Jmat.Real.erf

Percentage Accurate: 79.6% → 99.6%

Time: 28.5s

Alternatives: 12

Speedup: 279.5×

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: 79.6% 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: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 4 \cdot 10^{-11}:\\ \;\;\;\;\frac{{x_m}^{3} \cdot 1.436724444676459 + 10^{-27}}{{\left(x_m \cdot 1.128386358070218\right)}^{2} + \left(10^{-18} - \left(x_m \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{{\left(\sqrt[3]{1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x_m, 0.3275911, 1\right) \cdot e^{{x_m}^{2}}}}\right)}^{3}}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 4e-11)
   (/
    (+ (* (pow x_m 3.0) 1.436724444676459) 1e-27)
    (+
     (pow (* x_m 1.128386358070218) 2.0)
     (- 1e-18 (* (* x_m 1.128386358070218) 1e-9))))
   (pow
    (cbrt
     (pow
      (cbrt
       (-
        1.0
        (/
         (+
          0.254829592
          (/
           (+
            -0.284496736
            (/
             (+
              1.421413741
              (/
               (+ -1.453152027 (/ 1.061405429 (fma x_m 0.3275911 1.0)))
               (fma x_m 0.3275911 1.0)))
             (fma x_m 0.3275911 1.0)))
           (fma x_m 0.3275911 1.0)))
         (* (fma x_m 0.3275911 1.0) (exp (pow x_m 2.0))))))
      3.0))
    3.0)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 4e-11) {
		tmp = ((pow(x_m, 3.0) * 1.436724444676459) + 1e-27) / (pow((x_m * 1.128386358070218), 2.0) + (1e-18 - ((x_m * 1.128386358070218) * 1e-9)));
	} else {
		tmp = pow(cbrt(pow(cbrt((1.0 - ((0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / (fma(x_m, 0.3275911, 1.0) * exp(pow(x_m, 2.0)))))), 3.0)), 3.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 4e-11)
		tmp = Float64(Float64(Float64((x_m ^ 3.0) * 1.436724444676459) + 1e-27) / Float64((Float64(x_m * 1.128386358070218) ^ 2.0) + Float64(1e-18 - Float64(Float64(x_m * 1.128386358070218) * 1e-9))));
	else
		tmp = cbrt((cbrt(Float64(1.0 - Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / Float64(fma(x_m, 0.3275911, 1.0) * exp((x_m ^ 2.0)))))) ^ 3.0)) ^ 3.0;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 4e-11], N[(N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 1.436724444676459), $MachinePrecision] + 1e-27), $MachinePrecision] / N[(N[Power[N[(x$95$m * 1.128386358070218), $MachinePrecision], 2.0], $MachinePrecision] + N[(1e-18 - N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] * 1e-9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[Power[N[Power[N[(1.0 - N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision] * N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]]

Derivation

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

Alternative 2: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 4 \cdot 10^{-11}:\\ \;\;\;\;\frac{{x_m}^{3} \cdot 1.436724444676459 + 10^{-27}}{{\left(x_m \cdot 1.128386358070218\right)}^{2} + \left(10^{-18} - \left(x_m \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{e^{{x_m}^{2}} \cdot \mathsf{fma}\left(0.3275911, x_m, 1\right)}}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 4e-11)
   (/
    (+ (* (pow x_m 3.0) 1.436724444676459) 1e-27)
    (+
     (pow (* x_m 1.128386358070218) 2.0)
     (- 1e-18 (* (* x_m 1.128386358070218) 1e-9))))
   (pow
    (cbrt
     (-
      1.0
      (/
       (+
        0.254829592
        (/
         (+
          -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)))
         (fma 0.3275911 x_m 1.0)))
       (* (exp (pow x_m 2.0)) (fma 0.3275911 x_m 1.0)))))
    3.0)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 4e-11) {
		tmp = ((pow(x_m, 3.0) * 1.436724444676459) + 1e-27) / (pow((x_m * 1.128386358070218), 2.0) + (1e-18 - ((x_m * 1.128386358070218) * 1e-9)));
	} else {
		tmp = pow(cbrt((1.0 - ((0.254829592 + ((-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))) / fma(0.3275911, x_m, 1.0))) / (exp(pow(x_m, 2.0)) * fma(0.3275911, x_m, 1.0))))), 3.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 4e-11)
		tmp = Float64(Float64(Float64((x_m ^ 3.0) * 1.436724444676459) + 1e-27) / Float64((Float64(x_m * 1.128386358070218) ^ 2.0) + Float64(1e-18 - Float64(Float64(x_m * 1.128386358070218) * 1e-9))));
	else
		tmp = cbrt(Float64(1.0 - Float64(Float64(0.254829592 + 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))) / fma(0.3275911, x_m, 1.0))) / Float64(exp((x_m ^ 2.0)) * fma(0.3275911, x_m, 1.0))))) ^ 3.0;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 4e-11], N[(N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 1.436724444676459), $MachinePrecision] + 1e-27), $MachinePrecision] / N[(N[Power[N[(x$95$m * 1.128386358070218), $MachinePrecision], 2.0], $MachinePrecision] + N[(1e-18 - N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] * 1e-9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(1.0 - N[(N[(0.254829592 + 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[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision] * N[(0.3275911 * x$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]]

