Jmat.Real.erfi, branch x less than or equal to 0.5

Average Error: 0.1 → 0.1

Time: 7.5s

Precision: binary64

Cost: 33344

\[x \leq 0.5\]

Math

\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

↓

\[\left|{\pi}^{-0.5} \cdot \left(x \cdot \left(\left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right) + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right)\right| \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (/ 1.0 (sqrt PI))
   (+
    (+
     (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) (* (* (fabs x) (fabs x)) (fabs x))))
     (*
      (/ 1.0 5.0)
      (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x))))
    (*
     (/ 1.0 21.0)
     (*
      (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x))
      (fabs x)))))))

↓

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (pow PI -0.5)
   (*
    x
    (+
     (+ (* 0.6666666666666666 (* x x)) 2.0)
     (+ (* 0.2 (pow x 4.0)) (* 0.047619047619047616 (pow x 6.0))))))))

C

double code(double x) {
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * ((fabs(x) * fabs(x)) * fabs(x)))) + ((1.0 / 5.0) * ((((fabs(x) * fabs(x)) * fabs(x)) * fabs(x)) * fabs(x)))) + ((1.0 / 21.0) * ((((((fabs(x) * fabs(x)) * fabs(x)) * fabs(x)) * fabs(x)) * fabs(x)) * fabs(x))))));
}

↓

double code(double x) {
	return fabs((pow(((double) M_PI), -0.5) * (x * (((0.6666666666666666 * (x * x)) + 2.0) + ((0.2 * pow(x, 4.0)) + (0.047619047619047616 * pow(x, 6.0)))))));
}

Java

public static double code(double x) {
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * ((Math.abs(x) * Math.abs(x)) * Math.abs(x)))) + ((1.0 / 5.0) * ((((Math.abs(x) * Math.abs(x)) * Math.abs(x)) * Math.abs(x)) * Math.abs(x)))) + ((1.0 / 21.0) * ((((((Math.abs(x) * Math.abs(x)) * Math.abs(x)) * Math.abs(x)) * Math.abs(x)) * Math.abs(x)) * Math.abs(x))))));
}

↓

public static double code(double x) {
	return Math.abs((Math.pow(Math.PI, -0.5) * (x * (((0.6666666666666666 * (x * x)) + 2.0) + ((0.2 * Math.pow(x, 4.0)) + (0.047619047619047616 * Math.pow(x, 6.0)))))));
}

Python

def code(x):
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * ((math.fabs(x) * math.fabs(x)) * math.fabs(x)))) + ((1.0 / 5.0) * ((((math.fabs(x) * math.fabs(x)) * math.fabs(x)) * math.fabs(x)) * math.fabs(x)))) + ((1.0 / 21.0) * ((((((math.fabs(x) * math.fabs(x)) * math.fabs(x)) * math.fabs(x)) * math.fabs(x)) * math.fabs(x)) * math.fabs(x))))))

↓

def code(x):
	return math.fabs((math.pow(math.pi, -0.5) * (x * (((0.6666666666666666 * (x * x)) + 2.0) + ((0.2 * math.pow(x, 4.0)) + (0.047619047619047616 * math.pow(x, 6.0)))))))

Julia

function code(x)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * Float64(Float64(abs(x) * abs(x)) * abs(x)))) + Float64(Float64(1.0 / 5.0) * Float64(Float64(Float64(Float64(abs(x) * abs(x)) * abs(x)) * abs(x)) * abs(x)))) + Float64(Float64(1.0 / 21.0) * Float64(Float64(Float64(Float64(Float64(Float64(abs(x) * abs(x)) * abs(x)) * abs(x)) * abs(x)) * abs(x)) * abs(x))))))
end

↓

function code(x)
	return abs(Float64((pi ^ -0.5) * Float64(x * Float64(Float64(Float64(0.6666666666666666 * Float64(x * x)) + 2.0) + Float64(Float64(0.2 * (x ^ 4.0)) + Float64(0.047619047619047616 * (x ^ 6.0)))))))
end

