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

Percentage Accurate: 100.0% → 100.0%

Time: 15.4s

Alternatives: 24

Speedup: 1.9×

• 99.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \geq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \frac{{\left({\left(e^{x}\right)}^{2}\right)}^{\left(x \cdot 0.5\right)}}{\frac{1}{\sqrt{\frac{1}{\pi}}}} \cdot \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\frac{0.75}{t\_0} + \frac{1.875}{x \cdot \left(x \cdot t\_0\right)}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x (* x (* x x))))))
   (*
    (/ (pow (pow (exp x) 2.0) (* x 0.5)) (/ 1.0 (sqrt (/ 1.0 PI))))
    (+
     (/ (+ 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 = x * (x * (x * (x * x)));
	return (pow(pow(exp(x), 2.0), (x * 0.5)) / (1.0 / sqrt((1.0 / ((double) M_PI))))) * (((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 = x * (x * (x * (x * x)));
	return (Math.pow(Math.pow(Math.exp(x), 2.0), (x * 0.5)) / (1.0 / Math.sqrt((1.0 / Math.PI)))) * (((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 = x * (x * (x * (x * x)))
	return (math.pow(math.pow(math.exp(x), 2.0), (x * 0.5)) / (1.0 / math.sqrt((1.0 / math.pi)))) * (((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(x * Float64(x * Float64(x * Float64(x * x))))
	return Float64(Float64(((exp(x) ^ 2.0) ^ Float64(x * 0.5)) / Float64(1.0 / sqrt(Float64(1.0 / pi)))) * Float64(Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) / abs(x)) + Float64(Float64(0.75 / t_0) + Float64(1.875 / Float64(x * Float64(x * t_0))))))
end

MATLAB

function tmp = code(x)
	t_0 = x * (x * (x * (x * x)));
	tmp = (((exp(x) ^ 2.0) ^ (x * 0.5)) / (1.0 / sqrt((1.0 / pi)))) * (((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[(x * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[N[Power[N[Exp[x], $MachinePrecision], 2.0], $MachinePrecision], N[(x * 0.5), $MachinePrecision]], $MachinePrecision] / N[(1.0 / N[Sqrt[N[(1.0 / Pi), $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[(x * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

6. Applied egg-rr 99.9%

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

   1. pow-exp N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-down N/A

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

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

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

   5. pow2 N/A

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

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

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

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

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. *-lowering-*.f64 100.0

       \[\leadsto \frac{{\left({\left(e^{x}\right)}^{2}\right)}^{\color{blue}{\left(x \cdot 0.5\right)}}}{\sqrt{\pi}} \cdot \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\frac{0.75}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} + \frac{1.875}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right)\right) \]

8. Applied egg-rr 100.0%

   \[\leadsto \frac{\color{blue}{{\left({\left(e^{x}\right)}^{2}\right)}^{\left(x \cdot 0.5\right)}}}{\sqrt{\pi}} \cdot \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\frac{0.75}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} + \frac{1.875}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right)\right) \]
9. Step-by-step derivation

   1. /-rgt-identity N/A

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

   2. clear-num N/A

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

   3. metadata-eval N/A

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

   4. sqrt-div N/A

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

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

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

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

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

   7. /-lowering-/.f64 N/A

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

   8. PI-lowering-PI.f64 100.0

      \[\leadsto \frac{{\left({\left(e^{x}\right)}^{2}\right)}^{\left(x \cdot 0.5\right)}}{\frac{1}{\sqrt{\frac{1}{\color{blue}{\pi}}}}} \cdot \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\frac{0.75}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} + \frac{1.875}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right)\right) \]

10. Applied egg-rr 100.0%

    \[\leadsto \frac{{\left({\left(e^{x}\right)}^{2}\right)}^{\left(x \cdot 0.5\right)}}{\color{blue}{\frac{1}{\sqrt{\frac{1}{\pi}}}}} \cdot \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\frac{0.75}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} + \frac{1.875}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right)\right) \]
11. Add Preprocessing

Alternative 2: 100.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\frac{0.75}{t\_0} + \frac{1.875}{x \cdot \left(x \cdot t\_0\right)}\right)\right) \cdot \frac{{\left(e^{\mathsf{fma}\left(x, 2, x + x\right)}\right)}^{\left(x \cdot 0.25\right)}}{\sqrt{\pi}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x (* x (* x x))))))
   (*
    (+
     (/ (+ 1.0 (/ 0.5 (* x x))) (fabs x))
     (+ (/ 0.75 t_0) (/ 1.875 (* x (* x t_0)))))
    (/ (pow (exp (fma x 2.0 (+ x x))) (* x 0.25)) (sqrt PI)))))

C

double code(double x) {
	double t_0 = x * (x * (x * (x * x)));
	return (((1.0 + (0.5 / (x * x))) / fabs(x)) + ((0.75 / t_0) + (1.875 / (x * (x * t_0))))) * (pow(exp(fma(x, 2.0, (x + x))), (x * 0.25)) / sqrt(((double) M_PI)));
}

