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

Average Error: 0.2 → 0.2

Time: 9.4s

Precision: binary64

Cost: 46016

\[x \leq 0.5\]

Math

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

↓

\[\left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(2, x, 0.2 \cdot {x}^{5} + \left(0.6666666666666666 \cdot {x}^{3} + 0.047619047619047616 \cdot {x}^{7}\right)\right)\right| \]

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
  (*
   (sqrt (/ 1.0 PI))
   (fma
    2.0
    x
    (+
     (* 0.2 (pow x 5.0))
     (+
      (* 0.6666666666666666 (pow x 3.0))
      (* 0.047619047619047616 (pow x 7.0))))))))

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((sqrt((1.0 / ((double) M_PI))) * fma(2.0, x, ((0.2 * pow(x, 5.0)) + ((0.6666666666666666 * pow(x, 3.0)) + (0.047619047619047616 * pow(x, 7.0)))))));
}

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(sqrt(Float64(1.0 / pi)) * fma(2.0, x, Float64(Float64(0.2 * (x ^ 5.0)) + Float64(Float64(0.6666666666666666 * (x ^ 3.0)) + Float64(0.047619047619047616 * (x ^ 7.0)))))))
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[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(2.0 * x + N[(N[(0.2 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 0.2

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

2. Simplified 0.6

   \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(0.2, {x}^{5}, \mathsf{fma}\left(x, \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right), 0.047619047619047616 \cdot {x}^{7}\right)\right)}{\sqrt{\pi}}\right|} \]
   Proof
   (fabs.f64 (/.f64 (fma.f64 1/5 (pow.f64 x 5) (fma.f64 x (fma.f64 2/3 (*.f64 x x) 2) (*.f64 1/21 (pow.f64 x 7)))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (fma.f64 (Rewrite<= metadata-eval (/.f64 1 5)) (pow.f64 x 5) (fma.f64 x (fma.f64 2/3 (*.f64 x x) 2) (*.f64 1/21 (pow.f64 x 7)))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (fma.f64 (/.f64 1 5) (pow.f64 (Rewrite<= unpow1_binary64 (pow.f64 x 1)) 5) (fma.f64 x (fma.f64 2/3 (*.f64 x x) 2) (*.f64 1/21 (pow.f64 x 7)))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (fma.f64 (/.f64 1 5) (pow.f64 (Rewrite=> sqr-pow_binary64 (*.f64 (pow.f64 x (/.f64 1 2)) (pow.f64 x (/.f64 1 2)))) 5) (fma.f64 x (fma.f64 2/3 (*.f64 x x) 2) (*.f64 1/21 (pow.f64 x 7)))) (sqrt.f64 (PI.f64)))): 141 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (fma.f64 (/.f64 1 5) (pow.f64 (Rewrite<= fabs-sqr_binary64 (fabs.f64 (*.f64 (pow.f64 x (/.f64 1 2)) (pow.f64 x (/.f64 1 2))))) 5) (fma.f64 x (fma.f64 2/3 (*.f64 x x) 2) (*.f64 1/21 (pow.f64 x 7)))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (fma.f64 (/.f64 1 5) (pow.f64 (fabs.f64 (Rewrite<= sqr-pow_binary64 (pow.f64 x 1))) 5) (fma.f64 x (fma.f64 2/3 (*.f64 x x) 2) (*.f64 1/21 (pow.f64 x 7)))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 141 points decrease in error
   (fabs.f64 (/.f64 (fma.f64 (/.f64 1 5) (pow.f64 (fabs.f64 (Rewrite=> unpow1_binary64 x)) 5) (fma.f64 x (fma.f64 2/3 (*.f64 x x) 2) (*.f64 1/21 (pow.f64 x 7)))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (fma.f64 (/.f64 1 5) (pow.f64 (fabs.f64 x) (Rewrite<= metadata-eval (+.f64 4 1))) (fma.f64 x (fma.f64 2/3 (*.f64 x x) 2) (*.f64 1/21 (pow.f64 x 7)))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (fma.f64 (/.f64 1 5) (pow.f64 (fabs.f64 x) (+.f64 (Rewrite<= metadata-eval (+.f64 3 1)) 1)) (fma.f64 x (fma.f64 2/3 (*.f64 x x) 2) (*.f64 1/21 (pow.f64 x 7)))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (fma.f64 (/.f64 1 5) (Rewrite<= pow-plus_binary64 (*.f64 (pow.f64 (fabs.f64 x) (+.f64 3 1)) (fabs.f64 x))) (fma.f64 x (fma.f64 2/3 (*.f64 x x) 2) (*.f64 1/21 (pow.f64 x 7)))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (fma.f64 (/.f64 1 5) (*.f64 (Rewrite<= pow-plus_binary64 (*.f64 (pow.f64 (fabs.f64 x) 3) (fabs.f64 x))) (fabs.f64 x)) (fma.f64 x (fma.f64 2/3 (*.f64 x x) 2) (*.f64 1/21 (pow.f64 x 7)))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (fma.f64 (/.f64 1 5) (*.f64 (*.f64 (Rewrite=> unpow3_binary64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x))) (fabs.f64 x)) (fabs.f64 x)) (fma.f64 x (fma.f64 2/3 (*.f64 x x) 2) (*.f64 1/21 (pow.f64 x 7)))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (fma.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fma.f64 (Rewrite<= unpow1_binary64 (pow.f64 x 1)) (fma.f64 2/3 (*.f64 x x) 2) (*.f64 1/21 (pow.f64 x 7)))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (fma.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fma.f64 (Rewrite=> sqr-pow_binary64 (*.f64 (pow.f64 x (/.f64 1 2)) (pow.f64 x (/.f64 1 2)))) (fma.f64 2/3 (*.f64 x x) 2) (*.f64 1/21 (pow.f64 x 7)))) (sqrt.f64 (PI.f64)))): 169 points increase in error, 14 points decrease in error
