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

Percentage Accurate: 100.0% → 100.0%

Time: 12.6s

Alternatives: 12

Speedup: 1.4×

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

Specification

\[x \geq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 100.0%

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

6. Applied egg-rr 100.0%

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

   1. exp-prod N/A

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

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

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

   3. exp-lowering-exp.f64 100.0

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

8. Applied egg-rr 100.0%

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

   1. sqr-pow N/A

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

   2. pow-prod-down N/A

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

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

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

   4. prod-exp N/A

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

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

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

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

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

   7. div-inv N/A

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

   8. metadata-eval N/A

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

   9. *-lowering-*.f64 100.0

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

10. Applied egg-rr 100.0%

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

11. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x):
	t_0 = x * (x * x)
	return ((((1.0 + (0.5 / (x * x))) / math.fabs(x)) + ((0.75 / ((x * x) * t_0)) + (1.875 / (x * (t_0 * 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(Float64(1.0 + Float64(0.5 / Float64(x * x))) / abs(x)) + Float64(Float64(0.75 / Float64(Float64(x * x) * t_0)) + Float64(1.875 / Float64(x * Float64(t_0 * t_0))))) * (exp(x) ^ x)) / sqrt(pi))
end

MATLAB

function tmp = code(x)
	t_0 = x * (x * x);
	tmp = ((((1.0 + (0.5 / (x * x))) / abs(x)) + ((0.75 / ((x * x) * t_0)) + (1.875 / (x * (t_0 * 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[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(0.75 / N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[(x * N[(t$95$0 * t$95$0), $MachinePrecision]), $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%

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

6. Applied egg-rr 100.0%

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

   1. exp-prod N/A

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

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

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

   3. exp-lowering-exp.f64 100.0

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

8. Applied egg-rr 100.0%

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

Alternative 3: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	return Float64(Float64(Float64(Float64(Float64(1.0 + Float64(0.5 / Float64(x * x))) / abs(x)) + Float64(Float64(0.75 / Float64(Float64(x * x) * t_0)) + Float64(1.875 / Float64(x * Float64(t_0 * t_0))))) * (exp(1) ^ Float64(x * x))) / sqrt(pi))
end

MATLAB

function tmp = code(x)
	t_0 = x * (x * x);
	tmp = ((((1.0 + (0.5 / (x * x))) / abs(x)) + ((0.75 / ((x * x) * t_0)) + (1.875 / (x * (t_0 * t_0))))) * (2.71828182845904523536 ^ (x * x))) / sqrt(pi);
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 100.0%

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

6. Applied egg-rr 100.0%

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

   1. exp-prod N/A

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

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

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

   3. exp-lowering-exp.f64 100.0

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

8. Applied egg-rr 100.0%

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

   1. pow-exp N/A

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

   2. *-rgt-identity N/A

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

   3. *-commutative N/A

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

   4. exp-prod N/A

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

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

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

   6. exp-1-e N/A

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

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

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

   8. *-lowering-*.f64 100.0

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

10. Applied egg-rr 100.0%

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

Alternative 4: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x):
	t_0 = x * (x * x)
	return ((((1.0 + (0.5 / (x * x))) / math.fabs(x)) + ((0.75 / ((x * x) * t_0)) + (1.875 / (x * (t_0 * 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(Float64(1.0 + Float64(0.5 / Float64(x * x))) / abs(x)) + Float64(Float64(0.75 / Float64(Float64(x * x) * t_0)) + Float64(1.875 / Float64(x * Float64(t_0 * t_0))))) * exp(Float64(x * x))) / sqrt(pi))
end

MATLAB

function tmp = code(x)
	t_0 = x * (x * x);
	tmp = ((((1.0 + (0.5 / (x * x))) / abs(x)) + ((0.75 / ((x * x) * t_0)) + (1.875 / (x * (t_0 * 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[(N[(1.0 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(0.75 / N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(1.875 / N[(x * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 100.0%

