Jmat.Real.erf

Percentage Accurate: 79.3% → 99.7%

Time: 29.1s

Alternatives: 12

Speedup: 142.3×

Specification

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x))))))
   (-
    1.0
    (*
     (*
      t_0
      (+
       0.254829592
       (*
        t_0
        (+
         -0.284496736
         (*
          t_0
          (+ 1.421413741 (* t_0 (+ -1.453152027 (* t_0 1.061405429)))))))))
     (exp (- (* (fabs x) (fabs x))))))))

C

double code(double x) {
	double t_0 = 1.0 / (1.0 + (0.3275911 * fabs(x)));
	return 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * exp(-(fabs(x) * fabs(x))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = 1.0d0 / (1.0d0 + (0.3275911d0 * abs(x)))
    code = 1.0d0 - ((t_0 * (0.254829592d0 + (t_0 * ((-0.284496736d0) + (t_0 * (1.421413741d0 + (t_0 * ((-1.453152027d0) + (t_0 * 1.061405429d0))))))))) * exp(-(abs(x) * abs(x))))
end function

Java

public static double code(double x) {
	double t_0 = 1.0 / (1.0 + (0.3275911 * Math.abs(x)));
	return 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * Math.exp(-(Math.abs(x) * Math.abs(x))));
}

Python

def code(x):
	t_0 = 1.0 / (1.0 + (0.3275911 * math.fabs(x)))
	return 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * math.exp(-(math.fabs(x) * math.fabs(x))))

Julia

function code(x)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x))))
	return Float64(1.0 - Float64(Float64(t_0 * Float64(0.254829592 + Float64(t_0 * Float64(-0.284496736 + Float64(t_0 * Float64(1.421413741 + Float64(t_0 * Float64(-1.453152027 + Float64(t_0 * 1.061405429))))))))) * exp(Float64(-Float64(abs(x) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / (1.0 + (0.3275911 * abs(x)));
	tmp = 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * exp(-(abs(x) * abs(x))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(1.0 - N[(N[(t$95$0 * N[(0.254829592 + N[(t$95$0 * N[(-0.284496736 + N[(t$95$0 * N[(1.421413741 + N[(t$95$0 * N[(-1.453152027 + N[(t$95$0 * 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]]

Initial Program: 79.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x))))))
   (-
    1.0
    (*
     (*
      t_0
      (+
       0.254829592
       (*
        t_0
        (+
         -0.284496736
         (*
          t_0
          (+ 1.421413741 (* t_0 (+ -1.453152027 (* t_0 1.061405429)))))))))
     (exp (- (* (fabs x) (fabs x))))))))

C

double code(double x) {
	double t_0 = 1.0 / (1.0 + (0.3275911 * fabs(x)));
	return 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * exp(-(fabs(x) * fabs(x))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = 1.0d0 / (1.0d0 + (0.3275911d0 * abs(x)))
    code = 1.0d0 - ((t_0 * (0.254829592d0 + (t_0 * ((-0.284496736d0) + (t_0 * (1.421413741d0 + (t_0 * ((-1.453152027d0) + (t_0 * 1.061405429d0))))))))) * exp(-(abs(x) * abs(x))))
end function

Java

public static double code(double x) {
	double t_0 = 1.0 / (1.0 + (0.3275911 * Math.abs(x)));
	return 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * Math.exp(-(Math.abs(x) * Math.abs(x))));
}

Python

def code(x):
	t_0 = 1.0 / (1.0 + (0.3275911 * math.fabs(x)))
	return 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * math.exp(-(math.fabs(x) * math.fabs(x))))

Julia

function code(x)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x))))
	return Float64(1.0 - Float64(Float64(t_0 * Float64(0.254829592 + Float64(t_0 * Float64(-0.284496736 + Float64(t_0 * Float64(1.421413741 + Float64(t_0 * Float64(-1.453152027 + Float64(t_0 * 1.061405429))))))))) * exp(Float64(-Float64(abs(x) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / (1.0 + (0.3275911 * abs(x)));
	tmp = 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * exp(-(abs(x) * abs(x))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(1.0 - N[(N[(t$95$0 * N[(0.254829592 + N[(t$95$0 * N[(-0.284496736 + N[(t$95$0 * N[(1.421413741 + N[(t$95$0 * N[(-1.453152027 + N[(t$95$0 * 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]]

