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

Percentage Accurate: 100.0% → 100.0%

Time: 12.7s

Alternatives: 15

Speedup: 1.9×

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

Specification

\[x \geq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. associate-*l/ N/A

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

6. Applied rewrites 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-exp.f64 100.0

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

8. Applied rewrites 100.0%

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

9. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. associate-*l/ N/A

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

6. Applied rewrites 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-exp.f64 100.0

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

8. Applied rewrites 100.0%

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

10. Applied rewrites 100.0%

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

11. Final simplification 100.0%

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

Alternative 3: 100.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. associate-*l/ N/A

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

6. Applied rewrites 100.0%

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

7. Final simplification 100.0%

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

Alternative 4: 99.6% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. associate-*l/ N/A

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

6. Applied rewrites 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-exp.f64 100.0

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

8. Applied rewrites 100.0%

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

9. Taylor expanded in x around inf

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

    1. +-commutative N/A

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

    2. associate-*r/ N/A

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

    3. times-frac N/A

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

    4. unpow2 N/A

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

    5. sqr-abs N/A

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

    6. unpow2 N/A

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

    7. unpow2 N/A

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

    8. sqr-abs N/A

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

    9. unpow2 N/A

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

    10. distribute-lft1-in N/A

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

    11. associate-*r/ N/A

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

    12. times-frac N/A

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

11. Applied rewrites 99.6%

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

12. Final simplification 99.6%

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

Alternative 5: 99.5% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. distribute-rgt-out N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

7. Applied rewrites 99.5%

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

9. Applied rewrites 99.5%

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

11. Applied rewrites 99.5%

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

12. Final simplification 99.5%

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

Alternative 6: 99.5% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Taylor expanded in x around inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. distribute-rgt-out N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

7. Applied rewrites 99.5%

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

9. Applied rewrites 99.5%

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

Alternative 7: 99.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. distribute-rgt-out N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

7. Applied rewrites 99.5%

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

9. Applied rewrites 99.5%

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

10. Final simplification 99.5%

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

Alternative 8: 87.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.16666666666666666, x \cdot x, 0.5\right)\\ t_1 := t\_0 \cdot x\\ \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(t\_1 \cdot t\_1, x \cdot x, -1\right)}{\mathsf{fma}\left(t\_0, x \cdot x, -1\right)}, x \cdot x, 1\right) \cdot \frac{\sqrt{\frac{1}{\pi}}}{\left|x\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x \cdot x, 1\right), x \cdot x, 1\right)}{\sqrt{\pi} \cdot \left|x\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma 0.16666666666666666 (* x x) 0.5)) (t_1 (* t_0 x)))
   (if (<= (fabs x) 2e+77)
     (*
      (fma
       (/ (fma (* t_1 t_1) (* x x) -1.0) (fma t_0 (* x x) -1.0))
       (* x x)
       1.0)
      (/ (sqrt (/ 1.0 PI)) (fabs x)))
     (/ (fma (fma 0.5 (* x x) 1.0) (* x x) 1.0) (* (sqrt PI) (fabs x))))))

C

double code(double x) {
	double t_0 = fma(0.16666666666666666, (x * x), 0.5);
	double t_1 = t_0 * x;
	double tmp;
	if (fabs(x) <= 2e+77) {
		tmp = fma((fma((t_1 * t_1), (x * x), -1.0) / fma(t_0, (x * x), -1.0)), (x * x), 1.0) * (sqrt((1.0 / ((double) M_PI))) / fabs(x));
	} else {
		tmp = fma(fma(0.5, (x * x), 1.0), (x * x), 1.0) / (sqrt(((double) M_PI)) * fabs(x));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(0.16666666666666666, Float64(x * x), 0.5)
	t_1 = Float64(t_0 * x)
	tmp = 0.0
	if (abs(x) <= 2e+77)
		tmp = Float64(fma(Float64(fma(Float64(t_1 * t_1), Float64(x * x), -1.0) / fma(t_0, Float64(x * x), -1.0)), Float64(x * x), 1.0) * Float64(sqrt(Float64(1.0 / pi)) / abs(x)));
	else
		tmp = Float64(fma(fma(0.5, Float64(x * x), 1.0), Float64(x * x), 1.0) / Float64(sqrt(pi) * abs(x)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.16666666666666666 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * x), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 2e+77], N[(N[(N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] * N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision] / N[(t$95$0 * N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[(N[Sqrt[Pi], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.99999999999999997e77

