Jmat.Real.erf

Average Error: 13.4 → 12.6

Time: 24.6s

Precision: binary64

Cost: 164864

Math

\[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

↓

\[\begin{array}{l} t_0 := e^{-x \cdot x}\\ t_1 := 1 + 0.3275911 \cdot \left|x\right|\\ t_2 := \frac{1}{t_1}\\ t_3 := -1.453152027 + \frac{1.061405429}{t_1}\\ t_4 := t_2 \cdot \left(t_0 \cdot \left(0.254829592 + t_2 \cdot \left(-0.284496736 + t_2 \cdot \left(1.421413741 + t_2 \cdot t_3\right)\right)\right)\right)\\ \frac{\log \left(e^{1 - {\left(t_2 \cdot \left(\left(0.254829592 + t_2 \cdot \left(-0.284496736 + t_2 \cdot \left(1.421413741 + t_2 \cdot \sqrt[3]{t_3 \cdot {t_3}^{2}}\right)\right)\right) \cdot t_0\right)\right)}^{3}}\right)}{1 + t_4 \cdot \left(1 + t_4\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (-
  1.0
  (*
   (*
    (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x))))
    (+
     0.254829592
     (*
      (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x))))
      (+
       -0.284496736
       (*
        (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x))))
        (+
         1.421413741
         (*
          (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x))))
          (+
           -1.453152027
           (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) 1.061405429)))))))))
   (exp (- (* (fabs x) (fabs x)))))))

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- (* x x))))
        (t_1 (+ 1.0 (* 0.3275911 (fabs x))))
        (t_2 (/ 1.0 t_1))
        (t_3 (+ -1.453152027 (/ 1.061405429 t_1)))
        (t_4
         (*
          t_2
          (*
           t_0
           (+
            0.254829592
            (* t_2 (+ -0.284496736 (* t_2 (+ 1.421413741 (* t_2 t_3))))))))))
   (/
    (log
     (exp
      (-
       1.0
       (pow
        (*
         t_2
         (*
          (+
           0.254829592
           (*
            t_2
            (+
             -0.284496736
             (* t_2 (+ 1.421413741 (* t_2 (cbrt (* t_3 (pow t_3 2.0)))))))))
          t_0))
        3.0))))
    (+ 1.0 (* t_4 (+ 1.0 t_4))))))

C

double code(double x) {
	return 1.0 - (((1.0 / (1.0 + (0.3275911 * fabs(x)))) * (0.254829592 + ((1.0 / (1.0 + (0.3275911 * fabs(x)))) * (-0.284496736 + ((1.0 / (1.0 + (0.3275911 * fabs(x)))) * (1.421413741 + ((1.0 / (1.0 + (0.3275911 * fabs(x)))) * (-1.453152027 + ((1.0 / (1.0 + (0.3275911 * fabs(x)))) * 1.061405429))))))))) * exp(-(fabs(x) * fabs(x))));
}

↓

double code(double x) {
	double t_0 = exp(-(x * x));
	double t_1 = 1.0 + (0.3275911 * fabs(x));
	double t_2 = 1.0 / t_1;
	double t_3 = -1.453152027 + (1.061405429 / t_1);
	double t_4 = t_2 * (t_0 * (0.254829592 + (t_2 * (-0.284496736 + (t_2 * (1.421413741 + (t_2 * t_3)))))));
	return log(exp((1.0 - pow((t_2 * ((0.254829592 + (t_2 * (-0.284496736 + (t_2 * (1.421413741 + (t_2 * cbrt((t_3 * pow(t_3, 2.0))))))))) * t_0)), 3.0)))) / (1.0 + (t_4 * (1.0 + t_4)));
}

Java

public static double code(double x) {
	return 1.0 - (((1.0 / (1.0 + (0.3275911 * Math.abs(x)))) * (0.254829592 + ((1.0 / (1.0 + (0.3275911 * Math.abs(x)))) * (-0.284496736 + ((1.0 / (1.0 + (0.3275911 * Math.abs(x)))) * (1.421413741 + ((1.0 / (1.0 + (0.3275911 * Math.abs(x)))) * (-1.453152027 + ((1.0 / (1.0 + (0.3275911 * Math.abs(x)))) * 1.061405429))))))))) * Math.exp(-(Math.abs(x) * Math.abs(x))));
}

↓

public static double code(double x) {
	double t_0 = Math.exp(-(x * x));
	double t_1 = 1.0 + (0.3275911 * Math.abs(x));
	double t_2 = 1.0 / t_1;
	double t_3 = -1.453152027 + (1.061405429 / t_1);
	double t_4 = t_2 * (t_0 * (0.254829592 + (t_2 * (-0.284496736 + (t_2 * (1.421413741 + (t_2 * t_3)))))));
	return Math.log(Math.exp((1.0 - Math.pow((t_2 * ((0.254829592 + (t_2 * (-0.284496736 + (t_2 * (1.421413741 + (t_2 * Math.cbrt((t_3 * Math.pow(t_3, 2.0))))))))) * t_0)), 3.0)))) / (1.0 + (t_4 * (1.0 + t_4)));
}