   (fabs.f64 (/.f64 (fma.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fma.f64 (Rewrite<= fabs-sqr_binary64 (fabs.f64 (*.f64 (pow.f64 x (/.f64 1 2)) (pow.f64 x (/.f64 1 2))))) (fma.f64 2/3 (*.f64 x x) 2) (*.f64 1/21 (pow.f64 x 7)))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (fma.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fma.f64 (fabs.f64 (Rewrite<= sqr-pow_binary64 (pow.f64 x 1))) (fma.f64 2/3 (*.f64 x x) 2) (*.f64 1/21 (pow.f64 x 7)))) (sqrt.f64 (PI.f64)))): 14 points increase in error, 169 points decrease in error
   (fabs.f64 (/.f64 (fma.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fma.f64 (fabs.f64 (Rewrite=> unpow1_binary64 x)) (fma.f64 2/3 (*.f64 x x) 2) (*.f64 1/21 (pow.f64 x 7)))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (fma.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fma.f64 (fabs.f64 x) (fma.f64 (Rewrite<= metadata-eval (/.f64 2 3)) (*.f64 x x) 2) (*.f64 1/21 (pow.f64 x 7)))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (fma.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fma.f64 (fabs.f64 x) (fma.f64 (/.f64 2 3) (Rewrite<= sqr-abs_binary64 (*.f64 (fabs.f64 x) (fabs.f64 x))) 2) (*.f64 1/21 (pow.f64 x 7)))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (fma.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fma.f64 (fabs.f64 x) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) 2)) (*.f64 1/21 (pow.f64 x 7)))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (fma.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fma.f64 (fabs.f64 x) (Rewrite=> +-commutative_binary64 (+.f64 2 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))))) (*.f64 1/21 (pow.f64 x 7)))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (fma.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fma.f64 (fabs.f64 x) (+.f64 2 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x)))) (*.f64 (Rewrite<= metadata-eval (/.f64 1 21)) (pow.f64 x 7)))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (fma.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fma.f64 (fabs.f64 x) (+.f64 2 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x)))) (*.f64 (/.f64 1 21) (pow.f64 (Rewrite<= unpow1_binary64 (pow.f64 x 1)) 7)))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (fma.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fma.f64 (fabs.f64 x) (+.f64 2 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x)))) (*.f64 (/.f64 1 21) (pow.f64 (Rewrite=> sqr-pow_binary64 (*.f64 (pow.f64 x (/.f64 1 2)) (pow.f64 x (/.f64 1 2)))) 7)))) (sqrt.f64 (PI.f64)))): 141 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (fma.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fma.f64 (fabs.f64 x) (+.f64 2 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x)))) (*.f64 (/.f64 1 21) (pow.f64 (Rewrite<= fabs-sqr_binary64 (fabs.f64 (*.f64 (pow.f64 x (/.f64 1 2)) (pow.f64 x (/.f64 1 2))))) 7)))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (fma.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fma.f64 (fabs.f64 x) (+.f64 2 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x)))) (*.f64 (/.f64 1 21) (pow.f64 (fabs.f64 (Rewrite<= sqr-pow_binary64 (pow.f64 x 1))) 7)))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 141 points decrease in error
   (fabs.f64 (/.f64 (fma.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fma.f64 (fabs.f64 x) (+.f64 2 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x)))) (*.f64 (/.f64 1 21) (pow.f64 (fabs.f64 (Rewrite=> unpow1_binary64 x)) 7)))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (fma.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fma.f64 (fabs.f64 x) (+.f64 2 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x)))) (*.f64 (/.f64 1 21) (pow.f64 (fabs.f64 x) (Rewrite<= metadata-eval (+.f64 6 1)))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (fma.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fma.f64 (fabs.f64 x) (+.f64 2 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x)))) (*.f64 (/.f64 1 21) (pow.f64 (fabs.f64 x) (+.f64 (Rewrite<= metadata-eval (*.f64 2 3)) 1))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (fma.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fma.f64 (fabs.f64 x) (+.f64 2 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x)))) (*.f64 (/.f64 1 21) (Rewrite<= pow-plus_binary64 (*.f64 (pow.f64 (fabs.f64 x) (*.f64 2 3)) (fabs.f64 x)))))) (sqrt.f64 (PI.f64)))): 2 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (fma.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fma.f64 (fabs.f64 x) (+.f64 2 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x)))) (*.f64 (/.f64 1 21) (*.f64 (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 (fabs.f64 x) 3) (pow.f64 (fabs.f64 x) 3))) (fabs.f64 x))))) (sqrt.f64 (PI.f64)))): 1 points increase in error, 2 points decrease in error
