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

Average Error: 2.8 → 2.7

Time: 23.0s

Precision: binary64

Cost: 66368

\[x \geq 0.5\]

Math

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

↓

\[\frac{1}{\sqrt{\pi}} \cdot \left(e^{x \cdot x} \cdot \frac{1.5 \cdot \frac{1}{\left|x\right| \cdot {x}^{4}} + \left(2 \cdot \frac{1}{\left|x\right|} + \left(\frac{\frac{3.75}{\left|x\right|}}{{x}^{6}} + \frac{1}{\left|x\right| \cdot {x}^{2}}\right)\right)}{2}\right) \]

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
 (*
  (/ 1.0 (sqrt PI))
  (*
   (exp (* x x))
   (/
    (+
     (* 1.5 (/ 1.0 (* (fabs x) (pow x 4.0))))
     (+
      (* 2.0 (/ 1.0 (fabs x)))
      (+ (/ (/ 3.75 (fabs x)) (pow x 6.0)) (/ 1.0 (* (fabs x) (pow x 2.0))))))
    2.0))))

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) {
	return (1.0 / sqrt(((double) M_PI))) * (exp((x * x)) * (((1.5 * (1.0 / (fabs(x) * pow(x, 4.0)))) + ((2.0 * (1.0 / fabs(x))) + (((3.75 / fabs(x)) / pow(x, 6.0)) + (1.0 / (fabs(x) * pow(x, 2.0)))))) / 2.0));
}

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) {
	return (1.0 / Math.sqrt(Math.PI)) * (Math.exp((x * x)) * (((1.5 * (1.0 / (Math.abs(x) * Math.pow(x, 4.0)))) + ((2.0 * (1.0 / Math.abs(x))) + (((3.75 / Math.abs(x)) / Math.pow(x, 6.0)) + (1.0 / (Math.abs(x) * Math.pow(x, 2.0)))))) / 2.0));
}

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):
	return (1.0 / math.sqrt(math.pi)) * (math.exp((x * x)) * (((1.5 * (1.0 / (math.fabs(x) * math.pow(x, 4.0)))) + ((2.0 * (1.0 / math.fabs(x))) + (((3.75 / math.fabs(x)) / math.pow(x, 6.0)) + (1.0 / (math.fabs(x) * math.pow(x, 2.0)))))) / 2.0))

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)
	return Float64(Float64(1.0 / sqrt(pi)) * Float64(exp(Float64(x * x)) * Float64(Float64(Float64(1.5 * Float64(1.0 / Float64(abs(x) * (x ^ 4.0)))) + Float64(Float64(2.0 * Float64(1.0 / abs(x))) + Float64(Float64(Float64(3.75 / abs(x)) / (x ^ 6.0)) + Float64(1.0 / Float64(abs(x) * (x ^ 2.0)))))) / 2.0)))
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)
	tmp = (1.0 / sqrt(pi)) * (exp((x * x)) * (((1.5 * (1.0 / (abs(x) * (x ^ 4.0)))) + ((2.0 * (1.0 / abs(x))) + (((3.75 / abs(x)) / (x ^ 6.0)) + (1.0 / (abs(x) * (x ^ 2.0)))))) / 2.0));
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_] := N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(1.5 * N[(1.0 / N[(N[Abs[x], $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(3.75 / N[Abs[x], $MachinePrecision]), $MachinePrecision] / N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(N[Abs[x], $MachinePrecision] * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $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 2.8

   \[\leadsto \color{blue}{\frac{1}{\sqrt{\pi}} \cdot \left(e^{x \cdot x} \cdot \left(\left|\frac{1}{x}\right| \cdot \left(\frac{\frac{\frac{1}{x}}{x}}{x \cdot x} \cdot \left(0.75 + \frac{\frac{1}{x}}{x} \cdot 1.875\right) + \left(0.5 \cdot \frac{\frac{1}{x}}{x} + 1\right)\right)\right)\right)} \]
   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-2 [=>] 2.8	\[ \color{blue}{\left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \cdot \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right)} \]
   rational.json-simplify-43 [=>] 2.8	\[ \color{blue}{\frac{1}{\sqrt{\pi}} \cdot \left(e^{\left|x\right| \cdot \left|x\right|} \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\right)} \]

