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

Percentage Accurate: 100.0% → 100.0%

Time: 12.7s

Alternatives: 9

Speedup: 1.8×

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

Specification

\[x \geq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 99.5% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. sqr-abs N/A

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

   8. unpow2 N/A

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

   9. lower-exp.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-fabs.f64 99.7

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

7. Simplified 99.7%

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

   1. lift-PI.f64 N/A

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

   2. /-rgt-identity N/A

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

   3. sqrt-div N/A

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

   4. metadata-eval N/A

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

   5. /-rgt-identity N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-fabs.f64 N/A

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

   10. frac-times N/A

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

   11. *-lft-identity N/A

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

   12. lower-/.f64 N/A

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

   13. lower-*.f64 99.7

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

9. Applied egg-rr 99.7%

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

Alternative 3: 86.1% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(1.0 / x), $MachinePrecision] * N[(N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] + N[(N[(N[Abs[x], $MachinePrecision] * 0.5 + N[(0.75 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[Sqrt[Pi], $MachinePrecision] * 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 45.7%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{x \cdot x} - \frac{0.25}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}, \frac{1}{\frac{1 - \frac{0.5}{x \cdot x}}{\left|x\right|}}, \frac{0.75}{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} + \frac{1.875}{\left(x \cdot x\right) \cdot \left(\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}\right)} \]

5. Taylor expanded in x around inf

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

7. Simplified 45.5%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right) + 1\right)} \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

   2. lower-fma.f64 N/A

      \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\color{blue}{\mathsf{fma}\left({x}^{2}, 1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right), 1\right)} \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

   3. unpow2 N/A

      \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, 1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right), 1\right) \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, 1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right), 1\right) \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

   5. +-commutative N/A

      \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right) + 1}, 1\right) \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{6} \cdot {x}^{2}, 1\right)}, 1\right) \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

   7. unpow2 N/A

      \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{6} \cdot {x}^{2}, 1\right), 1\right) \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{6} \cdot {x}^{2}, 1\right), 1\right) \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

   9. +-commutative N/A

      \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{6} \cdot {x}^{2} + \frac{1}{2}}, 1\right), 1\right) \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

   10. *-commutative N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

   11. unpow2 N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{6} + \frac{1}{2}, 1\right), 1\right) \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

   12. associate-*l* N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{6}\right)} + \frac{1}{2}, 1\right), 1\right) \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

   13. lower-fma.f64 N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{6}, \frac{1}{2}\right)}, 1\right), 1\right) \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

   14. lower-*.f64 34.9

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

10. Simplified 34.9%

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

12. Applied egg-rr 89.2%

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

13. Final simplification 89.2%

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

Alternative 4: 87.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ t_1 := x \cdot \mathsf{fma}\left(x, x \cdot 0.5, 1\right)\\ \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{+61}:\\ \;\;\;\;t\_0 \cdot \frac{\mathsf{fma}\left(x \cdot x, t\_1 \cdot t\_1, -1\right)}{\left|x\right| \cdot \mathsf{fma}\left(x, t\_1, -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(t\_0 \cdot \left(\left|x\right| \cdot 0.16666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 PI))) (t_1 (* x (fma x (* x 0.5) 1.0))))
   (if (<= (fabs x) 5e+61)
     (* t_0 (/ (fma (* x x) (* t_1 t_1) -1.0) (* (fabs x) (fma x t_1 -1.0))))
     (* (* (* x x) (* x x)) (* t_0 (* (fabs x) 0.16666666666666666))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 / ((double) M_PI)));
	double t_1 = x * fma(x, (x * 0.5), 1.0);
	double tmp;
	if (fabs(x) <= 5e+61) {
		tmp = t_0 * (fma((x * x), (t_1 * t_1), -1.0) / (fabs(x) * fma(x, t_1, -1.0)));
	} else {
		tmp = ((x * x) * (x * x)) * (t_0 * (fabs(x) * 0.16666666666666666));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / pi))
	t_1 = Float64(x * fma(x, Float64(x * 0.5), 1.0))
	tmp = 0.0
	if (abs(x) <= 5e+61)
		tmp = Float64(t_0 * Float64(fma(Float64(x * x), Float64(t_1 * t_1), -1.0) / Float64(abs(x) * fma(x, t_1, -1.0))));
	else
		tmp = Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(t_0 * Float64(abs(x) * 0.16666666666666666)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(x * N[(x * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 5e+61], N[(t$95$0 * N[(N[(N[(x * x), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision] + -1.0), $MachinePrecision] / N[(N[Abs[x], $MachinePrecision] * N[(x * t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(N[Abs[x], $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 5.00000000000000018e61