Julia

function code(x)
	t_0 = Float64(x * Float64(x * Float64(x * Float64(x * x))))
	return Float64(Float64(Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) / abs(x)) + Float64(Float64(0.75 / t_0) + Float64(1.875 / Float64(x * Float64(x * t_0))))) * Float64((exp(fma(x, 2.0, Float64(x + x))) ^ Float64(x * 0.25)) / sqrt(pi)))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(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[(x * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Exp[N[(x * 2.0 + N[(x + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(x * 0.25), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

6. Applied egg-rr 99.9%

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

   1. pow-exp N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-down N/A

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

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

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

   5. pow2 N/A

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

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

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

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

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. *-lowering-*.f64 100.0

       \[\leadsto \frac{{\left({\left(e^{x}\right)}^{2}\right)}^{\color{blue}{\left(x \cdot 0.5\right)}}}{\sqrt{\pi}} \cdot \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\frac{0.75}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} + \frac{1.875}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right)\right) \]

8. Applied egg-rr 100.0%

   \[\leadsto \frac{\color{blue}{{\left({\left(e^{x}\right)}^{2}\right)}^{\left(x \cdot 0.5\right)}}}{\sqrt{\pi}} \cdot \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\frac{0.75}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} + \frac{1.875}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right)\right) \]
9. Step-by-step derivation

   1. sqr-pow N/A

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

   2. pow-prod-down N/A

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

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

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

10. Applied egg-rr 100.0%

    \[\leadsto \frac{\color{blue}{{\left(e^{\mathsf{fma}\left(x, 2, x + x\right)}\right)}^{\left(x \cdot 0.25\right)}}}{\sqrt{\pi}} \cdot \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\frac{0.75}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} + \frac{1.875}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right)\right) \]

11. Final simplification 100.0%

    \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\frac{0.75}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} + \frac{1.875}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right)\right) \cdot \frac{{\left(e^{\mathsf{fma}\left(x, 2, x + x\right)}\right)}^{\left(x \cdot 0.25\right)}}{\sqrt{\pi}} \]
12. Add Preprocessing

Alternative 3: 100.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

6. Applied egg-rr 99.9%

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

   1. pow-exp N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-down N/A

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

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

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

   5. pow2 N/A

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

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

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

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

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. *-lowering-*.f64 100.0

       \[\leadsto \frac{{\left({\left(e^{x}\right)}^{2}\right)}^{\color{blue}{\left(x \cdot 0.5\right)}}}{\sqrt{\pi}} \cdot \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\frac{0.75}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} + \frac{1.875}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right)\right) \]

8. Applied egg-rr 100.0%

   \[\leadsto \frac{\color{blue}{{\left({\left(e^{x}\right)}^{2}\right)}^{\left(x \cdot 0.5\right)}}}{\sqrt{\pi}} \cdot \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\frac{0.75}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} + \frac{1.875}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right)\right) \]
9. Step-by-step derivation

   1. pow-exp N/A

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

   2. rem-log-exp N/A

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

   3. pow-exp N/A

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

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

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

   5. unpow2 N/A

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

   6. log-prod N/A

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

   7. rem-log-exp N/A

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

   8. rem-log-exp N/A

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

   9. +-lowering-+.f64 100.0

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

10. Applied egg-rr 100.0%

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

11. Final simplification 100.0%

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

Alternative 4: 100.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{x}\right)}^{x}\right) \cdot \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\frac{0.75}{x \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x, 0\right)\right)\right)} + \frac{1.875}{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (* (/ 1.0 (sqrt PI)) (pow (exp x) x))
  (+
   (/ (+ 1.0 (/ 0.5 (* x x))) (fabs x))
   (+
    (/ 0.75 (* x (* x (* x (fma x x 0.0)))))
    (/ 1.875 (* (fabs x) (* (* x x) (* (* x x) (* x x)))))))))

C

double code(double x) {
	return ((1.0 / sqrt(((double) M_PI))) * pow(exp(x), x)) * (((1.0 + (0.5 / (x * x))) / fabs(x)) + ((0.75 / (x * (x * (x * fma(x, x, 0.0))))) + (1.875 / (fabs(x) * ((x * x) * ((x * x) * (x * x)))))));
}

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * (exp(x) ^ x)) * Float64(Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) / abs(x)) + Float64(Float64(0.75 / Float64(x * Float64(x * Float64(x * fma(x, x, 0.0))))) + Float64(1.875 / Float64(abs(x) * Float64(Float64(x * x) * Float64(Float64(x * x) * Float64(x * x))))))))
end

Wolfram

code[x_] := N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(0.75 / N[(x * N[(x * N[(x * N[(x * x + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[(N[Abs[x], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

