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

Average Error: 2.9 → 2.7

Time: 15.4s

Precision: binary64

Cost: 72320

\[x \geq 0.5\]

Math

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

↓

\[\left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot e^{{x}^{2}}\right) \]

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
 (*
  (+
   (* 1.875 (/ 1.0 (pow (fabs x) 7.0)))
   (+
    (* 0.75 (/ 1.0 (pow (fabs x) 5.0)))
    (+ (* 0.5 (/ 1.0 (pow (fabs x) 3.0))) (/ 1.0 (fabs x)))))
  (* (sqrt (/ 1.0 PI)) (exp (pow x 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 ((1.875 * (1.0 / pow(fabs(x), 7.0))) + ((0.75 * (1.0 / pow(fabs(x), 5.0))) + ((0.5 * (1.0 / pow(fabs(x), 3.0))) + (1.0 / fabs(x))))) * (sqrt((1.0 / ((double) M_PI))) * exp(pow(x, 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 ((1.875 * (1.0 / Math.pow(Math.abs(x), 7.0))) + ((0.75 * (1.0 / Math.pow(Math.abs(x), 5.0))) + ((0.5 * (1.0 / Math.pow(Math.abs(x), 3.0))) + (1.0 / Math.abs(x))))) * (Math.sqrt((1.0 / Math.PI)) * Math.exp(Math.pow(x, 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 ((1.875 * (1.0 / math.pow(math.fabs(x), 7.0))) + ((0.75 * (1.0 / math.pow(math.fabs(x), 5.0))) + ((0.5 * (1.0 / math.pow(math.fabs(x), 3.0))) + (1.0 / math.fabs(x))))) * (math.sqrt((1.0 / math.pi)) * math.exp(math.pow(x, 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(Float64(1.875 * Float64(1.0 / (abs(x) ^ 7.0))) + Float64(Float64(0.75 * Float64(1.0 / (abs(x) ^ 5.0))) + Float64(Float64(0.5 * Float64(1.0 / (abs(x) ^ 3.0))) + Float64(1.0 / abs(x))))) * Float64(sqrt(Float64(1.0 / pi)) * exp((x ^ 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 = ((1.875 * (1.0 / (abs(x) ^ 7.0))) + ((0.75 * (1.0 / (abs(x) ^ 5.0))) + ((0.5 * (1.0 / (abs(x) ^ 3.0))) + (1.0 / abs(x))))) * (sqrt((1.0 / pi)) * exp((x ^ 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[(N[(1.875 * N[(1.0 / N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.75 * N[(1.0 / N[Power[N[Abs[x], $MachinePrecision], 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[(1.0 / N[Power[N[Abs[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[Exp[N[Power[x, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 2.9

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

2. Simplified 2.9

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

   [Start] 2.9	\[ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
   rational_best_oopsla_all_46_json_45_simplify-80 [=>] 2.9	\[ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\color{blue}{\left|x \cdot x\right|}}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
   rational_best_oopsla_all_46_json_45_simplify-57 [=>] 2.9	\[ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\color{blue}{x \cdot x}}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
   rational_best_oopsla_all_46_json_45_simplify-35 [=>] 2.9	\[ \left(\frac{1}{\sqrt{\pi}} \cdot e^{x \cdot x}\right) \cdot \color{blue}{\left(\frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) + \left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\right)} \]

3. Taylor expanded in x around 0 2.8

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

4. Taylor expanded in x around 0 2.7

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

5. Taylor expanded in x around 0 2.7

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

6. Taylor expanded in x around inf 2.7

   \[\leadsto \color{blue}{\left(e^{{x}^{2}} \cdot \left(0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right)\right) \cdot \sqrt{\frac{1}{\pi}}} \]

7. Simplified 2.7

   \[\leadsto \color{blue}{\left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot e^{{x}^{2}}\right)} \]
   Proof

   [Start] 2.7	\[ \left(e^{{x}^{2}} \cdot \left(0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right)\right) \cdot \sqrt{\frac{1}{\pi}} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 2.7	\[ \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(e^{{x}^{2}} \cdot \left(0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 2.7	\[ \sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(\left(0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \cdot e^{{x}^{2}}\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-7 [=>] 2.7	\[ \color{blue}{\left(0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot e^{{x}^{2}}\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-82 [=>] 2.7	\[ \color{blue}{\left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right)} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot e^{{x}^{2}}\right) \]

8. Final simplification 2.7

   \[\leadsto \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot e^{{x}^{2}}\right) \]

Alternatives

Alternative 1
Error	2.7
Cost	72320

\[\left(e^{{x}^{2}} \cdot \left(0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right)\right) \cdot \sqrt{\frac{1}{\pi}} \]

Alternative 2
Error	2.7
Cost	72320

\[\left(\left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} + \left(\frac{1}{\left|x\right|} + 0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}}\right)\right)\right) \cdot e^{{x}^{2}}\right) \cdot \sqrt{\frac{1}{\pi}} \]

Alternative 3
Error	2.7
Cost	72320

\[\left(0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot e^{{x}^{2}}\right) \]

Alternative 4
Error	56.4
Cost	59328

\[\left(0.75 \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(1.875 \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(0.5 \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} + \frac{1}{\left|x\right|}\right)\right)\right) \cdot \sqrt{\frac{1}{\pi}} \]

Reproduce

herbie shell --seed 2023090 
(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)))))))