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

Average Error: 2.8 → 1.3

Time: 16.8s

Precision: binary64

Cost: 58880

\[x \geq 0.5\]

Math

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

↓

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

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
 (*
  (sqrt (/ 1.0 PI))
  (/
   (pow (exp x) x)
   (/
    x
    (+
     (fma 0.75 (pow x -4.0) (* 0.5 (pow x -2.0)))
     (fma 1.875 (pow x -6.0) 1.0))))))

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 sqrt((1.0 / ((double) M_PI))) * (pow(exp(x), x) / (x / (fma(0.75, pow(x, -4.0), (0.5 * pow(x, -2.0))) + fma(1.875, pow(x, -6.0), 1.0))));
}

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(sqrt(Float64(1.0 / pi)) * Float64((exp(x) ^ x) / Float64(x / Float64(fma(0.75, (x ^ -4.0), Float64(0.5 * (x ^ -2.0))) + fma(1.875, (x ^ -6.0), 1.0)))))
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[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] / N[(x / N[(N[(0.75 * N[Power[x, -4.0], $MachinePrecision] + N[(0.5 * N[Power[x, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.875 * N[Power[x, -6.0], $MachinePrecision] + 1.0), $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 1.3

   \[\leadsto \color{blue}{\frac{\frac{{\left(e^{x}\right)}^{x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right)} \]
   Proof
   (*.f64 (/.f64 (/.f64 (pow.f64 (exp.f64 x) x) (fabs.f64 x)) (sqrt.f64 (PI.f64))) (+.f64 1 (+.f64 (/.f64 15/8 (pow.f64 x 6)) (/.f64 (+.f64 1/2 (/.f64 3/4 (*.f64 x x))) (*.f64 x x))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (/.f64 (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 x x))) (fabs.f64 x)) (sqrt.f64 (PI.f64))) (+.f64 1 (+.f64 (/.f64 15/8 (pow.f64 x 6)) (/.f64 (+.f64 1/2 (/.f64 3/4 (*.f64 x x))) (*.f64 x x))))): 148 points increase in error, 30 points decrease in error
   (*.f64 (/.f64 (/.f64 (exp.f64 (Rewrite<= sqr-abs_binary64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (fabs.f64 x)) (sqrt.f64 (PI.f64))) (+.f64 1 (+.f64 (/.f64 15/8 (pow.f64 x 6)) (/.f64 (+.f64 1/2 (/.f64 3/4 (*.f64 x x))) (*.f64 x x))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (fabs.f64 x)) (sqrt.f64 (PI.f64))) (+.f64 1 (+.f64 (/.f64 15/8 (pow.f64 x 6)) (/.f64 (+.f64 1/2 (/.f64 3/4 (*.f64 x x))) (*.f64 x x))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (fabs.f64 x)) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (sqrt.f64 (PI.f64))) (+.f64 1 (+.f64 (/.f64 15/8 (pow.f64 x 6)) (/.f64 (+.f64 1/2 (/.f64 3/4 (*.f64 x x))) (*.f64 x x))))): 33 points increase in error, 17 points decrease in error
   (*.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (sqrt.f64 (PI.f64))))) (+.f64 1 (+.f64 (/.f64 15/8 (pow.f64 x 6)) (/.f64 (+.f64 1/2 (/.f64 3/4 (*.f64 x x))) (*.f64 x x))))): 33 points increase in error, 42 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (sqrt.f64 (PI.f64)))) (+.f64 1 (+.f64 (/.f64 15/8 (pow.f64 x 6)) (/.f64 (+.f64 1/2 (/.f64 3/4 (*.f64 x x))) (*.f64 x x))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))))) (+.f64 1 (+.f64 (/.f64 15/8 (pow.f64 x 6)) (/.f64 (+.f64 1/2 (/.f64 3/4 (*.f64 x x))) (*.f64 x x))))): 54 points increase in error, 79 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 15/8 1)) (pow.f64 x 6)) (/.f64 (+.f64 1/2 (/.f64 3/4 (*.f64 x x))) (*.f64 x x))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (/.f64 (*.f64 (Rewrite<= metadata-eval (/.f64 15 8)) 1) (pow.f64 x 6)) (/.f64 (+.f64 1/2 (/.f64 3/4 (*.f64 x x))) (*.f64 x x))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (/.f64 (*.f64 (/.f64 15 8) 1) (pow.f64 x (Rewrite<= metadata-eval (*.f64 2 3)))) (/.f64 (+.f64 1/2 (/.f64 3/4 (*.f64 x x))) (*.f64 x x))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (/.f64 (*.f64 (/.f64 15 8) 1) (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 x 3) (pow.f64 x 3)))) (/.f64 (+.f64 1/2 (/.f64 3/4 (*.f64 x x))) (*.f64 x x))))): 9 points increase in error, 9 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (/.f64 (*.f64 (/.f64 15 8) 1) (Rewrite<= cube-prod_binary64 (pow.f64 (*.f64 x x) 3))) (/.f64 (+.f64 1/2 (/.f64 3/4 (*.f64 x x))) (*.f64 x x))))): 16 points increase in error, 7 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (/.f64 (*.f64 (/.f64 15 8) 1) (pow.f64 (Rewrite<= sqr-abs_binary64 (*.f64 (fabs.f64 x) (fabs.f64 x))) 3)) (/.f64 (+.f64 1/2 (/.f64 3/4 (*.f64 x x))) (*.f64 x x))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (/.f64 (*.f64 (/.f64 15 8) 1) (pow.f64 (*.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 (fabs.f64 x) 1)) (fabs.f64 x)) 3)) (/.f64 (+.f64 1/2 (/.f64 3/4 (*.f64 x x))) (*.f64 x x))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (/.f64 (*.f64 (/.f64 15 8) 1) (pow.f64 (Rewrite<= associate-/r/_binary64 (/.f64 (fabs.f64 x) (/.f64 1 (fabs.f64 x)))) 3)) (/.f64 (+.f64 1/2 (/.f64 3/4 (*.f64 x x))) (*.f64 x x))))): 14 points increase in error, 1 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (/.f64 (*.f64 (/.f64 15 8) 1) (Rewrite=> cube-div_binary64 (/.f64 (pow.f64 (fabs.f64 x) 3) (pow.f64 (/.f64 1 (fabs.f64 x)) 3)))) (/.f64 (+.f64 1/2 (/.f64 3/4 (*.f64 x x))) (*.f64 x x))))): 9 points increase in error, 15 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (/.f64 (*.f64 (/.f64 15 8) 1) (/.f64 (Rewrite=> unpow3_binary64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x))) (pow.f64 (/.f64 1 (fabs.f64 x)) 3))) (/.f64 (+.f64 1/2 (/.f64 3/4 (*.f64 x x))) (*.f64 x x))))): 6 points increase in error, 4 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (/.f64 (*.f64 (/.f64 15 8) 1) (/.f64 (*.f64 (*.