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

Average Error: 0.2 → 0.1

Time: 8.5s

Precision: binary64

Cost: 39552

\[x \leq 0.5\]

Math

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

↓

\[\left|x \cdot \frac{\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

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
  (*
   x
   (/
    (+
     (+ (* 0.2 (pow x 4.0)) (* 0.047619047619047616 (pow x 6.0)))
     (fma 0.6666666666666666 (* x x) 2.0))
    (sqrt PI)))))

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((x * ((((0.2 * pow(x, 4.0)) + (0.047619047619047616 * pow(x, 6.0))) + fma(0.6666666666666666, (x * x), 2.0)) / sqrt(((double) M_PI)))));
}

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(x * Float64(Float64(Float64(Float64(0.2 * (x ^ 4.0)) + Float64(0.047619047619047616 * (x ^ 6.0))) + fma(0.6666666666666666, Float64(x * x), 2.0)) / sqrt(pi))))
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[(x * N[(N[(N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $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 0.5

   \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|}} \]
   Proof
   (/.f64 (fabs.f64 x) (fabs.f64 (/.f64 (sqrt.f64 (PI.f64)) (+.f64 (fma.f64 1/5 (pow.f64 x 4) (*.f64 1/21 (pow.f64 x 6))) (fma.f64 2/3 (*.f64 x x) 2))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= fabs-fabs_binary64 (fabs.f64 (fabs.f64 x))) (fabs.f64 (/.f64 (sqrt.f64 (PI.f64)) (+.f64 (fma.f64 1/5 (pow.f64 x 4) (*.f64 1/21 (pow.f64 x 6))) (fma.f64 2/3 (*.f64 x x) 2))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fabs.f64 (fabs.f64 x)) (fabs.f64 (/.f64 (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 2/3 (*.f64 x x) 2))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fabs.f64 (fabs.f64 x)) (fabs.f64 (/.f64 (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 2/3 (*.f64 x x) 2))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fabs.f64 (fabs.f64 x)) (fabs.f64 (/.f64 (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 2/3 (*.f64 x x) 2))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fabs.f64 (fabs.f64 x)) (fabs.f64 (/.f64 (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 2/3 (*.f64 x x) 2))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fabs.f64 (fabs.f64 x)) (fabs.f64 (/.f64 (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 2/3 (*.f64 x x) 2))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fabs.f64 (fabs.f64 x)) (fabs.f64 (/.f64 (sqrt.f64 (PI.f64)) (+.f64 (fma.f64 (/.f64 1 5) (*.f64 (*.f64 x x) (Rewrite<= sqr-abs_binary64 (*.f64 (fabs.f64 x) (fabs.f64 x)))) (*.f64 1/21 (pow.f64 x 6))) (fma.f64 2/3 (*.f64 x x) 2))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fabs.f64 (fabs.f64 x)) (fabs.f64 (/.f64 (sqrt.f64 (PI.f64)) (+.f64 (fma.f64 (/.f64 1 5) (*.f64 (Rewrite<= sqr-abs_binary64 (*.f64 (fabs.f64 x) (fabs.f64 x))) (*.f64 (fabs.f64 x) (fabs.f64 x))) (*.f64 1/21 (pow.f64 x 6))) (fma.f64 2/3 (*.f64 x x) 2))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fabs.f64 (fabs.f64 x)) (fabs.f64 (/.f64 (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 2/3 (*.f64 x x) 2))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fabs.f64 (fabs.f64 x)) (fabs.f64 (/.f64 (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 2/3 (*.f64 x x) 2))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fabs.f64 (fabs.f64 x)) (fabs.f64 (/.f64 (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 2/3 (*.f64 x x) 2))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fabs.f64 (fabs.f64 x)) (fabs.f64 (/.f64 (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 2/3 (*.f64 x x) 2))))): 6 points increase in error, 3 points decrease in error
   (/.f64 (fabs.f64 (fabs.f64 x)) (fabs.f64 (/.f64 (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 2/3 (*.f64 x x) 2))))): 13 points increase in error, 6 points decrease in error
   (/.f64 (fabs.f64 (fabs.f64 x)) (fabs.f64 (/.f64 (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 2/3 (*.f64 x x) 2))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fabs.f64 (fabs.f64 x)) (fabs.f64 (/.f64 (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 2/3 (*.f64 x x) 2))))): 0 points increase in error, 2 points decrease in error
   (/.f64 (fabs.f64 (fabs.f64 x)) (fabs.f64 (/.f64 (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 2/3 (*.f64 x x) 2))))): 4 points increase in error, 7 points decrease in error
   (/.f64 (fabs.f64 (fabs.f64 x)) (fabs.f64 (/.f64 (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 2/3 (*.f64 x x) 2))))): 6 points increase in error, 5 points decrease in error