Julia

function code(x)
	return Float64(1.0 - Float64(Float64(Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x)))) * Float64(0.254829592 + Float64(Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x)))) * Float64(-0.284496736 + Float64(Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x)))) * Float64(1.421413741 + Float64(Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x)))) * Float64(-1.453152027 + Float64(Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x)))) * 1.061405429))))))))) * exp(Float64(-Float64(abs(x) * abs(x))))))
end

↓

function code(x)
	t_0 = exp(Float64(-Float64(x * x)))
	t_1 = Float64(1.0 + Float64(0.3275911 * abs(x)))
	t_2 = Float64(1.0 / t_1)
	t_3 = Float64(-1.453152027 + Float64(1.061405429 / t_1))
	t_4 = Float64(t_2 * Float64(t_0 * Float64(0.254829592 + Float64(t_2 * Float64(-0.284496736 + Float64(t_2 * Float64(1.421413741 + Float64(t_2 * t_3))))))))
	return Float64(log(exp(Float64(1.0 - (Float64(t_2 * Float64(Float64(0.254829592 + Float64(t_2 * Float64(-0.284496736 + Float64(t_2 * Float64(1.421413741 + Float64(t_2 * cbrt(Float64(t_3 * (t_3 ^ 2.0))))))))) * t_0)) ^ 3.0)))) / Float64(1.0 + Float64(t_4 * Float64(1.0 + t_4))))
end

Wolfram

code[x_] := N[(1.0 - N[(N[(N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.254829592 + N[(N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.284496736 + N[(N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.421413741 + N[(N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.453152027 + N[(N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.061405429), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := Block[{t$95$0 = N[Exp[(-N[(x * x), $MachinePrecision])], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(-1.453152027 + N[(1.061405429 / t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * N[(t$95$0 * N[(0.254829592 + N[(t$95$2 * N[(-0.284496736 + N[(t$95$2 * N[(1.421413741 + N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[Log[N[Exp[N[(1.0 - N[Power[N[(t$95$2 * N[(N[(0.254829592 + N[(t$95$2 * N[(-0.284496736 + N[(t$95$2 * N[(1.421413741 + N[(t$95$2 * N[Power[N[(t$95$3 * N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[(1.0 + N[(t$95$4 * N[(1.0 + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Initial program 13.4

   \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

2. Simplified 13.4

   \[\leadsto \color{blue}{1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)} \]
   Proof

   [Start] 13.4

   \[ 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   associate-*l* [=>] 13.4

   \[ 1 - \color{blue}{\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)} \]

3. Applied egg-rr 13.4

   \[\leadsto \color{blue}{\frac{1 - {\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)\right)}^{3}}{1 + \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)\right) \cdot \left(1 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)\right)}} \]

4. Applied egg-rr 12.6

   \[\leadsto \frac{\color{blue}{\log \left(e^{1 - {\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)\right)}^{3}}\right)}}{1 + \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)\right) \cdot \left(1 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)\right)} \]

5. Applied egg-rr 12.6

   \[\leadsto \frac{\log \left(e^{1 - {\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\sqrt[3]{\left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right) \cdot {\left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)}^{2}}}\right)\right)\right) \cdot e^{-x \cdot x}\right)\right)}^{3}}\right)}{1 + \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)\right) \cdot \left(1 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)\right)} \]

6. Final simplification 12.6

   \[\leadsto \frac{\log \left(e^{1 - {\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \sqrt[3]{\left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right) \cdot {\left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)}^{2}}\right)\right)\right) \cdot e^{-x \cdot x}\right)\right)}^{3}}\right)}{1 + \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(e^{-x \cdot x} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right)\right) \cdot \left(1 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(e^{-x \cdot x} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right)\right)\right)} \]

Alternatives

Alternative 1
Error	12.6
Cost	144960

\[\begin{array}{l} t_0 := 1 + 0.3275911 \cdot \left|x\right|\\ t_1 := \frac{1}{t_0}\\ t_2 := t_1 \cdot \left(e^{-x \cdot x} \cdot \left(0.254829592 + t_1 \cdot \left(-0.284496736 + t_1 \cdot \left(1.421413741 + t_1 \cdot \left(-1.453152027 + \frac{1.061405429}{t_0}\right)\right)\right)\right)\right)\\ \frac{\log \left(e^{1 - {t_2}^{3}}\right)}{1 + t_2 \cdot \left(1 + t_2\right)} \end{array} \]

Alternative 2
Error	13.4
Cost	138624

\[\begin{array}{l} t_0 := 1 + 0.3275911 \cdot \left|x\right|\\ t_1 := \frac{1}{t_0}\\ t_2 := t_1 \cdot \left(e^{-x \cdot x} \cdot \left(0.254829592 + t_1 \cdot \left(-0.284496736 + t_1 \cdot \left(1.421413741 + t_1 \cdot \left(-1.453152027 + \frac{1.061405429}{t_0}\right)\right)\right)\right)\right)\\ \frac{\frac{1 - {t_2}^{4}}{1 + {t_2}^{2}}}{1 + t_2} \end{array} \]