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 (fabs.f64 x) 1)) (fabs.f64 x)) (fabs.f64 x)) (pow.f64 (/.f64 1 (fabs.f64 x)) 3))) (/.f64 (+.f64 1/2 (/.f64 3/4 (*.f64 x x))) (*.f64 x x))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (/.f64 (*.f64 (/.f64 15 8) 1) (/.f64 (*.f64 (Rewrite<= associate-/r/_binary64 (/.f64 (fabs.f64 x) (/.f64 1 (fabs.f64 x)))) (fabs.f64 x)) (pow.f64 (/.f64 1 (fabs.f64 x)) 3))) (/.f64 (+.f64 1/2 (/.f64 3/4 (*.f64 x x))) (*.f64 x x))))): 6 points increase in error, 1 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (/.f64 (*.f64 (/.f64 15 8) 1) (/.f64 (Rewrite<= associate-/r/_binary64 (/.f64 (fabs.f64 x) (/.f64 (/.f64 1 (fabs.f64 x)) (fabs.f64 x)))) (pow.f64 (/.f64 1 (fabs.f64 x)) 3))) (/.f64 (+.f64 1/2 (/.f64 3/4 (*.f64 x x))) (*.f64 x x))))): 10 points increase in error, 7 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (/.f64 (*.f64 (/.f64 15 8) 1) (/.f64 (/.f64 (fabs.f64 x) (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 1 (fabs.f64 x)))) (fabs.f64 x))) (pow.f64 (/.f64 1 (fabs.f64 x)) 3))) (/.f64 (+.f64 1/2 (/.f64 3/4 (*.f64 x x))) (*.f64 x x))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (/.f64 (*.f64 (/.f64 15 8) 1) (/.f64 (/.f64 (fabs.f64 x) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))))) (pow.f64 (/.f64 1 (fabs.f64 x)) 3))) (/.f64 (+.f64 1/2 (/.f64 3/4 (*.f64 x x))) (*.f64 x x))))): 10 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (/.f64 (*.f64 (/.f64 15 8) 1) (/.f64 (/.f64 (fabs.f64 x) (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x)))) (Rewrite=> unpow3_binary64 (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x)))))) (/.f64 (+.f64 1/2 (/.f64 3/4 (*.f64 x x))) (*.f64 x x))))): 5 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (/.f64 (*.f64 (/.f64 15 8) 1) (Rewrite<= associate-/r*_binary64 (/.f64 (fabs.f64 x) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))))))) (/.f64 (+.f64 1/2 (/.f64 3/4 (*.f64 x x))) (*.f64 x x))))): 3 points increase in error, 6 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (/.f64 (*.f64 (/.f64 15 8) 1) (/.f64 (fabs.f64 x) (Rewrite<= *-commutative_binary64 (*.f64 (*.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 (+.f64 1/2 (/.f64 3/4 (*.f64 x x))) (*.f64 x x))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (/.f64 (*.f64 (/.f64 15 8) 1) (/.f64 (fabs.f64 x) (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 (+.f64 1/2 (/.f64 3/4 (*.f64 x x))) (*.f64 x x))))): 3 points increase in error, 7 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (/.f64 15 8) (/.f64 1 (/.f64 (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))))))) (/.f64 (+.f64 1/2 (/.f64 3/4 (*.f64 x x))) (*.f64 x x))))): 4 points increase in error, 3 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (*.f64 (/.f64 15 8) (Rewrite=> associate-/r/_binary64 (*.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)))))) (/.f64 (+.f64 1/2 (/.f64 3/4 (*.f64 x x))) (*.f64 x x))))): 7 points increase in error, 3 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (*.f64 (/.f64 15 8) (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))))) (/.f64 (+.f64 1/2 (/.f64 3/4 (*.f64 x x))) (*.f64 x x))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (*.f64 (/.f64 15 8) (*.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 (+.f64 (Rewrite<= metadata-eval (/.f64 1 2)) (/.f64 3/4 (*.f64 x x))) (*.f64 x x))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (*.f64 (/.f64 15 8) (*.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 (+.f64 (/.f64 1 2) (/.f64 (Rewrite<= metadata-eval (*.f64 1 3/4)) (*.f64 x x))) (*.f64 x x))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (*.f64 (/.f64 15 8) (*.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 (+.f64 (/.f64 1 2) (/.f64 (*.f64 1 (Rewrite<= metadata-eval (/.f64 3 4))) (*.f64 x x))) (*.f64 x x))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (*.f64 (/.f64 15 8) (*.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 (+.f64 (/.f64 1 2) (/.f64 (*.f64 1 (/.f64 3 4)) (Rewrite<= sqr-abs_binary64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (*.f64 x x))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (*.f64 (/.f64 15 8) (*.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 (+.f64 (/.f64 1 2) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (*.f64 (fabs.f64 x) (fabs.f64 x))) (/.f64 3 4)))) (*.f64 x x))))): 3 points increase in error, 3 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (*.f64 (/.f64 15 8) (*.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 (+.f64 (/.f64 1 2) (*.f64 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 1 (fabs.f64 x)) (fabs.f64 x))) (/.f64 3 4))) (*.f64 x x))))): 5 points increase in error, 1 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (*.f64 (/.f64 15 8) (*.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 (+.f64 (/.f64 1 2) (*.f64 (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 1 (fabs.f64 x)))) (fabs.f64 x)) (/.f64 3 4))) (*.f64 x x))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (*.f64 (/.f64 15 8) (*.