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

Average Error: 2.8 → 1.2

Time: 14.0s

Precision: binary64

Cost: 46336

\[x \geq 0.5\]

Math

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

↓

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

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

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

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

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

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((exp(x) ^ x) * Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(Float64(1.875 * (x ^ -7.0)) + Float64(0.5 / (x ^ 3.0))) + Float64(Float64(1.0 / x) + Float64(0.75 / (x ^ 5.0))))))
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 = (exp(x) ^ x) * (sqrt((1.0 / pi)) * (((1.875 * (x ^ -7.0)) + (0.5 / (x ^ 3.0))) + ((1.0 / x) + (0.75 / (x ^ 5.0)))));
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[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(1.875 * N[Power[x, -7.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] + N[(0.75 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 1.4

   \[\leadsto \color{blue}{{\left(e^{x}\right)}^{x} \cdot \frac{\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + {\left(\frac{1}{\left|x\right|}\right)}^{5} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)}{\sqrt{\pi}}} \]
   Proof
   (*.f64 (pow.f64 (exp.f64 x) x) (/.f64 (+.f64 (/.f64 (+.f64 1 (/.f64 1/2 (*.f64 x x))) (fabs.f64 x)) (*.f64 (pow.f64 (/.f64 1 (fabs.f64 x)) 5) (+.f64 3/4 (/.f64 15/8 (*.f64 x x))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 x x))) (/.f64 (+.f64 (/.f64 (+.f64 1 (/.f64 1/2 (*.f64 x x))) (fabs.f64 x)) (*.f64 (pow.f64 (/.f64 1 (fabs.f64 x)) 5) (+.f64 3/4 (/.f64 15/8 (*.f64 x x))))) (sqrt.f64 (PI.f64)))): 152 points increase in error, 29 points decrease in error
   (*.f64 (exp.f64 (Rewrite<= sqr-abs_binary64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (/.f64 (+.f64 (/.f64 (+.f64 1 (/.f64 1/2 (*.f64 x x))) (fabs.f64 x)) (*.f64 (pow.f64 (/.f64 1 (fabs.f64 x)) 5) (+.f64 3/4 (/.f64 15/8 (*.f64 x x))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (+.f64 (/.f64 (+.f64 1 (/.f64 (Rewrite<= metadata-eval (*.f64 1/2 1)) (*.f64 x x))) (fabs.f64 x)) (*.f64 (pow.f64 (/.f64 1 (fabs.f64 x)) 5) (+.f64 3/4 (/.f64 15/8 (*.f64 x x))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (+.f64 (/.f64 (+.f64 1 (/.f64 (*.f64 (Rewrite<= metadata-eval (/.f64 1 2)) 1) (*.f64 x x))) (fabs.f64 x)) (*.f64 (pow.f64 (/.f64 1 (fabs.f64 x)) 5) (+.f64 3/4 (/.f64 15/8 (*.f64 x x))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (+.f64 (/.f64 (+.f64 1 (/.f64 (*.f64 (/.f64 1 2) 1) (Rewrite<= sqr-abs_binary64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (fabs.f64 x)) (*.f64 (pow.f64 (/.f64 1 (fabs.f64 x)) 5) (+.f64 3/4 (/.f64 15/8 (*.f64 x x))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (+.f64 (/.f64 (+.f64 1 (Rewrite<= associate-*r/_binary64 (*.f64 (/.f64 1 2) (/.f64 1 (*.f64 (fabs.f64 x) (fabs.f64 x)))))) (fabs.f64 x)) (*.f64 (pow.f64 (/.f64 1 (fabs.f64 x)) 5) (+.f64 3/4 (/.f64 15/8 (*.f64 x x))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (+.f64 (/.f64 (+.f64 1 (*.f64 (/.f64 1 2) (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 1 (fabs.f64 x)) (fabs.f64 x))))) (fabs.f64 x)) (*.f64 (pow.f64 (/.f64 1 (fabs.f64 x)) 5) (+.f64 3/4 (/.f64 15/8 (*.f64 x x))))) (sqrt.f64 (PI.f64)))): 1 points increase in error, 1 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (+.f64 (/.f64 (+.f64 1 (*.f64 (/.f64 1 2) (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 1 (fabs.f64 x)))) (fabs.f64 x)))) (fabs.f64 x)) (*.f64 (pow.f64 (/.f64 1 (fabs.f64 x)) 5) (+.f64 3/4 (/.f64 15/8 (*.f64 x x))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (+.f64 (/.f64 (+.f64 1 (*.f64 (/.f64 1 2) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x)))))) (fabs.f64 x)) (*.f64 (pow.f64 (/.f64 1 (fabs.f64 x)) 5) (+.f64 3/4 (/.f64 15/8 (*.f64 x x))))) (sqrt.f64 (PI.f64)))): 1 points increase in error, 0 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (+.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (/.f64 1 2) (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x)))) 1)) (fabs.f64 x)) (*.f64 (pow.f64 (/.f64 1 (fabs.f64 x)) 5) (+.f64 3/4 (/.f64 15/8 (*.f64 x x))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (+.f64 (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (+.f64 (*.f64 (/.f64 1 2) (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x)))) 1) 1)) (fabs.f64 x)) (*.f64 (pow.f64 (/.f64 1 (fabs.f64 x)) 5) (+.f64 3/4 (/.f64 15/8 (*.f64 x x))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (+.f64 (*.f64 (/.f64 1 2) (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x)))) 1) (/.f64 1 (fabs.f64 x)))) (*.f64 (pow.f64 (/.f64 1 (fabs.f64 x)) 5) (+.f64 3/4 (/.f64 15/8 (*.f64 x x))))) (sqrt.f64 (PI.f64)))): 18 points increase in error, 25 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (+.f64 (Rewrite<= distribute-rgt1-in_binary64 (+.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (*.f64 (/.f64 1 2) (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x)))) (/.f64 1 (fabs.f64 x))))) (*.f64 (pow.f64 (/.f64 1 (fabs.f64 x)) 5) (+.f64 3/4 (/.f64 15/8 (*.f64 x x))))) (sqrt.f64 (PI.f64)))): 29 points increase in error, 31 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (+.f64 (+.f64 (/.f64 1 (fabs.f64 x)) (Rewrite<= associate-*r*_binary64 (*.f64 (/.f64 1 2) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x)))))) (*.f64 (pow.f64 (/.f64 1 (fabs.f64 x)) 5) (+.f64 3/4 (/.f64 15/8 (*.f64 x x))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (+.f64 (+.