   (/.f64 (fabs.f64 (fabs.f64 x)) (fabs.f64 (/.f64 (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 2/3 (*.f64 x x) 2))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fabs.f64 (fabs.f64 x)) (fabs.f64 (/.f64 (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<= metadata-eval (/.f64 2 3)) (*.f64 x x) 2))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fabs.f64 (fabs.f64 x)) (fabs.f64 (/.f64 (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 2 3) (Rewrite<= sqr-abs_binary64 (*.f64 (fabs.f64 x) (fabs.f64 x))) 2))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fabs.f64 (fabs.f64 x)) (fabs.f64 (/.f64 (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 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) 2)))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fabs.f64 (fabs.f64 x)) (fabs.f64 (/.f64 (sqrt.f64 (PI.f64)) (Rewrite=> +-commutative_binary64 (+.f64 (+.f64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) 2) (+.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))))))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fabs.f64 (fabs.f64 x)) (fabs.f64 (/.f64 (sqrt.f64 (PI.f64)) (Rewrite<= *-rgt-identity_binary64 (*.f64 (+.f64 (+.f64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) 2) (+.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))))) 1))))): 0 points increase in error, 0 points decrease in error
   (/.f64 (fabs.f64 (fabs.f64 x)) (fabs.f64 (/.f64 (sqrt.f64 (PI.f64)) (Rewrite=> *-rgt-identity_binary64 (+.f64 (+.f64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) 2) (+.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))))))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fabs-div_binary64 (fabs.f64 (/.f64 (fabs.f64 x) (/.f64 (sqrt.f64 (PI.f64)) (+.f64 (+.f64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) 2) (+.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))))))))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (fabs.f64 x) (+.f64 (+.f64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) 2) (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))))) (sqrt.f64 (PI.f64))))): 7 points increase in error, 3 points decrease in error
   (fabs.f64 (/.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (fabs.f64 x) (+.f64 (*.f64 (/.f64 2 3) (*.f64 (fabs.f64 x) (fabs.f64 x))) 2)) (*.f64 (fabs.f64 x) (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))))))) (sqrt.f64 (PI.f64)))): 1 points increase in error, 0 points decrease in error
   (fabs.f64 (/.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 (fabs.f64 x) (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))))) (sqrt.f64 (PI.f64)))): 1 points increase in error, 0 points decrease in error
   (fabs.f64 (/.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 (fabs.f64 x) (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (+.f64 (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 (fabs.f64 x) (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (+.f64 (+.f64 (*.f64 2 (fabs.f64 x)) (*.f64 (/.f64 2 3) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (fabs.f64 x)) (*.f64 (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (fabs.f64 x))))) (sqrt.f64 (PI.f64)))): 1 points increase in error, 1 points decrease in error
   (fabs.f64 (/.f64 (+.f64 (+.f64 (*.f64 2 (fabs.f64 x)) (*.f64 (/.f64 2 3) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (fabs.f64 x)))) (sqrt.f64 (PI.f64)))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (/.f64 (+.f64 (+.f64 (*.f64 2 (fabs.f64 x)) (*.f64 (/.f64 2 3) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (+.f64 (*.f64 (/.f64 1 5) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))) (Rewrite<= associate-*r*_binary64 (*.f64 (/.f64 1 21) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))))) (sqrt.f64 (PI.f64)))): 1 points increase in error, 2 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)))): 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))))))): 24 points increase in error, 132 points decrease in error

3. Applied egg-rr 0.1

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

4. Taylor expanded in x around 0 0.1

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

5. Final simplification 0.1

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

Alternatives

Alternative 1
Error	1.0
Cost	26052

\[\begin{array}{l} \mathbf{if}\;x \leq -1.85:\\ \;\;\;\;0.047619047619047616 \cdot \left|\frac{{x}^{7}}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 2
Error	0.9
Cost	26052

\[\begin{array}{l} \mathbf{if}\;x \leq -1.85:\\ \;\;\;\;\left|\frac{\frac{-0.047619047619047616}{\sqrt{\pi}}}{{x}^{-7}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 3
Error	4.6
Cost	19456

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

Reproduce

herbie shell --seed 2022325 
(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)))))))