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 (+.f64 (/.f64 1 2) (*.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x)))) (/.f64 3 4))) (*.f64 x x))))): 6 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (*.f64 (/.f64 15 8) (*.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 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (+.f64 (/.f64 1 2) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 3 4))))) (*.f64 x x))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (*.f64 (/.f64 15 8) (*.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 (*.f64 1 (+.f64 (/.f64 1 2) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 3 4)))) (Rewrite<= sqr-abs_binary64 (*.f64 (fabs.f64 x) (fabs.f64 x))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (*.f64 (/.f64 15 8) (*.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)))) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (*.f64 (fabs.f64 x) (fabs.f64 x))) (+.f64 (/.f64 1 2) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 3 4)))))))): 1 points increase in error, 4 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (*.f64 (/.f64 15 8) (*.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 (Rewrite<= associate-/l/_binary64 (/.f64 (/.f64 1 (fabs.f64 x)) (fabs.f64 x))) (+.f64 (/.f64 1 2) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 3 4))))))): 11 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (*.f64 (/.f64 15 8) (*.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 (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 1 (fabs.f64 x)))) (fabs.f64 x)) (+.f64 (/.f64 1 2) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 3 4))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (*.f64 (/.f64 15 8) (*.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 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x)))) (+.f64 (/.f64 1 2) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 3 4))))))): 3 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (*.f64 (/.f64 15 8) (*.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)))) (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 2)) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 3 4)))))))): 2 points increase in error, 4 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (*.f64 (/.f64 15 8) (*.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 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 1 2) (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))))) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 3 4))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (*.f64 (/.f64 15 8) (*.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 (*.f64 (/.f64 1 2) (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x)))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.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))))))): 2 points increase in error, 1 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (*.f64 (/.f64 15 8) (*.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 (*.f64 (/.f64 1 2) (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x)))) (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x)))) (/.f64 3 4)))))): 2 points increase in error, 1 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (+.f64 (*.f64 (/.f64 15 8) (*.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 (*.f64 (/.f64 1 2) (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x)))) (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 3 4) (*.f64 (*.f64 (*.f64 (/.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
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 (*.f64 (/.f64 1 2) (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x)))) (*.f64 (/.f64 3 4) (*.f64 (*.f64 (*.f64 (/.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 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
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 1 (Rewrite=> associate-+l+_binary64 (+.f64 (*.f64 (/.f64 1 2) (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x)))) (+.f64 (*.f64 (/.f64 3 4) (*.f64 (*.f64 (*.f64 (/.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 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))))))))): 3 points increase in error, 9 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 1 (*.f64 (/.f64 1 2) (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))))) (+.f64 (*.f64 (/.f64 3 4) (*.f64 (*.f64 (*.f64 (/.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 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)))))))): 30 points increase in error, 21 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (/.f64 1 2) (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x)))) 1)) (+.f64 (*.f64 (/.f64 3 4) (*.f64 (*.f64 (*.f64 (/.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 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<= distribute-rgt-out_binary64 (+.f64 (*.f64 (+.f64 (*.f64 (/.f64 1 2) (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x)))) 1) (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))))) (*.f64 (+.f64 (*.f64 (/.f64 3 4) (*.f64 (*.f64 (*.f64 (/.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 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 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))))))): 52 points increase in error, 52 points decrease in error
