/A

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

    11. associate-*r* N/A

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

    12. distribute-rgt-out N/A

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

    13. *-commutative N/A

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

    14. *-lowering-*.f64 N/A

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

13. Simplified 80.5%

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

Alternative 10: 80.4% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * 0.16666666666666666), $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 egg-rr 100.0%

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

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. associate-/l* N/A

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

   4. *-lowering-*.f64 N/A

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

   5. unpow2 N/A

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

   6. sqr-abs N/A

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

   7. unpow2 N/A

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

   8. exp-lowering-exp.f64 N/A

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. /-lowering-/.f64 N/A

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

   12. sqrt-lowering-sqrt.f64 N/A

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

   13. *-inverses N/A

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

   14. /-lowering-/.f64 N/A

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

   15. *-inverses N/A

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

   16. PI-lowering-PI.f64 N/A

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

   17. fabs-lowering-fabs.f64 99.7

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

7. Simplified 99.7%

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

8. Taylor expanded in x around 0

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

10. Simplified 80.5%

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

11. Taylor expanded in x around inf

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

    1. *-commutative N/A

       \[\leadsto \color{blue}{\left(\left({x}^{4} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) \cdot \frac{1}{6}} \]

    2. associate-*l* N/A

       \[\leadsto \color{blue}{\left({x}^{4} \cdot \left(\left|x\right| \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)} \cdot \frac{1}{6} \]

    3. *-commutative N/A

       \[\leadsto \left({x}^{4} \cdot \color{blue}{\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)}\right) \cdot \frac{1}{6} \]

    4. associate-*r* N/A

       \[\leadsto \color{blue}{{x}^{4} \cdot \left(\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right) \cdot \frac{1}{6}\right)} \]

    5. *-commutative N/A

       \[\leadsto {x}^{4} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right)} \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \color{blue}{{x}^{4} \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right)} \]

    7. metadata-eval N/A

       \[\leadsto {x}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right) \]

    8. pow-sqr N/A

       \[\leadsto \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)} \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right) \]

    9. *-lowering-*.f64 N/A

       \[\leadsto \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)} \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right) \]

    10. unpow2 N/A

        \[\leadsto \left(\color{blue}{\left(x \cdot x\right)} \cdot {x}^{2}\right) \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \left(\color{blue}{\left(x \cdot x\right)} \cdot {x}^{2}\right) \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right) \]

    12. unpow2 N/A

        \[\leadsto \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right) \]

    13. *-lowering-*.f64 N/A

        \[\leadsto \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{1}{6} \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)\right) \]

    14. *-commutative N/A

        \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right) \cdot \frac{1}{6}\right)} \]

    15. associate-*l* N/A

        \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \frac{1}{6}\right)\right)} \]

    16. *-lowering-*.f64 N/A

        \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \frac{1}{6}\right)\right)} \]

    17. sqrt-lowering-sqrt.f64 N/A

        \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}}} \cdot \left(\left|x\right| \cdot \frac{1}{6}\right)\right) \]

    18. /-lowering-/.f64 N/A

        \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{\color{blue}{\frac{1}{\mathsf{PI}\left(\right)}}} \cdot \left(\left|x\right| \cdot \frac{1}{6}\right)\right) \]

    19. PI-lowering-PI.f64 N/A

        \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{\frac{1}{\color{blue}{\mathsf{PI}\left(\right)}}} \cdot \left(\left|x\right| \cdot \frac{1}{6}\right)\right) \]

    20. *-lowering-*.f64 N/A

        \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(\left|x\right| \cdot \frac{1}{6}\right)}\right) \]

    21. fabs-lowering-fabs.f64 80.5

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

13. Simplified 80.5%

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

Alternative 11: 75.5% accurate, 9.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] * N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $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 egg-rr 100.0%

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

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. associate-/l* N/A

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

   4. *-lowering-*.f64 N/A

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

   5. unpow2 N/A

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

   6. sqr-abs N/A

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

   7. unpow2 N/A

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

   8. exp-lowering-exp.f64 N/A

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. /-lowering-/.f64 N/A

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

   12. sqrt-lowering-sqrt.f64 N/A

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

   13. *-inverses N/A

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

   14. /-lowering-/.f64 N/A

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

   15. *-inverses N/A

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

   16. PI-lowering-PI.f64 N/A

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

   17. fabs-lowering-fabs.f64 99.7

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

7. Simplified 99.7%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. accelerator-lowering-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. *-lowering-*.f64 74.6