   1. sqr-abs N/A

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

   2. exp-prod N/A

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

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

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

   4. exp-lowering-exp.f64 100.0

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

6. Applied egg-rr 100.0%

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

   1. associate-*r* N/A

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

   2. sqr-abs N/A

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

   3. cube-unmult N/A

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

   4. sqr-pow N/A

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

   5. unpow-prod-down N/A

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

   6. sqr-abs N/A

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

   7. unpow-prod-down N/A

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

   8. sqr-pow N/A

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

   9. cube-unmult N/A

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

   10. *-commutative N/A

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

   11. associate-*r* N/A

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

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

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

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

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

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

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

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

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

   16. sqr-abs N/A

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

   17. +-lft-identity N/A

       \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot {\left(e^{x}\right)}^{x}\right) \cdot \left(\frac{\frac{\frac{1}{2}}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{\frac{3}{4}}{x \cdot \left(x \cdot \left(x \cdot \left(\left|x\right| \cdot \color{blue}{\left(0 + \left|x\right|\right)}\right)\right)\right)} + \frac{\frac{15}{8}}{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]

   18. +-commutative N/A

       \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot {\left(e^{x}\right)}^{x}\right) \cdot \left(\frac{\frac{\frac{1}{2}}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{\frac{3}{4}}{x \cdot \left(x \cdot \left(x \cdot \left(\left|x\right| \cdot \color{blue}{\left(\left|x\right| + 0\right)}\right)\right)\right)} + \frac{\frac{15}{8}}{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]

   19. distribute-rgt-out N/A

       \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot {\left(e^{x}\right)}^{x}\right) \cdot \left(\frac{\frac{\frac{1}{2}}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{\frac{3}{4}}{x \cdot \left(x \cdot \left(x \cdot \color{blue}{\left(\left|x\right| \cdot \left|x\right| + 0 \cdot \left|x\right|\right)}\right)\right)} + \frac{\frac{15}{8}}{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]

   20. sqr-abs N/A

       \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot {\left(e^{x}\right)}^{x}\right) \cdot \left(\frac{\frac{\frac{1}{2}}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{\frac{3}{4}}{x \cdot \left(x \cdot \left(x \cdot \left(\color{blue}{x \cdot x} + 0 \cdot \left|x\right|\right)\right)\right)} + \frac{\frac{15}{8}}{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]

   21. mul0-lft N/A

       \[\leadsto \left(\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot {\left(e^{x}\right)}^{x}\right) \cdot \left(\frac{\frac{\frac{1}{2}}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{\frac{3}{4}}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x + \color{blue}{0}\right)\right)\right)} + \frac{\frac{15}{8}}{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]

   22. accelerator-lowering-fma.f64 100.0

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{x}\right)}^{x}\right) \cdot \left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \left(\frac{0.75}{x \cdot \left(x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(x, x, 0\right)}\right)\right)} + \frac{1.875}{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]

8. Applied egg-rr 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{x}\right)}^{x}\right) \cdot \left(\frac{\frac{0.5}{x \cdot x} + 1}{\left|x\right|} + \left(\color{blue}{\frac{0.75}{x \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x, 0\right)\right)\right)}} + \frac{1.875}{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]

9. Final simplification 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{x}\right)}^{x}\right) \cdot \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\frac{0.75}{x \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x, 0\right)\right)\right)} + \frac{1.875}{\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)}\right)\right) \]
10. Add Preprocessing

Alternative 5: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(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[(x * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

6. Applied egg-rr 99.9%

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

   1. pow-exp N/A

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

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

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

   3. exp-lowering-exp.f64 100.0

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

8. Applied egg-rr 100.0%

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

9. Final simplification 100.0%

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

Alternative 6: 100.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\\ \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\frac{0.75}{t\_0} + \frac{1.875}{x \cdot \left(x \cdot t\_0\right)}\right)\right) \cdot e^{\mathsf{fma}\left(0 - \log \pi, 0.5, x \cdot x\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x (* x (* x x))))))
   (*
    (+
     (/ (+ 1.0 (/ 0.5 (* x x))) (fabs x))
     (+ (/ 0.75 t_0) (/ 1.875 (* x (* x t_0)))))
    (exp (fma (- 0.0 (log PI)) 0.5 (* x x))))))

C

double code(double x) {
	double t_0 = x * (x * (x * (x * x)));
	return (((1.0 + (0.5 / (x * x))) / fabs(x)) + ((0.75 / t_0) + (1.875 / (x * (x * t_0))))) * exp(fma((0.0 - log(((double) M_PI))), 0.5, (x * x)));
}

Julia

function code(x)
	t_0 = Float64(x * Float64(x * Float64(x * Float64(x * x))))
	return Float64(Float64(Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) / abs(x)) + Float64(Float64(0.75 / t_0) + Float64(1.875 / Float64(x * Float64(x * t_0))))) * exp(fma(Float64(0.0 - log(pi)), 0.5, Float64(x * x))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(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[(x * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(0.0 - N[Log[Pi], $MachinePrecision]), $MachinePrecision] * 0.5 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