MATLAB

function tmp = code(x)
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + ((2.0 / 3.0) * ((abs(x) * abs(x)) * abs(x)))) + ((1.0 / 5.0) * ((((abs(x) * abs(x)) * abs(x)) * abs(x)) * abs(x)))) + ((1.0 / 21.0) * ((((((abs(x) * abs(x)) * abs(x)) * abs(x)) * abs(x)) * abs(x)) * abs(x))))));
end

↓

function tmp = code(x)
	tmp = abs(((pi ^ -0.5) * (x * (((0.6666666666666666 * (x * x)) + 2.0) + ((0.2 * (x ^ 4.0)) + (0.047619047619047616 * (x ^ 6.0)))))));
end

Wolfram

code[x_] := N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * N[(N[(N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

code[x_] := N[Abs[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x * N[(N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] + N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 0.1

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

2. Simplified 0.5

   \[\leadsto \color{blue}{\left|\frac{\left|x\right| \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {\left(\left|x\right|\right)}^{4}, 0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{6}\right)\right)}{\sqrt{\pi}}\right|} \]
   Proof
   (fabs.f64 (/.f64 (*.f64 (fabs.f64 x) (+.f64 (fma.f64 2/3 (*.f64 x x) 2) (fma.f64 1/5 (pow.f64 (fabs.f64 x) 4) (*.f64 1/21 (pow.f64 (fabs.f64 x) 6))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (*.f64 (fabs.f64 x) (+.f64 (fma.f64 (Rewrite<= metadata-eval (/.f64 2 3)) (*.f64 x x) 2) (fma.f64 1/5 (pow.f64 (fabs.f64 x) 4) (*.f64 1/21 (pow.f64 (fabs.f64 x) 6))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (*.f64 (fabs.f64 x) (+.f64 (fma.f64 (/.f64 2 3) (Rewrite<= sqr-abs_binary64 (*.f64 (fabs.f64 x) (fabs.f64 x))) 2) (fma.f64 1/5 (pow.f64 (fabs.f64 x) 4) (*.f64 1/21 (pow.f64 (fabs.f64 x) 6))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (*.f64 (fabs.f64 x) (+.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) 2)) (fma.f64 1/5 (pow.f64 (fabs.f64 x) 4) (*.f64 1/21 (pow.f64 (fabs.f64 x) 6))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (*.f64 (fabs.f64 x) (+.f64 (Rewrite=> +-commutative_binary64 (+.f64 2 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))))) (fma.f64 1/5 (pow.f64 (fabs.f64 x) 4) (*.f64 1/21 (pow.f64 (fabs.f64 x) 6))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (*.f64 (fabs.f64 x) (+.f64 (+.f64 2 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x)))) (fma.f64 (Rewrite<= metadata-eval (/.f64 1 5)) (pow.f64 (fabs.f64 x) 4) (*.f64 1/21 (pow.f64 (fabs.f64 x) 6))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (*.f64 (fabs.f64 x) (+.f64 (+.f64 2 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x)))) (fma.f64 (/.f64 1 5) (pow.f64 (fabs.f64 x) (Rewrite<= metadata-eval (+.f64 3 1))) (*.f64 1/21 (pow.f64 (fabs.f64 x) 6))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (*.f64 (fabs.f64 x) (+.f64 (+.f64 2 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x)))) (fma.f64 (/.f64 1 5) (Rewrite<= pow-plus_binary64 (*.f64 (pow.f64 (fabs.f64 x) 3) (fabs.f64 x))) (*.f64 1/21 (pow.f64 (fabs.f64 x) 6))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (*.f64 (fabs.f64 x) (+.f64 (+.f64 2 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x)))) (fma.f64 (/.f64 1 5) (*.f64 (Rewrite=> unpow3_binary64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x))) (fabs.f64 x)) (*.f64 1/21 (pow.f64 (fabs.f64 x) 6))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (*.f64 (fabs.f64 x) (+.f64 (+.f64 2 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x)))) (fma.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (*.f64 (Rewrite<= metadata-eval (/.f64 1 21)) (pow.f64 (fabs.f64 x) 6))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (*.f64 (fabs.f64 x) (+.f64 (+.f64 2 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x)))) (fma.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (*.f64 (/.f64 1 21) (pow.f64 (fabs.f64 x) (Rewrite<= metadata-eval (*.f64 2 3))))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (*.f64 (fabs.f64 x) (+.f64 (+.f64 2 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x)))) (fma.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (*.f64 (/.f64 1 21) (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 (fabs.f64 x) 3) (pow.f64 (fabs.f64 x) 3))))))) (sqrt.f64 (PI.f64)))): 5 points increase in error, 3 points decrease in error
   (fabs.f64 (/.f64 (*.f64 (fabs.f64 x) (+.f64 (+.f64 2 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x)))) (fma.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (*.f64 (/.f64 1 21) (Rewrite<= cube-prod_binary64 (pow.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) 3)))))) (sqrt.f64 (PI.f64)))): 8 points increase in error, 3 points decrease in error
