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

Average Accuracy: 95.6% → 98.1%

Time: 18.1s

Precision: binary64

Cost: 46400

\[x \geq 0.5\]

Math

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

↓

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

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
 (*
  (*
   (pow (exp x) x)
   (+
    (/ 1.875 (pow x 7.0))
    (+ (/ 1.0 x) (+ (/ 0.75 (pow x 5.0)) (/ 0.5 (pow x 3.0))))))
  (/ 1.0 (pow PI 0.5))))

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 (pow(exp(x), x) * ((1.875 / pow(x, 7.0)) + ((1.0 / x) + ((0.75 / pow(x, 5.0)) + (0.5 / pow(x, 3.0)))))) * (1.0 / pow(((double) M_PI), 0.5));
}

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 (Math.pow(Math.exp(x), x) * ((1.875 / Math.pow(x, 7.0)) + ((1.0 / x) + ((0.75 / Math.pow(x, 5.0)) + (0.5 / Math.pow(x, 3.0)))))) * (1.0 / Math.pow(Math.PI, 0.5));
}

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 (math.pow(math.exp(x), x) * ((1.875 / math.pow(x, 7.0)) + ((1.0 / x) + ((0.75 / math.pow(x, 5.0)) + (0.5 / math.pow(x, 3.0)))))) * (1.0 / math.pow(math.pi, 0.5))

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((exp(x) ^ x) * Float64(Float64(1.875 / (x ^ 7.0)) + Float64(Float64(1.0 / x) + Float64(Float64(0.75 / (x ^ 5.0)) + Float64(0.5 / (x ^ 3.0)))))) * Float64(1.0 / (pi ^ 0.5)))
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 = ((exp(x) ^ x) * ((1.875 / (x ^ 7.0)) + ((1.0 / x) + ((0.75 / (x ^ 5.0)) + (0.5 / (x ^ 3.0)))))) * (1.0 / (pi ^ 0.5));
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[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] * N[(N[(1.875 / N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] + N[(N[(0.75 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Power[Pi, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.6%

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

2. Simplified 95.7%

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

   [Start] 95.6	\[ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
   *-commutative [=>] 95.6	\[ \color{blue}{\left(e^{\left|x\right| \cdot \left|x\right|} \cdot \frac{1}{\sqrt{\pi}}\right)} \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
   associate-*l* [=>] 95.6	\[ \color{blue}{e^{\left|x\right| \cdot \left|x\right|} \cdot \left(\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\right)} \]

3. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{\frac{\frac{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left({x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right), \frac{\mathsf{fma}\left(0.5, {x}^{-2}, 1\right)}{x}\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}}} \]
   Proof

   [Start] 95.7	\[ e^{x \cdot x} \cdot \frac{\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)}{\sqrt{\pi}} \]
   associate-*r/ [=>] 95.7	\[ \color{blue}{\frac{e^{x \cdot x} \cdot \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\pi}}} \]
   add-sqr-sqrt [=>] 95.7	\[ \frac{e^{x \cdot x} \cdot \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\color{blue}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}} \]
   associate-/r* [=>] 95.7	\[ \color{blue}{\frac{\frac{e^{x \cdot x} \cdot \left(\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)\right)}{\sqrt{\sqrt{\pi}}}}{\sqrt{\sqrt{\pi}}}} \]

4. Taylor expanded in x around 0 98.0%

   \[\leadsto \frac{\frac{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left({x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right), \color{blue}{\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}}\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}} \]

5. Simplified 98.0%

   \[\leadsto \frac{\frac{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left({x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right), \color{blue}{\frac{1}{x} + \frac{0.5}{{x}^{3}}}\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}} \]
   Proof

   [Start] 98.0	\[ \frac{\frac{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left({x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right), \frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}} \]
   associate-*r/ [=>] 98.0	\[ \frac{\frac{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left({x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right), \frac{1}{x} + \color{blue}{\frac{0.5 \cdot 1}{{x}^{3}}}\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}} \]
   metadata-eval [=>] 98.0	\[ \frac{\frac{{\left(e^{x}\right)}^{x} \cdot \mathsf{fma}\left({x}^{-5}, \mathsf{fma}\left(1.875, {x}^{-2}, 0.75\right), \frac{1}{x} + \frac{\color{blue}{0.5}}{{x}^{3}}\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}} \]

6. Taylor expanded in x around 0 98.0%

   \[\leadsto \frac{\frac{{\left(e^{x}\right)}^{x} \cdot \color{blue}{\left(0.75 \cdot \frac{1}{{x}^{5}} + \left(\frac{1}{x} + \left(0.5 \cdot \frac{1}{{x}^{3}} + 1.875 \cdot \frac{1}{{x}^{7}}\right)\right)\right)}}{{\pi}^{0.25}}}{{\pi}^{0.25}} \]