6. Applied egg-rr 99.9%

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

   1. pow-exp N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-down N/A

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

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

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

   5. pow2 N/A

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

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

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

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

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. *-lowering-*.f64 100.0

       \[\leadsto \frac{{\left({\left(e^{x}\right)}^{2}\right)}^{\color{blue}{\left(x \cdot 0.5\right)}}}{\sqrt{\pi}} \cdot \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\frac{0.75}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} + \frac{1.875}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right)\right) \]

8. Applied egg-rr 100.0%

   \[\leadsto \frac{\color{blue}{{\left({\left(e^{x}\right)}^{2}\right)}^{\left(x \cdot 0.5\right)}}}{\sqrt{\pi}} \cdot \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\frac{0.75}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} + \frac{1.875}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right)\right) \]
9. Step-by-step derivation

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. metadata-eval N/A

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

   4. sqrt-div N/A

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

   5. pow1/2 N/A

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

   6. pow-to-exp N/A

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

   7. pow-pow N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. metadata-eval N/A

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

   11. pow-pow N/A

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

   12. unpow1 N/A

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

   13. exp-prod N/A

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

   14. prod-exp N/A

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

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

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

   16. neg-log N/A

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

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

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

10. Applied egg-rr 99.9%

    \[\leadsto \color{blue}{e^{\mathsf{fma}\left(0 - \log \pi, 0.5, x \cdot x\right)}} \cdot \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\frac{0.75}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} + \frac{1.875}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right)\right) \]

11. Final simplification 99.9%

    \[\leadsto \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \left(\frac{0.75}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)} + \frac{1.875}{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right)\right) \cdot e^{\mathsf{fma}\left(0 - \log \pi, 0.5, x \cdot x\right)} \]
12. Add Preprocessing

Alternative 7: 100.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

6. Applied egg-rr 99.9%

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

   1. /-rgt-identity N/A

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

   2. clear-num N/A

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

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

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

   4. metadata-eval N/A

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

   5. sqrt-div N/A

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

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

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

   7. /-lowering-/.f64 N/A

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

   8. PI-lowering-PI.f64 99.9

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

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

Alternative 8: 100.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

   1. associate-*l/ N/A

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

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

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

   3. *-lft-identity N/A

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

   4. sqr-abs N/A

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

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

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

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

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

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

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

   8. PI-lowering-PI.f64 99.9

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 9: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

6. Applied egg-rr 99.9%

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

Alternative 10: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 11: 99.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around inf

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

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

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

   2. fabs-lowering-fabs.f64 99.1

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

7. Simplified 99.1%

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

   1. unpow1 N/A

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

   2. metadata-eval N/A

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

   3. pow-prod-up N/A

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

   4. pow2 N/A

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

   5. sqr-abs N/A

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

   6. inv-pow N/A

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

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

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

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

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

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

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

   10. fabs-lowering-fabs.f64 99.1

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

9. Applied egg-rr 99.1%

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

10. Final simplification 99.1%

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

Alternative 12: 99.5% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around inf

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

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

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

   2. fabs-lowering-fabs.f64 99.1

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

7. Simplified 99.1%

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

9. Applied egg-rr 99.1%

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

10. Final simplification 99.1%

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

Alternative 13: 99.5% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around inf

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

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

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

   2. fabs-lowering-fabs.f64 99.1

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

7. Simplified 99.1%

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

8. Taylor expanded in x around inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

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

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

   5. unpow2 N/A

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

   6. sqr-abs N/A

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

   7. unpow2 N/A

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

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

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

   9. unpow2 N/A

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

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

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

   11. distribute-rgt-in N/A

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

   12. *-commutative N/A

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

   13. *-lft-identity N/A

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

10. Simplified 99.0%

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

11. Final simplification 99.0%

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

Alternative 14: 99.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\left|x\right|}{\frac{e^{\mathsf{fma}\left(x, x, 0\right)}}{\sqrt{\pi}}}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 1.0 (/ (fabs x) (/ (exp (fma x x 0.0)) (sqrt PI)))))

C

double code(double x) {
	return 1.0 / (fabs(x) / (exp(fma(x, x, 0.0)) / sqrt(((double) M_PI))));
}

Julia

function code(x)
	return Float64(1.0 / Float64(abs(x) / Float64(exp(fma(x, x, 0.0)) / sqrt(pi))))
end

Wolfram

code[x_] := N[(1.0 / N[(N[Abs[x], $MachinePrecision] / N[(N[Exp[N[(x * x + 0.0), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around inf

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

   1. *-lowering-*.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. sqrt-lowering-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. /-lowering-/.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. PI-lowering-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. /-lowering-/.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. exp-lowering-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. *-lowering-*.f64 N/A