Alternative 3
Error	13.4
Cost	103488

\[\begin{array}{l} t_0 := e^{-x \cdot x}\\ t_1 := 1 + 0.3275911 \cdot \left|x\right|\\ t_2 := \frac{1.061405429}{t_1}\\ t_3 := \frac{1}{t_1}\\ \frac{1 - {\left(\frac{t_0 \cdot \left(-0.254829592 + t_3 \cdot \left(0.284496736 - t_3 \cdot \left(1.421413741 + t_3 \cdot \frac{1.126581484710674 \cdot {t_1}^{-2} + -2.111650813574209}{t_2 + 1.453152027}\right)\right)\right)}{t_1}\right)}^{2}}{1 - \frac{t_0 \cdot \left(-0.254829592 + t_3 \cdot \left(0.284496736 + \left(1.421413741 + t_3 \cdot \left(-1.453152027 + t_2\right)\right) \cdot \frac{-1}{t_1}\right)\right)}{t_1}} \end{array} \]

Alternative 4
Error	13.4
Cost	55360

\[\begin{array}{l} t_0 := 1 + 0.3275911 \cdot \left|x\right|\\ t_1 := \frac{1}{t_0}\\ 1 + \left(e^{-x \cdot x} \cdot \left(0.254829592 + t_1 \cdot \left(-0.284496736 + t_1 \cdot \left(1.421413741 + t_1 \cdot \frac{1.126581484710674 \cdot {t_0}^{-2} + -2.111650813574209}{\frac{1.061405429}{t_0} + 1.453152027}\right)\right)\right)\right) \cdot \frac{-1}{t_0} \end{array} \]

Alternative 5
Error	13.4
Cost	54528

\[\begin{array}{l} t_0 := 0.3275911 \cdot \left|x\right|\\ t_1 := 1 + t_0\\ t_2 := \frac{1}{t_1}\\ 1 + \frac{e^{-x \cdot x} \cdot \left(-0.254829592 + t_2 \cdot \left(0.284496736 + \left(1.421413741 + t_2 \cdot \left(-1.453152027 + \frac{1.061405429}{t_1}\right)\right) \cdot \frac{-1}{t_1}\right)\right)}{1 + \log \left(e^{t_0}\right)} \end{array} \]

Alternative 6
Error	13.4
Cost	48768

\[\begin{array}{l} t_0 := 1 + 0.3275911 \cdot \left|x\right|\\ t_1 := \frac{1}{t_0}\\ 1 + \frac{e^{-x \cdot x} \cdot \left(-0.254829592 + t_1 \cdot \left(0.284496736 + t_1 \cdot \left(t_1 \cdot 1.453152027 - \left(1.421413741 + 1.061405429 \cdot \left(t_1 \cdot t_1\right)\right)\right)\right)\right)}{t_0} \end{array} \]

Alternative 7
Error	13.4
Cost	48192

\[\begin{array}{l} t_0 := 1 + 0.3275911 \cdot \left|x\right|\\ 1 + \frac{e^{-x \cdot x} \cdot \left(-0.254829592 + \frac{0.284496736 + \left(\left(1.421413741 + 1.061405429 \cdot \frac{1}{{t_0}^{2}}\right) + \frac{1}{t_0} \cdot -1.453152027\right) \cdot \frac{-1}{t_0}}{t_0}\right)}{t_0} \end{array} \]

Alternative 8
Error	13.4
Cost	41728

\[\begin{array}{l} t_0 := 1 + 0.3275911 \cdot \left|x\right|\\ t_1 := \frac{1}{t_0}\\ 1 + \frac{e^{-x \cdot x} \cdot \left(-0.254829592 + t_1 \cdot \left(0.284496736 + \left(1.421413741 + t_1 \cdot \left(-1.453152027 + \frac{1.061405429}{t_0}\right)\right) \cdot \frac{-1}{t_0}\right)\right)}{t_0} \end{array} \]

Alternative 9
Error	13.8
Cost	35328

\[\begin{array}{l} t_0 := 1 + 0.3275911 \cdot \left|x\right|\\ t_1 := \frac{1}{t_0}\\ 1 + \frac{e^{-x \cdot x} \cdot \left(-0.254829592 + t_1 \cdot \left(0.284496736 + \left(1.421413741 + t_1 \cdot \left(-1.453152027 + \frac{1.061405429}{t_0}\right)\right) \cdot \frac{-1}{t_0}\right)\right)}{1 + 0.3275911 \cdot x} \end{array} \]

Alternative 10
Error	28.6
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023187 
(FPCore (x)
  :name "Jmat.Real.erf"
  :precision binary64
  (- 1.0 (* (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ 0.254829592 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ -0.284496736 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ 1.421413741 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ -1.453152027 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) 1.061405429))))))))) (exp (- (* (fabs x) (fabs x)))))))