Alternative 1: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 0.254829592 + \mathsf{fma}\left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}, {\left(\mathsf{fma}\left(0.3275911, x_m, 1\right)\right)}^{-2}, \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}\right)\\ t_1 := e^{{x_m}^{2}}\\ \mathbf{if}\;\left|x_m\right| \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{10^{-9} + x_m \cdot -1.128386358070218}{10^{-18} - {x_m}^{2} \cdot 1.2732557730789702}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - {\left(\frac{t_0}{\mathsf{fma}\left(0.3275911, x_m, 1\right) \cdot t_1}\right)}^{2}}{1 + \frac{t_0}{t_1 \cdot \left(1 + x_m \cdot 0.3275911\right)}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0
         (+
          0.254829592
          (fma
           (+
            1.421413741
            (/
             (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
             (fma 0.3275911 x_m 1.0)))
           (pow (fma 0.3275911 x_m 1.0) -2.0)
           (/ -0.284496736 (fma 0.3275911 x_m 1.0)))))
        (t_1 (exp (pow x_m 2.0))))
   (if (<= (fabs x_m) 2e-7)
     (/
      1.0
      (/
       (+ 1e-9 (* x_m -1.128386358070218))
       (- 1e-18 (* (pow x_m 2.0) 1.2732557730789702))))
     (/
      (- 1.0 (pow (/ t_0 (* (fma 0.3275911 x_m 1.0) t_1)) 2.0))
      (+ 1.0 (/ t_0 (* t_1 (+ 1.0 (* x_m 0.3275911)))))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 0.254829592 + fma((1.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))), pow(fma(0.3275911, x_m, 1.0), -2.0), (-0.284496736 / fma(0.3275911, x_m, 1.0)));
	double t_1 = exp(pow(x_m, 2.0));
	double tmp;
	if (fabs(x_m) <= 2e-7) {
		tmp = 1.0 / ((1e-9 + (x_m * -1.128386358070218)) / (1e-18 - (pow(x_m, 2.0) * 1.2732557730789702)));
	} else {
		tmp = (1.0 - pow((t_0 / (fma(0.3275911, x_m, 1.0) * t_1)), 2.0)) / (1.0 + (t_0 / (t_1 * (1.0 + (x_m * 0.3275911)))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(0.254829592 + fma(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))), (fma(0.3275911, x_m, 1.0) ^ -2.0), Float64(-0.284496736 / fma(0.3275911, x_m, 1.0))))
	t_1 = exp((x_m ^ 2.0))
	tmp = 0.0
	if (abs(x_m) <= 2e-7)
		tmp = Float64(1.0 / Float64(Float64(1e-9 + Float64(x_m * -1.128386358070218)) / Float64(1e-18 - Float64((x_m ^ 2.0) * 1.2732557730789702))));
	else
		tmp = Float64(Float64(1.0 - (Float64(t_0 / Float64(fma(0.3275911, x_m, 1.0) * t_1)) ^ 2.0)) / Float64(1.0 + Float64(t_0 / Float64(t_1 * Float64(1.0 + Float64(x_m * 0.3275911))))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(0.254829592 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(0.3275911 * x$95$m + 1.0), $MachinePrecision], -2.0], $MachinePrecision] + N[(-0.284496736 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 2e-7], N[(1.0 / N[(N[(1e-9 + N[(x$95$m * -1.128386358070218), $MachinePrecision]), $MachinePrecision] / N[(1e-18 - N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 1.2732557730789702), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Power[N[(t$95$0 / N[(N[(0.3275911 * x$95$m + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$0 / N[(t$95$1 * N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 2: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \sqrt{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}\\ t_1 := 1 + \left|x_m\right| \cdot 0.3275911\\ t_2 := \frac{1}{t_1}\\ \mathbf{if}\;\left|x_m\right| \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{10^{-9} + x_m \cdot -1.128386358070218}{10^{-18} - {x_m}^{2} \cdot 1.2732557730789702}}\\ \mathbf{else}:\\ \;\;\;\;1 + t_2 \cdot \left(e^{x_m \cdot \left(-x_m\right)} \cdot \left(t_2 \cdot \left(\left(t_0 \cdot t_0\right) \cdot \frac{-1}{t_1} - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0
