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

Percentage Accurate: 100.0% → 100.0%

Time: 14.9s

Alternatives: 15

Speedup: 2.0×

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

Specification

\[x \geq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around inf

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

7. Simplified 100.0%

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

9. Applied egg-rr 100.0%

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

    1. exp-prod N/A

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

    2. lower-pow.f64 N/A

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

    3. lower-exp.f64 100.0

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

11. Applied egg-rr 100.0%

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

Alternative 2: 100.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around inf

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

7. Simplified 100.0%

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

9. Applied egg-rr 100.0%

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

11. Applied egg-rr 100.0%

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

Alternative 3: 100.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around inf

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

7. Simplified 100.0%

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

9. Applied egg-rr 100.0%

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

11. Applied egg-rr 100.0%

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

12. Final simplification 100.0%

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

Alternative 4: 99.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = x * (x * x);
	return (exp((x * x)) * ((1.0 / fabs(x)) + ((1.875 / (fabs(x) * ((x * x) * (x * t_0)))) + (0.5 / fabs(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)) + ((1.875 / (Math.abs(x) * ((x * x) * (x * t_0)))) + (0.5 / Math.abs(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)) + ((1.875 / (math.fabs(x) * ((x * x) * (x * t_0)))) + (0.5 / math.fabs(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(1.875 / Float64(abs(x) * Float64(Float64(x * x) * Float64(x * t_0)))) + Float64(0.5 / abs(t_0))))) / sqrt(pi))
end

MATLAB

function tmp = code(x)
	t_0 = x * (x * x);
	tmp = (exp((x * x)) * ((1.0 / abs(x)) + ((1.875 / (abs(x) * ((x * x) * (x * t_0)))) + (0.5 / abs(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[(1.875 / N[(N[Abs[x], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-fabs.f64 N/A

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

   4. unpow2 N/A

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

   5. sqr-abs N/A

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

   6. lower-*.f64 N/A

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

   7. cube-mult N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. cube-mult N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 98.6

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

7. Simplified 98.6%

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

9. Applied egg-rr 98.6%

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

Alternative 5: 99.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around inf

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

7. Simplified 100.0%

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

9. Applied egg-rr 100.0%

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

10. Taylor expanded in x around inf

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

    1. +-commutative N/A

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

    2. +-commutative N/A

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

    3. *-commutative N/A

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

    4. associate-*l/ N/A

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

    5. associate-/l* N/A

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

    6. distribute-lft-out N/A

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

    7. lower-*.f64 N/A

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

    8. lower-exp.f64 N/A

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

    9. unpow2 N/A

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

    10. lower-*.f64 N/A

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

    11. lower-+.f64 N/A

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

12. Simplified 98.6%

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

Alternative 6: 99.6% accurate, 2.4× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = x * (x * x);
	return (exp((x * x)) * ((1.0 / fabs(x)) + (1.875 / (fabs(x) * (t_0 * 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)) + (1.875 / (Math.abs(x) * (t_0 * 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)) + (1.875 / (math.fabs(x) * (t_0 * 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(1.875 / Float64(abs(x) * Float64(t_0 * t_0))))) / sqrt(pi))
end

MATLAB

function tmp = code(x)
	t_0 = x * (x * x);
	tmp = (exp((x * x)) * ((1.0 / abs(x)) + (1.875 / (abs(x) * (t_0 * 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[(1.875 / N[(N[Abs[x], $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around inf

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

   1. lower-+.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-fabs.f64 N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-fabs.f64 N/A

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

   9. unpow2 N/A

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

   10. sqr-abs N/A

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

   11. lower-*.f64 N/A

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

   12. cube-mult N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 N/A

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

   17. cube-mult N/A

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

   18. unpow2 N/A

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

   19. lower-*.f64 N/A

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

   20. unpow2 N/A

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

   21. lower-*.f64 98.5

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

7. Simplified 98.5%

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

8. Final simplification 98.5%

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

Alternative 7: 99.6% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	t_0 = x * (x * x);
	tmp = (exp((x * x)) / sqrt(pi)) * ((1.0 / abs(x)) + (1.875 / (abs(x) * (t_0 * t_0))));
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[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[(N[Abs[x], $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around inf

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

   1. lower-+.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-fabs.f64 N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-fabs.f64 N/A

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

   9. unpow2 N/A

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

   10. sqr-abs N/A

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

   11. lower-*.f64 N/A

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

   12. cube-mult N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 N/A

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

   17. cube-mult N/A

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

   18. unpow2 N/A

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

   19. lower-*.f64 N/A

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

   20. unpow2 N/A

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

   21. lower-*.f64 98.5

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

7. Simplified 98.5%

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

8. Final simplification 98.5%

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

Alternative 8: 99.7% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around inf