7. Simplified 98.1%

   \[\leadsto \frac{\frac{{\left(e^{x}\right)}^{x} \cdot \color{blue}{\left(\frac{1.875}{{x}^{7}} + \left(\frac{1}{x} + \left(\frac{0.75}{{x}^{5}} + \frac{0.5}{{x}^{3}}\right)\right)\right)}}{{\pi}^{0.25}}}{{\pi}^{0.25}} \]
   Proof

   [Start] 98.0	\[ \frac{\frac{{\left(e^{x}\right)}^{x} \cdot \left(0.75 \cdot \frac{1}{{x}^{5}} + \left(\frac{1}{x} + \left(0.5 \cdot \frac{1}{{x}^{3}} + 1.875 \cdot \frac{1}{{x}^{7}}\right)\right)\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}} \]
   associate-+r+ [=>] 98.0	\[ \frac{\frac{{\left(e^{x}\right)}^{x} \cdot \left(0.75 \cdot \frac{1}{{x}^{5}} + \color{blue}{\left(\left(\frac{1}{x} + 0.5 \cdot \frac{1}{{x}^{3}}\right) + 1.875 \cdot \frac{1}{{x}^{7}}\right)}\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}} \]
   +-commutative [<=] 98.0	\[ \frac{\frac{{\left(e^{x}\right)}^{x} \cdot \left(0.75 \cdot \frac{1}{{x}^{5}} + \left(\color{blue}{\left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)} + 1.875 \cdot \frac{1}{{x}^{7}}\right)\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}} \]
   associate-+r+ [=>] 98.0	\[ \frac{\frac{{\left(e^{x}\right)}^{x} \cdot \color{blue}{\left(\left(0.75 \cdot \frac{1}{{x}^{5}} + \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)\right) + 1.875 \cdot \frac{1}{{x}^{7}}\right)}}{{\pi}^{0.25}}}{{\pi}^{0.25}} \]
   +-commutative [<=] 98.0	\[ \frac{\frac{{\left(e^{x}\right)}^{x} \cdot \color{blue}{\left(1.875 \cdot \frac{1}{{x}^{7}} + \left(0.75 \cdot \frac{1}{{x}^{5}} + \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)\right)\right)}}{{\pi}^{0.25}}}{{\pi}^{0.25}} \]
   associate-*r/ [=>] 98.0	\[ \frac{\frac{{\left(e^{x}\right)}^{x} \cdot \left(\color{blue}{\frac{1.875 \cdot 1}{{x}^{7}}} + \left(0.75 \cdot \frac{1}{{x}^{5}} + \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)\right)\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}} \]
   metadata-eval [=>] 98.0	\[ \frac{\frac{{\left(e^{x}\right)}^{x} \cdot \left(\frac{\color{blue}{1.875}}{{x}^{7}} + \left(0.75 \cdot \frac{1}{{x}^{5}} + \left(0.5 \cdot \frac{1}{{x}^{3}} + \frac{1}{x}\right)\right)\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}} \]
   associate-+r+ [=>] 98.1	\[ \frac{\frac{{\left(e^{x}\right)}^{x} \cdot \left(\frac{1.875}{{x}^{7}} + \color{blue}{\left(\left(0.75 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{{x}^{3}}\right) + \frac{1}{x}\right)}\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}} \]
   +-commutative [=>] 98.1	\[ \frac{\frac{{\left(e^{x}\right)}^{x} \cdot \left(\frac{1.875}{{x}^{7}} + \color{blue}{\left(\frac{1}{x} + \left(0.75 \cdot \frac{1}{{x}^{5}} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)}\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}} \]
   associate-*r/ [=>] 98.1	\[ \frac{\frac{{\left(e^{x}\right)}^{x} \cdot \left(\frac{1.875}{{x}^{7}} + \left(\frac{1}{x} + \left(\color{blue}{\frac{0.75 \cdot 1}{{x}^{5}}} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}} \]
   metadata-eval [=>] 98.1	\[ \frac{\frac{{\left(e^{x}\right)}^{x} \cdot \left(\frac{1.875}{{x}^{7}} + \left(\frac{1}{x} + \left(\frac{\color{blue}{0.75}}{{x}^{5}} + 0.5 \cdot \frac{1}{{x}^{3}}\right)\right)\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}} \]
   associate-*r/ [=>] 98.1	\[ \frac{\frac{{\left(e^{x}\right)}^{x} \cdot \left(\frac{1.875}{{x}^{7}} + \left(\frac{1}{x} + \left(\frac{0.75}{{x}^{5}} + \color{blue}{\frac{0.5 \cdot 1}{{x}^{3}}}\right)\right)\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}} \]
   metadata-eval [=>] 98.1	\[ \frac{\frac{{\left(e^{x}\right)}^{x} \cdot \left(\frac{1.875}{{x}^{7}} + \left(\frac{1}{x} + \left(\frac{0.75}{{x}^{5}} + \frac{\color{blue}{0.5}}{{x}^{3}}\right)\right)\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}} \]