   (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (+.f64 (*.f64 (/.f64 1 2) (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x)))) 1) (/.f64 1 (fabs.f64 x))) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))))) (*.f64 (+.f64 (*.f64 (/.f64 3 4) (*.f64 (*.f64 (*.f64 (/.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 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 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))))): 25 points increase in error, 39 points decrease in error
   (+.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 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (*.f64 (+.f64 (*.f64 (/.f64 3 4) (*.f64 (*.f64 (*.f64 (/.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 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 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))))): 22 points increase in error, 24 points decrease in error
   (+.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 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (*.f64 (+.f64 (*.f64 (/.f64 3 4) (*.f64 (*.f64 (*.f64 (/.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 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 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))))): 0 points increase in error, 0 points decrease in error
   (+.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 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 (*.f64 (/.f64 3 4) (*.f64 (*.f64 (*.f64 (/.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 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
   (+.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 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (*.f64 (/.f64 3 4) (*.f64 (*.f64 (*.f64 (/.f64 1 (fabs.f64 x)) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x))) (/.f64 1 (fabs.f64 x)))) (*.f64 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))))) (*.f64 (*.f64 (/.f64 15 8) (*.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 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))))))): 8 points increase in error, 11 points decrease in error
   (+.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 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 (/.f64 3 4) (*.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 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))))) (*.f64 (*.f64 (/.f64 15 8) (*.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 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))))))): 5 points increase in error, 2 points decrease in error
   (+.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 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 (*.f64 (Rewrite<= associate-*r*_binary64 (*.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 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (*.f64 (*.f64 (/.f64 15 8) (*.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 (/.f64 1 (fabs.f64 x)) (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))))))): 1 points increase in error, 6 points decrease in error
   (+.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 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 (*.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 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 (/.f64 15 8) (*.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 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))))))): 8 points increase in error, 6 points decrease in error
   (+.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 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (+.f64 (*.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 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (*.f64 (Rewrite<= associate-*r*_binary64 (*.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 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))))): 6 points increase in error, 7 points decrease in error
   (+.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 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (Rewrite<= distribute-rgt-in_binary64 (*.f64 (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (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) (*.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)))))))): 6 points increase in error, 9 points decrease in error
   (+.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 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))) (Rewrite=> *-commutative_binary64 (*.f64 (+.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) (*.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 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x))))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-rgt-in_binary64 (*.f64 (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (+.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) (*.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, 36 points decrease in error
   (*.f64 (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (exp.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (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))))))): 25 points increase in error, 35 points decrease in error

