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

Percentage Accurate: 100.0% → 100.0%

Time: 13.7s

Alternatives: 13

Speedup: 2.0×

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

Specification

\[x \geq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

7. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

9. Applied rewrites 100.0%

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

    1. lift-/.f64 N/A

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

    2. clear-num N/A

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

    3. lift-+.f64 N/A

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

    4. +-commutative N/A

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

    5. lift-/.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. inv-pow N/A

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

    8. pow-to-exp N/A

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

    9. lower-exp.f64 N/A

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

    10. lower-*.f64 N/A

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

    11. lower-log.f64 N/A

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

    12. lower-/.f64 N/A

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

    13. lift-*.f64 N/A

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

    14. lift-/.f64 N/A

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

    15. +-commutative N/A

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

    16. lift-+.f64 100.0

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

11. Applied rewrites 100.0%

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

12. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

7. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

9. Applied rewrites 100.0%

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

10. Final simplification 100.0%

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

Alternative 3: 100.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

7. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-lft-identity 100.0

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

   6. lift-*.f64 N/A

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

   7. lift-fabs.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. sqr-abs N/A

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

   10. lift-*.f64 100.0

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

9. Applied rewrites 100.0%

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

10. Final simplification 100.0%

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

Alternative 4: 99.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 58.2%

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

5. Taylor expanded in x around inf

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

7. Applied rewrites 99.5%

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

9. Applied rewrites 99.5%

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

10. Final simplification 99.5%

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

Alternative 5: 99.6% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 58.2%

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

5. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. fabs-sqr N/A

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

   3. unpow2 N/A

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

   4. fabs-mul N/A

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

   5. unpow2 N/A

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

   6. unpow3 N/A

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

   7. *-rgt-identity N/A

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

   8. metadata-eval N/A

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

   9. distribute-rgt-neg-in N/A

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

   10. sub-neg N/A

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

   11. lower--.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-fabs.f64 N/A

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

   14. *-commutative N/A

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

   15. associate-*r/ N/A

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

   16. metadata-eval N/A

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

   17. associate-*r/ N/A

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

7. Applied rewrites 99.4%

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

9. Applied rewrites 99.4%

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

    1. lift-/.f64 N/A

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

11. Applied rewrites 99.4%

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

12. Final simplification 99.4%

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

Alternative 6: 99.6% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 58.2%

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

5. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. fabs-sqr N/A

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

   3. unpow2 N/A

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

   4. fabs-mul N/A

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

   5. unpow2 N/A

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

   6. unpow3 N/A

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

   7. *-rgt-identity N/A

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

   8. metadata-eval N/A

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

   9. distribute-rgt-neg-in N/A

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

   10. sub-neg N/A

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

   11. lower--.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-fabs.f64 N/A

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

   14. *-commutative N/A

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

   15. associate-*r/ N/A

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

   16. metadata-eval N/A

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

   17. associate-*r/ N/A

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

7. Applied rewrites 99.4%

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

9. Applied rewrites 99.4%

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

10. Final simplification 99.4%

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

Alternative 7: 99.6% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $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 rewrites 58.2%

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

5. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. fabs-sqr N/A

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

   3. unpow2 N/A

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

   4. fabs-mul N/A

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

   5. unpow2 N/A

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

   6. unpow3 N/A

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

   7. *-rgt-identity N/A

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

   8. metadata-eval N/A

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

   9. distribute-rgt-neg-in N/A

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

   10. sub-neg N/A

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

   11. lower--.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-fabs.f64 N/A

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

   14. *-commutative N/A

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

   15. associate-*r/ N/A

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

   16. metadata-eval N/A

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

   17. associate-*r/ N/A

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

7. Applied rewrites 99.4%

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

9. Applied rewrites 99.4%

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

10. Taylor expanded in x around inf

    \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{e^{{x}^{2}}}{{x}^{2} \cdot \left|x\right|} + \frac{e^{{x}^{2}}}{\left|x\right|}}}{\sqrt{\mathsf{PI}\left(\right)}} \]
11. 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. associate-*r/ N/A

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

    5. distribute-lft1-in N/A

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

    6. lower-*.f64 N/A

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

    7. lower-+.f64 N/A

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

    8. associate-*r/ N/A

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

    9. metadata-eval N/A

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

    10. lower-/.f64 N/A

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

    11. unpow2 N/A

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

    12. lower-*.f64 N/A

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

    13. lower-/.f64 N/A

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

    14. lower-exp.f64 N/A

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

    15. unpow2 N/A

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

    16. lower-*.f64 N/A

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

    17. lower-fabs.f64 99.4

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

12. Applied rewrites 99.4%

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

Alternative 8: 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 rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. sqr-abs N/A

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

   8. unpow2 N/A

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

   9. lower-exp.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-fabs.f64 99.4

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

7. Applied rewrites 99.4%

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

9. Applied rewrites 99.4%

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

10. Final simplification 99.4%

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

Alternative 9: 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 rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. sqr-abs N/A

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

   8. unpow2 N/A

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

   9. lower-exp.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-fabs.f64 99.4

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 81.4%

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

Alternative 10: 81.0% accurate, 8.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * N[Sqrt[N[(1.0 / Pi), $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 rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. sqr-abs N/A

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

   8. unpow2 N/A

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

   9. lower-exp.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-fabs.f64 99.4

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 77.8%

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

11. Taylor expanded in x around inf

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

13. Applied rewrites 77.8%

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

14. Final simplification 77.8%

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

Alternative 11: 81.0% accurate, 8.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. sqr-abs N/A

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

   8. unpow2 N/A

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

   9. lower-exp.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-fabs.f64 99.4

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 77.8%

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

11. Taylor expanded in x around inf

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

13. Applied rewrites 77.8%

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

Alternative 12: 52.2% 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 rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. sqr-abs N/A

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

   8. unpow2 N/A

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

   9. lower-exp.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-fabs.f64 99.4

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 47.5%

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

12. Applied rewrites 47.5%

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

13. Final simplification 47.5%

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

Alternative 13: 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 rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-PI.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. sqr-abs N/A

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

   8. unpow2 N/A

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

   9. lower-exp.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-fabs.f64 99.4

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 2.4%

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

12. Applied rewrites 2.4%

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

13. Final simplification 2.4%

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

Reproduce

herbie shell --seed 2024234 
(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)))))))