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

6. Applied egg-rr 100.0%

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

Alternative 5: 99.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around inf

   \[\leadsto \frac{\color{blue}{\left(\frac{\frac{1}{2}}{{x}^{2} \cdot \left|x\right|} + \left(\frac{\frac{3}{4}}{{x}^{5}} + \frac{1}{\left|x\right|}\right)\right)} \cdot e^{x \cdot x}}{\sqrt{\mathsf{PI}\left(\right)}} \]
8. Step-by-step derivation

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

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

   2. unpow2 N/A

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

   3. fabs-sqr N/A

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

   4. unpow2 N/A

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

   5. fabs-mul N/A

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

   6. unpow2 N/A

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

   7. unpow3 N/A

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

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

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

   9. cube-mult N/A

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

   10. unpow2 N/A

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

   11. fabs-mul N/A

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

   12. unpow2 N/A

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

   13. fabs-sqr N/A

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

   14. associate-*r* N/A

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

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

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

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

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

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

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

   18. +-commutative N/A

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

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

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

9. Simplified 99.6%

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

11. Applied egg-rr 99.6%

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

Alternative 6: 99.6% 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%

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

6. Applied egg-rr 100.0%

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

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. metadata-eval N/A

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

   4. unpow2 N/A

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

   5. sqr-abs N/A

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

   6. unpow2 N/A

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

   7. associate-*r/ N/A

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

   8. distribute-lft1-in N/A

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

   9. *-inverses N/A

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

   10. unpow2 N/A

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

   11. sqr-abs N/A

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

   12. unpow2 N/A

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

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

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

9. Simplified 99.5%

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

Alternative 7: 99.5% accurate, 3.5× speedup

Math

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

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)) / Float64(abs(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[(N[Abs[x], $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%

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

5. Taylor expanded in x around inf

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

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

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

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

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

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

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

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

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

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

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

   6. unpow2 N/A

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

   7. sqr-abs N/A

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

   8. unpow2 N/A

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

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

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

   10. unpow2 N/A

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

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

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

   12. fabs-lowering-fabs.f64 99.5

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

7. Simplified 99.5%

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

   1. sqrt-div N/A

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

   2. metadata-eval N/A

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

   3. frac-times N/A

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

   4. *-lft-identity N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. fabs-lowering-fabs.f64 99.5

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

9. Applied egg-rr 99.5%

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

10. Final simplification 99.5%

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

Alternative 8: 83.9% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around inf

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

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

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

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

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

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

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

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

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

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

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

   6. unpow2 N/A

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

   7. sqr-abs N/A

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

   8. unpow2 N/A

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

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

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

   10. unpow2 N/A

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

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

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

   12. fabs-lowering-fabs.f64 99.5

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

7. Simplified 99.5%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

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

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

   3. unpow2 N/A

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

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

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

   5. +-commutative N/A

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

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

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

   7. unpow2 N/A

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

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

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

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

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

   12. unpow2 N/A

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

   13. *-lowering-*.f64 78.4

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

10. Simplified 78.4%

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

Alternative 9: 59.3% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| \cdot \sqrt{\pi}\\ \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, x \cdot x, -1\right)}{\mathsf{fma}\left(x, x, -1\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, 1\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) (sqrt PI))))
   (if (<= (fabs x) 5e+102)
     (/ (fma (* x x) (* x x) -1.0) (* (fma x x -1.0) t_0))
     (/ (fma x x 1.0) t_0))))

C

double code(double x) {
	double t_0 = fabs(x) * sqrt(((double) M_PI));
	double tmp;
	if (fabs(x) <= 5e+102) {
		tmp = fma((x * x), (x * x), -1.0) / (fma(x, x, -1.0) * t_0);
	} else {
		tmp = fma(x, x, 1.0) / t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(abs(x) * sqrt(pi))
	tmp = 0.0
	if (abs(x) <= 5e+102)
		tmp = Float64(fma(Float64(x * x), Float64(x * x), -1.0) / Float64(fma(x, x, -1.0) * t_0));
	else
		tmp = Float64(fma(x, x, 1.0) / t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 5e+102], N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision] / N[(N[(x * x + -1.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x * x + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 5e102

   1. Initial program 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

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

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. sqr-abs N/A

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. fabs-lowering-fabs.f64 98.5

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

   7. Simplified 98.5%

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

   8. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt1-in N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