   1. Initial program 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-abs N/A

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

      8. unpow2 N/A

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

      9. lower-exp.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-fabs.f64 98.4

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

   7. Simplified 98.4%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 4.2

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

   10. Simplified 4.2%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-fma.f64 N/A

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

       4. flip-+ N/A

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

       5. lift-fabs.f64 N/A

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

       6. associate-/l/ N/A

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

       7. lower-/.f64 N/A

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

   12. Applied egg-rr 41.9%

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

   if 5.00000000000000018e61 < (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 34.7%

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{x \cdot x} - \frac{0.25}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}, \frac{1}{\frac{1 - \frac{0.5}{x \cdot x}}{\left|x\right|}}, \frac{0.75}{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} + \frac{1.875}{\left(x \cdot x\right) \cdot \left(\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}\right)} \]

   5. Taylor expanded in x around inf

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

   7. Simplified 34.7%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right) + 1\right)} \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\color{blue}{\mathsf{fma}\left({x}^{2}, 1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right), 1\right)} \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

      3. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, 1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right), 1\right) \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, 1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right), 1\right) \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

      5. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right) + 1}, 1\right) \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{6} \cdot {x}^{2}, 1\right)}, 1\right) \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

      7. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{6} \cdot {x}^{2}, 1\right), 1\right) \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{6} \cdot {x}^{2}, 1\right), 1\right) \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

      9. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{6} \cdot {x}^{2} + \frac{1}{2}}, 1\right), 1\right) \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

      10. *-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

      11. unpow2 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{6} + \frac{1}{2}, 1\right), 1\right) \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

      12. associate-*l* N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{6}\right)} + \frac{1}{2}, 1\right), 1\right) \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{6}, \frac{1}{2}\right)}, 1\right), 1\right) \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

      14. lower-*.f64 34.7

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

   10. Simplified 34.7%

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

   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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-/.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. lower-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. lower-*.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. lower-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. Add Preprocessing

Alternative 5: 83.8% accurate, 7.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/
  (fma x (* x (fma x (* x (fma x (* x 0.16666666666666666) 0.5)) 1.0)) 1.0)
  (* (sqrt 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(((double) M_PI)) * fabs(x));
}

Julia

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

Wolfram

code[x_] := N[(N[(x * N[(x * N[(x * N[(x * N[(x * N[(x * 0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $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%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. sqr-abs N/A

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

   8. unpow2 N/A

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

   9. lower-exp.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-fabs.f64 99.7

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

7. Simplified 99.7%

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

   1. lift-PI.f64 N/A

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

   2. /-rgt-identity N/A

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

   3. sqrt-div N/A

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

   4. metadata-eval N/A

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

   5. /-rgt-identity N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-fabs.f64 N/A

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

   10. frac-times N/A

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

   11. *-lft-identity N/A

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

   12. lower-/.f64 N/A

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

   13. lower-*.f64 99.7

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

9. Applied egg-rr 99.7%

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

10. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. unpow2 N/A

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

    3. associate-*l* N/A

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

    4. lower-fma.f64 N/A

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

12. Simplified 87.0%

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

Alternative 6: 81.0% 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 45.7%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{x \cdot x} - \frac{0.25}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}, \frac{1}{\frac{1 - \frac{0.5}{x \cdot x}}{\left|x\right|}}, \frac{0.75}{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} + \frac{1.875}{\left(x \cdot x\right) \cdot \left(\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}\right)} \]

5. Taylor expanded in x around inf

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

7. Simplified 45.5%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\color{blue}{\left({x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right) + 1\right)} \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

   2. lower-fma.f64 N/A

      \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\color{blue}{\mathsf{fma}\left({x}^{2}, 1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right), 1\right)} \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

   3. unpow2 N/A

      \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, 1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right), 1\right) \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(\color{blue}{x \cdot x}, 1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right), 1\right) \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