3. Applied egg-rr 2.7

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

4. Taylor expanded in x around 0 2.7

   \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left(e^{x \cdot x} \cdot \frac{\color{blue}{1.5 \cdot \frac{1}{\left|x\right| \cdot {x}^{4}} + \left(2 \cdot \frac{1}{\left|x\right|} + \left(3.75 \cdot \frac{1}{\left|x\right| \cdot {x}^{6}} + \frac{1}{\left|x\right| \cdot {x}^{2}}\right)\right)}}{2}\right) \]

5. Taylor expanded in x around 0 2.7

   \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left(e^{x \cdot x} \cdot \frac{1.5 \cdot \frac{1}{\left|x\right| \cdot {x}^{4}} + \left(2 \cdot \frac{1}{\left|x\right|} + \left(\color{blue}{\frac{3.75}{\left|x\right| \cdot {x}^{6}}} + \frac{1}{\left|x\right| \cdot {x}^{2}}\right)\right)}{2}\right) \]

6. Simplified 2.7

   \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left(e^{x \cdot x} \cdot \frac{1.5 \cdot \frac{1}{\left|x\right| \cdot {x}^{4}} + \left(2 \cdot \frac{1}{\left|x\right|} + \left(\color{blue}{\frac{\frac{3.75}{\left|x\right|}}{{x}^{6}}} + \frac{1}{\left|x\right| \cdot {x}^{2}}\right)\right)}{2}\right) \]
   Proof

   [Start] 2.7	\[ \frac{1}{\sqrt{\pi}} \cdot \left(e^{x \cdot x} \cdot \frac{1.5 \cdot \frac{1}{\left|x\right| \cdot {x}^{4}} + \left(2 \cdot \frac{1}{\left|x\right|} + \left(\frac{3.75}{\left|x\right| \cdot {x}^{6}} + \frac{1}{\left|x\right| \cdot {x}^{2}}\right)\right)}{2}\right) \]
   rational.json-simplify-46 [=>] 2.7	\[ \frac{1}{\sqrt{\pi}} \cdot \left(e^{x \cdot x} \cdot \frac{1.5 \cdot \frac{1}{\left|x\right| \cdot {x}^{4}} + \left(2 \cdot \frac{1}{\left|x\right|} + \left(\color{blue}{\frac{\frac{3.75}{\left|x\right|}}{{x}^{6}}} + \frac{1}{\left|x\right| \cdot {x}^{2}}\right)\right)}{2}\right) \]

7. Final simplification 2.7

   \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left(e^{x \cdot x} \cdot \frac{1.5 \cdot \frac{1}{\left|x\right| \cdot {x}^{4}} + \left(2 \cdot \frac{1}{\left|x\right|} + \left(\frac{\frac{3.75}{\left|x\right|}}{{x}^{6}} + \frac{1}{\left|x\right| \cdot {x}^{2}}\right)\right)}{2}\right) \]

Alternatives

Alternative 1
Error	2.7
Cost	66368

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

Alternative 2
Error	2.7
Cost	66368

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

Alternative 3
Error	2.7
Cost	34944

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

Alternative 4
Error	2.7
Cost	27776

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

Alternative 5
Error	2.7
Cost	27520

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

Alternative 6
Error	2.7
Cost	27520

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

Alternative 7
Error	48.3
Cost	26240

\[\frac{1}{\sqrt{\pi}} \cdot \left(e^{x \cdot x} \cdot \frac{1}{\left|x\right|}\right) \]

Alternative 8
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 9
Error	56.9
Cost	19456

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

Reproduce

herbie shell --seed 2023073 
(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)))))))