3. Taylor expanded in x around inf 2.7

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

4. Simplified 1.3

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\frac{{\left(e^{x}\right)}^{x}}{x} \cdot \left(\frac{1.875}{{x}^{6}} + \left(\frac{0.75}{{x}^{4}} + \left(1 + \frac{0.5}{x \cdot x}\right)\right)\right)\right)} \]
   Proof
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (*.f64 (/.f64 (pow.f64 (exp.f64 x) x) x) (+.f64 (/.f64 15/8 (pow.f64 x 6)) (+.f64 (/.f64 3/4 (pow.f64 x 4)) (+.f64 1 (/.f64 1/2 (*.f64 x x))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (*.f64 (/.f64 (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 x x))) x) (+.f64 (/.f64 15/8 (pow.f64 x 6)) (+.f64 (/.f64 3/4 (pow.f64 x 4)) (+.f64 1 (/.f64 1/2 (*.f64 x x))))))): 150 points increase in error, 24 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (*.f64 (/.f64 (exp.f64 (Rewrite<= unpow2_binary64 (pow.f64 x 2))) x) (+.f64 (/.f64 15/8 (pow.f64 x 6)) (+.f64 (/.f64 3/4 (pow.f64 x 4)) (+.f64 1 (/.f64 1/2 (*.f64 x x))))))): 0 points increase in error, 1 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (*.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (Rewrite<= unpow1_binary64 (pow.f64 x 1))) (+.f64 (/.f64 15/8 (pow.f64 x 6)) (+.f64 (/.f64 3/4 (pow.f64 x 4)) (+.f64 1 (/.f64 1/2 (*.f64 x x))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (*.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (Rewrite=> sqr-pow_binary64 (*.f64 (pow.f64 x (/.f64 1 2)) (pow.f64 x (/.f64 1 2))))) (+.f64 (/.f64 15/8 (pow.f64 x 6)) (+.f64 (/.f64 3/4 (pow.f64 x 4)) (+.f64 1 (/.f64 1/2 (*.f64 x x))))))): 58 points increase in error, 38 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (*.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (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 6)) (+.f64 (/.f64 3/4 (pow.f64 x 4)) (+.f64 1 (/.f64 1/2 (*.f64 x x))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (*.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 (Rewrite<= sqr-pow_binary64 (pow.f64 x 1)))) (+.f64 (/.f64 15/8 (pow.f64 x 6)) (+.f64 (/.f64 3/4 (pow.f64 x 4)) (+.f64 1 (/.f64 1/2 (*.f64 x x))))))): 38 points increase in error, 58 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (*.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 (Rewrite=> unpow1_binary64 x))) (+.f64 (/.f64 15/8 (pow.f64 x 6)) (+.f64 (/.f64 3/4 (pow.f64 x 4)) (+.f64 1 (/.f64 1/2 (*.f64 x x))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (*.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x)) (+.f64 (/.f64 15/8 (pow.f64 x 6)) (+.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 3/4 1)) (pow.f64 x 4)) (+.f64 1 (/.f64 1/2 (*.f64 x x))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (*.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x)) (+.f64 (/.f64 15/8 (pow.f64 x 6)) (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 3/4 (/.f64 1 (pow.f64 x 4)))) (+.f64 1 (/.f64 1/2 (*.f64 x x))))))): 3 points increase in error, 2 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (*.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x)) (+.f64 (/.f64 15/8 (pow.f64 x 6)) (+.f64 (*.f64 3/4 (/.f64 1 (pow.f64 x 4))) (+.f64 1 (/.f64 1/2 (Rewrite<= unpow2_binary64 (pow.f64 x 2)))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (*.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x)) (+.f64 (/.f64 15/8 (pow.f64 x 6)) (+.f64 (*.f64 3/4 (/.f64 1 (pow.f64 x 4))) (+.f64 1 (/.f64 (Rewrite<= metadata-eval (*.f64 1/2 1)) (pow.f64 x 2))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (*.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x)) (+.f64 (/.f64 15/8 (pow.f64 x 6)) (+.f64 (*.f64 3/4 (/.f64 1 (pow.f64 x 4))) (+.f64 1 (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 1 (pow.f64 x 2))))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (*.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x)) (+.f64 (/.f64 15/8 (pow.f64 x 6)) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 3/4 (/.f64 1 (pow.f64 x 4))) 1) (*.f64 1/2 (/.f64 1 (pow.f64 x 2)))))))): 5 points increase in error, 6 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (/.f64 15/8 (pow.f64 x 6)) (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x))) (*.f64 (+.f64 (+.f64 (*.f64 3/4 (/.f64 1 (pow.f64 x 4))) 1) (*.f64 1/2 (/.f64 1 (pow.f64 x 2)))) (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x)))))): 41 points increase in error, 41 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (Rewrite<= associate-/r/_binary64 (/.f64 15/8 (/.f64 (pow.f64 x 6) (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x))))) (*.f64 (+.f64 (+.f64 (*.f64 3/4 (/.f64 1 (pow.f64 x 4))) 1) (*.f64 1/2 (/.f64 1 (pow.f64 x 2)))) (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x))))): 8 points increase in error, 4 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 15/8 (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x))) (pow.f64 x 6))) (*.f64 (+.f64 (+.f64 (*.f64 3/4 (/.f64 1 (pow.f64 x 4))) 1) (*.f64 1/2 (/.f64 1 (pow.f64 x 2)))) (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x))))): 6 points increase in error, 7 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 15/8 (/.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x)) (pow.f64 x 6)))) (*.f64 (+.f64 (+.f64 (*.f64 3/4 (/.f64 1 (pow.f64 x 4))) 1) (*.f64 1/2 (/.f64 1 (pow.f64 x 2)))) (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x))))): 6 points increase in error, 6 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 15/8 (Rewrite<= associate-/r*_binary64 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 6))))) (*.f64 (+.f64 (+.f64 (*.f64 3/4 (/.f64 1 (pow.f64 x 4))) 1) (*.f64 1/2 (/.f64 1 (pow.f64 x 2)))) (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x))))): 5 points increase in error, 8 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 15/8 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 6)))) (*.f64 (Rewrite=> associate-+l+_binary64 (+.f64 (*.f64 3/4 (/.f64 1 (pow.f64 x 4))) (+.f64 1 (*.f64 1/2 (/.f64 1 (pow.f64 x 2)))))) (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x))))): 6 points increase in error, 5 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 15/8 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 6)))) (*.f64 (+.f64 (*.f64 3/4 (/.f64 1 (pow.f64 x 4))) (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 1/2 (/.f64 1 (pow.f64 x 2))) 1))) (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 15/8 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 