         (sqrt
          (+
           1.421413741
           (/
            (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
            (fma 0.3275911 x_m 1.0)))))
        (t_1 (+ 1.0 (* (fabs x_m) 0.3275911)))
        (t_2 (/ 1.0 t_1)))
   (if (<= (fabs x_m) 2e-7)
     (/
      1.0
      (/
       (+ 1e-9 (* x_m -1.128386358070218))
       (- 1e-18 (* (pow x_m 2.0) 1.2732557730789702))))
     (+
      1.0
      (*
       t_2
       (*
        (exp (* x_m (- x_m)))
        (-
         (* t_2 (- (* (* t_0 t_0) (/ -1.0 t_1)) -0.284496736))
         0.254829592)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = sqrt((1.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))));
	double t_1 = 1.0 + (fabs(x_m) * 0.3275911);
	double t_2 = 1.0 / t_1;
	double tmp;
	if (fabs(x_m) <= 2e-7) {
		tmp = 1.0 / ((1e-9 + (x_m * -1.128386358070218)) / (1e-18 - (pow(x_m, 2.0) * 1.2732557730789702)));
	} else {
		tmp = 1.0 + (t_2 * (exp((x_m * -x_m)) * ((t_2 * (((t_0 * t_0) * (-1.0 / t_1)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = sqrt(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))))
	t_1 = Float64(1.0 + Float64(abs(x_m) * 0.3275911))
	t_2 = Float64(1.0 / t_1)
	tmp = 0.0
	if (abs(x_m) <= 2e-7)
		tmp = Float64(1.0 / Float64(Float64(1e-9 + Float64(x_m * -1.128386358070218)) / Float64(1e-18 - Float64((x_m ^ 2.0) * 1.2732557730789702))));
	else
		tmp = Float64(1.0 + Float64(t_2 * Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(Float64(t_2 * Float64(Float64(Float64(t_0 * t_0) * Float64(-1.0 / t_1)) - -0.284496736)) - 0.254829592))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Sqrt[N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / t$95$1), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 2e-7], N[(1.0 / N[(N[(1e-9 + N[(x$95$m * -1.128386358070218), $MachinePrecision]), $MachinePrecision] / N[(1e-18 - N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 1.2732557730789702), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(t$95$2 * N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$2 * N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] - -0.284496736), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 3: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{10^{-9} + x_m \cdot -1.128386358070218}{10^{-18} - {x_m}^{2} \cdot 1.2732557730789702}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{\frac{\left(0.284496736 + {\left(\mathsf{fma}\left(0.3275911, x_m, 1\right)\right)}^{-2} \cdot 1.453152027\right) - \mathsf{fma}\left(1.061405429, {\left(\mathsf{fma}\left(0.3275911, x_m, 1\right)\right)}^{-3}, \frac{1.421413741}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, x_m, 1\right)} - 0.254829592}{e^{x_m \cdot x_m}}}{1 + x_m \cdot 0.3275911}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 2e-7)
   (/
    1.0
    (/
     (+ 1e-9 (* x_m -1.128386358070218))
     (- 1e-18 (* (pow x_m 2.0) 1.2732557730789702))))
   (+
    1.0
    (/
     (/
      (-
       (/
        (-
         (+ 0.284496736 (* (pow (fma 0.3275911 x_m 1.0) -2.0) 1.453152027))
         (fma
          1.061405429
          (pow (fma 0.3275911 x_m 1.0) -3.0)
          (/ 1.421413741 (fma 0.3275911 x_m 1.0))))
        (fma 0.3275911 x_m 1.0))
       0.254829592)
      (exp (* x_m x_m)))
     (+ 1.0 (* x_m 0.3275911))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 2e-7) {
		tmp = 1.0 / ((1e-9 + (x_m * -1.128386358070218)) / (1e-18 - (pow(x_m, 2.0) * 1.2732557730789702)));
	} else {
		tmp = 1.0 + ((((((0.284496736 + (pow(fma(0.3275911, x_m, 1.0), -2.0) * 1.453152027)) - fma(1.061405429, pow(fma(0.3275911, x_m, 1.0), -3.0), (1.421413741 / fma(0.3275911, x_m, 1.0)))) / fma(0.3275911, x_m, 1.0)) - 0.254829592) / exp((x_m * x_m))) / (1.0 + (x_m * 0.3275911)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 2e-7)