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   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. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-PI.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. unpow2 N/A

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

      10. sqr-abs N/A

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

      11. unpow2 N/A

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

      12. lower-exp.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 97.9

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

   7. Applied rewrites 97.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 37.0%

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

   12. Applied rewrites 49.4%

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

   if 1.99999999999999997e77 < (fabs.f64 x)

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

   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. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-PI.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. unpow2 N/A

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

      10. sqr-abs N/A

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

      11. unpow2 N/A

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

      12. lower-exp.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 100.0%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x \cdot x, 1\right), x \cdot x, 1\right)}{\color{blue}{\left|x\right|} \cdot \sqrt{\pi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(0.16666666666666666, x \cdot x, 0.5\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(0.16666666666666666, x \cdot x, 0.5\right) \cdot x\right), x \cdot x, -1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x \cdot x, 0.5\right), x \cdot x, -1\right)}, x \cdot x, 1\right) \cdot \frac{\sqrt{\frac{1}{\pi}}}{\left|x\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x \cdot x, 1\right), x \cdot x, 1\right)}{\sqrt{\pi} \cdot \left|x\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.5% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

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. associate-*r/ N/A

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-PI.f64 N/A

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

   8. lower-fabs.f64 N/A

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

   9. unpow2 N/A

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

   10. sqr-abs N/A

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

   11. unpow2 N/A

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

   12. lower-exp.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 99.4

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

7. Applied rewrites 99.4%

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

9. Applied rewrites 99.4%

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

Alternative 10: 99.4% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

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. associate-*r/ N/A

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-PI.f64 N/A

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

   8. lower-fabs.f64 N/A

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

   9. unpow2 N/A

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

   10. sqr-abs N/A

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

   11. unpow2 N/A

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

   12. lower-exp.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 99.4

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

7. Applied rewrites 99.4%

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

9. Applied rewrites 99.4%

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

10. Final simplification 99.4%

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

Alternative 11: 83.7% accurate, 5.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

5. Taylor expanded in x around inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. distribute-rgt-out N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

7. Applied rewrites 99.5%

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

9. Applied rewrites 99.5%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 83.3%

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

Alternative 12: 83.6% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

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. associate-*r/ N/A

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-PI.f64 N/A

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

   8. lower-fabs.f64 N/A

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

   9. unpow2 N/A

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

   10. sqr-abs N/A

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

   11. unpow2 N/A

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

   12. lower-exp.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 99.4

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 83.3%

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

12. Applied rewrites 83.3%

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

Alternative 13: 75.9% accurate, 9.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

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. associate-*r/ N/A

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-PI.f64 N/A

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

   8. lower-fabs.f64 N/A

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

   9. unpow2 N/A

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

   10. sqr-abs N/A

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

   11. unpow2 N/A

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

   12. lower-exp.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 99.4

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

7. Applied rewrites 99.4%

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

9. Applied rewrites 99.4%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 74.6%

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

13. Final simplification 74.6%

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

Alternative 14: 52.3% accurate, 13.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

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. associate-*r/ N/A

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-PI.f64 N/A

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

   8. lower-fabs.f64 N/A

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

   9. unpow2 N/A

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

   10. sqr-abs N/A

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

   11. unpow2 N/A

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

   12. lower-exp.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 99.4

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

7. Applied rewrites 99.4%

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

9. Applied rewrites 99.4%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 49.7%

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

13. Final simplification 49.7%

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

Alternative 15: 2.3% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

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. associate-*r/ N/A

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-PI.f64 N/A

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

   8. lower-fabs.f64 N/A

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

   9. unpow2 N/A

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

   10. sqr-abs N/A

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

   11. unpow2 N/A

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

   12. lower-exp.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 99.4

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

7. Applied rewrites 99.4%

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

9. Applied rewrites 99.4%

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

10. Taylor expanded in x around 0

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

12. Applied rewrites 2.3%

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

13. Final simplification 2.3%

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

Reproduce

herbie shell --seed 2024235 
(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)))))))