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 2) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) (*.f64 (pow.f64 (/.f64 1 (fabs.f64 x)) (Rewrite<= metadata-eval (*.f64 2 5/2))) (+.f64 3/4 (/.f64 15/8 (*.f64 x x))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (+.f64 (+.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 2) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) (*.f64 (pow.f64 (/.f64 1 (fabs.f64 x)) (*.f64 2 (Rewrite<= metadata-eval (+.f64 3/2 1)))) (+.f64 3/4 (/.f64 15/8 (*.f64 x x))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (+.f64 (+.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 2) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) (*.f64 (pow.f64 (/.f64 1 (fabs.f64 x)) (*.f64 2 (+.f64 (Rewrite<= metadata-eval (/.f64 3 2)) 1))) (+.f64 3/4 (/.f64 15/8 (*.f64 x x))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (+.f64 (+.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 2) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) (*.f64 (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 (/.f64 1 (fabs.f64 x)) (+.f64 (/.f64 3 2) 1)) (pow.f64 (/.f64 1 (fabs.f64 x)) (+.f64 (/.f64 3 2) 1)))) (+.f64 3/4 (/.f64 15/8 (*.f64 x x))))) (sqrt.f64 (PI.f64)))): 8 points increase in error, 10 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (+.f64 (+.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 2) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) (*.f64 (*.f64 (Rewrite<= pow-plus_binary64 (*.f64 (pow.f64 (/.f64 1 (fabs.f64 x)) (/.f64 3 2)) (/.f64 1 (fabs.f64 x)))) (pow.f64 (/.f64 1 (fabs.f64 x)) (+.f64 (/.f64 3 2) 1))) (+.f64 3/4 (/.f64 15/8 (*.f64 x x))))) (sqrt.f64 (PI.f64)))): 6 points increase in error, 5 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (+.f64 (+.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 2) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) (*.f64 (*.f64 (*.f64 (pow.f64 (/.f64 1 (fabs.f64 x)) (/.f64 3 2)) (/.f64 1 (fabs.f64 x))) (Rewrite<= pow-plus_binary64 (*.f64 (pow.f64 (/.f64 1 (fabs.f64 x)) (/.f64 3 2)) (/.f64 1 (fabs.f64 x))))) (+.f64 3/4 (/.f64 15/8 (*.f64 x x))))) (sqrt.f64 (PI.f64)))): 11 points increase in error, 1 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (+.f64 (+.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 2) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) (*.f64 (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 (pow.f64 (/.f64 1 (fabs.f64 x)) (/.f64 3 2)) (pow.f64 (/.f64 1 (fabs.f64 x)) (/.f64 3 2))) (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))))) (+.f64 3/4 (/.f64 15/8 (*.f64 x x))))) (sqrt.f64 (PI.f64)))): 4 points increase in error, 8 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (+.f64 (+.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 2) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) (*.f64 (*.f64 (Rewrite<= sqr-pow_binary64 (pow.f64 (/.f64 1 (fabs.f64 x)) 3)) (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x)))) (+.f64 3/4 (/.f64 15/8 (*.f64 x x))))) (sqrt.f64 (PI.f64)))): 5 points increase in error, 13 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (+.f64 (+.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 2) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) (*.f64 (*.f64 (Rewrite<= cube-unmult_binary64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))))) (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x)))) (+.f64 3/4 (/.f64 15/8 (*.f64 x x))))) (sqrt.f64 (PI.f64)))): 8 points increase in error, 5 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (+.f64 (+.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 2) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) (*.f64 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x)))) (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x)))) (+.f64 3/4 (/.f64 15/8 (*.f64 x x))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (+.f64 (+.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 2) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x)))) (+.f64 3/4 (/.f64 15/8 (*.f64 x x))))) (sqrt.f64 (PI.f64)))): 5 points increase in error, 7 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (+.f64 (+.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 2) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (+.f64 (Rewrite<= metadata-eval (/.f64 3 4)) (/.f64 15/8 (*.f64 x x))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (+.f64 (+.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 2) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (+.f64 (/.f64 3 4) (/.f64 (Rewrite<= metadata-eval (/.f64 15 8)) (*.f64 x x))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (+.f64 (+.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 2) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (+.f64 (/.f64 3 4) (/.f64 (/.f64 15 8) (Rewrite<= sqr-abs_binary64 (*.f64 (fabs.f64 x) (fabs.f64 x))))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (+.f64 (+.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 2) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (+.f64 (/.f64 3 4) (/.f64 (/.f64 15 8) (*.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 (fabs.f64 x) 1)) (fabs.f64 x)))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (+.f64 (+.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 2) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (+.f64 (/.f64 3 4) (/.f64 (/.f64 15 8) (Rewrite<= associate-/r/_binary64 (/.f64 (fabs.f64 x) (/.f64 1 (fabs.f64 x)))))))) (sqrt.f64 (PI.f64)))): 11 points increase in error, 0 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (+.f64 (+.