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

Percentage Accurate: 100.0% → 100.0%

Time: 13.4s

Alternatives: 12

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, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.9%

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

6. Applied rewrites 99.9%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. sqr-pow N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-exp.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-/.f64 100.0

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

8. Applied rewrites 100.0%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. pow-prod-down N/A

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

   5. lower-pow.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. prod-exp N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-+.f64 100.0

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

   11. lift-/.f64 N/A

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

   12. div-inv N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 100.0

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

10. Applied rewrites 100.0%

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

11. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

4. Applied rewrites 100.0%

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

6. Applied rewrites 100.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-exp.f64 100.0

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

8. Applied rewrites 100.0%

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

Reproduce

herbie shell --seed 2024223 
(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)))))))