   (fabs.f64 (/.f64 (*.f64 (fabs.f64 x) (+.f64 (+.f64 2 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x)))) (fma.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (*.f64 (/.f64 1 21) (Rewrite=> unpow3_binary64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (*.f64 (fabs.f64 x) (fabs.f64 x))) (*.f64 (fabs.f64 x) (fabs.f64 x)))))))) (sqrt.f64 (PI.f64)))): 1 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (*.f64 (fabs.f64 x) (+.f64 (+.f64 2 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x)))) (fma.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (*.f64 (/.f64 1 21) (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (*.f64 (fabs.f64 x) (fabs.f64 x))))))) (sqrt.f64 (PI.f64)))): 1 points increase in error, 5 points decrease in error
   (fabs.f64 (/.f64 (*.f64 (fabs.f64 x) (+.f64 (+.f64 2 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x)))) (fma.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (*.f64 (/.f64 1 21) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 4 points decrease in error
   (fabs.f64 (/.f64 (*.f64 (fabs.f64 x) (+.f64 (+.f64 2 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x)))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (fabs.f64 x) (+.f64 2 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))))) (*.f64 (fabs.f64 x) (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))))))) (sqrt.f64 (PI.f64)))): 2 points increase in error, 1 points decrease in error
   (fabs.f64 (/.f64 (+.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 2 (fabs.f64 x)) (*.f64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) (fabs.f64 x)))) (*.f64 (fabs.f64 x) (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 1 points decrease in error
   (fabs.f64 (/.f64 (+.f64 (+.f64 (*.f64 2 (fabs.f64 x)) (Rewrite<= associate-*r*_binary64 (*.f64 (/.f64 2 3) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x))))) (*.f64 (fabs.f64 x) (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (+.f64 (+.f64 (*.f64 2 (fabs.f64 x)) (*.f64 (/.f64 2 3) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (fabs.f64 x)) (*.f64 (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (fabs.f64 x))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 3 points decrease in error
   (fabs.f64 (/.f64 (+.f64 (+.f64 (*.f64 2 (fabs.f64 x)) (*.f64 (/.f64 2 3) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (fabs.f64 x)))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (+.f64 (+.f64 (*.f64 2 (fabs.f64 x)) (*.f64 (/.f64 2 3) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (Rewrite<= associate-*r*_binary64 (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))))) (sqrt.f64 (PI.f64)))): 3 points increase in error, 1 points decrease in error
   (fabs.f64 (/.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 (*.f64 2 (fabs.f64 x)) (*.f64 (/.f64 2 3) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))))) (sqrt.f64 (PI.f64)))): 1 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (+.f64 (+.f64 (+.f64 (*.f64 2 (fabs.f64 x)) (*.f64 (/.f64 2 3) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (+.f64 (+.f64 (+.f64 (*.f64 2 (fabs.f64 x)) (*.f64 (/.f64 2 3) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))))))): 28 points increase in error, 134 points decrease in error
   (fabs.f64 (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (Rewrite=> associate-+l+_binary64 (+.f64 (+.f64 (*.f64 2 (fabs.f64 x)) (*.f64 (/.f64 2 3) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))))))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (Rewrite=> distribute-rgt-in_binary64 (+.f64 (*.f64 (+.f64 (*.f64 2 (fabs.f64 x)) (*.f64 (/.f64 2 3) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (/.f64 1 (sqrt.f64 (PI.f64)))) (*.f64 (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (/.f64 1 (sqrt.f64 (PI.f64))))))): 2 points increase in error, 2 points decrease in error
   (fabs.f64 (+.f64 (*.f64 (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 (/.f64 2 3) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x))) (*.f64 2 (fabs.f64 x)))) (/.f64 1 (sqrt.f64 (PI.f64)))) (*.f64 (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (/.f64 1 (sqrt.f64 (PI.f64)))))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (+.f64 (*.f64 (+.f64 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) (fabs.f64 x))) (*.f64 2 (fabs.f64 x))) (/.f64 1 (sqrt.f64 (PI.f64)))) (*.f64 (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (/.f64 1 (sqrt.f64 (PI.f64)))))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (+.f64 (*.f64 (Rewrite=> distribute-rgt-out_binary64 (*.f64 (fabs.f64 x) (+.f64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) 2))) (/.f64 1 (sqrt.f64 (PI.f64)))) (*.f64 (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (/.f64 1 (sqrt.f64 (PI.f64)))))): 1 points increase in error, 0 points decrease in error