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

   12. fabs-lowering-fabs.f64 98.8

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

7. Simplified 98.8%

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

   1. sqrt-div N/A

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

   2. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. div-inv N/A

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

   6. clear-num N/A

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

   7. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

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

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

   12. sqr-abs N/A

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

   13. +-lft-identity N/A

       \[\leadsto \frac{1}{\frac{\left|x\right|}{\frac{e^{\left|x\right| \cdot \color{blue}{\left(0 + \left|x\right|\right)}}}{\sqrt{\mathsf{PI}\left(\right)}}}} \]

   14. +-commutative N/A

       \[\leadsto \frac{1}{\frac{\left|x\right|}{\frac{e^{\left|x\right| \cdot \color{blue}{\left(\left|x\right| + 0\right)}}}{\sqrt{\mathsf{PI}\left(\right)}}}} \]

   15. distribute-rgt-out N/A

       \[\leadsto \frac{1}{\frac{\left|x\right|}{\frac{e^{\color{blue}{\left|x\right| \cdot \left|x\right| + 0 \cdot \left|x\right|}}}{\sqrt{\mathsf{PI}\left(\right)}}}} \]

   16. sqr-abs N/A

       \[\leadsto \frac{1}{\frac{\left|x\right|}{\frac{e^{\color{blue}{x \cdot x} + 0 \cdot \left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)}}}} \]

   17. mul0-lft N/A

       \[\leadsto \frac{1}{\frac{\left|x\right|}{\frac{e^{x \cdot x + \color{blue}{0}}}{\sqrt{\mathsf{PI}\left(\right)}}}} \]

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

       \[\leadsto \frac{1}{\frac{\left|x\right|}{\frac{e^{\color{blue}{\mathsf{fma}\left(x, x, 0\right)}}}{\sqrt{\mathsf{PI}\left(\right)}}}} \]

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

       \[\leadsto \frac{1}{\frac{\left|x\right|}{\frac{e^{\mathsf{fma}\left(x, x, 0\right)}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}}}} \]

   20. PI-lowering-PI.f64 98.8

       \[\leadsto \frac{1}{\frac{\left|x\right|}{\frac{e^{\mathsf{fma}\left(x, x, 0\right)}}{\sqrt{\color{blue}{\pi}}}}} \]

9. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{1}{\frac{\left|x\right|}{\frac{e^{\mathsf{fma}\left(x, x, 0\right)}}{\sqrt{\pi}}}}} \]
10. Add Preprocessing

Alternative 15: 99.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\mathsf{fma}\left(x, x, 0\right)}}{\left|x\right| \cdot \sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (exp (fma x x 0.0)) (* (fabs x) (sqrt PI))))

C

double code(double x) {
	return exp(fma(x, x, 0.0)) / (fabs(x) * sqrt(((double) M_PI)));
}

Julia

function code(x)
	return Float64(exp(fma(x, x, 0.0)) / Float64(abs(x) * sqrt(pi)))
end

Wolfram

code[x_] := N[(N[Exp[N[(x * x + 0.0), $MachinePrecision]], $MachinePrecision] / N[(N[Abs[x], $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around inf

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

   1. *-lowering-*.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. sqrt-lowering-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. /-lowering-/.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. PI-lowering-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. /-lowering-/.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. exp-lowering-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. *-lowering-*.f64 N/A

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

   12. fabs-lowering-fabs.f64 98.8

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

7. Simplified 98.8%

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

   1. sqrt-div N/A

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

   2. metadata-eval N/A

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

   3. frac-times N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

   7. sqr-abs N/A

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

   8. +-lft-identity N/A

      \[\leadsto \frac{e^{\left|x\right| \cdot \color{blue}{\left(0 + \left|x\right|\right)}}}{\sqrt{\mathsf{PI}\left(\right)} \cdot \left|x\right|} \]

   9. +-commutative N/A

      \[\leadsto \frac{e^{\left|x\right| \cdot \color{blue}{\left(\left|x\right| + 0\right)}}}{\sqrt{\mathsf{PI}\left(\right)} \cdot \left|x\right|} \]

   10. distribute-rgt-out N/A

       \[\leadsto \frac{e^{\color{blue}{\left|x\right| \cdot \left|x\right| + 0 \cdot \left|x\right|}}}{\sqrt{\mathsf{PI}\left(\right)} \cdot \left|x\right|} \]

   11. sqr-abs N/A

       \[\leadsto \frac{e^{\color{blue}{x \cdot x} + 0 \cdot \left|x\right|}}{\sqrt{\mathsf{PI}\left(\right)} \cdot \left|x\right|} \]

   12. mul0-lft N/A

       \[\leadsto \frac{e^{x \cdot x + \color{blue}{0}}}{\sqrt{\mathsf{PI}\left(\right)} \cdot \left|x\right|} \]

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

       \[\leadsto \frac{e^{\color{blue}{\mathsf{fma}\left(x, x, 0\right)}}}{\sqrt{\mathsf{PI}\left(\right)} \cdot \left|x\right|} \]

