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

Average Error: 2.8 → 2.6

Time: 25.5s

Precision: binary64

Cost: 150656

\[x \geq 0.5\]

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ t_1 := \frac{1}{\left|x\right|}\\ t_2 := e^{{x}^{2}}\\ \left(t_0 \cdot \frac{\left|x\right|}{\frac{x \cdot x}{t_2}} + \left(0.5 \cdot \left(t_0 \cdot \frac{\frac{t_2}{\left|x\right|}}{{x}^{2}}\right) + 0.75 \cdot \left(t_0 \cdot \frac{t_1 \cdot t_2}{{x}^{4}}\right)\right)\right) + \left(t_2 \cdot \frac{t_1}{{x}^{6}}\right) \cdot \left(t_0 \cdot 1.875\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (* (/ 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)))))))

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 PI))) (t_1 (/ 1.0 (fabs x))) (t_2 (exp (pow x 2.0))))
   (+
    (+
     (* t_0 (/ (fabs x) (/ (* x x) t_2)))
     (+
      (* 0.5 (* t_0 (/ (/ t_2 (fabs x)) (pow x 2.0))))
      (* 0.75 (* t_0 (/ (* t_1 t_2) (pow x 4.0))))))
    (* (* t_2 (/ t_1 (pow x 6.0))) (* t_0 1.875)))))

C

double code(double x) {
	return ((1.0 / sqrt(((double) M_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)))));
}

↓

double code(double x) {
	double t_0 = sqrt((1.0 / ((double) M_PI)));
	double t_1 = 1.0 / fabs(x);
	double t_2 = exp(pow(x, 2.0));
	return ((t_0 * (fabs(x) / ((x * x) / t_2))) + ((0.5 * (t_0 * ((t_2 / fabs(x)) / pow(x, 2.0)))) + (0.75 * (t_0 * ((t_1 * t_2) / pow(x, 4.0)))))) + ((t_2 * (t_1 / pow(x, 6.0))) * (t_0 * 1.875));
}

Java

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

↓

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 / Math.PI));
	double t_1 = 1.0 / Math.abs(x);
	double t_2 = Math.exp(Math.pow(x, 2.0));
	return ((t_0 * (Math.abs(x) / ((x * x) / t_2))) + ((0.5 * (t_0 * ((t_2 / Math.abs(x)) / Math.pow(x, 2.0)))) + (0.75 * (t_0 * ((t_1 * t_2) / Math.pow(x, 4.0)))))) + ((t_2 * (t_1 / Math.pow(x, 6.0))) * (t_0 * 1.875));
}

Python

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

↓

def code(x):
	t_0 = math.sqrt((1.0 / math.pi))
	t_1 = 1.0 / math.fabs(x)
	t_2 = math.exp(math.pow(x, 2.0))
	return ((t_0 * (math.fabs(x) / ((x * x) / t_2))) + ((0.5 * (t_0 * ((t_2 / math.fabs(x)) / math.pow(x, 2.0)))) + (0.75 * (t_0 * ((t_1 * t_2) / math.pow(x, 4.0)))))) + ((t_2 * (t_1 / math.pow(x, 6.0))) * (t_0 * 1.875))

Julia

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

↓

function code(x)
	t_0 = sqrt(Float64(1.0 / pi))
	t_1 = Float64(1.0 / abs(x))
	t_2 = exp((x ^ 2.0))
	return Float64(Float64(Float64(t_0 * Float64(abs(x) / Float64(Float64(x * x) / t_2))) + Float64(Float64(0.5 * Float64(t_0 * Float64(Float64(t_2 / abs(x)) / (x ^ 2.0)))) + Float64(0.75 * Float64(t_0 * Float64(Float64(t_1 * t_2) / (x ^ 4.0)))))) + Float64(Float64(t_2 * Float64(t_1 / (x ^ 6.0))) * Float64(t_0 * 1.875)))
end

MATLAB

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

↓

function tmp = code(x)
	t_0 = sqrt((1.0 / pi));
	t_1 = 1.0 / abs(x);
	t_2 = exp((x ^ 2.0));
	tmp = ((t_0 * (abs(x) / ((x * x) / t_2))) + ((0.5 * (t_0 * ((t_2 / abs(x)) / (x ^ 2.0)))) + (0.75 * (t_0 * ((t_1 * t_2) / (x ^ 4.0)))))) + ((t_2 * (t_1 / (x ^ 6.0))) * (t_0 * 1.875));
end

Wolfram

code[x_] := 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[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 2.0), $MachinePrecision] * N[(N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 / 4.0), $MachinePrecision] * N[(N[(N[(N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(15.0 / 8.0), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[Power[x, 2.0], $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(t$95$0 * N[(N[Abs[x], $MachinePrecision] / N[(N[(x * x), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[(t$95$0 * N[(N[(t$95$2 / N[Abs[x], $MachinePrecision]), $MachinePrecision] / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.75 * N[(t$95$0 * N[(N[(t$95$1 * t$95$2), $MachinePrecision] / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 * N[(t$95$1 / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * 1.875), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 2.8