   (fabs.f64 (+.f64 (Rewrite=> associate-*l*_binary64 (*.f64 (fabs.f64 x) (*.f64 (+.f64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) 2) (/.f64 1 (sqrt.f64 (PI.f64)))))) (*.f64 (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (/.f64 1 (sqrt.f64 (PI.f64)))))): 0 points increase in error, 1 points decrease in error
   (fabs.f64 (+.f64 (*.f64 (fabs.f64 x) (*.f64 (+.f64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) 2) (/.f64 1 (sqrt.f64 (PI.f64))))) (*.f64 (+.f64 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (fabs.f64 x))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (/.f64 1 (sqrt.f64 (PI.f64)))))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (+.f64 (*.f64 (fabs.f64 x) (*.f64 (+.f64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) 2) (/.f64 1 (sqrt.f64 (PI.f64))))) (*.f64 (+.f64 (*.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (fabs.f64 x)) (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (fabs.f64 x)))) (/.f64 1 (sqrt.f64 (PI.f64)))))): 2 points increase in error, 2 points decrease in error
   (fabs.f64 (+.f64 (*.f64 (fabs.f64 x) (*.f64 (+.f64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) 2) (/.f64 1 (sqrt.f64 (PI.f64))))) (*.f64 (Rewrite=> distribute-rgt-out_binary64 (*.f64 (fabs.f64 x) (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))))) (/.f64 1 (sqrt.f64 (PI.f64)))))): 2 points increase in error, 1 points decrease in error
   (fabs.f64 (+.f64 (*.f64 (fabs.f64 x) (*.f64 (+.f64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) 2) (/.f64 1 (sqrt.f64 (PI.f64))))) (Rewrite=> associate-*l*_binary64 (*.f64 (fabs.f64 x) (*.f64 (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (/.f64 1 (sqrt.f64 (PI.f64)))))))): 4 points increase in error, 3 points decrease in error
   (fabs.f64 (Rewrite=> distribute-lft-out_binary64 (*.f64 (fabs.f64 x) (+.f64 (*.f64 (+.f64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) 2) (/.f64 1 (sqrt.f64 (PI.f64)))) (*.f64 (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (/.f64 1 (sqrt.f64 (PI.f64)))))))): 2 points increase in error, 1 points decrease in error
   (fabs.f64 (*.f64 (fabs.f64 (Rewrite<= unpow1_binary64 (pow.f64 x 1))) (+.f64 (*.f64 (+.f64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) 2) (/.f64 1 (sqrt.f64 (PI.f64)))) (*.f64 (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (/.f64 1 (sqrt.f64 (PI.f64))))))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (*.f64 (fabs.f64 (Rewrite=> sqr-pow_binary64 (*.f64 (pow.f64 x (/.f64 1 2)) (pow.f64 x (/.f64 1 2))))) (+.f64 (*.f64 (+.f64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) 2) (/.f64 1 (sqrt.f64 (PI.f64)))) (*.f64 (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (/.f64 1 (sqrt.f64 (PI.f64))))))): 183 points increase in error, 0 points decrease in error
   (fabs.f64 (*.f64 (Rewrite=> fabs-sqr_binary64 (*.f64 (pow.f64 x (/.f64 1 2)) (pow.f64 x (/.f64 1 2)))) (+.f64 (*.f64 (+.f64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) 2) (/.f64 1 (sqrt.f64 (PI.f64)))) (*.f64 (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (/.f64 1 (sqrt.f64 (PI.f64))))))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (*.f64 (Rewrite<= sqr-pow_binary64 (pow.f64 x 1)) (+.f64 (*.f64 (+.f64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) 2) (/.f64 1 (sqrt.f64 (PI.f64)))) (*.f64 (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (/.f64 1 (sqrt.f64 (PI.f64))))))): 0 points increase in error, 183 points decrease in error
   (fabs.f64 (*.f64 (Rewrite=> unpow1_binary64 x) (+.f64 (*.f64 (+.f64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) 2) (/.f64 1 (sqrt.f64 (PI.f64)))) (*.f64 (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (/.f64 1 (sqrt.f64 (PI.f64))))))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> fabs-mul_binary64 (*.f64 (fabs.f64 x) (fabs.f64 (+.f64 (*.f64 (+.f64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) 2) (/.f64 1 (sqrt.f64 (PI.f64)))) (*.f64 (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (/.f64 1 (sqrt.f64 (PI.f64)))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= fabs-fabs_binary64 (fabs.f64 (fabs.f64 x))) (fabs.f64 (+.f64 (*.f64 (+.f64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) 2) (/.f64 1 (sqrt.f64 (PI.f64)))) (*.f64 (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (/.f64 1 (sqrt.f64 (PI.f64))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (fabs.f64 (fabs.f64 x)) (fabs.