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

       \[\leadsto \frac{e^{\mathsf{fma}\left(x, x, 0\right)}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)} \cdot \left|x\right|}} \]

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

       \[\leadsto \frac{e^{\mathsf{fma}\left(x, x, 0\right)}}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left|x\right|} \]

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

       \[\leadsto \frac{e^{\mathsf{fma}\left(x, x, 0\right)}}{\sqrt{\color{blue}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|} \]

   17. fabs-lowering-fabs.f64 98.8

       \[\leadsto \frac{e^{\mathsf{fma}\left(x, x, 0\right)}}{\sqrt{\pi} \cdot \color{blue}{\left|x\right|}} \]

9. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{e^{\mathsf{fma}\left(x, x, 0\right)}}{\sqrt{\pi} \cdot \left|x\right|}} \]

10. Final simplification 98.8%

    \[\leadsto \frac{e^{\mathsf{fma}\left(x, x, 0\right)}}{\left|x\right| \cdot \sqrt{\pi}} \]
11. Add Preprocessing

Alternative 16: 84.4% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around inf

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

   1. *-lowering-*.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. sqrt-lowering-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. /-lowering-/.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. PI-lowering-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. /-lowering-/.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. exp-lowering-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. *-lowering-*.f64 N/A

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

   12. fabs-lowering-fabs.f64 98.8

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

7. Simplified 98.8%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

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

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

   3. unpow2 N/A

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

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

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

   5. +-commutative N/A

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

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

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

   7. unpow2 N/A

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

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

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

   9. +-commutative N/A

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

   10. unpow2 N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

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

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

   14. *-commutative N/A

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

   15. *-lowering-*.f64 85.9

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

10. Simplified 85.9%

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

Alternative 17: 81.4% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around inf

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

   1. *-lowering-*.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. sqrt-lowering-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. /-lowering-/.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. PI-lowering-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. /-lowering-/.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. exp-lowering-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. *-lowering-*.f64 N/A

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

   12. fabs-lowering-fabs.f64 98.8

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

7. Simplified 98.8%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-+r+ N/A

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

   4. associate-*r/ N/A

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

   5. *-rgt-identity N/A

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

   6. +-commutative N/A

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

10. Simplified 83.4%

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

    1. *-commutative N/A

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

    2. sqrt-div N/A

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

    3. metadata-eval N/A

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

    4. un-div-inv N/A

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

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

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

12. Applied egg-rr 83.4%

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

Alternative 18: 81.3% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around inf

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

   1. *-lowering-*.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. sqrt-lowering-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. /-lowering-/.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. PI-lowering-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. /-lowering-/.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. exp-lowering-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. *-lowering-*.f64 N/A

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

   12. fabs-lowering-fabs.f64 98.8

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

7. Simplified 98.8%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-+r+ N/A

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

   4. associate-*r/ N/A

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

   5. *-rgt-identity N/A

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

   6. +-commutative N/A

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

10. Simplified 83.4%

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

11. Taylor expanded in x around inf

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

    1. +-commutative N/A

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

    2. metadata-eval N/A

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

    3. pow-sqr N/A

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

    4. associate-*r/ N/A

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

    5. unpow2 N/A

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

    6. sqr-abs N/A

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

    7. times-frac N/A

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

    8. metadata-eval N/A

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

    9. associate-*r/ N/A

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

    10. *-inverses N/A

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

    11. *-rgt-identity N/A

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

    12. *-rgt-identity N/A

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

    13. *-inverses N/A

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

    14. associate-/l* N/A

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

    15. sqr-abs N/A

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

    16. unpow2 N/A

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

    17. +-commutative N/A

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

13. Simplified 83.4%

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

Alternative 19: 81.3% accurate, 8.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around inf

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

   1. *-lowering-*.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. sqrt-lowering-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. /-lowering-/.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. PI-lowering-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. /-lowering-/.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. exp-lowering-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. *-lowering-*.f64 N/A

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

   12. fabs-lowering-fabs.f64 98.8

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

7. Simplified 98.8%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-+r+ N/A

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

   4. associate-*r/ N/A

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

   5. *-rgt-identity N/A

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

   6. +-commutative N/A

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

10. Simplified 83.4%

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

11. Taylor expanded in x around inf

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

    1. +-commutative N/A

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

    2. metadata-eval N/A

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

    3. pow-sqr N/A

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

    4. associate-*r/ N/A

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

    5. unpow2 N/A

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

    6. sqr-abs N/A

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

    7. times-frac N/A

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

    8. metadata-eval N/A

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

    9. associate-*r/ N/A

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

    10. *-inverses N/A

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

    11. *-rgt-identity N/A

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

    12. *-rgt-identity N/A

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

    13. *-inverses N/A

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

    14. associate-/l* N/A

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

    15. sqr-abs N/A

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

    16. unpow2 N/A

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

    17. +-commutative N/A

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

13. Simplified 83.4%

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

Alternative 20: 81.3% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around inf

