Jmat.Real.erf

Average Error: 13.4 → 0.2

Time: 13.4s

Precision: binary64

Cost: 54916

Math

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

↓

\[\begin{array}{l} t_0 := e^{x \cdot \left(-x\right)}\\ t_1 := 1 + 0.3275911 \cdot \left|x\right|\\ t_2 := \frac{1}{t_1}\\ \mathbf{if}\;x \leq -0.0013079059934821581:\\ \;\;\;\;1 + t_0 \cdot \left(t_2 \cdot \left(-0.254829592 + \frac{\left(0.284496736 - \frac{1}{{t_1}^{2}} \cdot -1.453152027\right) + \left(1.061405429 \cdot \frac{-1}{{t_1}^{3}} + t_2 \cdot -1.421413741\right)}{t_1}\right)\right)\\ \mathbf{elif}\;x \leq 2.3179839018548173 \cdot 10^{-8}:\\ \;\;\;\;10^{-9} + \sqrt{\left(x \cdot x\right) \cdot 1.2732557730789702}\\ \mathbf{else}:\\ \;\;\;\;1 + t_0 \cdot \left(t_2 \cdot \left(-0.254829592 + t_2 \cdot \left(0.284496736 + t_2 \cdot \left(-1.421413741 + \frac{1.453152027 + \frac{-1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right)\\ \end{array} \]

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)))
   (if (<= x -0.0013079059934821581)
     (+
      1.0
      (*
       t_0
       (*
        t_2
        (+
         -0.254829592
         (/
          (+
           (- 0.284496736 (* (/ 1.0 (pow t_1 2.0)) -1.453152027))
           (+ (* 1.061405429 (/ -1.0 (pow t_1 3.0))) (* t_2 -1.421413741)))
          t_1)))))
     (if (<= x 2.3179839018548173e-8)
       (+ 1e-9 (sqrt (* (* x x) 1.2732557730789702)))
       (+
        1.0
        (*
         t_0
         (*
          t_2
          (+
           -0.254829592
           (*
            t_2
            (+
             0.284496736
             (*
              t_2
              (+
               -1.421413741
               (/
                (+ 1.453152027 (/ -1.061405429 (fma 0.3275911 x 1.0)))
                (fma 0.3275911 x 1.0))))))))))))))

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 tmp;
	if (x <= -0.0013079059934821581) {
		tmp = 1.0 + (t_0 * (t_2 * (-0.254829592 + (((0.284496736 - ((1.0 / pow(t_1, 2.0)) * -1.453152027)) + ((1.061405429 * (-1.0 / pow(t_1, 3.0))) + (t_2 * -1.421413741))) / t_1))));
	} else if (x <= 2.3179839018548173e-8) {
		tmp = 1e-9 + sqrt(((x * x) * 1.2732557730789702));
	} else {
		tmp = 1.0 + (t_0 * (t_2 * (-0.254829592 + (t_2 * (0.284496736 + (t_2 * (-1.421413741 + ((1.453152027 + (-1.061405429 / fma(0.3275911, x, 1.0))) / fma(0.3275911, x, 1.0)))))))));
	}
	return tmp;
}

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(x * Float64(-x)))
	t_1 = Float64(1.0 + Float64(0.3275911 * abs(x)))
	t_2 = Float64(1.0 / t_1)
	tmp = 0.0
	if (x <= -0.0013079059934821581)
		tmp = Float64(1.0 + Float64(t_0 * Float64(t_2 * Float64(-0.254829592 + Float64(Float64(Float64(0.284496736 - Float64(Float64(1.0 / (t_1 ^ 2.0)) * -1.453152027)) + Float64(Float64(1.061405429 * Float64(-1.0 / (t_1 ^ 3.0))) + Float64(t_2 * -1.421413741))) / t_1)))));
	elseif (x <= 2.3179839018548173e-8)
		tmp = Float64(1e-9 + sqrt(Float64(Float64(x * x) * 1.2732557730789702)));
	else
		tmp = Float64(1.0 + Float64(t_0 * Float64(t_2 * Float64(-0.254829592 + Float64(t_2 * Float64(0.284496736 + Float64(t_2 * Float64(-1.421413741 + Float64(Float64(1.453152027 + Float64(-1.061405429 / fma(0.3275911, x, 1.0))) / fma(0.3275911, x, 1.0))))))))));
	end
	return tmp
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]}, If[LessEqual[x, -0.0013079059934821581], N[(1.0 + N[(t$95$0 * N[(t$95$2 * N[(-0.254829592 + N[(N[(N[(0.284496736 - N[(N[(1.0 / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] * -1.453152027), $MachinePrecision]), $MachinePrecision] + N[(N[(1.061405429 * N[(-1.0 / N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * -1.421413741), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.3179839018548173e-8], N[(1e-9 + N[Sqrt[N[(N[(x * x), $MachinePrecision] * 1.2732557730789702), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(t$95$0 * N[(t$95$2 * N[(-0.254829592 + N[(t$95$2 * N[(0.284496736 + N[(t$95$2 * N[(-1.421413741 + N[(N[(1.453152027 + N[(-1.061405429 / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00130790599348215813