   (fabs.f64 (/.f64 (fma.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fma.f64 (fabs.f64 x) (+.f64 2 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x)))) (*.f64 (/.f64 1 21) (*.f64 (Rewrite<= cube-prod_binary64 (pow.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) 3)) (fabs.f64 x))))) (sqrt.f64 (PI.f64)))): 3 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (fma.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fma.f64 (fabs.f64 x) (+.f64 2 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x)))) (*.f64 (/.f64 1 21) (*.f64 (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)))) (fabs.f64 x))))) (sqrt.f64 (PI.f64)))): 1 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (fma.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fma.f64 (fabs.f64 x) (+.f64 2 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x)))) (*.f64 (/.f64 1 21) (*.f64 (*.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))) (fabs.f64 x))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 4 points decrease in error
   (fabs.f64 (/.f64 (fma.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fma.f64 (fabs.f64 x) (+.f64 2 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x)))) (*.f64 (/.f64 1 21) (*.f64 (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))) (fabs.f64 x))))) (sqrt.f64 (PI.f64)))): 1 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (fma.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (fabs.f64 x) (+.f64 2 (*.f64 (/.f64 2 3) (*.f64 (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 (/.f64 (fma.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (+.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 (/.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)))): 2 points increase in error, 1 points decrease in error
   (fabs.f64 (/.f64 (fma.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 2 (fabs.f64 x)) (Rewrite<= associate-*r*_binary64 (*.f64 (/.f64 2 3) (*.f64 (*.f64 (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 (/.f64 (Rewrite<= fma-def_binary64 (+.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 (*.f64 2 (fabs.f64 x)) (*.f64 (/.f64 2 3) (*.f64 (*.f64 (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 (/.f64 (Rewrite<= associate-+l+_binary64 (+.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 2 (fabs.f64 x)) (*.f64 (/.f64 2 3) (*.f64 (*.f64 (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, 1 points decrease in error
   (fabs.f64 (/.f64 (+.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 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 (/.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))))))): 30 points increase in error, 136 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)))))))): 1 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))))))): 0 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)))))): 2 points increase in error, 1 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, 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 (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)))))): 1 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 (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)))))): 3 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)))))))): 1 points increase in error, 1 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))))))): 1 points increase in error, 1 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, 3 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))))))): 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))) (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))))))): 0 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))))))): 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<= 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)))))): 1 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)))) (*.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, 0 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, 1 points decrease in error