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

2. Simplified 2.7

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

   [Start] 2.8	\[ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
   rational.json-simplify-38 [=>] 2.8	\[ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\color{blue}{\left|x \cdot x\right|}}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
   rational.json-simplify-21 [<=] 2.8	\[ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\color{blue}{x \cdot x}}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
   rational.json-simplify-1 [=>] 2.8	\[ \left(\frac{1}{\sqrt{\pi}} \cdot e^{x \cdot x}\right) \cdot \color{blue}{\left(\frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) + \left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\right)} \]

3. Taylor expanded in x around inf 2.7

   \[\leadsto \color{blue}{1.875 \cdot \left(\frac{\left|\frac{1}{x}\right| \cdot e^{{x}^{2}}}{{x}^{6}} \cdot \sqrt{\frac{1}{\pi}}\right) + \left(0.75 \cdot \left(\frac{\left|\frac{1}{x}\right| \cdot e^{{x}^{2}}}{{x}^{4}} \cdot \sqrt{\frac{1}{\pi}}\right) + \left(\left(\left|\frac{1}{x}\right| \cdot e^{{x}^{2}}\right) \cdot \sqrt{\frac{1}{\pi}} + 0.5 \cdot \left(\frac{\left|\frac{1}{x}\right| \cdot e^{{x}^{2}}}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)} \]

4. Simplified 2.7

   \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{\left|x\right|} \cdot e^{{x}^{2}}\right) + \left(0.5 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \frac{\frac{1}{\left|x\right|} \cdot e^{{x}^{2}}}{{x}^{2}}\right) + 0.75 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \frac{\frac{1}{\left|x\right|} \cdot e^{{x}^{2}}}{{x}^{4}}\right)\right)\right) + \left(e^{{x}^{2}} \cdot \frac{\frac{1}{\left|x\right|}}{{x}^{6}}\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 1.875\right)} \]
   Proof

   [Start] 2.7

   \[ 1.875 \cdot \left(\frac{\left|\frac{1}{x}\right| \cdot e^{{x}^{2}}}{{x}^{6}} \cdot \sqrt{\frac{1}{\pi}}\right) + \left(0.75 \cdot \left(\frac{\left|\frac{1}{x}\right| \cdot e^{{x}^{2}}}{{x}^{4}} \cdot \sqrt{\frac{1}{\pi}}\right) + \left(\left(\left|\frac{1}{x}\right| \cdot e^{{x}^{2}}\right) \cdot \sqrt{\frac{1}{\pi}} + 0.5 \cdot \left(\frac{\left|\frac{1}{x}\right| \cdot e^{{x}^{2}}}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) \]

   rational.json-simplify-1 [=>] 2.7

   \[ \color{blue}{\left(0.75 \cdot \left(\frac{\left|\frac{1}{x}\right| \cdot e^{{x}^{2}}}{{x}^{4}} \cdot \sqrt{\frac{1}{\pi}}\right) + \left(\left(\left|\frac{1}{x}\right| \cdot e^{{x}^{2}}\right) \cdot \sqrt{\frac{1}{\pi}} + 0.5 \cdot \left(\frac{\left|\frac{1}{x}\right| \cdot e^{{x}^{2}}}{{x}^{2}} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right) + 1.875 \cdot \left(\frac{\left|\frac{1}{x}\right| \cdot e^{{x}^{2}}}{{x}^{6}} \cdot \sqrt{\frac{1}{\pi}}\right)} \]

5. Applied egg-rr 2.6

   \[\leadsto \left(\sqrt{\frac{1}{\pi}} \cdot \color{blue}{\frac{\left|x\right|}{\frac{x \cdot x}{e^{{x}^{2}}}}} + \left(0.5 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \frac{\frac{1}{\left|x\right|} \cdot e^{{x}^{2}}}{{x}^{2}}\right) + 0.75 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \frac{\frac{1}{\left|x\right|} \cdot e^{{x}^{2}}}{{x}^{4}}\right)\right)\right) + \left(e^{{x}^{2}} \cdot \frac{\frac{1}{\left|x\right|}}{{x}^{6}}\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 1.875\right) \]

6. Taylor expanded in x around inf 2.6

   \[\leadsto \left(\sqrt{\frac{1}{\pi}} \cdot \frac{\left|x\right|}{\frac{x \cdot x}{e^{{x}^{2}}}} + \left(0.5 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \frac{\color{blue}{\frac{e^{{x}^{2}}}{\left|x\right|}}}{{x}^{2}}\right) + 0.75 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \frac{\frac{1}{\left|x\right|} \cdot e^{{x}^{2}}}{{x}^{4}}\right)\right)\right) + \left(e^{{x}^{2}} \cdot \frac{\frac{1}{\left|x\right|}}{{x}^{6}}\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 1.875\right) \]

