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

Average Error: 0.1 → 0.1

Time: 9.1s

Precision: binary64

Cost: 33280

\[x \leq 0.5\]

Math

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

↓

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

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)
   (+
    (* (pow x 4.0) (+ (* (* x x) 0.047619047619047616) 0.2))
    (fma (* x 0.6666666666666666) x 2.0)))))

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) * ((pow(x, 4.0) * (((x * x) * 0.047619047619047616) + 0.2)) + fma((x * 0.6666666666666666), x, 2.0))));
}

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(Float64((pi ^ -0.5) * x) * Float64(Float64((x ^ 4.0) * Float64(Float64(Float64(x * x) * 0.047619047619047616) + 0.2)) + fma(Float64(x * 0.6666666666666666), x, 2.0))))
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[(N[Power[Pi, -0.5], $MachinePrecision] * x), $MachinePrecision] * N[(N[(N[Power[x, 4.0], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * 0.047619047619047616), $MachinePrecision] + 0.2), $MachinePrecision]), $MachinePrecision] + N[(N[(x * 0.6666666666666666), $MachinePrecision] * x + 2.0), $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

   \[\leadsto \color{blue}{\left|\frac{\left|x\right|}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(x \cdot 0.6666666666666666, x, 2\right)\right)\right|} \]
   Proof
   (fabs.f64 (*.f64 (/.f64 (fabs.f64 x) (sqrt.f64 (PI.f64))) (+.f64 (fma.f64 1/5 (pow.f64 x 4) (*.f64 1/21 (pow.f64 x 6))) (fma.f64 (*.f64 x 2/3) x 2)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (*.f64 (/.f64 (fabs.f64 x) (sqrt.f64 (PI.f64))) (+.f64 (fma.f64 (Rewrite<= metadata-eval (/.f64 1 5)) (pow.f64 x 4) (*.f64 1/21 (pow.f64 x 6))) (fma.f64 (*.f64 x 2/3) x 2)))): 4 points increase in error, 39 points decrease in error
   (fabs.f64 (*.f64 (/.f64 (fabs.f64 x) (sqrt.f64 (PI.f64))) (+.f64 (fma.f64 (/.f64 1 5) (pow.f64 x (Rewrite<= metadata-eval (*.f64 2 2))) (*.f64 1/21 (pow.f64 x 6))) (fma.f64 (*.f64 x 2/3) x 2)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (*.f64 (/.f64 (fabs.f64 x) (sqrt.f64 (PI.f64))) (+.f64 (fma.f64 (/.f64 1 5) (Rewrite<= pow-sqr_binary64 (*.f64 (pow.f64 x 2) (pow.f64 x 2))) (*.f64 1/21 (pow.f64 x 6))) (fma.f64 (*.f64 x 2/3) x 2)))): 39 points increase in error, 4 points decrease in error
   (fabs.f64 (*.f64 (/.f64 (fabs.f64 x) (sqrt.f64 (PI.f64))) (+.f64 (fma.f64 (/.f64 1 5) (*.f64 (Rewrite=> unpow2_binary64 (*.f64 x x)) (pow.f64 x 2)) (*.f64 1/21 (pow.f64 x 6))) (fma.f64 (*.f64 x 2/3) x 2)))): 4 points increase in error, 0 points decrease in error
   (fabs.f64 (*.f64 (/.f64 (fabs.f64 x) (sqrt.f64 (PI.f64))) (+.f64 (fma.f64 (/.f64 1 5) (*.f64 (*.f64 x x) (Rewrite=> unpow2_binary64 (*.f64 x x))) (*.f64 1/21 (pow.f64 x 6))) (fma.f64 (*.f64 x 2/3) x 2)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (*.f64 (/.f64 (fabs.f64 x) (sqrt.f64 (PI.f64))) (+.f64 (fma.f64 (/.f64 1 5) (*.f64 (Rewrite<= sqr-abs_binary64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (*.f64 x x)) (*.f64 1/21 (pow.f64 x 6))) (fma.f64 (*.f64 x 2/3) x 2)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (*.f64 (/.f64 (fabs.f64 x) (sqrt.f64 (PI.f64))) (+.f64 (fma.f64 (/.f64 1 5) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (Rewrite<= sqr-abs_binary64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (*.f64 1/21 (pow.f64 x 6))) (fma.f64 (*.f64 x 2/3) x 2)))): 0 points increase in error, 4 points decrease in error
   (fabs.f64 (*.f64 (/.f64 (fabs.f64 x) (sqrt.f64 (PI.f64))) (+.f64 (fma.f64 (/.f64 1 5) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (*.f64 1/21 (pow.f64 x 6))) (fma.f64 (*.f64 x 2/3) x 2)))): 5 points increase in error, 39 points decrease in error
   (fabs.f64 (*.f64 (/.f64 (fabs.f64 x) (sqrt.f64 (PI.f64))) (+.f64 (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 x 6))) (fma.f64 (*.f64 x 2/3) x 2)))): 39 points increase in error, 5 points decrease in error
   (fabs.f64 (*.f64 (/.f64 (fabs.f64 x) (sqrt.f64 (PI.f64))) (+.f64 (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 x (Rewrite<= metadata-eval (*.f64 2 3))))) (fma.f64 (*.f64 x 2/3) x 2)))): 0 points increase in error, 43 points decrease in error
   (fabs.f64 (*.f64 (/.f64 (fabs.f64 x) (sqrt.f64 (PI.f64))) (+.f64 (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 x 3) (pow.f64 x 3))))) (fma.f64 (*.f64 x 2/3) x 2)))): 43 points increase in error, 0 points decrease in error
   (fabs.f64 (*.f64 (/.f64 (fabs.f64 x) (sqrt.f64 (PI.f64))) (+.f64 (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 x x) 3)))) (fma.f64 (*.f64 x 2/3) x 2)))): 1 points increase in error, 8 points decrease in error
   (fabs.f64 (*.f64 (/.f64 (fabs.f64 x) (sqrt.f64 (PI.f64))) (+.f64 (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 (Rewrite<= sqr-abs_binary64 (*.f64 (fabs.f64 x) (fabs.f64 x))) 3))) (fma.f64 (*.f64 x 2/3) x 2)))): 0 points increase in error, 40 points decrease in error
   (fabs.f64 (*.f64 (/.f64 (fabs.f64 x) (sqrt.f64 (PI.f64))) (+.f64 (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)))))) (fma.f64 (*.f64 x 2/3) x 2)))): 5 points increase in error, 0 points decrease in error
   (fabs.f64 (*.f64 (/.f64 (fabs.f64 x) (sqrt.f64 (PI.f64))) (+.f64 (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))))) (fma.f64 (*.f64 x 2/3) x 2)))): 39 points increase in error, 5 points decrease in error
   (fabs.f64 (*.f64 (/.f64 (fabs.f64 x) (sqrt.f64 (PI.f64))) (+.f64 (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))))) (fma.f64 (*.f64 x 2/3) x 2)))): 0 points increase in error, 43 points decrease in error