6)))) (*.f64 (Rewrite=> associate-+r+_binary64 (+.f64 (+.f64 (*.f64 3/4 (/.f64 1 (pow.f64 x 4))) (*.f64 1/2 (/.f64 1 (pow.f64 x 2)))) 1)) (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x))))): 23 points increase in error, 29 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 15/8 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 6)))) (Rewrite<= distribute-lft1-in_binary64 (+.f64 (*.f64 (+.f64 (*.f64 3/4 (/.f64 1 (pow.f64 x 4))) (*.f64 1/2 (/.f64 1 (pow.f64 x 2)))) (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x))) (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x)))))): 37 points increase in error, 31 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 15/8 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 6)))) (+.f64 (*.f64 (+.f64 (*.f64 3/4 (/.f64 1 (pow.f64 x 4))) (*.f64 1/2 (/.f64 1 (pow.f64 x 2)))) (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x))) (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 15/8 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 6)))) (Rewrite<= distribute-rgt-in_binary64 (*.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x)) (+.f64 (+.f64 (*.f64 3/4 (/.f64 1 (pow.f64 x 4))) (*.f64 1/2 (/.f64 1 (pow.f64 x 2)))) 1))))): 31 points increase in error, 37 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 15/8 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 6)))) (*.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x)) (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 3/4 (/.f64 1 (pow.f64 x 4))) (+.f64 (*.f64 1/2 (/.f64 1 (pow.f64 x 2))) 1)))))): 29 points increase in error, 23 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 15/8 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 6)))) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (*.f64 3/4 (/.f64 1 (pow.f64 x 4))) (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x))) (*.f64 (+.f64 (*.f64 1/2 (/.f64 1 (pow.f64 x 2))) 1) (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x))))))): 37 points increase in error, 36 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 15/8 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 6)))) (+.f64 (*.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 3/4 1) (pow.f64 x 4))) (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x))) (*.f64 (+.f64 (*.f64 1/2 (/.f64 1 (pow.f64 x 2))) 1) (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x)))))): 0 points increase in error, 1 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 15/8 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 6)))) (+.f64 (*.f64 (/.f64 (Rewrite=> metadata-eval 3/4) (pow.f64 x 4)) (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x))) (*.f64 (+.f64 (*.f64 1/2 (/.f64 1 (pow.f64 x 2))) 1) (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 15/8 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 6)))) (+.f64 (Rewrite<= associate-/r/_binary64 (/.f64 3/4 (/.f64 (pow.f64 x 4) (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x))))) (*.f64 (+.f64 (*.f64 1/2 (/.f64 1 (pow.f64 x 2))) 1) (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x)))))): 2 points increase in error, 1 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 15/8 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 6)))) (+.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 3/4 (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x))) (pow.f64 x 4))) (*.f64 (+.f64 (*.f64 1/2 (/.f64 1 (pow.f64 x 2))) 1) (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x)))))): 2 points increase in error, 2 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 15/8 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 6)))) (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 3/4 (/.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x)) (pow.f64 x 4)))) (*.f64 (+.f64 (*.f64 1/2 (/.f64 1 (pow.f64 x 2))) 1) (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x)))))): 2 points increase in error, 3 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 15/8 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 6)))) (+.f64 (*.f64 3/4 (Rewrite<= associate-/r*_binary64 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 4))))) (*.f64 (+.f64 (*.f64 1/2 (/.f64 1 (pow.f64 x 2))) 1) (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x)))))): 2 points increase in error, 2 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 15/8 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 6)))) (+.f64 (*.f64 3/4 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 4)))) (Rewrite<= distribute-rgt1-in_binary64 (+.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x)) (*.f64 (*.f64 1/2 (/.f64 1 (pow.f64 x 2))) (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x)))))))): 31 points increase in error, 33 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 15/8 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 6)))) (+.f64 (*.f64 3/4 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 4)))) (+.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x)) (*.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 1/2 1) (pow.f64 x 2))) (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 15/8 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 6)))) (+.f64 (*.f64 3/4 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 4)))) (+.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x)) (*.f64 (/.f64 (Rewrite=> metadata-eval 1/2) (pow.f64 x 2)) (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 15/8 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 6)))) (+.f64 (*.f64 3/4 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 4)))) (+.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x)) (Rewrite<= associate-/r/_binary64 (/.f64 1/2 (/.f64 (pow.f64 x 2) (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x))))))))): 0 points increase in error, 3 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 15/8 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 6)))) (+.f64 (*.f64 3/4 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 4)))) (+.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x)) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 1/2 (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x))) (pow.f64 x 2))))))): 1 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 15/8 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 6)))) (+.f64 (*.f64 3/4 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 4)))) (+.