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 2) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (+.f64 (/.f64 3 4) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (/.f64 15 8) (/.f64 1 (fabs.f64 x))) (fabs.f64 x)))))) (sqrt.f64 (PI.f64)))): 4 points increase in error, 6 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (+.f64 (+.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 2) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (+.f64 (/.f64 3 4) (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (*.f64 (/.f64 15 8) (/.f64 1 (fabs.f64 x))) 1)) (fabs.f64 x))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (+.f64 (+.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 2) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (+.f64 (/.f64 3 4) (Rewrite<= associate-*r/_binary64 (*.f64 (*.f64 (/.f64 15 8) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))))) (sqrt.f64 (PI.f64)))): 5 points increase in error, 0 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (+.f64 (+.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 2) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (/.f64 3 4) (*.f64 (*.f64 (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x)))) (*.f64 (*.f64 (*.f64 (/.f64 15 8) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (*.f64 (*.f64 (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))))) (sqrt.f64 (PI.f64)))): 7 points increase in error, 6 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (+.f64 (+.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 2) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) (+.f64 (*.f64 (/.f64 3 4) (*.f64 (*.f64 (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x)))) (Rewrite<= associate-*r*_binary64 (*.f64 (*.f64 (/.f64 15 8) (/.f64 1 (fabs.f64 x))) (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (*.f64 (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x)))))))) (sqrt.f64 (PI.f64)))): 6 points increase in error, 1 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (+.f64 (+.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 2) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) (+.f64 (*.f64 (/.f64 3 4) (*.f64 (*.f64 (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x)))) (*.f64 (*.f64 (/.f64 15 8) (/.f64 1 (fabs.f64 x))) (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (+.f64 (+.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 2) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) (+.f64 (*.f64 (/.f64 3 4) (*.f64 (*.f64 (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x)))) (Rewrite<= associate-*r*_binary64 (*.f64 (/.f64 15 8) (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x)))))))) (sqrt.f64 (PI.f64)))): 1 points increase in error, 3 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (+.f64 (+.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 2) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) (+.f64 (*.f64 (/.f64 3 4) (*.f64 (*.f64 (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x)))) (*.f64 (/.f64 15 8) (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 2) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) (*.f64 (/.f64 3 4) (*.f64 (*.f64 (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) (*.f64 (/.f64 15 8) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x)))))) (sqrt.f64 (PI.f64)))): 24 points increase in error, 24 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 (+.f64 (+.f64 (+.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 2) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) (*.f64 (/.f64 3 4) (*.f64 (*.f64 (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) (*.f64 (/.f64 15 8) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) 1)) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (Rewrite<= associate-*r/_binary64 (*.f64 (+.f64 (+.f64 (+.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 2) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) (*.f64 (/.f64 3 4) (*.f64 (*.f64 (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) (*.f64 (/.f64 15 8) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) (/.f64 1 (sqrt.f64 (PI.f64)))))): 65 points increase in error, 69 points decrease in error
   (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (+.f64 (+.f64 (+.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 2) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) (*.f64 (/.f64 3 4) (*.f64 (*.f64 (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) (*.f64 (/.f64 15 8) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x)))))) (/.f64 1 (sqrt.f64 (PI.f64))))): 52 points increase in error, 44 points decrease in error
   (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (*.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (+.f64 (+.f64 (+.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 2) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) (*.f64 (/.f64 3 4) (*.f64 (*.f64 (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) (*.f64 (/.f64 15 8) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x)))))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (+.f64 (+.f64 (+.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 2) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) (*.f64 (/.f64 3 4) (*.f64 (*.f64 (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))) (*.f64 (/.f64 15 8) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))))): 48 points increase in error, 57 points decrease in error