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

   1. *-lowering-*.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. sqrt-lowering-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. /-lowering-/.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. PI-lowering-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. /-lowering-/.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. exp-lowering-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. *-lowering-*.f64 N/A

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

   12. fabs-lowering-fabs.f64 98.8

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

7. Simplified 98.8%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-+r+ N/A

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

   4. associate-*r/ N/A

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

   5. *-rgt-identity N/A

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

   6. +-commutative N/A

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

10. Simplified 83.4%

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

11. Taylor expanded in x around inf

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

    1. associate-*r* N/A

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

    2. *-commutative N/A

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

    3. metadata-eval N/A

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

    4. pow-sqr N/A

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

    5. associate-*l* N/A

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

    6. *-commutative N/A

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

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

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

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

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

    9. unpow2 N/A

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

    10. associate-*l* N/A

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

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

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

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

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

    13. *-commutative N/A

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

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

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

    15. unpow2 N/A

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

    16. *-lowering-*.f64 83.4

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

13. Simplified 83.4%

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

Alternative 21: 51.3% accurate, 10.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around inf

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

   1. *-lowering-*.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. sqrt-lowering-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. /-lowering-/.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. PI-lowering-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. /-lowering-/.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. exp-lowering-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. *-lowering-*.f64 N/A

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

   12. fabs-lowering-fabs.f64 98.8

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

7. Simplified 98.8%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. accelerator-lowering-fma.f64 53.5

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

10. Simplified 53.5%

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

Alternative 22: 5.4% accurate, 10.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around inf

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

   1. *-lowering-*.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. sqrt-lowering-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. /-lowering-/.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. PI-lowering-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. /-lowering-/.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. exp-lowering-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. *-lowering-*.f64 N/A

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

   12. fabs-lowering-fabs.f64 98.8

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

7. Simplified 98.8%

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

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. distribute-lft-out N/A

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

   3. +-commutative N/A

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

   4. *-lft-identity N/A

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

   5. associate-*l/ N/A

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

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

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

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

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

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

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

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

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

   10. associate-*l/ N/A

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

   11. *-lft-identity N/A

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

10. Simplified 5.7%

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

Alternative 23: 5.4% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot 0.125\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (sqrt (/ 1.0 PI)) (* (fabs x) 0.125)))

C

double code(double x) {
	return sqrt((1.0 / ((double) M_PI))) * (fabs(x) * 0.125);
}

Java

public static double code(double x) {
	return Math.sqrt((1.0 / Math.PI)) * (Math.abs(x) * 0.125);
}

Python

def code(x):
	return math.sqrt((1.0 / math.pi)) * (math.fabs(x) * 0.125)

Julia

function code(x)
	return Float64(sqrt(Float64(1.0 / pi)) * Float64(abs(x) * 0.125))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((1.0 / pi)) * (abs(x) * 0.125);
end

Wolfram

code[x_] := N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * 0.125), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around 0

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

7. Simplified 13.5%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto \frac{\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \frac{\color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right)\right) + 1}}{\left|x\right|}\right) \cdot \mathsf{fma}\left(\frac{3}{4}, x \cdot x, \frac{15}{8}\right)}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \]

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

      \[\leadsto \frac{\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, 1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right), 1\right)}}{\left|x\right|}\right) \cdot \mathsf{fma}\left(\frac{3}{4}, x \cdot x, \frac{15}{8}\right)}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \]

   3. unpow2 N/A

      \[\leadsto \frac{\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, 1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right), 1\right)}{\left|x\right|}\right) \cdot \mathsf{fma}\left(\frac{3}{4}, x \cdot x, \frac{15}{8}\right)}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \]

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

      \[\leadsto \frac{\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, 1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right), 1\right)}{\left|x\right|}\right) \cdot \mathsf{fma}\left(\frac{3}{4}, x \cdot x, \frac{15}{8}\right)}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \]

   5. +-commutative N/A

      \[\leadsto \frac{\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {x}^{2}\right) + 1}, 1\right)}{\left|x\right|}\right) \cdot \mathsf{fma}\left(\frac{3}{4}, x \cdot x, \frac{15}{8}\right)}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \]

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

      \[\leadsto \frac{\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{6} \cdot {x}^{2}, 1\right)}, 1\right)}{\left|x\right|}\right) \cdot \mathsf{fma}\left(\frac{3}{4}, x \cdot x, \frac{15}{8}\right)}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \]

   7. unpow2 N/A

      \[\leadsto \frac{\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{6} \cdot {x}^{2}, 1\right), 1\right)}{\left|x\right|}\right) \cdot \mathsf{fma}\left(\frac{3}{4}, x \cdot x, \frac{15}{8}\right)}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \]

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

      \[\leadsto \frac{\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + \frac{1}{6} \cdot {x}^{2}, 1\right), 1\right)}{\left|x\right|}\right) \cdot \mathsf{fma}\left(\frac{3}{4}, x \cdot x, \frac{15}{8}\right)}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \]