		tmp = Float64(1.0 / Float64(Float64(1e-9 + Float64(x_m * -1.128386358070218)) / Float64(1e-18 - Float64((x_m ^ 2.0) * 1.2732557730789702))));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(Float64(Float64(Float64(0.284496736 + Float64((fma(0.3275911, x_m, 1.0) ^ -2.0) * 1.453152027)) - fma(1.061405429, (fma(0.3275911, x_m, 1.0) ^ -3.0), Float64(1.421413741 / fma(0.3275911, x_m, 1.0)))) / fma(0.3275911, x_m, 1.0)) - 0.254829592) / exp(Float64(x_m * x_m))) / Float64(1.0 + Float64(x_m * 0.3275911))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 2e-7], N[(1.0 / N[(N[(1e-9 + N[(x$95$m * -1.128386358070218), $MachinePrecision]), $MachinePrecision] / N[(1e-18 - N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 1.2732557730789702), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(N[(N[(N[(0.284496736 + N[(N[Power[N[(0.3275911 * x$95$m + 1.0), $MachinePrecision], -2.0], $MachinePrecision] * 1.453152027), $MachinePrecision]), $MachinePrecision] - N[(1.061405429 * N[Power[N[(0.3275911 * x$95$m + 1.0), $MachinePrecision], -3.0], $MachinePrecision] + N[(1.421413741 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision] / N[Exp[N[(x$95$m * x$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 4: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{10^{-9} + x_m \cdot -1.128386358070218}{10^{-18} - {x_m}^{2} \cdot 1.2732557730789702}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{0.254829592 + \left(\frac{-0.284496736}{\mathsf{fma}\left(0.3275911, x_m, 1\right)} + \left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}\right) \cdot {\left(\mathsf{fma}\left(0.3275911, x_m, 1\right)\right)}^{-2}\right)}{e^{x_m \cdot x_m}}}{\mathsf{fma}\left(0.3275911, \left|x_m\right|, 1\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 2e-7)
   (/
    1.0
    (/
     (+ 1e-9 (* x_m -1.128386358070218))
     (- 1e-18 (* (pow x_m 2.0) 1.2732557730789702))))
   (-
    1.0
    (/
     (/
      (+
       0.254829592
       (+
        (/ -0.284496736 (fma 0.3275911 x_m 1.0))
        (*
         (+
          1.421413741
          (/
           (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
           (fma 0.3275911 x_m 1.0)))
         (pow (fma 0.3275911 x_m 1.0) -2.0))))
      (exp (* x_m x_m)))
     (fma 0.3275911 (fabs x_m) 1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 2e-7) {
		tmp = 1.0 / ((1e-9 + (x_m * -1.128386358070218)) / (1e-18 - (pow(x_m, 2.0) * 1.2732557730789702)));
	} else {
		tmp = 1.0 - (((0.254829592 + ((-0.284496736 / fma(0.3275911, x_m, 1.0)) + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) * pow(fma(0.3275911, x_m, 1.0), -2.0)))) / exp((x_m * x_m))) / fma(0.3275911, fabs(x_m), 1.0));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 2e-7)
		tmp = Float64(1.0 / Float64(Float64(1e-9 + Float64(x_m * -1.128386358070218)) / Float64(1e-18 - Float64((x_m ^ 2.0) * 1.2732557730789702))));
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 / fma(0.3275911, x_m, 1.0)) + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) * (fma(0.3275911, x_m, 1.0) ^ -2.0)))) / exp(Float64(x_m * x_m))) / fma(0.3275911, abs(x_m), 1.0)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 2e-7], N[(1.0 / N[(N[(1e-9 + N[(x$95$m * -1.128386358070218), $MachinePrecision]), $MachinePrecision] / N[(1e-18 - N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 1.2732557730789702), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(N[(0.254829592 + N[(N[(-0.284496736 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(0.3275911 * x$95$m + 1.0), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Exp[N[(x$95$m * x$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 5: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{10^{-9} + x_m \cdot -1.128386358070218}{10^{-18} - {x_m}^{2} \cdot 1.2732557730789702}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{\left(-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}\right) \cdot \frac{-1}{\mathsf{fma}\left(0.3275911, x_m, 1\right)} - 0.254829592}{e^{x_m \cdot x_m}}}{\mathsf{fma}\left(0.3275911, \left|x_m\right|, 1\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 2e-7)