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x)) (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x)) (pow.f64 x 2)))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 15/8 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 6)))) (+.f64 (*.f64 3/4 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 4)))) (+.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x)) (*.f64 1/2 (Rewrite<= associate-/r*_binary64 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 2))))))))): 2 points increase in error, 4 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 15/8 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 6)))) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 3/4 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 4)))) (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x))) (*.f64 1/2 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 2)))))))): 6 points increase in error, 4 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 15/8 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 6)))) (+.f64 (*.f64 3/4 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 4)))) (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x)))) (*.f64 1/2 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 2))))))): 9 points increase in error, 9 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 (*.f64 3/4 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 4)))) (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x))) (*.f64 15/8 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 6)))))) (*.f64 1/2 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 1/2 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 2)))) (+.f64 (+.f64 (*.f64 3/4 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 4)))) (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x))) (*.f64 15/8 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 6)))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 1/2 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 2)))) (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 15/8 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 6)))) (+.f64 (*.f64 3/4 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 4)))) (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 1/2 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 2)))) (*.f64 15/8 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 6))))) (+.f64 (*.f64 3/4 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 4)))) (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x)))))): 22 points increase in error, 25 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (Rewrite=> +-commutative_binary64 (+.f64 (+.f64 (*.f64 3/4 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 4)))) (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x))) (+.f64 (*.f64 1/2 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 2)))) (*.f64 15/8 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 6)))))))): 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 3/4 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 4)))) (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x)))) (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 1/2 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 2)))) (*.f64 15/8 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 6)))))))): 48 points increase in error, 47 points decrease in error
   (+.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (*.f64 3/4 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 4)))) (sqrt.f64 (/.f64 1 (PI.f64)))) (*.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x)) (sqrt.f64 (/.f64 1 (PI.f64)))))) (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 1/2 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 2)))) (*.f64 15/8 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 6))))))): 25 points increase in error, 32 points decrease in error
   (+.f64 (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 3/4 (*.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 4))) (sqrt.f64 (/.f64 1 (PI.f64)))))) (*.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x)) (sqrt.f64 (/.f64 1 (PI.f64))))) (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 1/2 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 2)))) (*.f64 15/8 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 6))))))): 4 points increase in error, 3 points decrease in error
   (+.f64 (+.f64 (*.f64 3/4 (*.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 4))) (sqrt.f64 (/.f64 1 (PI.f64))))) (*.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x)) (sqrt.f64 (/.f64 1 (PI.f64))))) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (*.f64 1/2 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 2)))) (sqrt.f64 (/.f64 1 (PI.f64)))) (*.f64 (*.f64 15/8 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 6)))) (sqrt.f64 (/.f64 1 (PI.f64))))))): 4 points increase in error, 15 points decrease in error
   (+.f64 (+.f64 (*.f64 3/4 (*.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 4))) (sqrt.f64 (/.f64 1 (PI.f64))))) (*.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x)) (sqrt.f64 (/.f64 1 (PI.f64))))) (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 1/2 (*.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 2))) (sqrt.f64 (/.f64 1 (PI.f64)))))) (*.f64 (*.f64 15/8 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 6)))) (sqrt.f64 (/.f64 1 (PI.f64)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (+.f64 (*.f64 3/4 (*.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 4))) (sqrt.f64 (/.f64 1 (PI.f64))))) (*.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x)) (sqrt.f64 (/.f64 1 (PI.f64))))) (+.f64 (*.f64 1/2 (*.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 2))) (sqrt.f64 (/.f64 1 (PI.f64))))) (Rewrite<= associate-*r*_binary64 (*.f64 15/8 (*.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 6))) (sqrt.f64 (/.f64 1 (PI.f64)))))))): 8 points increase in error, 5 points decrease in error
   (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 3/4 (*.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 4))) (sqrt.f64 (/.f64 1 (PI.f64))))) (+.f64 (*.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (fabs.f64 x)) (sqrt.f64 (/.f64 1 (PI.f64)))) (+.f64 (*.f64 1/2 (*.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 2))) (sqrt.f64 (/.f64 1 (PI.f64))))) (*.f64 15/8 (*.f64 (/.f64 (exp.f64 (pow.f64 x 2)) (*.f64 (fabs.f64 x) (pow.f64 x 6))) (sqrt.f64 (/.f64 1 (PI.f64))))))))): 16 points increase in error, 15 points decrease in error