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

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

   10. Simplified 3.6%

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

       1. flip-+ N/A

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

       2. clear-num N/A

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

       3. frac-times N/A

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

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

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

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

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

       6. metadata-eval N/A

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

       7. sub-neg N/A

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

       8. metadata-eval N/A

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

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

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

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

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

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

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

       12. sqrt-div N/A

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

       13. metadata-eval N/A

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

       14. associate-/r/ N/A

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

       15. /-rgt-identity N/A

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

   12. Applied egg-rr 18.6%

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

   if 5e102 < (fabs.f64 x)

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf

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

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

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. sqr-abs N/A

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

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

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

      12. fabs-lowering-fabs.f64 100.0

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt1-in N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

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

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

   10. Simplified 74.3%

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

       1. clear-num N/A

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

       2. un-div-inv N/A

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

       3. sqrt-div N/A

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

       4. metadata-eval N/A

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

       5. associate-/r/ N/A

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

       6. /-rgt-identity N/A

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

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

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

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

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

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

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

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

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

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

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

       12. PI-lowering-PI.f64 74.3

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

   12. Applied egg-rr 74.3%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, x \cdot x, -1\right)}{\mathsf{fma}\left(x, x, -1\right) \cdot \left(\left|x\right| \cdot \sqrt{\pi}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, 1\right)}{\left|x\right| \cdot \sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.3% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around inf

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

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

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

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

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

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

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

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

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

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

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

   6. unpow2 N/A

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

   7. sqr-abs N/A

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

   8. unpow2 N/A

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

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

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

   10. unpow2 N/A

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

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

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

   12. fabs-lowering-fabs.f64 99.5

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

7. Simplified 99.5%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

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

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

   3. unpow2 N/A

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

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

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. unpow2 N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

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

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

   11. *-commutative N/A

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

   12. *-lowering-*.f64 69.7

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

10. Simplified 69.7%

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

11. Final simplification 69.7%

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

Alternative 11: 51.6% accurate, 13.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

5. Taylor expanded in x around inf

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

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

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

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

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

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

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

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

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

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

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

   6. unpow2 N/A

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

   7. sqr-abs N/A

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

   8. unpow2 N/A

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

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

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

   10. unpow2 N/A

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

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

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

   12. fabs-lowering-fabs.f64 99.5

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

7. Simplified 99.5%

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

8. Taylor expanded in x around 0

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

   1. *-rgt-identity N/A

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

   2. associate-*r/ N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. distribute-rgt1-in N/A

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

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

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

   7. unpow2 N/A

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

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

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

   9. associate-*r/ N/A

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

   10. *-rgt-identity N/A

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

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

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

10. Simplified 47.8%

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

    1. clear-num N/A

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

    2. un-div-inv N/A

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

    3. sqrt-div N/A

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

    4. metadata-eval N/A

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

    5. associate-/r/ N/A

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

    6. /-rgt-identity N/A

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

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

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

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

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

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

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

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

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

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

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

    12. PI-lowering-PI.f64 47.8

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

12. Applied egg-rr 47.8%

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

Alternative 12: 2.3% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around inf

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

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

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

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

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

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

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

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

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

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

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

   6. unpow2 N/A

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

   7. sqr-abs N/A

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

   8. unpow2 N/A

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

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

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

   10. unpow2 N/A

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

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

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

   12. fabs-lowering-fabs.f64 99.5

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

7. Simplified 99.5%

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

8. Taylor expanded in x around 0

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

   1. associate-*r/ N/A

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

   2. *-rgt-identity N/A

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

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

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

   4. rem-exp-log N/A

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

   5. rec-exp N/A

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

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

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

   7. rec-exp N/A

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

   8. rem-exp-log N/A

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

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

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

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

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

   11. fabs-lowering-fabs.f64 2.3

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

10. Simplified 2.3%

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

    1. sqrt-div N/A

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

    2. metadata-eval N/A

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

    3. associate-/l/ N/A

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

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

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

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

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

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

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

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

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

    8. PI-lowering-PI.f64 2.3

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

12. Applied egg-rr 2.3%

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

Reproduce

herbie shell --seed 2024198 
(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)))))))