   (fabs.f64 (*.f64 (/.f64 (fabs.f64 x) (sqrt.f64 (PI.f64))) (+.f64 (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))))) (fma.f64 (*.f64 x 2/3) x 2)))): 43 points increase in error, 0 points decrease in error
   (fabs.f64 (*.f64 (/.f64 (fabs.f64 x) (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)))) (fma.f64 (*.f64 (Rewrite<= unpow1_binary64 (pow.f64 x 1)) 2/3) x 2)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (*.f64 (/.f64 (fabs.f64 x) (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)))) (fma.f64 (*.f64 (Rewrite=> sqr-pow_binary64 (*.f64 (pow.f64 x (/.f64 1 2)) (pow.f64 x (/.f64 1 2)))) 2/3) x 2)))): 4 points increase in error, 39 points decrease in error
   (fabs.f64 (*.f64 (/.f64 (fabs.f64 x) (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)))) (fma.f64 (*.f64 (Rewrite<= fabs-sqr_binary64 (fabs.f64 (*.f64 (pow.f64 x (/.f64 1 2)) (pow.f64 x (/.f64 1 2))))) 2/3) x 2)))): 39 points increase in error, 4 points decrease in error
   (fabs.f64 (*.f64 (/.f64 (fabs.f64 x) (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)))) (fma.f64 (*.f64 (fabs.f64 (Rewrite<= sqr-pow_binary64 (pow.f64 x 1))) 2/3) x 2)))): 14 points increase in error, 26 points decrease in error
   (fabs.f64 (*.f64 (/.f64 (fabs.f64 x) (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)))) (fma.f64 (*.f64 (fabs.f64 (Rewrite=> unpow1_binary64 x)) 2/3) x 2)))): 0 points increase in error, 14 points decrease in error
   (fabs.f64 (*.f64 (/.f64 (fabs.f64 x) (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)))) (fma.f64 (*.f64 (fabs.f64 x) (Rewrite<= metadata-eval (/.f64 2 3))) x 2)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (*.f64 (/.f64 (fabs.f64 x) (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)))) (fma.f64 (Rewrite=> *-commutative_binary64 (*.f64 (/.f64 2 3) (fabs.f64 x))) x 2)))): 39 points increase in error, 4 points decrease in error
   (fabs.f64 (*.f64 (/.f64 (fabs.f64 x) (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)))) (fma.f64 (*.f64 (/.f64 2 3) (fabs.f64 x)) (Rewrite<= unpow1_binary64 (pow.f64 x 1)) 2)))): 0 points increase in error, 43 points decrease in error
   (fabs.f64 (*.f64 (/.f64 (fabs.f64 x) (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)))) (fma.f64 (*.f64 (/.f64 2 3) (fabs.f64 x)) (Rewrite=> sqr-pow_binary64 (*.f64 (pow.f64 x (/.f64 1 2)) (pow.f64 x (/.f64 1 2)))) 2)))): 4 points increase in error, 0 points decrease in error
   (fabs.f64 (*.f64 (/.f64 (fabs.f64 x) (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)))) (fma.f64 (*.f64 (/.f64 2 3) (fabs.f64 x)) (Rewrite<= fabs-sqr_binary64 (fabs.f64 (*.f64 (pow.f64 x (/.f64 1 2)) (pow.f64 x (/.f64 1 2))))) 2)))): 39 points increase in error, 4 points decrease in error
   (fabs.f64 (*.f64 (/.f64 (fabs.f64 x) (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)))) (fma.f64 (*.f64 (/.f64 2 3) (fabs.f64 x)) (fabs.f64 (Rewrite<= sqr-pow_binary64 (pow.f64 x 1))) 2)))): 5 points increase in error, 39 points decrease in error
   (fabs.f64 (*.f64 (/.f64 (fabs.f64 x) (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)))) (fma.f64 (*.f64 (/.f64 2 3) (fabs.f64 x)) (fabs.f64 (Rewrite=> unpow1_binary64 x)) 2)))): 39 points increase in error, 1 points decrease in error
   (fabs.f64 (*.f64 (/.f64 (fabs.f64 x) (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)))) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (*.f64 (/.f64 2 3) (fabs.f64 x)) (fabs.f64 x)) 2))))): 0 points increase in error, 39 points decrease in error
   (fabs.f64 (*.f64 (/.f64 (fabs.f64 x) (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 (Rewrite<= associate-*r*_binary64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x)))) 2)))): 39 points increase in error, 0 points decrease in error
   (fabs.f64 (Rewrite=> associate-*l/_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 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) 2))) (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 (*.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 (fabs.f64 x) (+.f64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) 2)))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 43 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 (fabs.f64 x) (+.f64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) 2))) (sqrt.f64 (PI.f64)))): 5 points increase in error, 0 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 (fabs.f64 x) (+.f64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) 2))) (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))) (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 (fabs.f64 x) (+.f64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) 2))) (sqrt.f64 (PI.f64)))): 39 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)))) (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))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 39 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 (Rewrite<= associate-*r*_binary64 (*.f64 (/.f64 2 3) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 2 (fabs.f64 x)))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 4 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)))) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 2 (fabs.f64 x)) (*.f64 (/.f64 2 3) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))))) (sqrt.f64 (PI.f64)))): 43 points increase in error, 0 points decrease in error
   (fabs.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 (/.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, 39 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)))): 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)))): 39 points increase in error, 4 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))))))): 0 points increase in error, 0 points decrease in error