5. Taylor expanded in x around 0 1.3

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

6. Simplified 1.3

   \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\frac{{\left(e^{x}\right)}^{x}}{x} \cdot \left(\frac{1.875}{{x}^{6}} + \color{blue}{\left(1 + \left(\frac{0.75}{{x}^{4}} + \frac{0.5}{x \cdot x}\right)\right)}\right)\right) \]
   Proof
   (+.f64 1 (+.f64 (/.f64 3/4 (pow.f64 x 4)) (/.f64 1/2 (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (+.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 3/4 1)) (pow.f64 x 4)) (/.f64 1/2 (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 3/4 (/.f64 1 (pow.f64 x 4)))) (/.f64 1/2 (*.f64 x x)))): 6 points increase in error, 5 points decrease in error
   (+.f64 1 (+.f64 (*.f64 3/4 (/.f64 1 (pow.f64 x 4))) (/.f64 (Rewrite<= metadata-eval (*.f64 1/2 1)) (*.f64 x x)))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (+.f64 (*.f64 3/4 (/.f64 1 (pow.f64 x 4))) (/.f64 (*.f64 1/2 1) (Rewrite<= unpow2_binary64 (pow.f64 x 2))))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (+.f64 (*.f64 3/4 (/.f64 1 (pow.f64 x 4))) (Rewrite<= associate-*r/_binary64 (*.f64 1/2 (/.f64 1 (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 1 (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 1/2 (/.f64 1 (pow.f64 x 2))) (*.f64 3/4 (/.f64 1 (pow.f64 x 4)))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 1 (*.f64 1/2 (/.f64 1 (pow.f64 x 2)))) (*.f64 3/4 (/.f64 1 (pow.f64 x 4))))): 28 points increase in error, 28 points decrease in error
   (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 3/4 (/.f64 1 (pow.f64 x 4))) (+.f64 1 (*.f64 1/2 (/.f64 1 (pow.f64 x 2)))))): 0 points increase in error, 0 points decrease in error

7. Applied egg-rr 1.3

   \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\frac{{\left(e^{x}\right)}^{x}}{x} \cdot \left(\color{blue}{\log \left(1 + \mathsf{expm1}\left(1.875 \cdot {x}^{-6}\right)\right)} + \left(1 + \left(\frac{0.75}{{x}^{4}} + \frac{0.5}{x \cdot x}\right)\right)\right)\right) \]

8. Applied egg-rr 1.3

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

9. Final simplification 1.3

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

Alternatives

Alternative 1
Error	1.3
Cost	40000

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

Alternative 2
Error	1.3
Cost	33600

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

Alternative 3
Error	1.3
Cost	33536

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

Alternative 4
Error	41.4
Cost	33280

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

Alternative 5
Error	43.7
Cost	33216

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

Alternative 6
Error	44.7
Cost	26432

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

Alternative 7
Error	48.3
Cost	26048

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

Alternative 8
Error	48.3
Cost	26048

\[\sqrt{\frac{1}{\pi}} \cdot \frac{e^{{x}^{2}}}{x} \]

Alternative 9
Error	55.6
Cost	20928

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

Alternative 10
Error	56.3
Cost	20672

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

Alternative 11
Error	56.4
Cost	19904

\[\sqrt{\frac{1}{\pi}} \cdot \left(\frac{2.1875}{{x}^{3}} + \frac{2.1875}{x}\right) \]

Alternative 12
Error	56.6
Cost	13184

\[\sqrt{\frac{1}{\pi}} \cdot \frac{2.1875}{x} \]

Reproduce

herbie shell --seed 2022291 
(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)))))))