3. Taylor expanded in x around 0 0.2

   \[\leadsto \left|\color{blue}{0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right) + \left(0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right) + \left(0.2 \cdot \left({x}^{5} \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)}\right| \]

4. Simplified 0.2

   \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.2, {x}^{5}, 0.047619047619047616 \cdot {x}^{7}\right)\right)\right)}\right| \]
   Proof
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (fma.f64 2 x (fma.f64 2/3 (pow.f64 x 3) (fma.f64 1/5 (pow.f64 x 5) (*.f64 1/21 (pow.f64 x 7)))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (fma.f64 2 x (fma.f64 2/3 (pow.f64 x 3) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 1/5 (pow.f64 x 5)) (*.f64 1/21 (pow.f64 x 7))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (fma.f64 2 x (fma.f64 2/3 (pow.f64 x 3) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 1/21 (pow.f64 x 7)) (*.f64 1/5 (pow.f64 x 5))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (fma.f64 2 x (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2/3 (pow.f64 x 3)) (+.f64 (*.f64 1/21 (pow.f64 x 7)) (*.f64 1/5 (pow.f64 x 5))))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 x) (+.f64 (*.f64 2/3 (pow.f64 x 3)) (+.f64 (*.f64 1/21 (pow.f64 x 7)) (*.f64 1/5 (pow.f64 x 5))))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (*.f64 2 x)) (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 2/3 (pow.f64 x 3)) (+.f64 (*.f64 1/21 (pow.f64 x 7)) (*.f64 1/5 (pow.f64 x 5))))))): 2 points increase in error, 3 points decrease in error
   (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 2 x) (sqrt.f64 (/.f64 1 (PI.f64))))) (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 2/3 (pow.f64 x 3)) (+.f64 (*.f64 1/21 (pow.f64 x 7)) (*.f64 1/5 (pow.f64 x 5)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 x (sqrt.f64 (/.f64 1 (PI.f64)))))) (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 2/3 (pow.f64 x 3)) (+.f64 (*.f64 1/21 (pow.f64 x 7)) (*.f64 1/5 (pow.f64 x 5)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 2 (*.f64 x (sqrt.f64 (/.f64 1 (PI.f64))))) (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (*.f64 2/3 (pow.f64 x 3))) (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 1/21 (pow.f64 x 7)) (*.f64 1/5 (pow.f64 x 5))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 2 (*.f64 x (sqrt.f64 (/.f64 1 (PI.f64))))) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 2/3 (pow.f64 x 3)) (sqrt.f64 (/.f64 1 (PI.f64))))) (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 1/21 (pow.f64 x 7)) (*.f64 1/5 (pow.f64 x 5)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 2 (*.f64 x (sqrt.f64 (/.f64 1 (PI.f64))))) (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 2/3 (*.f64 (pow.f64 x 3) (sqrt.f64 (/.f64 1 (PI.f64)))))) (*.f64 (sqrt.f64 (/.f64 1 (PI.f64))) (+.f64 (*.f64 1/21 (pow.f64 x 7)) (*.f64 1/5 (pow.f64 x 5)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 2 (*.f64 x (sqrt.f64 (/.f64 1 (PI.f64))))) (+.f64 (*.f64 2/3 (*.f64 (pow.f64 x 3) (sqrt.f64 (/.f64 1 (PI.f64))))) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (*.f64 1/21 (pow.f64 x 7)) (sqrt.f64 (/.f64 1 (PI.f64)))) (*.f64 (*.f64 1/5 (pow.f64 x 5)) (sqrt.f64 (/.f64 1 (PI.f64)))))))): 1 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 2 (*.f64 x (sqrt.f64 (/.f64 1 (PI.f64))))) (+.f64 (*.f64 2/3 (*.f64 (pow.f64 x 3) (sqrt.f64 (/.f64 1 (PI.f64))))) (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 1/21 (*.f64 (pow.f64 x 7) (sqrt.f64 (/.f64 1 (PI.f64)))))) (*.f64 (*.f64 1/5 (pow.f64 x 5)) (sqrt.f64 (/.f64 1 (PI.f64))))))): 2 points increase in error, 1 points decrease in error
   (+.f64 (*.f64 2 (*.f64 x (sqrt.f64 (/.f64 1 (PI.f64))))) (+.f64 (*.f64 2/3 (*.f64 (pow.f64 x 3) (sqrt.f64 (/.f64 1 (PI.f64))))) (+.f64 (*.f64 1/21 (*.f64 (pow.f64 x 7) (sqrt.f64 (/.f64 1 (PI.f64))))) (Rewrite<= associate-*r*_binary64 (*.f64 1/5 (*.f64 (pow.f64 x 5) (sqrt.f64 (/.f64 1 (PI.f64))))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 2 (*.f64 x (sqrt.f64 (/.f64 1 (PI.f64))))) (+.f64 (*.f64 2/3 (*.f64 (pow.f64 x 3) (sqrt.f64 (/.f64 1 (PI.f64))))) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 1/5 (*.f64 (pow.f64 x 5) (sqrt.f64 (/.f64 1 (PI.f64))))) (*.f64 1/21 (*.f64 (pow.f64 x 7) (sqrt.f64 (/.f64 1 (PI.f64))))))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 (*.f64 2/3 (*.f64 (pow.f64 x 3) (sqrt.f64 (/.f64 1 (PI.f64))))) (+.f64 (*.f64 1/5 (*.f64 (pow.f64 x 5) (sqrt.f64 (/.f64 1 (PI.f64))))) (*.f64 1/21 (*.f64 (pow.f64 x 7) (sqrt.f64 (/.f64 1 (PI.f64))))))) (*.f64 2 (*.f64 x (sqrt.f64 (/.f64 1 (PI.f64))))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 2/3 (*.f64 (pow.f64 x 3) (sqrt.f64 (/.f64 1 (PI.f64))))) (+.f64 (+.f64 (*.f64 1/5 (*.f64 (pow.f64 x 5) (sqrt.f64 (/.f64 1 (PI.f64))))) (*.f64 1/21 (*.f64 (pow.f64 x 7) (sqrt.f64 (/.f64 1 (PI.f64)))))) (*.f64 2 (*.f64 x (sqrt.f64 (/.f64 1 (PI.f64)))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 2/3 (*.f64 (pow.f64 x 3) (sqrt.f64 (/.f64 1 (PI.f64))))) (+.f64 (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 1/21 (*.f64 (pow.f64 x 7) (sqrt.f64 (/.f64 1 (PI.f64))))) (*.f64 1/5 (*.f64 (pow.f64 x 5) (sqrt.f64 (/.f64 1 (PI.f64))))))) (*.f64 2 (*.f64 x (sqrt.f64 (/.f64 1 (PI.f64))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 2/3 (*.f64 (pow.f64 x 3) (sqrt.f64 (/.f64 1 (PI.f64))))) (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 1/21 (*.f64 (pow.f64 x 7) (sqrt.f64 (/.f64 1 (PI.f64))))) (+.f64 (*.f64 1/5 (*.f64 (pow.f64 x 5) (sqrt.f64 (/.f64 1 (PI.f64))))) (*.f64 2 (*.f64 x (sqrt.f64 (/.f64 1 (PI.f64))))))))): 1 points increase in error, 1 points decrease in error