   (/
    1.0
    (/
     (+ 1e-9 (* x_m -1.128386358070218))
     (- 1e-18 (* (pow x_m 2.0) 1.2732557730789702))))
   (+
    1.0
    (/
     (/
      (-
       (*
        (+
         -0.284496736
         (/
          (+
           1.421413741
           (/
            (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
            (fma 0.3275911 x_m 1.0)))
          (fma 0.3275911 x_m 1.0)))
        (/ -1.0 (fma 0.3275911 x_m 1.0)))
       0.254829592)
      (exp (* x_m x_m)))
     (fma 0.3275911 (fabs x_m) 1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 2e-7) {
		tmp = 1.0 / ((1e-9 + (x_m * -1.128386358070218)) / (1e-18 - (pow(x_m, 2.0) * 1.2732557730789702)));
	} else {
		tmp = 1.0 + (((((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) * (-1.0 / fma(0.3275911, x_m, 1.0))) - 0.254829592) / exp((x_m * x_m))) / fma(0.3275911, fabs(x_m), 1.0));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 2e-7)
		tmp = Float64(1.0 / Float64(Float64(1e-9 + Float64(x_m * -1.128386358070218)) / Float64(1e-18 - Float64((x_m ^ 2.0) * 1.2732557730789702))));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) * Float64(-1.0 / fma(0.3275911, x_m, 1.0))) - 0.254829592) / exp(Float64(x_m * x_m))) / fma(0.3275911, abs(x_m), 1.0)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 2e-7], N[(1.0 / N[(N[(1e-9 + N[(x$95$m * -1.128386358070218), $MachinePrecision]), $MachinePrecision] / N[(1e-18 - N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 1.2732557730789702), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision] / N[Exp[N[(x$95$m * x$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 6: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{10^{-9} + x_m \cdot -1.128386358070218}{10^{-18} - {x_m}^{2} \cdot 1.2732557730789702}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\frac{\frac{\frac{-1.421413741 - \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)} - -0.284496736}{\mathsf{fma}\left(0.3275911, x_m, 1\right)} - 0.254829592}{e^{x_m \cdot x_m}}}{\mathsf{fma}\left(0.3275911, \left|x_m\right|, 1\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 2e-7)
   (/
    1.0
    (/
     (+ 1e-9 (* x_m -1.128386358070218))
     (- 1e-18 (* (pow x_m 2.0) 1.2732557730789702))))
   (+
    1.0
    (/
     (/
      (-
       (/
        (-
         (/
          (-
           -1.421413741
           (/
            (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
            (fma 0.3275911 x_m 1.0)))
          (fma 0.3275911 x_m 1.0))
         -0.284496736)
        (fma 0.3275911 x_m 1.0))
       0.254829592)
      (exp (* x_m x_m)))
     (fma 0.3275911 (fabs x_m) 1.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 2e-7) {
		tmp = 1.0 / ((1e-9 + (x_m * -1.128386358070218)) / (1e-18 - (pow(x_m, 2.0) * 1.2732557730789702)));
	} else {
		tmp = 1.0 + (((((((-1.421413741 - ((-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0)) - -0.284496736) / fma(0.3275911, x_m, 1.0)) - 0.254829592) / exp((x_m * x_m))) / fma(0.3275911, fabs(x_m), 1.0));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 2e-7)