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

   1. associate-*r/ N/A

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

   2. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. unpow2 N/A

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

   5. fabs-sqr N/A

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

   6. unpow2 N/A

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

   7. fabs-mul N/A

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

   8. unpow2 N/A

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

   9. unpow3 N/A

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

   10. associate-*r/ N/A

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

   11. lower-+.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-fabs.f64 N/A

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

   14. associate-*r/ N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f64 N/A

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

   17. unpow3 N/A

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

   18. unpow2 N/A

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

   19. fabs-mul N/A

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

9. Simplified 98.5%

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

10. Final simplification 98.5%

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

Alternative 9: 99.7% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around inf

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

   1. associate-*r/ N/A

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

   2. times-frac N/A

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

   3. distribute-lft1-in N/A

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

   4. lower-*.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-fabs.f64 98.5

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

9. Simplified 98.5%

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

10. Final simplification 98.5%

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

Alternative 10: 99.6% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-exp.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-fabs.f64 98.5

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

9. Simplified 98.5%

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

    1. lift-*.f64 N/A

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

    2. lift-exp.f64 N/A

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

    3. lift-fabs.f64 N/A

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

    4. lift-PI.f64 N/A

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

    5. lift-sqrt.f64 N/A

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

    6. associate-/l/ N/A

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

    7. lower-/.f64 N/A

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

    8. *-commutative N/A

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

    9. lift-fabs.f64 N/A

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

    10. rem-square-sqrt N/A

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

    11. sqrt-prod N/A

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

    12. rem-sqrt-square N/A

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

    13. mul-fabs N/A

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

    14. lower-fabs.f64 N/A

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

    15. lower-*.f64 98.5

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

11. Applied egg-rr 98.5%

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

Alternative 11: 68.8% accurate, 8.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-exp.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-fabs.f64 98.5

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

9. Simplified 98.5%

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

10. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. +-commutative N/A

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

    4. lower-fma.f64 N/A

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

    5. +-commutative N/A

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

    6. *-commutative N/A

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

    7. lower-fma.f64 70.6

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

12. Simplified 70.6%

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

Alternative 12: 52.1% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-exp.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-fabs.f64 98.5

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

9. Simplified 98.5%

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

10. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. +-commutative N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f64 54.4

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

12. Simplified 54.4%

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

Alternative 13: 3.2% accurate, 12.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-exp.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-fabs.f64 98.5

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

9. Simplified 98.5%

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

10. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. lower-+.f64 3.3

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

12. Simplified 3.3%

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

Alternative 14: 2.3% accurate, 13.3× speedup

Math

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

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(Float64(1.0 / abs(x)) / sqrt(pi))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-exp.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-fabs.f64 98.5

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

9. Simplified 98.5%

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

10. Taylor expanded in x around 0

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

    1. lower-/.f64 N/A

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

    2. lower-fabs.f64 2.4

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

12. Simplified 2.4%

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

Alternative 15: 1.9% accurate, 13.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \frac{1}{\left|x\right|} + \frac{3}{4} \cdot \frac{1}{\left|{x}^{3}\right|}}{{x}^{5}} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}} \]

7. Simplified 1.9%

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

8. Taylor expanded in x around inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-PI.f64 N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-fabs.f64 2.1

       \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \frac{0.5}{x \cdot \color{blue}{\left|x\right|}} \]

10. Simplified 2.1%

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

    1. lift-PI.f64 N/A

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

    2. /-rgt-identity N/A

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

    3. clear-num N/A

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

    4. lift-/.f64 N/A

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

    5. lift-sqrt.f64 N/A

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

    6. lift-fabs.f64 N/A

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

    7. lift-*.f64 N/A

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

    8. lift-/.f64 N/A

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

    9. *-commutative N/A

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

    10. lift-/.f64 N/A

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

    11. lift-sqrt.f64 N/A

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

    12. lift-/.f64 N/A

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

    13. sqrt-div N/A

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

    14. metadata-eval N/A

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

    15. lift-sqrt.f64 N/A

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

    16. frac-times N/A

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

    17. metadata-eval N/A

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

    18. lower-/.f64 N/A

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

    19. lower-*.f64 2.1

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

12. Applied egg-rr 2.1%

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

13. Final simplification 2.1%

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

Reproduce

herbie shell --seed 2024214 
(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)))))))