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (/.f64 1 (sqrt.f64 (PI.f64)))) (*.f64 (+.f64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) 2) (/.f64 1 (sqrt.f64 (PI.f64)))))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fabs-mul_binary64 (fabs.f64 (*.f64 (fabs.f64 x) (+.f64 (*.f64 (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (/.f64 1 (sqrt.f64 (PI.f64)))) (*.f64 (+.f64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) 2) (/.f64 1 (sqrt.f64 (PI.f64)))))))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (fabs.f64 x) (*.f64 (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (/.f64 1 (sqrt.f64 (PI.f64))))) (*.f64 (fabs.f64 x) (*.f64 (+.f64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) 2) (/.f64 1 (sqrt.f64 (PI.f64)))))))): 1 points increase in error, 2 points decrease in error
   (fabs.f64 (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (fabs.f64 x) (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))))) (/.f64 1 (sqrt.f64 (PI.f64))))) (*.f64 (fabs.f64 x) (*.f64 (+.f64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) 2) (/.f64 1 (sqrt.f64 (PI.f64))))))): 3 points increase in error, 4 points decrease in error
   (fabs.f64 (+.f64 (*.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (fabs.f64 x)) (*.f64 (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (fabs.f64 x)))) (/.f64 1 (sqrt.f64 (PI.f64)))) (*.f64 (fabs.f64 x) (*.f64 (+.f64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) 2) (/.f64 1 (sqrt.f64 (PI.f64))))))): 1 points increase in error, 2 points decrease in error
   (fabs.f64 (+.f64 (*.f64 (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (fabs.f64 x))) (/.f64 1 (sqrt.f64 (PI.f64)))) (*.f64 (fabs.f64 x) (*.f64 (+.f64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) 2) (/.f64 1 (sqrt.f64 (PI.f64))))))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (+.f64 (*.f64 (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (Rewrite<= associate-*r*_binary64 (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))))) (/.f64 1 (sqrt.f64 (PI.f64)))) (*.f64 (fabs.f64 x) (*.f64 (+.f64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) 2) (/.f64 1 (sqrt.f64 (PI.f64))))))): 2 points increase in error, 2 points decrease in error
   (fabs.f64 (+.f64 (*.f64 (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (/.f64 1 (sqrt.f64 (PI.f64)))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (fabs.f64 x) (+.f64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) 2)) (/.f64 1 (sqrt.f64 (PI.f64))))))): 1 points increase in error, 0 points decrease in error
   (fabs.f64 (+.f64 (*.f64 (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (/.f64 1 (sqrt.f64 (PI.f64)))) (*.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) (fabs.f64 x)) (*.f64 2 (fabs.f64 x)))) (/.f64 1 (sqrt.f64 (PI.f64)))))): 0 points increase in error, 1 points decrease in error
   (fabs.f64 (+.f64 (*.f64 (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (/.f64 1 (sqrt.f64 (PI.f64)))) (*.f64 (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (/.f64 2 3) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 2 (fabs.f64 x))) (/.f64 1 (sqrt.f64 (PI.f64)))))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (+.f64 (*.f64 (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (/.f64 1 (sqrt.f64 (PI.f64)))) (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 2 (fabs.f64 x)) (*.f64 (/.f64 2 3) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x))))) (/.f64 1 (sqrt.f64 (PI.f64)))))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (Rewrite<= distribute-rgt-in_binary64 (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (+.f64 (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (+.f64 (*.f64 2 (fabs.f64 x)) (*.f64 (/.f64 2 3) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))))))): 2 points increase in error, 2 points decrease in error
   (fabs.f64 (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 (*.f64 2 (fabs.f64 x)) (*.f64 (/.f64 2 3) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))))))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (*.f64 (/.f64 1 (sqrt.f64 (PI.f64))) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 (*.f64 2 (fabs.f64 x)) (*.f64 (/.f64 2 3) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))))))): 0 points increase in error, 0 points decrease in error