   1. Initial program 0.1

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

   2. Taylor expanded in x around 0 0.1

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

   3. Taylor expanded in x around inf 0.1

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

   if -0.00130790599348215813 < x < 2.3179839018548173e-8

   1. Initial program 27.0

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

   2. Applied egg-rr 27.5

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

   3. Taylor expanded in x around 0 1.3

      \[\leadsto \color{blue}{10^{-9} + 1.128386358070218 \cdot x} \]

   4. Applied egg-rr 32.7

      \[\leadsto 10^{-9} + \color{blue}{e^{\log \left(1.128386358070218 \cdot x\right)}} \]

   5. Applied egg-rr 0.2

      \[\leadsto 10^{-9} + \color{blue}{\sqrt{1.2732557730789702 \cdot \left(x \cdot x\right)}} \]

   if 2.3179839018548173e-8 < x

   1. Initial program 0.3

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

   2. Applied egg-rr 0.3

      \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\left(1 \cdot \left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0013079059934821581:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.254829592 + \frac{\left(0.284496736 - \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}} \cdot -1.453152027\right) + \left(1.061405429 \cdot \frac{-1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot -1.421413741\right)}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\\ \mathbf{elif}\;x \leq 2.3179839018548173 \cdot 10^{-8}:\\ \;\;\;\;10^{-9} + \sqrt{\left(x \cdot x\right) \cdot 1.2732557730789702}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.421413741 + \frac{1.453152027 + \frac{-1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	48516

\[\begin{array}{l} t_0 := 1 + 0.3275911 \cdot \left|x\right|\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;\left|x\right| \leq 4 \cdot 10^{-8}:\\ \;\;\;\;10^{-9} + \sqrt{\left(x \cdot x\right) \cdot 1.2732557730789702}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(t_1 \cdot \left(-0.254829592 + t_1 \cdot \left(0.284496736 + \left(1.421413741 + t_1 \cdot \left(-1.453152027 + t_1 \cdot 1.061405429\right)\right) \cdot \frac{-1}{t_0}\right)\right)\right)\\ \end{array} \]

Alternative 2
Error	0.4
Cost	48004

\[\begin{array}{l} t_0 := \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\\ \mathbf{if}\;\left|x\right| \leq 0.002:\\ \;\;\;\;10^{-9} + \sqrt{\left(x \cdot x\right) \cdot 1.2732557730789702}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(t_0 \cdot \left(-0.254829592 + t_0 \cdot \left(0.284496736 + t_0 \cdot \left(-1.421413741 + \frac{1.453152027 + \frac{-1.061405429}{\mathsf{fma}\left(0.3275911, x, 1\right)}}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)\right)\right)\right)\\ \end{array} \]

Alternative 3
Error	0.5
Cost	46532

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.002:\\ \;\;\;\;10^{-9} + \sqrt{\left(x \cdot x\right) \cdot 1.2732557730789702}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.254829592 + \left(1.029667143 \cdot \frac{-1}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}} + \frac{0.284496736}{\mathsf{fma}\left(0.3275911, x, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}\\ \end{array} \]

Alternative 4
Error	0.7
Cost	33480

\[\begin{array}{l} \mathbf{if}\;x \leq -2825946.7282828824:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.6180735205572689:\\ \;\;\;\;10^{-9} + \sqrt{\left(x \cdot x\right) \cdot 1.2732557730789702}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{0.284496736 + \left(-0.254829592 + \frac{-1.421413741}{{\left(\mathsf{fma}\left(0.3275911, x, 1\right)\right)}^{2}}\right)}{\mathsf{fma}\left(0.3275911, x, 1\right) \cdot {\left(e^{x}\right)}^{x}}\\ \end{array} \]

Alternative 5
Error	0.7
Cost	13576

\[\begin{array}{l} \mathbf{if}\;x \leq -2825946.7282828824:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.6180735205572689:\\ \;\;\;\;10^{-9} + \sqrt{\left(x \cdot x\right) \cdot 1.2732557730789702}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.7778892405807117}{x \cdot {\left(e^{x}\right)}^{x}}\\ \end{array} \]

Alternative 6
Error	0.7
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -2825946.7282828824:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.6180735205572689:\\ \;\;\;\;10^{-9} + \sqrt{\left(x \cdot x\right) \cdot 1.2732557730789702}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	1.0
Cost	1096

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0013079059934821581:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.6180735205572689:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot 1.2732557730789702 + -1 \cdot 10^{-18}}{x \cdot 1.128386358070218 + -1 \cdot 10^{-9}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Error	1.0
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0013079059934821581:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 0.6180735205572689:\\ \;\;\;\;10^{-9} + x \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Error	1.5
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0013079059934821581:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.3179839018548173 \cdot 10^{-8}:\\ \;\;\;\;10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10
Error	29.8
Cost	64

\[10^{-9} \]

Reproduce

herbie shell --seed 2022300 
(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)))))))