3. Taylor expanded in x around 0 1.3

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

4. Simplified 1.3

   \[\leadsto {\left(e^{x}\right)}^{x} \cdot \frac{\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|} + \color{blue}{\frac{1}{{x}^{5}}} \cdot \left(0.75 + \frac{1.875}{x \cdot x}\right)}{\sqrt{\pi}} \]
   Proof
   (/.f64 1 (pow.f64 x 5)): 0 points increase in error, 0 points decrease in error
   (/.f64 1 (pow.f64 (Rewrite<= unpow1_binary64 (pow.f64 x 1)) 5)): 0 points increase in error, 0 points decrease in error
   (/.f64 1 (pow.f64 (Rewrite=> sqr-pow_binary64 (*.f64 (pow.f64 x (/.f64 1 2)) (pow.f64 x (/.f64 1 2)))) 5)): 63 points increase in error, 53 points decrease in error
   (/.f64 1 (pow.f64 (Rewrite<= fabs-sqr_binary64 (fabs.f64 (*.f64 (pow.f64 x (/.f64 1 2)) (pow.f64 x (/.f64 1 2))))) 5)): 0 points increase in error, 0 points decrease in error
   (/.f64 1 (pow.f64 (fabs.f64 (Rewrite<= sqr-pow_binary64 (pow.f64 x 1))) 5)): 53 points increase in error, 63 points decrease in error
   (/.f64 1 (pow.f64 (fabs.f64 (Rewrite=> unpow1_binary64 x)) 5)): 0 points increase in error, 0 points decrease in error