7. Final simplification 2.6

   \[\leadsto \left(\sqrt{\frac{1}{\pi}} \cdot \frac{\left|x\right|}{\frac{x \cdot x}{e^{{x}^{2}}}} + \left(0.5 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \frac{\frac{e^{{x}^{2}}}{\left|x\right|}}{{x}^{2}}\right) + 0.75 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \frac{\frac{1}{\left|x\right|} \cdot e^{{x}^{2}}}{{x}^{4}}\right)\right)\right) + \left(e^{{x}^{2}} \cdot \frac{\frac{1}{\left|x\right|}}{{x}^{6}}\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot 1.875\right) \]

Alternatives

Alternative 1
Error	2.7
Cost	150528

\[\begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ t_1 := \frac{1}{\left|x\right|}\\ t_2 := e^{{x}^{2}}\\ t_3 := t_1 \cdot t_2\\ \left(t_0 \cdot t_3 + \left(0.5 \cdot \left(t_0 \cdot \frac{\frac{t_2}{\left|x\right|}}{{x}^{2}}\right) + 0.75 \cdot \left(t_0 \cdot \frac{t_3}{{x}^{4}}\right)\right)\right) + \left(t_2 \cdot \frac{t_1}{{x}^{6}}\right) \cdot \left(t_0 \cdot 1.875\right) \end{array} \]

Alternative 2
Error	2.6
Cost	150528

\[\begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ t_1 := e^{{x}^{2}}\\ t_2 := \frac{t_1}{\left|x\right|}\\ \left(t_0 \cdot \frac{\left|x\right|}{\frac{x \cdot x}{t_1}} + \left(0.5 \cdot \left(t_0 \cdot \frac{t_2}{{x}^{2}}\right) + 0.75 \cdot \left(t_0 \cdot \frac{t_2}{{x}^{4}}\right)\right)\right) + \left(t_1 \cdot \frac{\frac{1}{\left|x\right|}}{{x}^{6}}\right) \cdot \left(t_0 \cdot 1.875\right) \end{array} \]

Alternative 3
Error	2.7
Cost	124416

\[\begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ t_1 := \left|\frac{1}{x}\right|\\ t_2 := e^{{x}^{2}}\\ 1.875 \cdot \left(\frac{t_1 \cdot t_2}{{x}^{6}} \cdot t_0\right) + \left(\frac{\left(0.75 \cdot t_1 + 0.5 \cdot \left|x\right|\right) \cdot t_2}{{x}^{4}} \cdot t_0 + \frac{\left|x\right| \cdot t_2}{{x}^{2}} \cdot t_0\right) \end{array} \]

Alternative 4
Error	2.6
Cost	98432

\[\begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \sqrt{\frac{1}{\pi}}\\ t_2 := e^{{x}^{2}}\\ t_1 \cdot \left(t_2 \cdot \left(\frac{t_0 \cdot 0.75 + 0.5 \cdot \left|x\right|}{{x}^{4}} + \frac{\left|x\right|}{{x}^{2}}\right)\right) + \left(t_0 \cdot \frac{t_2}{{x}^{6}}\right) \cdot \left(1.875 \cdot t_1\right) \end{array} \]

Alternative 5
Error	2.7
Cost	85440

\[\begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := e^{{x}^{2}}\\ \sqrt{\frac{1}{\pi}} \cdot \left(1.875 \cdot \left(t_0 \cdot \frac{t_1}{{x}^{6}}\right) + t_1 \cdot \left(\frac{t_0 \cdot 0.75 + 0.5 \cdot \left|x\right|}{{x}^{4}} + \frac{\left|x\right|}{{x}^{2}}\right)\right) \end{array} \]

Alternative 6
Error	2.7
Cost	49856

\[\begin{array}{l} t_0 := \frac{1}{x \cdot x}\\ t_1 := \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{\frac{0.75 + 1.875 \cdot t_0}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + 0.5 \cdot \left(2 + t_0\right)}{x \cdot x}\\ \frac{t_1 + t_1}{2 \cdot \frac{1}{\left|x\right|}} \end{array} \]

Alternative 7
Error	2.7
Cost	27776

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

Alternative 8
Error	2.7
Cost	27776

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

Alternative 9
Error	56.3
Cost	21120

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

Alternative 10
Error	56.3
Cost	21120

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

Alternative 11
Error	56.9
Cost	19456

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

Reproduce

herbie shell --seed 2023075 
(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)))))))