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

10. Simplified 74.6%

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

    1. clear-num N/A

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

    2. un-div-inv N/A

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

    3. /-lowering-/.f64 N/A

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

    4. accelerator-lowering-fma.f64 N/A

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

    5. *-lowering-*.f64 N/A

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

    6. associate-*r* N/A

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

    7. *-commutative N/A

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

    8. accelerator-lowering-fma.f64 N/A

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

    9. *-lowering-*.f64 N/A

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

    10. sqrt-div N/A

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

    11. metadata-eval N/A

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

    12. associate-/r/ N/A

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

    13. /-rgt-identity N/A

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

    14. *-lowering-*.f64 N/A

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

    15. fabs-lowering-fabs.f64 N/A

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

    16. sqrt-lowering-sqrt.f64 N/A

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

    17. PI-lowering-PI.f64 74.6

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

12. Applied egg-rr 74.6%

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

13. Final simplification 74.6%

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

Alternative 12: 68.5% accurate, 10.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $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 egg-rr 100.0%

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

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. associate-/l* N/A

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

   4. *-lowering-*.f64 N/A

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

   5. unpow2 N/A

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

   6. sqr-abs N/A

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

   7. unpow2 N/A

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

   8. exp-lowering-exp.f64 N/A

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. /-lowering-/.f64 N/A

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

   12. sqrt-lowering-sqrt.f64 N/A

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

   13. *-inverses N/A

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

   14. /-lowering-/.f64 N/A

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

   15. *-inverses N/A

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

   16. PI-lowering-PI.f64 N/A

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

   17. fabs-lowering-fabs.f64 99.7

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

7. Simplified 99.7%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. accelerator-lowering-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. *-lowering-*.f64 74.6

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

10. Simplified 74.6%

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

11. Taylor expanded in x around inf

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

13. Simplified 68.1%

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

14. Final simplification 68.1%

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

Alternative 13: 68.5% accurate, 10.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 100.0%

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

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. associate-/l* N/A

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

   4. *-lowering-*.f64 N/A

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

   5. unpow2 N/A

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

   6. sqr-abs N/A

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

   7. unpow2 N/A

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

   8. exp-lowering-exp.f64 N/A

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. /-lowering-/.f64 N/A

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

   12. sqrt-lowering-sqrt.f64 N/A

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

   13. *-inverses N/A

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

   14. /-lowering-/.f64 N/A

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

   15. *-inverses N/A

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

   16. PI-lowering-PI.f64 N/A

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

   17. fabs-lowering-fabs.f64 99.7

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

7. Simplified 99.7%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. accelerator-lowering-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. *-lowering-*.f64 74.6

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

10. Simplified 74.6%

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

11. Taylor expanded in x around inf

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

    1. associate-*r* N/A

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

    2. *-commutative N/A

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

    3. associate-*r/ N/A

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

    4. metadata-eval N/A

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

    5. pow-sqr N/A

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

    6. associate-*l* N/A

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

    7. associate-*l/ N/A

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

    8. associate-*r/ N/A

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

    9. *-commutative N/A

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

    10. *-lowering-*.f64 N/A

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

    11. sqrt-lowering-sqrt.f64 N/A

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

    12. /-lowering-/.f64 N/A

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

    13. PI-lowering-PI.f64 N/A

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

    14. *-commutative N/A

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

    15. associate-*r/ N/A

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

13. Simplified 68.1%

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

Alternative 14: 2.3% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(1.0 / 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 egg-rr 100.0%

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

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. associate-/l* N/A

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

   4. *-lowering-*.f64 N/A

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

   5. unpow2 N/A

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

   6. sqr-abs N/A

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

   7. unpow2 N/A

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

   8. exp-lowering-exp.f64 N/A

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. /-lowering-/.f64 N/A

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

   12. sqrt-lowering-sqrt.f64 N/A

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

   13. *-inverses N/A

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

   14. /-lowering-/.f64 N/A

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

   15. *-inverses N/A

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

   16. PI-lowering-PI.f64 N/A

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

   17. fabs-lowering-fabs.f64 99.7

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

7. Simplified 99.7%

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

8. Taylor expanded in x around 0

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

   1. associate-*r/ N/A

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

   2. *-rgt-identity N/A

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

   3. /-lowering-/.f64 N/A

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

   4. sqrt-lowering-sqrt.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. PI-lowering-PI.f64 N/A

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

   7. fabs-lowering-fabs.f64 2.3

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

10. Simplified 2.3%

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

    1. div-inv N/A

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

    2. sqrt-div N/A

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

    3. metadata-eval N/A

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

    4. frac-times N/A

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

    5. metadata-eval N/A

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

    6. /-lowering-/.f64 N/A

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

    7. *-lowering-*.f64 N/A

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

    8. sqrt-lowering-sqrt.f64 N/A

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

    9. PI-lowering-PI.f64 N/A

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

    10. fabs-lowering-fabs.f64 2.3

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

12. Applied egg-rr 2.3%

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

Reproduce

herbie shell --seed 2024194 
(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)))))))