5. Taylor expanded in x around 0 1.2

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

6. Simplified 1.2

   \[\leadsto {\left(e^{x}\right)}^{x} \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{0.5}{{x}^{3}} + \frac{1}{x}\right) + \left(\frac{1.875}{{x}^{7}} + \frac{0.75}{{x}^{5}}\right)\right)\right)} \]
   Proof
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (+.f64 (/.f64 1/2 (pow.f64 x 3)) (/.f64 1 x)) (+.f64 (/.f64 15/8 (pow.f64 x 7)) (/.f64 3/4 (pow.f64 x 5))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (+.f64 (/.f64 1/2 (Rewrite=> cube-mult_binary64 (*.f64 x (*.f64 x x)))) (/.f64 1 x)) (+.f64 (/.f64 15/8 (pow.f64 x 7)) (/.f64 3/4 (pow.f64 x 5))))): 1 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (+.f64 (/.f64 1/2 (*.f64 (Rewrite<= unpow1_binary64 (pow.f64 x 1)) (*.f64 x x))) (/.f64 1 x)) (+.f64 (/.f64 15/8 (pow.f64 x 7)) (/.f64 3/4 (pow.f64 x 5))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (+.f64 (/.f64 1/2 (*.f64 (Rewrite=> sqr-pow_binary64 (*.f64 (pow.f64 x (/.f64 1 2)) (pow.f64 x (/.f64 1 2)))) (*.f64 x x))) (/.f64 1 x)) (+.f64 (/.f64 15/8 (pow.f64 x 7)) (/.f64 3/4 (pow.f64 x 5))))): 4 points increase in error, 2 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (+.f64 (/.f64 1/2 (*.f64 (Rewrite<= fabs-sqr_binary64 (fabs.f64 (*.f64 (pow.f64 x (/.f64 1 2)) (pow.f64 x (/.f64 1 2))))) (*.f64 x x))) (/.f64 1 x)) (+.f64 (/.f64 15/8 (pow.f64 x 7)) (/.f64 3/4 (pow.f64 x 5))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (+.f64 (/.f64 1/2 (*.f64 (fabs.f64 (Rewrite<= sqr-pow_binary64 (pow.f64 x 1))) (*.f64 x x))) (/.f64 1 x)) (+.f64 (/.f64 15/8 (pow.f64 x 7)) (/.f64 3/4 (pow.f64 x 5))))): 2 points increase in error, 4 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (+.f64 (/.f64 1/2 (*.f64 (fabs.f64 (Rewrite=> unpow1_binary64 x)) (*.f64 x x))) (/.f64 1 x)) (+.f64 (/.f64 15/8 (pow.f64 x 7)) (/.f64 3/4 (pow.f64 x 5))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (+.f64 (/.f64 1/2 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 (fabs.f64 x) x) x))) (/.f64 1 x)) (+.f64 (/.f64 15/8 (pow.f64 x 7)) (/.f64 3/4 (pow.f64 x 5))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (+.f64 (/.f64 1/2 (Rewrite<= associate-*r*_binary64 (*.f64 (fabs.f64 x) (*.f64 x x)))) (/.f64 1 x)) (+.f64 (/.f64 15/8 (pow.f64 x 7)) (/.f64 3/4 (pow.f64 x 5))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (+.f64 (/.f64 1/2 (*.f64 (fabs.f64 x) (Rewrite<= unpow2_binary64 (pow.f64 x 2)))) (/.f64 1 x)) (+.f64 (/.f64 15/8 (pow.f64 x 7)) (/.f64 3/4 (pow.f64 x 5))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (+.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 1/2 1)) (*.f64 (fabs.f64 x) (pow.f64 x 2))) (/.f64 1 x)) (+.f64 (/.f64 15/8 (pow.f64 x 7)) (/.f64 3/4 (pow.f64 x 5))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 1 (*.f64 (fabs.f64 x) (pow.f64 x 2))))) (/.f64 1 x)) (+.f64 (/.f64 15/8 (pow.f64 x 7)) (/.f64 3/4 (pow.f64 x 5))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (+.f64 (*.f64 1/2 (/.f64 1 (*.f64 (fabs.f64 x) (pow.f64 x 2)))) (/.f64 1 (Rewrite<= unpow1_binary64 (pow.f64 x 1)))) (+.f64 (/.f64 15/8 (pow.f64 x 7)) (/.f64 3/4 (pow.f64 x 5))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (+.f64 (*.f64 1/2 (/.f64 1 (*.f64 (fabs.f64 x) (pow.f64 x 2)))) (/.f64 1 (Rewrite=> sqr-pow_binary64 (*.f64 (pow.f64 x (/.f64 1 2)) (pow.f64 x (/.f64 1 2)))))) (+.f64 (/.f64 15/8 (pow.f64 x 7)) (/.f64 3/4 (pow.f64 x 5))))): 42 points increase in error, 30 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (+.f64 (*.f64 1/2 (/.f64 1 (*.f64 (fabs.f64 x) (pow.f64 x 2)))) (/.f64 1 (Rewrite<= fabs-sqr_binary64 (fabs.f64 (*.f64 (pow.f64 x (/.f64 1 2)) (pow.f64 x (/.f64 1 2))))))) (+.f64 (/.f64 15/8 (pow.f64 x 7)) (/.f64 3/4 (pow.f64 x 5))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (+.f64 (*.f64 1/2 (/.f64 1 (*.f64 (fabs.f64 x) (pow.f64 x 2)))) (/.f64 1 (fabs.f64 (Rewrite<= sqr-pow_binary64 (pow.f64 x 1))))) (+.f64 (/.f64 15/8 (pow.f64 x 7)) (/.f64 3/4 (pow.f64 x 5))))): 30 points increase in error, 42 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (+.f64 (*.f64 1/2 (/.f64 1 (*.f64 (fabs.f64 x) (pow.f64 x 2)))) (/.f64 1 (fabs.f64 (Rewrite=> unpow1_binary64 x)))) (+.f64 (/.f64 15/8 (pow.f64 x 7)) (/.f64 3/4 (pow.f64 x 5))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (+.f64 (*.f64 1/2 (/.f64 1 (*.f64 (fabs.f64 x) (pow.f64 x 2)))) (/.f64 1 (fabs.f64 x))) (+.