		tmp = Float64(1.0 / Float64(Float64(1e-9 + Float64(x_m * -1.128386358070218)) / Float64(1e-18 - Float64((x_m ^ 2.0) * 1.2732557730789702))));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(-1.421413741 - Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0)) - -0.284496736) / fma(0.3275911, x_m, 1.0)) - 0.254829592) / exp(Float64(x_m * x_m))) / fma(0.3275911, abs(x_m), 1.0)));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 2e-7], N[(1.0 / N[(N[(1e-9 + N[(x$95$m * -1.128386358070218), $MachinePrecision]), $MachinePrecision] / N[(1e-18 - N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 1.2732557730789702), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(N[(N[(N[(N[(-1.421413741 - N[(N[(-1.453152027 + N[(1.061405429 / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] - -0.284496736), $MachinePrecision] / N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision] / N[Exp[N[(x$95$m * x$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(0.3275911 * N[Abs[x$95$m], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 7: 99.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + \left|x_m\right| \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;\left|x_m\right| \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{10^{-9} + x_m \cdot -1.128386358070218}{10^{-18} - {x_m}^{2} \cdot 1.2732557730789702}}\\ \mathbf{else}:\\ \;\;\;\;1 + t_1 \cdot \left(e^{x_m \cdot \left(-x_m\right)} \cdot \left(t_1 \cdot \left(t_1 \cdot \left(\left(-1.453152027 + \frac{1.061405429}{1 + x_m \cdot 0.3275911}\right) \cdot \frac{-1}{t_0} - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x_m) 0.3275911))) (t_1 (/ 1.0 t_0)))
   (if (<= (fabs x_m) 2e-7)
     (/
      1.0
      (/
       (+ 1e-9 (* x_m -1.128386358070218))
       (- 1e-18 (* (pow x_m 2.0) 1.2732557730789702))))
     (+
      1.0
      (*
       t_1
       (*
        (exp (* x_m (- x_m)))
        (-
         (*
          t_1
          (-
           (*
            t_1
            (-
             (*
              (+ -1.453152027 (/ 1.061405429 (+ 1.0 (* x_m 0.3275911))))
              (/ -1.0 t_0))
             1.421413741))
           -0.284496736))
         0.254829592)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 + (fabs(x_m) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (fabs(x_m) <= 2e-7) {
		tmp = 1.0 / ((1e-9 + (x_m * -1.128386358070218)) / (1e-18 - (pow(x_m, 2.0) * 1.2732557730789702)));
	} else {
		tmp = 1.0 + (t_1 * (exp((x_m * -x_m)) * ((t_1 * ((t_1 * (((-1.453152027 + (1.061405429 / (1.0 + (x_m * 0.3275911)))) * (-1.0 / t_0)) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (abs(x_m) * 0.3275911d0)
    t_1 = 1.0d0 / t_0
    if (abs(x_m) <= 2d-7) then
        tmp = 1.0d0 / ((1d-9 + (x_m * (-1.128386358070218d0))) / (1d-18 - ((x_m ** 2.0d0) * 1.2732557730789702d0)))
    else
        tmp = 1.0d0 + (t_1 * (exp((x_m * -x_m)) * ((t_1 * ((t_1 * ((((-1.453152027d0) + (1.061405429d0 / (1.0d0 + (x_m * 0.3275911d0)))) * ((-1.0d0) / t_0)) - 1.421413741d0)) - (-0.284496736d0))) - 0.254829592d0)))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = 1.0 + (Math.abs(x_m) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (Math.abs(x_m) <= 2e-7) {
		tmp = 1.0 / ((1e-9 + (x_m * -1.128386358070218)) / (1e-18 - (Math.pow(x_m, 2.0) * 1.2732557730789702)));
	} else {
		tmp = 1.0 + (t_1 * (Math.exp((x_m * -x_m)) * ((t_1 * ((t_1 * (((-1.453152027 + (1.061405429 / (1.0 + (x_m * 0.3275911)))) * (-1.0 / t_0)) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = 1.0 + (math.fabs(x_m) * 0.3275911)
	t_1 = 1.0 / t_0
	tmp = 0
	if math.fabs(x_m) <= 2e-7:
		tmp = 1.0 / ((1e-9 + (x_m * -1.128386358070218)) / (1e-18 - (math.pow(x_m, 2.0) * 1.2732557730789702)))
	else:
		tmp = 1.0 + (t_1 * (math.exp((x_m * -x_m)) * ((t_1 * ((t_1 * (((-1.453152027 + (1.061405429 / (1.0 + (x_m * 0.3275911)))) * (-1.0 / t_0)) - 1.421413741)) - -0.284496736)) - 0.254829592)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 + Float64(abs(x_m) * 0.3275911))