8. Applied egg-rr 98.1%

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

   [Start] 98.1	\[ \frac{\frac{{\left(e^{x}\right)}^{x} \cdot \left(\frac{1.875}{{x}^{7}} + \left(\frac{1}{x} + \left(\frac{0.75}{{x}^{5}} + \frac{0.5}{{x}^{3}}\right)\right)\right)}{{\pi}^{0.25}}}{{\pi}^{0.25}} \]
   associate-/l/ [=>] 98.1	\[ \color{blue}{\frac{{\left(e^{x}\right)}^{x} \cdot \left(\frac{1.875}{{x}^{7}} + \left(\frac{1}{x} + \left(\frac{0.75}{{x}^{5}} + \frac{0.5}{{x}^{3}}\right)\right)\right)}{{\pi}^{0.25} \cdot {\pi}^{0.25}}} \]
   div-inv [=>] 98.1	\[ \color{blue}{\left({\left(e^{x}\right)}^{x} \cdot \left(\frac{1.875}{{x}^{7}} + \left(\frac{1}{x} + \left(\frac{0.75}{{x}^{5}} + \frac{0.5}{{x}^{3}}\right)\right)\right)\right) \cdot \frac{1}{{\pi}^{0.25} \cdot {\pi}^{0.25}}} \]
   pow-sqr [=>] 98.1	\[ \left({\left(e^{x}\right)}^{x} \cdot \left(\frac{1.875}{{x}^{7}} + \left(\frac{1}{x} + \left(\frac{0.75}{{x}^{5}} + \frac{0.5}{{x}^{3}}\right)\right)\right)\right) \cdot \frac{1}{\color{blue}{{\pi}^{\left(2 \cdot 0.25\right)}}} \]
   metadata-eval [=>] 98.1	\[ \left({\left(e^{x}\right)}^{x} \cdot \left(\frac{1.875}{{x}^{7}} + \left(\frac{1}{x} + \left(\frac{0.75}{{x}^{5}} + \frac{0.5}{{x}^{3}}\right)\right)\right)\right) \cdot \frac{1}{{\pi}^{\color{blue}{0.5}}} \]

9. Final simplification 98.1%

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

Alternatives

Alternative 1
Accuracy	98.1%
Cost	40128

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

Alternative 2
Accuracy	98.0%
Cost	40064

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

Alternative 3
Accuracy	98.0%
Cost	39936

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

Alternative 4
Accuracy	98.0%
Cost	39872

\[\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}} \cdot \left(\frac{1 + \frac{1.875}{{x}^{6}}}{x} + \frac{0.5 + \frac{0.75}{x \cdot x}}{{x}^{3}}\right) \]

Alternative 5
Accuracy	98.0%
Cost	39872

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

Alternative 6
Accuracy	95.8%
Cost	33600

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

Alternative 7
Accuracy	95.8%
Cost	33600

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

Alternative 8
Accuracy	98.0%
Cost	33600

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

Alternative 9
Accuracy	31.7%
Cost	33216

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

Alternative 10
Accuracy	31.7%
Cost	33152

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

Alternative 11
Accuracy	30.1%
Cost	32960

\[\begin{array}{l} t_0 := e^{x \cdot x}\\ \frac{\frac{t_0}{x} + 0.5 \cdot \frac{t_0}{{x}^{3}}}{\sqrt{\pi}} \end{array} \]

Alternative 12
Accuracy	25.0%
Cost	26432

\[\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{1}{x} + \frac{0.75}{{x}^{5}}\right) \]

Alternative 13
Accuracy	30.1%
Cost	26432

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

Alternative 14
Accuracy	24.5%
Cost	25920

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

Alternative 15
Accuracy	24.5%
Cost	19584

\[\frac{\frac{e^{x \cdot x}}{x}}{\sqrt{\pi}} \]

Alternative 16
Accuracy	11.3%
Cost	19520

\[\frac{\frac{1.875}{{x}^{7}}}{\sqrt{\pi}} \]

Alternative 17
Accuracy	10.9%
Cost	13312

\[\frac{\frac{0.5}{x}}{x \cdot \left(x \cdot \sqrt{\pi}\right)} \]

Reproduce

herbie shell --seed 2023131 
(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)))))))