5. Taylor expanded in x around 0 0.2

   \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(2, x, \color{blue}{0.2 \cdot {x}^{5} + \left(0.6666666666666666 \cdot {x}^{3} + 0.047619047619047616 \cdot {x}^{7}\right)}\right)\right| \]

6. Final simplification 0.2

   \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(2, x, 0.2 \cdot {x}^{5} + \left(0.6666666666666666 \cdot {x}^{3} + 0.047619047619047616 \cdot {x}^{7}\right)\right)\right| \]

Alternatives

Alternative 1
Error	0.4
Cost	45828

\[\begin{array}{l} \mathbf{if}\;x \leq -3.717311750876522:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(0.2, {x}^{5}, \mathsf{fma}\left(x, x \cdot \left(x \cdot 0.6666666666666666\right), 0.047619047619047616 \cdot {x}^{7}\right)\right)}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(2, x, 0.2 \cdot {x}^{5} + 0.6666666666666666 \cdot {x}^{3}\right)\right|\\ \end{array} \]

Alternative 2
Error	0.6
Cost	45824

\[\left|\frac{\mathsf{fma}\left(0.2, {x}^{5}, \mathsf{fma}\left(x, 2 + 0.6666666666666666 \cdot \left(x \cdot x\right), 0.047619047619047616 \cdot {x}^{7}\right)\right)}{\sqrt{\pi}}\right| \]

Alternative 3
Error	1.1
Cost	32576

\[\left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(2, x, 0.047619047619047616 \cdot {x}^{7}\right)\right| \]

Alternative 4
Error	1.5
Cost	32448

\[\left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, 2 \cdot x\right)}{\sqrt{\pi}}\right| \]

Alternative 5
Error	0.8
Cost	26052

\[\begin{array}{l} \mathbf{if}\;x \leq -3.717311750876522:\\ \;\;\;\;\left|\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot x\right)\right|\\ \end{array} \]

Alternative 6
Error	4.6
Cost	19968

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

Alternative 7
Error	5.3
Cost	19456

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

Alternative 8
Error	4.9
Cost	19456

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

Reproduce

herbie shell --seed 2022296 
(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)))))))