	t_1 = Float64(1.0 / t_0)
	tmp = 0.0
	if (abs(x_m) <= 2e-7)
		tmp = Float64(1.0 / Float64(Float64(1e-9 + Float64(x_m * -1.128386358070218)) / Float64(1e-18 - Float64((x_m ^ 2.0) * 1.2732557730789702))));
	else
		tmp = Float64(1.0 + Float64(t_1 * Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(Float64(t_1 * Float64(Float64(t_1 * Float64(Float64(Float64(-1.453152027 + Float64(1.061405429 / Float64(1.0 + Float64(x_m * 0.3275911)))) * Float64(-1.0 / t_0)) - 1.421413741)) - -0.284496736)) - 0.254829592))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = 1.0 + (abs(x_m) * 0.3275911);
	t_1 = 1.0 / t_0;
	tmp = 0.0;
	if (abs(x_m) <= 2e-7)
		tmp = 1.0 / ((1e-9 + (x_m * -1.128386358070218)) / (1e-18 - ((x_m ^ 2.0) * 1.2732557730789702)));
	else
		tmp = 1.0 + (t_1 * (exp((x_m * -x_m)) * ((t_1 * ((t_1 * (((-1.453152027 + (1.061405429 / (1.0 + (x_m * 0.3275911)))) * (-1.0 / t_0)) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 2e-7], N[(1.0 / N[(N[(1e-9 + N[(x$95$m * -1.128386358070218), $MachinePrecision]), $MachinePrecision] / N[(1e-18 - N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 1.2732557730789702), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(t$95$1 * N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(N[(t$95$1 * N[(N[(t$95$1 * N[(N[(N[(-1.453152027 + N[(1.061405429 / N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] - 1.421413741), $MachinePrecision]), $MachinePrecision] - -0.284496736), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 8: 99.2% accurate, 3.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 0.04:\\ \;\;\;\;\frac{1}{\frac{10^{-9} + x_m \cdot -1.128386358070218}{10^{-18} - {x_m}^{2} \cdot 1.2732557730789702}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.04)
   (/
    1.0
    (/
     (+ 1e-9 (* x_m -1.128386358070218))
     (- 1e-18 (* (pow x_m 2.0) 1.2732557730789702))))
   1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.04) {
		tmp = 1.0 / ((1e-9 + (x_m * -1.128386358070218)) / (1e-18 - (pow(x_m, 2.0) * 1.2732557730789702)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (abs(x_m) <= 0.04d0) then
        tmp = 1.0d0 / ((1d-9 + (x_m * (-1.128386358070218d0))) / (1d-18 - ((x_m ** 2.0d0) * 1.2732557730789702d0)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.abs(x_m) <= 0.04) {
		tmp = 1.0 / ((1e-9 + (x_m * -1.128386358070218)) / (1e-18 - (Math.pow(x_m, 2.0) * 1.2732557730789702)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if math.fabs(x_m) <= 0.04:
		tmp = 1.0 / ((1e-9 + (x_m * -1.128386358070218)) / (1e-18 - (math.pow(x_m, 2.0) * 1.2732557730789702)))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.04)
		tmp = Float64(1.0 / Float64(Float64(1e-9 + Float64(x_m * -1.128386358070218)) / Float64(1e-18 - Float64((x_m ^ 2.0) * 1.2732557730789702))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (abs(x_m) <= 0.04)
		tmp = 1.0 / ((1e-9 + (x_m * -1.128386358070218)) / (1e-18 - ((x_m ^ 2.0) * 1.2732557730789702)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.04], N[(1.0 / N[(N[(1e-9 + N[(x$95$m * -1.128386358070218), $MachinePrecision]), $MachinePrecision] / N[(1e-18 - N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 1.2732557730789702), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 9: 99.2% accurate, 3.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 0.04:\\ \;\;\;\;\frac{{x_m}^{2} \cdot 1.2732557730789702 - 10^{-18}}{x_m \cdot 1.128386358070218 - 10^{-9}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.04)
   (/
    (- (* (pow x_m 2.0) 1.2732557730789702) 1e-18)
    (- (* x_m 1.128386358070218) 1e-9))
   1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.04) {