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 15/8 1)) (pow.f64 x 7)) (/.f64 3/4 (pow.f64 x 5))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (+.f64 (*.f64 1/2 (/.f64 1 (*.f64 (fabs.f64 x) (pow.f64 x 2)))) (/.f64 1 (fabs.f64 x))) (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 15/8 (/.f64 1 (pow.f64 x 7)))) (/.f64 3/4 (pow.f64 x 5))))): 7 points increase in error, 1 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (+.f64 (*.f64 1/2 (/.f64 1 (*.f64 (fabs.f64 x) (pow.f64 x 2)))) (/.f64 1 (fabs.f64 x))) (+.f64 (*.f64 15/8 (/.f64 1 (pow.f64 x 7))) (/.f64 (Rewrite<= metadata-eval (*.f64 3/4 1)) (pow.f64 x 5))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (+.f64 (*.f64 1/2 (/.f64 1 (*.f64 (fabs.f64 x) (pow.f64 x 2)))) (/.f64 1 (fabs.f64 x))) (+.f64 (*.f64 15/8 (/.f64 1 (pow.f64 x 7))) (Rewrite<= associate-*r/_binary64 (*.f64 3/4 (/.f64 1 (pow.f64 x 5))))))): 2 points increase in error, 2 points decrease in error
   (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 1/2 (/.f64 1 (*.f64 (fabs.f64 x) (pow.f64 x 2)))) (/.f64 1 (fabs.f64 x)))) (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 15/8 (/.f64 1 (pow.f64 x 7))) (*.f64 3/4 (/.f64 1 (pow.f64 x 5))))))): 48 points increase in error, 46 points decrease in error
   (+.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (*.f64 1/2 (/.f64 1 (*.f64 (fabs.f64 x) (pow.f64 x 2)))) (sqrt.f64 (/.f64 1 (PI.f64)))) (*.f64 (/.f64 1 (fabs.f64 x)) (sqrt.f64 (/.f64 1 (PI.f64)))))) (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 15/8 (/.f64 1 (pow.f64 x 7))) (*.f64 3/4 (/.f64 1 (pow.f64 x 5)))))): 22 points increase in error, 40 points decrease in error
   (+.f64 (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 1/2 (*.f64 (/.f64 1 (*.f64 (fabs.f64 x) (pow.f64 x 2))) (sqrt.f64 (/.f64 1 (PI.f64)))))) (*.f64 (/.f64 1 (fabs.f64 x)) (sqrt.f64 (/.f64 1 (PI.f64))))) (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 15/8 (/.f64 1 (pow.f64 x 7))) (*.f64 3/4 (/.f64 1 (pow.f64 x 5)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (sqrt.f64 (/.f64 1 (PI.f64)))) (*.f64 1/2 (*.f64 (/.f64 1 (*.f64 (fabs.f64 x) (pow.f64 x 2))) (sqrt.f64 (/.f64 1 (PI.f64))))))) (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 15/8 (/.f64 1 (pow.f64 x 7))) (*.f64 3/4 (/.f64 1 (pow.f64 x 5)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (sqrt.f64 (/.f64 1 (PI.f64)))) (*.f64 1/2 (*.f64 (/.f64 1 (*.f64 (fabs.f64 x) (pow.f64 x 2))) (sqrt.f64 (/.f64 1 (PI.f64)))))) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (*.f64 15/8 (/.f64 1 (pow.f64 x 7))) (sqrt.f64 (/.f64 1 (PI.f64)))) (*.f64 (*.f64 3/4 (/.f64 1 (pow.f64 x 5))) (sqrt.f64 (/.f64 1 (PI.f64))))))): 4 points increase in error, 10 points decrease in error
   (+.f64 (+.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (sqrt.f64 (/.f64 1 (PI.f64)))) (*.f64 1/2 (*.f64 (/.f64 1 (*.f64 (fabs.f64 x) (pow.f64 x 2))) (sqrt.f64 (/.f64 1 (PI.f64)))))) (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 15/8 (*.f64 (/.f64 1 (pow.f64 x 7)) (sqrt.f64 (/.f64 1 (PI.f64)))))) (*.f64 (*.f64 3/4 (/.f64 1 (pow.f64 x 5))) (sqrt.f64 (/.f64 1 (PI.f64)))))): 6 points increase in error, 3 points decrease in error
   (+.f64 (+.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (sqrt.f64 (/.f64 1 (PI.f64)))) (*.f64 1/2 (*.f64 (/.f64 1 (*.f64 (fabs.f64 x) (pow.f64 x 2))) (sqrt.f64 (/.f64 1 (PI.f64)))))) (+.f64 (*.f64 15/8 (*.f64 (/.f64 1 (pow.f64 x 7)) (sqrt.f64 (/.f64 1 (PI.f64))))) (Rewrite<= associate-*r*_binary64 (*.f64 3/4 (*.f64 (/.f64 1 (pow.f64 x 5)) (sqrt.f64 (/.f64 1 (PI.f64)))))))): 4 points increase in error, 3 points decrease in error
   (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (sqrt.f64 (/.f64 1 (PI.f64)))) (*.f64 1/2 (*.f64 (/.f64 1 (*.f64 (fabs.f64 x) (pow.f64 x 2))) (sqrt.f64 (/.f64 1 (PI.f64)))))) (*.f64 15/8 (*.f64 (/.f64 1 (pow.f64 x 7)) (sqrt.f64 (/.f64 1 (PI.f64)))))) (*.f64 3/4 (*.f64 (/.f64 1 (pow.f64 x 5)) (sqrt.f64 (/.f64 1 (PI.f64))))))): 38 points increase in error, 28 points decrease in error
   (+.f64 (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (sqrt.f64 (/.f64 1 (PI.f64)))) (+.f64 (*.f64 1/2 (*.f64 (/.f64 1 (*.f64 (fabs.f64 x) (pow.f64 x 2))) (sqrt.f64 (/.f64 1 (PI.f64))))) (*.f64 15/8 (*.f64 (/.f64 1 (pow.f64 x 7)) (sqrt.f64 (/.f64 1 (PI.f64)))))))) (*.f64 3/4 (*.f64 (/.f64 1 (pow.f64 x 5)) (sqrt.f64 (/.f64 1 (PI.f64)))))): 38 points increase in error, 23 points decrease in error
   (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 3/4 (*.f64 (/.f64 1 (pow.f64 x 5)) (sqrt.f64 (/.f64 1 (PI.f64))))) (+.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (sqrt.f64 (/.f64 1 (PI.f64)))) (+.f64 (*.f64 1/2 (*.f64 (/.f64 1 (*.f64 (fabs.f64 x) (pow.f64 x 2))) (sqrt.f64 (/.f64 1 (PI.f64))))) (*.f64 15/8 (*.f64 (/.f64 1 (pow.f64 x 7)) (sqrt.f64 (/.f64 1 (PI.f64))))))))): 0 points increase in error, 0 points decrease in error