   9. +-commutative N/A

      \[\leadsto \frac{\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{6} \cdot {x}^{2} + \frac{1}{2}}, 1\right), 1\right)}{\left|x\right|}\right) \cdot \mathsf{fma}\left(\frac{3}{4}, x \cdot x, \frac{15}{8}\right)}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \]

   10. unpow2 N/A

       \[\leadsto \frac{\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{6} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{1}{2}, 1\right), 1\right)}{\left|x\right|}\right) \cdot \mathsf{fma}\left(\frac{3}{4}, x \cdot x, \frac{15}{8}\right)}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \]

   11. associate-*r* N/A

       \[\leadsto \frac{\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{1}{6} \cdot x\right) \cdot x} + \frac{1}{2}, 1\right), 1\right)}{\left|x\right|}\right) \cdot \mathsf{fma}\left(\frac{3}{4}, x \cdot x, \frac{15}{8}\right)}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \]

   12. *-commutative N/A

       \[\leadsto \frac{\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{1}{6} \cdot x\right)} + \frac{1}{2}, 1\right), 1\right)}{\left|x\right|}\right) \cdot \mathsf{fma}\left(\frac{3}{4}, x \cdot x, \frac{15}{8}\right)}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \]

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

       \[\leadsto \frac{\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{6} \cdot x, \frac{1}{2}\right)}, 1\right), 1\right)}{\left|x\right|}\right) \cdot \mathsf{fma}\left(\frac{3}{4}, x \cdot x, \frac{15}{8}\right)}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \]

   14. *-commutative N/A

       \[\leadsto \frac{\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{6}}, \frac{1}{2}\right), 1\right), 1\right)}{\left|x\right|}\right) \cdot \mathsf{fma}\left(\frac{3}{4}, x \cdot x, \frac{15}{8}\right)}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \]

   15. *-lowering-*.f64 1.8

       \[\leadsto \frac{\left(\sqrt{\frac{1}{\pi}} \cdot \frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.16666666666666666}, 0.5\right), 1\right), 1\right)}{\left|x\right|}\right) \cdot \mathsf{fma}\left(0.75, x \cdot x, 1.875\right)}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \]

10. Simplified 1.8%

    \[\leadsto \frac{\left(\sqrt{\frac{1}{\pi}} \cdot \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.16666666666666666, 0.5\right), 1\right), 1\right)}}{\left|x\right|}\right) \cdot \mathsf{fma}\left(0.75, x \cdot x, 1.875\right)}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \]

11. Taylor expanded in x around inf

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

    1. associate-*r* N/A

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

    2. *-commutative N/A

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

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

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

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

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

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

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

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

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

    7. *-commutative N/A

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

    8. unpow2 N/A

       \[\leadsto \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\frac{\color{blue}{x \cdot x}}{\left|x\right|} \cdot \frac{1}{8}\right) \]

    9. sqr-abs N/A

       \[\leadsto \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{\left|x\right|} \cdot \frac{1}{8}\right) \]

    10. associate-/l* N/A

        \[\leadsto \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\color{blue}{\left(\left|x\right| \cdot \frac{\left|x\right|}{\left|x\right|}\right)} \cdot \frac{1}{8}\right) \]

    11. *-inverses N/A

        \[\leadsto \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\left(\left|x\right| \cdot \color{blue}{1}\right) \cdot \frac{1}{8}\right) \]

    12. *-rgt-identity N/A

        \[\leadsto \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(\color{blue}{\left|x\right|} \cdot \frac{1}{8}\right) \]

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

        \[\leadsto \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(\left|x\right| \cdot \frac{1}{8}\right)} \]

    14. fabs-lowering-fabs.f64 5.6

        \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\left|x\right|} \cdot 0.125\right) \]

13. Simplified 5.6%

    \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot 0.125\right)} \]
14. Add Preprocessing

Alternative 24: 2.3% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around inf

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

   1. *-lowering-*.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. sqrt-lowering-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. /-lowering-/.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. PI-lowering-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. /-lowering-/.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. exp-lowering-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. *-lowering-*.f64 N/A

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

   12. fabs-lowering-fabs.f64 98.8

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

7. Simplified 98.8%

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

8. Taylor expanded in x around 0

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

   1. associate-*r/ N/A

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

   2. *-rgt-identity N/A

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

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

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

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

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

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

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

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

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

   7. fabs-lowering-fabs.f64 2.3

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

10. Simplified 2.3%

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

    1. div-inv N/A

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

    2. sqrt-div N/A

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

    3. metadata-eval N/A

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

    4. frac-times N/A

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

    5. metadata-eval N/A

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

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

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

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

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

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

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

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

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

    10. fabs-lowering-fabs.f64 2.3

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

12. Applied egg-rr 2.3%

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

13. Final simplification 2.3%

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

Reproduce

herbie shell --seed 2024197 
(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)))))))