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

Average Error: 0.2 → 0.1

Time: 9.5s

Precision: binary64

Cost: 39616

\[x \leq 0.5\]

Math

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

↓

\[\left|\mathsf{fma}\left(0.2, {x}^{5}, x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right) + 0.047619047619047616 \cdot {x}^{7}\right) \cdot {\pi}^{-0.5}\right| \]

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
  (*
   (fma
    0.2
    (pow x 5.0)
    (+
     (* x (+ (* 0.6666666666666666 (* x x)) 2.0))
     (* 0.047619047619047616 (pow x 7.0))))
   (pow PI -0.5))))

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((fma(0.2, pow(x, 5.0), ((x * ((0.6666666666666666 * (x * x)) + 2.0)) + (0.047619047619047616 * pow(x, 7.0)))) * pow(((double) M_PI), -0.5)));
}

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(fma(0.2, (x ^ 5.0), Float64(Float64(x * Float64(Float64(0.6666666666666666 * Float64(x * x)) + 2.0)) + Float64(0.047619047619047616 * (x ^ 7.0)))) * (pi ^ -0.5)))
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[(0.2 * N[Power[x, 5.0], $MachinePrecision] + N[(N[(x * N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[Pi, -0.5], $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)))): 118 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, 118 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)))): 173 points increase in error, 10 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)))): 10 points increase in error, 173 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)))): 118 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, 118 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)))): 0 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)) (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)))): 5 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<= cube-prod_binary64 (pow.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) 3)) (fabs.f64 x))))) (sqrt.f64 (PI.f64)))): 4 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=> 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)))): 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)) (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)))): 2 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)) (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, 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 (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)))): 0 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, 0 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))))))): 31 points increase in error, 137 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))))))): 1 points increase in error, 0 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)))))): 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 (+.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, 3 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)))))): 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))))) (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)))))))): 0 points increase in error, 2 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)))))))): 1 points increase in error, 0 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))))))): 175 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, 175 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)))))))): 0 points increase in error, 1 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))))))): 2 points increase in error, 0 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))))))): 0 points increase in error, 1 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))))))): 3 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, 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 (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)))))))): 0 points increase in error, 1 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

   \[\leadsto \left|\color{blue}{\mathsf{fma}\left(0.2, {x}^{5}, \mathsf{fma}\left(x, \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right), 0.047619047619047616 \cdot {x}^{7}\right)\right) \cdot {\pi}^{-0.5}}\right| \]

4. Applied egg-rr 0.1

   \[\leadsto \left|\mathsf{fma}\left(0.2, {x}^{5}, \color{blue}{x \cdot \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + 0.047619047619047616 \cdot {x}^{7}}\right) \cdot {\pi}^{-0.5}\right| \]

5. Applied egg-rr 0.1

   \[\leadsto \left|\mathsf{fma}\left(0.2, {x}^{5}, x \cdot \color{blue}{\left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right)} + 0.047619047619047616 \cdot {x}^{7}\right) \cdot {\pi}^{-0.5}\right| \]

6. Final simplification 0.1

   \[\leadsto \left|\mathsf{fma}\left(0.2, {x}^{5}, x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right) + 0.047619047619047616 \cdot {x}^{7}\right) \cdot {\pi}^{-0.5}\right| \]

Alternatives

Alternative 1
Error	0.8
Cost	39232

\[\left|{\pi}^{-0.5} \cdot \mathsf{fma}\left(0.2, {x}^{5}, 0.047619047619047616 \cdot {x}^{7} + x \cdot 2\right)\right| \]

Alternative 2
Error	0.4
Cost	39108

\[\begin{array}{l} \mathbf{if}\;x \leq -6.922269893730041:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \mathsf{fma}\left(0.2, {x}^{5}, 0.047619047619047616 \cdot {x}^{7}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right) + 0.2 \cdot {x}^{5}\right)\right|\\ \end{array} \]

Alternative 3
Error	0.6
Cost	33028

\[\begin{array}{l} \mathbf{if}\;x \leq -6.922269893730041:\\ \;\;\;\;\left|\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right) + 0.2 \cdot {x}^{5}\right)\right|\\ \end{array} \]

Alternative 4
Error	0.7
Cost	26372

\[\begin{array}{l} \mathbf{if}\;x \leq -6.922269893730041:\\ \;\;\;\;\left|\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)\right|\\ \end{array} \]

Alternative 5
Error	0.7
Cost	26308

\[\begin{array}{l} \mathbf{if}\;x \leq -6.922269893730041:\\ \;\;\;\;\left|\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right)\right)\right|\\ \end{array} \]

Alternative 6
Error	0.7
Cost	26244

\[\begin{array}{l} \mathbf{if}\;x \leq -6.922269893730041:\\ \;\;\;\;\left|\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{\mathsf{fma}\left(x, x \cdot 0.6666666666666666, 2\right)}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 7
Error	0.9
Cost	26052

\[\begin{array}{l} \mathbf{if}\;x \leq -6.922269893730041:\\ \;\;\;\;\left|\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 8
Error	5.0
Cost	19456

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

Alternative 9
Error	4.6
Cost	19456

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

Reproduce

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