		tmp = ((pow(x_m, 2.0) * 1.2732557730789702) - 1e-18) / ((x_m * 1.128386358070218) - 1e-9);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (abs(x_m) <= 0.04d0) then
        tmp = (((x_m ** 2.0d0) * 1.2732557730789702d0) - 1d-18) / ((x_m * 1.128386358070218d0) - 1d-9)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.abs(x_m) <= 0.04) {
		tmp = ((Math.pow(x_m, 2.0) * 1.2732557730789702) - 1e-18) / ((x_m * 1.128386358070218) - 1e-9);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if math.fabs(x_m) <= 0.04:
		tmp = ((math.pow(x_m, 2.0) * 1.2732557730789702) - 1e-18) / ((x_m * 1.128386358070218) - 1e-9)
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.04)
		tmp = Float64(Float64(Float64((x_m ^ 2.0) * 1.2732557730789702) - 1e-18) / Float64(Float64(x_m * 1.128386358070218) - 1e-9));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (abs(x_m) <= 0.04)
		tmp = (((x_m ^ 2.0) * 1.2732557730789702) - 1e-18) / ((x_m * 1.128386358070218) - 1e-9);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.04], N[(N[(N[(N[Power[x$95$m, 2.0], $MachinePrecision] * 1.2732557730789702), $MachinePrecision] - 1e-18), $MachinePrecision] / N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] - 1e-9), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 10: 99.2% accurate, 7.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 0.04:\\ \;\;\;\;10^{-9} + x_m \cdot 1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 0.04) (+ 1e-9 (* x_m 1.128386358070218)) 1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 0.04) {
		tmp = 1e-9 + (x_m * 1.128386358070218);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (abs(x_m) <= 0.04d0) then
        tmp = 1d-9 + (x_m * 1.128386358070218d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.abs(x_m) <= 0.04) {
		tmp = 1e-9 + (x_m * 1.128386358070218);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if math.fabs(x_m) <= 0.04:
		tmp = 1e-9 + (x_m * 1.128386358070218)
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 0.04)
		tmp = Float64(1e-9 + Float64(x_m * 1.128386358070218));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (abs(x_m) <= 0.04)
		tmp = 1e-9 + (x_m * 1.128386358070218);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 0.04], N[(1e-9 + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 11: 97.6% accurate, 142.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 2.8 \cdot 10^{-5}:\\ \;\;\;\;10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 2.8e-5) 1e-9 1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.8e-5) {
		tmp = 1e-9;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 2.8d-5) then
        tmp = 1d-9
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 2.8e-5) {
		tmp = 1e-9;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 2.8e-5:
		tmp = 1e-9
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.8e-5)
		tmp = 1e-9;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 2.8e-5)
		tmp = 1e-9;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.8e-5], 1e-9, 1.0]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 12: 52.8% accurate, 856.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 10^{-9} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 1e-9)

C

x_m = fabs(x);
double code(double x_m) {
	return 1e-9;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = 1d-9
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 1e-9;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 1e-9

Julia

x_m = abs(x)
function code(x_m)
	return 1e-9
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 1e-9;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 1e-9

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Reproduce

herbie shell --seed 2024006 
(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)))))))