3. Applied egg-rr 0.1

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

4. Applied egg-rr 0.1

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

5. Final simplification 0.1

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

Alternatives

Alternative 1
Error	1.1
Cost	32832

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

Alternative 2
Error	0.5
Cost	32772

\[\begin{array}{l} \mathbf{if}\;x \leq -1.85:\\ \;\;\;\;\left|\frac{0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)\right|\\ \end{array} \]

Alternative 3
Error	0.6
Cost	26436

\[\begin{array}{l} \mathbf{if}\;x \leq -2.2:\\ \;\;\;\;\left|\frac{0.047619047619047616}{\frac{\sqrt{\pi}}{{x}^{7}}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)\right|\\ \end{array} \]

Alternative 4
Error	0.8
Cost	26052

\[\begin{array}{l} \mathbf{if}\;x \leq -2.2:\\ \;\;\;\;\left|\frac{0.047619047619047616}{\frac{\sqrt{\pi}}{{x}^{7}}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot 2 + x \cdot \left(x \cdot \left(x \cdot 0.6666666666666666\right)\right)}{\frac{1}{{\pi}^{-0.5}}}\right|\\ \end{array} \]

Alternative 5
Error	4.4
Cost	20160

\[\left|\frac{x \cdot 2 + x \cdot \left(x \cdot \left(x \cdot 0.6666666666666666\right)\right)}{\frac{1}{{\pi}^{-0.5}}}\right| \]

Alternative 6
Error	4.6
Cost	19840

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

Alternative 7
Error	4.5
Cost	19456

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

Reproduce

herbie shell --seed 2022343 
(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)))))))