7. Taylor expanded in x around 0 1.2

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

8. Simplified 1.2

   \[\leadsto {\left(e^{x}\right)}^{x} \cdot \color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{1.875}{{x}^{7}} + \frac{0.5}{{x}^{3}}\right) + \left(\frac{1}{x} + \frac{0.75}{{x}^{5}}\right)\right)\right)} \]
   Proof
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (+.f64 (/.f64 15/8 (pow.f64 x 7)) (/.f64 1/2 (pow.f64 x 3))) (+.f64 (/.f64 1 x) (/.f64 3/4 (pow.f64 x 5))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (+.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 15/8 1)) (pow.f64 x 7)) (/.f64 1/2 (pow.f64 x 3))) (+.f64 (/.f64 1 x) (/.f64 3/4 (pow.f64 x 5))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 15/8 (/.f64 1 (pow.f64 x 7)))) (/.f64 1/2 (pow.f64 x 3))) (+.f64 (/.f64 1 x) (/.f64 3/4 (pow.f64 x 5))))): 8 points increase in error, 3 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (+.f64 (*.f64 15/8 (/.f64 1 (pow.f64 x 7))) (/.f64 (Rewrite<= metadata-eval (*.f64 1/2 1)) (pow.f64 x 3))) (+.f64 (/.f64 1 x) (/.f64 3/4 (pow.f64 x 5))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (+.f64 (*.f64 15/8 (/.f64 1 (pow.f64 x 7))) (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 1 (pow.f64 x 3))))) (+.f64 (/.f64 1 x) (/.f64 3/4 (pow.f64 x 5))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (+.f64 (*.f64 15/8 (/.f64 1 (pow.f64 x 7))) (*.f64 1/2 (/.f64 1 (pow.f64 x 3)))) (+.f64 (/.f64 1 x) (/.f64 (Rewrite<= metadata-eval (*.f64 3/4 1)) (pow.f64 x 5))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (+.f64 (*.f64 15/8 (/.f64 1 (pow.f64 x 7))) (*.f64 1/2 (/.f64 1 (pow.f64 x 3)))) (+.f64 (/.f64 1 x) (Rewrite<= associate-*r/_binary64 (*.f64 3/4 (/.f64 1 (pow.f64 x 5))))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 15/8 (/.f64 1 (pow.f64 x 7))) (*.f64 1/2 (/.f64 1 (pow.f64 x 3))))) (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (/.f64 1 x) (*.f64 3/4 (/.f64 1 (pow.f64 x 5))))))): 54 points increase in error, 35 points decrease in error
   (+.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (*.f64 15/8 (/.f64 1 (pow.f64 x 7))) (sqrt.f64 (/.f64 1 (PI.f64)))) (*.f64 (*.f64 1/2 (/.f64 1 (pow.f64 x 3))) (sqrt.f64 (/.f64 1 (PI.f64)))))) (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (/.f64 1 x) (*.f64 3/4 (/.f64 1 (pow.f64 x 5)))))): 6 points increase in error, 5 points decrease in error
   (+.f64 (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 15/8 (*.f64 (/.f64 1 (pow.f64 x 7)) (sqrt.f64 (/.f64 1 (PI.f64)))))) (*.f64 (*.f64 1/2 (/.f64 1 (pow.f64 x 3))) (sqrt.f64 (/.f64 1 (PI.f64))))) (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (/.f64 1 x) (*.f64 3/4 (/.f64 1 (pow.f64 x 5)))))): 2 points increase in error, 5 points decrease in error
   (+.f64 (+.f64 (*.f64 15/8 (*.f64 (/.f64 1 (pow.f64 x 7)) (sqrt.f64 (/.f64 1 (PI.f64))))) (Rewrite<= associate-*r*_binary64 (*.f64 1/2 (*.f64 (/.f64 1 (pow.f64 x 3)) (sqrt.f64 (/.f64 1 (PI.f64))))))) (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (/.f64 1 x) (*.f64 3/4 (/.f64 1 (pow.f64 x 5)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 1/2 (*.f64 (/.f64 1 (pow.f64 x 3)) (sqrt.f64 (/.f64 1 (PI.f64))))) (*.f64 15/8 (*.f64 (/.f64 1 (pow.f64 x 7)) (sqrt.f64 (/.f64 1 (PI.f64))))))) (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (/.f64 1 x) (*.f64 3/4 (/.f64 1 (pow.f64 x 5)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (*.f64 1/2 (*.f64 (/.f64 1 (pow.f64 x 3)) (sqrt.f64 (/.f64 1 (PI.f64))))) (*.f64 15/8 (*.f64 (/.f64 1 (pow.f64 x 7)) (sqrt.f64 (/.f64 1 (PI.f64)))))) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (/.f64 1 x) (sqrt.f64 (/.f64 1 (PI.f64)))) (*.f64 (*.f64 3/4 (/.f64 1 (pow.f64 x 5))) (sqrt.f64 (/.f64 1 (PI.f64))))))): 34 points increase in error, 31 points decrease in error
   (+.f64 (+.f64 (*.f64 1/2 (*.f64 (/.f64 1 (pow.f64 x 3)) (sqrt.f64 (/.f64 1 (PI.f64))))) (*.f64 15/8 (*.f64 (/.f64 1 (pow.f64 x 7)) (sqrt.f64 (/.f64 1 (PI.f64)))))) (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (/.f64 1 x))) (*.f64 (*.f64 3/4 (/.f64 1 (pow.f64 x 5))) (sqrt.f64 (/.f64 1 (PI.f64)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (*.f64 1/2 (*.f64 (/.f64 1 (pow.f64 x 3)) (sqrt.f64 (/.f64 1 (PI.f64))))) (*.f64 15/8 (*.f64 (/.f64 1 (pow.f64 x 7)) (sqrt.f64 (/.f64 1 (PI.f64)))))) (+.f64 (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (/.f64 1 x)) (Rewrite<= associate-*r*_binary64 (*.f64 3/4 (*.f64 (/.f64 1 (pow.f64 x 5)) (sqrt.f64 (/.f64 1 (PI.f64)))))))): 1 points increase in error, 1 points decrease in error
   (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (/.f64 1 x)) (*.f64 3/4 (*.f64 (/.f64 1 (pow.f64 x 5)) (sqrt.f64 (/.f64 1 (PI.f64)))))) (+.f64 (*.f64 1/2 (*.f64 (/.f64 1 (pow.f64 x 3)) (sqrt.f64 (/.f64 1 (PI.f64))))) (*.f64 15/8 (*.f64 (/.f64 1 (pow.f64 x 7)) (sqrt.f64 (/.f64 1 (PI.f64)))))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (/.f64 1 x)) (+.f64 (*.f64 3/4 (*.f64 (/.f64 1 (pow.f64 x 5)) (sqrt.f64 (/.f64 1 (PI.f64))))) (+.f64 (*.f64 1/2 (*.f64 (/.f64 1 (pow.f64 x 3)) (sqrt.f64 (/.f64 1 (PI.f64))))) (*.f64 15/8 (*.f64 (/.f64 1 (pow.f64 x 7)) (sqrt.f64 (/.f64 1 (PI.f64))))))))): 25 points increase in error, 37 points decrease in error