3. Applied egg-rr 0.1

   \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \left(x \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right)}\right| \]

4. Taylor expanded in x around 0 0.1

   \[\leadsto \left|{\pi}^{-0.5} \cdot \left(x \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \color{blue}{\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)}\right)\right)\right| \]

5. Applied egg-rr 0.1

   \[\leadsto \left|{\pi}^{-0.5} \cdot \left(x \cdot \left(\color{blue}{\left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right)} + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right)\right| \]

6. Final simplification 0.1

   \[\leadsto \left|{\pi}^{-0.5} \cdot \left(x \cdot \left(\left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right) + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right)\right| \]

Alternatives

Alternative 1
Error	0.9
Cost	33024

\[\left|x \cdot \left(\left(2 + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

Alternative 2
Error	0.8
Cost	32960

\[\left|{\pi}^{-0.5} \cdot \left(x \cdot \left(2 + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right)\right| \]

Alternative 3
Error	0.6
Cost	32900

\[\begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \mathbf{if}\;x \leq -1.85:\\ \;\;\;\;\left|t_0 \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t_0 \cdot \left(x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right) + x \cdot 2\right)\right|\\ \end{array} \]

Alternative 4
Error	1.1
Cost	32832

\[\left|\frac{x}{\sqrt{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{6} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right)\right| \]

Alternative 5
Error	2.9
Cost	26052

\[\begin{array}{l} \mathbf{if}\;x \leq -2.2:\\ \;\;\;\;\left|\sqrt{\frac{{x}^{14}}{\frac{\pi}{0.0022675736961451248}}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right) + x \cdot 2\right)\right|\\ \end{array} \]

Alternative 6
Error	0.7
Cost	26052

\[\begin{array}{l} \mathbf{if}\;x \leq -2.2:\\ \;\;\;\;\left|{x}^{7} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right) + x \cdot 2\right)\right|\\ \end{array} \]

Alternative 7
Error	4.3
Cost	20096

\[\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right) + x \cdot 2\right)\right| \]

Alternative 8
Error	4.3
Cost	19968

\[\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right)\right)\right| \]

Alternative 9
Error	4.7
Cost	19456

\[\left|x \cdot \frac{2}{\sqrt{\pi}}\right| \]

Reproduce

herbie shell --seed 2022326 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x less than or equal to 0.5"
  :precision binary64
  :pre (<= x 0.5)
  (fabs (* (/ 1.0 (sqrt PI)) (+ (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) (* (* (fabs x) (fabs x)) (fabs x)))) (* (/ 1.0 5.0) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))) (* (/ 1.0 21.0) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)))))))