   5. +-commutative N/A

      \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right) + 1}, 1\right) \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{6} \cdot {x}^{2}, 1\right)}, 1\right) \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

   7. unpow2 N/A

      \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{6} \cdot {x}^{2}, 1\right), 1\right) \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{6} \cdot {x}^{2}, 1\right), 1\right) \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

   9. +-commutative N/A

      \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{6} \cdot {x}^{2} + \frac{1}{2}}, 1\right), 1\right) \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

   10. *-commutative N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{6}} + \frac{1}{2}, 1\right), 1\right) \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

   11. unpow2 N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{6} + \frac{1}{2}, 1\right), 1\right) \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

   12. associate-*l* N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{1}{6}\right)} + \frac{1}{2}, 1\right), 1\right) \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

   13. lower-fma.f64 N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{6}, \frac{1}{2}\right)}, 1\right), 1\right) \cdot \left(\left|x\right| + \frac{\mathsf{fma}\left(\left|x\right|, \frac{1}{2}, \frac{\frac{3}{4}}{\left|x\right|}\right)}{x \cdot x}\right)\right)}{x \cdot x} \]

   14. lower-*.f64 34.9

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

10. Simplified 34.9%

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

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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-*.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. lower-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. lower-/.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. lower-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. lower-*.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. lower-fabs.f64 84.2

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

13. Simplified 84.2%

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

Alternative 7: 76.6% accurate, 9.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. sqr-abs N/A

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

   8. unpow2 N/A

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

   9. lower-exp.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-fabs.f64 99.7

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

7. Simplified 99.7%

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

   1. lift-PI.f64 N/A

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

   2. /-rgt-identity N/A

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

   3. sqrt-div N/A

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

   4. metadata-eval N/A

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

   5. /-rgt-identity N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-fabs.f64 N/A

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

   10. frac-times N/A

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

   11. *-lft-identity N/A

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

   12. lower-/.f64 N/A

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

   13. lower-*.f64 99.7

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

9. Applied egg-rr 99.7%

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

10. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. unpow2 N/A

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

    3. associate-*l* N/A

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

    4. lower-fma.f64 N/A

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

    5. +-commutative N/A

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

    6. distribute-lft-in N/A

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

    7. *-rgt-identity N/A

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

    8. lower-fma.f64 N/A

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

    9. *-commutative N/A

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

    10. unpow2 N/A

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

    11. associate-*l* N/A

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

    12. *-commutative N/A

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

    13. lower-*.f64 N/A

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

    14. *-commutative N/A

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

    15. lower-*.f64 79.1

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

12. Simplified 79.1%

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

Alternative 8: 52.0% accurate, 13.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(x * x + 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%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. sqr-abs N/A

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

   8. unpow2 N/A

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

   9. lower-exp.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-fabs.f64 99.7

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

7. Simplified 99.7%

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

   1. lift-PI.f64 N/A

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

   2. /-rgt-identity N/A

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

   3. sqrt-div N/A

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

   4. metadata-eval N/A

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

   5. /-rgt-identity N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-fabs.f64 N/A

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

   10. frac-times N/A

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

   11. *-lft-identity N/A

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

   12. lower-/.f64 N/A

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

   13. lower-*.f64 99.7

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

9. Applied egg-rr 99.7%

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

10. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. unpow2 N/A

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

    3. lower-fma.f64 56.1

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

12. Simplified 56.1%

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

Alternative 9: 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%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. sqr-abs N/A

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

   8. unpow2 N/A

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

   9. lower-exp.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-fabs.f64 99.7

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

7. Simplified 99.7%

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

   1. lift-PI.f64 N/A

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

   2. /-rgt-identity N/A

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

   3. sqrt-div N/A

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

   4. metadata-eval N/A

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

   5. /-rgt-identity N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-fabs.f64 N/A

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

   10. frac-times N/A

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

   11. *-lft-identity N/A

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

   12. lower-/.f64 N/A

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

   13. lower-*.f64 99.7

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

9. Applied egg-rr 99.7%

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

10. Taylor expanded in x around 0

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

12. Simplified 2.2%

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

Reproduce

herbie shell --seed 2024207 
(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)))))))