9. Applied egg-rr 1.5

   \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\left(\color{blue}{\left(e^{\mathsf{log1p}\left(1.875 \cdot {x}^{-7}\right)} - 1\right)} + \frac{0.5}{{x}^{3}}\right) + \left(\frac{1}{x} + \frac{0.75}{{x}^{5}}\right)\right)\right) \]

10. Simplified 1.2

    \[\leadsto {\left(e^{x}\right)}^{x} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(\left(\color{blue}{1.875 \cdot {x}^{-7}} + \frac{0.5}{{x}^{3}}\right) + \left(\frac{1}{x} + \frac{0.75}{{x}^{5}}\right)\right)\right) \]
    Proof
    (*.f64 15/8 (pow.f64 x -7)): 0 points increase in error, 0 points decrease in error
    (Rewrite<= expm1-log1p_binary64 (expm1.f64 (log1p.f64 (*.f64 15/8 (pow.f64 x -7))))): 20 points increase in error, 13 points decrease in error
    (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 (log1p.f64 (*.f64 15/8 (pow.f64 x -7)))) 1)): 103 points increase in error, 109 points decrease in error

11. Final simplification 1.2

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

Alternatives

Alternative 1
Error	1.3
Cost	40192

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

Alternative 2
Error	1.3
Cost	39936

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

Alternative 3
Error	2.7
Cost	33728

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

Alternative 4
Error	2.7
Cost	33600

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

Alternative 5
Error	2.7
Cost	33600

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

Alternative 6
Error	43.6
Cost	33476

\[\begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \mathbf{if}\;x \leq 1.2:\\ \;\;\;\;t_0 \cdot \left(\left(\frac{1.875}{{x}^{7}} + \frac{2.625}{{x}^{5}}\right) + \left(\frac{2.1875}{x} + \frac{2.1875}{{x}^{3}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{x}\right)}^{x} \cdot \left(t_0 \cdot \left(\frac{0.5}{{x}^{3}} + \frac{1}{x}\right)\right)\\ \end{array} \]

Alternative 7
Error	48.0
Cost	32896

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

Alternative 8
Error	44.7
Cost	32896

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

Alternative 9
Error	48.3
Cost	26048

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

Alternative 10
Error	48.3
Cost	26048

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

Alternative 11
Error	48.3
Cost	19712

\[\sqrt{\frac{1}{\pi}} \cdot \frac{e^{x \cdot x}}{x} \]

Alternative 12
Error	56.5
Cost	13568

\[\sqrt{\frac{1}{\pi}} \cdot \left(\frac{x}{\frac{x}{x}} + \frac{1.5}{x}\right) \]

Alternative 13
Error	56.5
Cost	13440

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

Alternative 14
Error	56.7
Cost	13312

\[\sqrt{\frac{1}{\pi}} \cdot \frac{x}{\frac{x}{x}} \]

Reproduce

herbie shell --seed 2022332 
(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)))))))