Derivation

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

Alternative 3: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + \left|x_m\right| \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;\left|x_m\right| \leq 4 \cdot 10^{-11}:\\ \;\;\;\;\frac{{x_m}^{3} \cdot 1.436724444676459 + 10^{-27}}{{\left(x_m \cdot 1.128386358070218\right)}^{2} + \left(10^{-18} - \left(x_m \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x_m \cdot \left(-x_m\right)} \cdot \left(\left(0.254829592 + t_1 \cdot \left(-0.284496736 + t_1 \cdot {\left(\sqrt{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}{\mathsf{fma}\left(0.3275911, x_m, 1\right)}}\right)}^{2}\right)\right) \cdot \frac{-1}{t_0}\right)\\ \end{array} \end{array} \]

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) 4e-11)
     (/
      (+ (* (pow x_m 3.0) 1.436724444676459) 1e-27)
      (+
       (pow (* x_m 1.128386358070218) 2.0)
       (- 1e-18 (* (* x_m 1.128386358070218) 1e-9))))
     (+
      1.0
      (*
       (exp (* x_m (- x_m)))
       (*
        (+
         0.254829592
         (*
          t_1
          (+
           -0.284496736
           (*
            t_1
            (pow
             (sqrt
              (+
               1.421413741
               (/
                (+ -1.453152027 (/ 1.061405429 (fma 0.3275911 x_m 1.0)))
                (fma 0.3275911 x_m 1.0))))
             2.0)))))
        (/ -1.0 t_0)))))))

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) <= 4e-11) {
		tmp = ((pow(x_m, 3.0) * 1.436724444676459) + 1e-27) / (pow((x_m * 1.128386358070218), 2.0) + (1e-18 - ((x_m * 1.128386358070218) * 1e-9)));
	} else {
		tmp = 1.0 + (exp((x_m * -x_m)) * ((0.254829592 + (t_1 * (-0.284496736 + (t_1 * pow(sqrt((1.421413741 + ((-1.453152027 + (1.061405429 / fma(0.3275911, x_m, 1.0))) / fma(0.3275911, x_m, 1.0)))), 2.0))))) * (-1.0 / t_0)));
	}
	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) <= 4e-11)
		tmp = Float64(Float64(Float64((x_m ^ 3.0) * 1.436724444676459) + 1e-27) / Float64((Float64(x_m * 1.128386358070218) ^ 2.0) + Float64(1e-18 - Float64(Float64(x_m * 1.128386358070218) * 1e-9))));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(Float64(0.254829592 + Float64(t_1 * Float64(-0.284496736 + Float64(t_1 * (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)))) ^ 2.0))))) * Float64(-1.0 / t_0))));
	end
	return 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], 4e-11], N[(N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 1.436724444676459), $MachinePrecision] + 1e-27), $MachinePrecision] / N[(N[Power[N[(x$95$m * 1.128386358070218), $MachinePrecision], 2.0], $MachinePrecision] + N[(1e-18 - N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] * 1e-9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[N[(x$95$m * (-x$95$m)), $MachinePrecision]], $MachinePrecision] * N[(N[(0.254829592 + N[(t$95$1 * N[(-0.284496736 + N[(t$95$1 * N[Power[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], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

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

Alternative 4: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x_m\right| \leq 4 \cdot 10^{-11}:\\ \;\;\;\;\frac{{x_m}^{3} \cdot 1.436724444676459 + 10^{-27}}{{\left(x_m \cdot 1.128386358070218\right)}^{2} + \left(10^{-18} - \left(x_m \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x_m, 0.3275911, 1\right)}}{\mathsf{fma}\left(x_m, 0.3275911, 1\right) \cdot e^{{x_m}^{2}}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 4e-11)
   (/
    (+ (* (pow x_m 3.0) 1.436724444676459) 1e-27)
    (+
     (pow (* x_m 1.128386358070218) 2.0)
     (- 1e-18 (* (* x_m 1.128386358070218) 1e-9))))
   (-
    1.0
    (/
     (+
      0.254829592
      (/
       (+
        -0.284496736
        (/
         (+
          1.421413741
          (/
           (+ -1.453152027 (/ 1.061405429 (fma x_m 0.3275911 1.0)))
           (fma x_m 0.3275911 1.0)))
         (fma x_m 0.3275911 1.0)))
       (fma x_m 0.3275911 1.0)))
     (* (fma x_m 0.3275911 1.0) (exp (pow x_m 2.0)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 4e-11) {
		tmp = ((pow(x_m, 3.0) * 1.436724444676459) + 1e-27) / (pow((x_m * 1.128386358070218), 2.0) + (1e-18 - ((x_m * 1.128386358070218) * 1e-9)));
	} else {
		tmp = 1.0 - ((0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / (fma(x_m, 0.3275911, 1.0) * exp(pow(x_m, 2.0))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 4e-11)
		tmp = Float64(Float64(Float64((x_m ^ 3.0) * 1.436724444676459) + 1e-27) / Float64((Float64(x_m * 1.128386358070218) ^ 2.0) + Float64(1e-18 - Float64(Float64(x_m * 1.128386358070218) * 1e-9))));
	else
		tmp = Float64(1.0 - Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / fma(x_m, 0.3275911, 1.0))) / Float64(fma(x_m, 0.3275911, 1.0) * exp((x_m ^ 2.0)))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 4e-11], N[(N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 1.436724444676459), $MachinePrecision] + 1e-27), $MachinePrecision] / N[(N[Power[N[(x$95$m * 1.128386358070218), $MachinePrecision], 2.0], $MachinePrecision] + N[(1e-18 - N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] * 1e-9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x$95$m * 0.3275911 + 1.0), $MachinePrecision] * N[Exp[N[Power[x$95$m, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

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

Alternative 5: 99.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + x_m \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ t_2 := 1 + \left|x_m\right| \cdot 0.3275911\\ \mathbf{if}\;x_m \leq 1.42 \cdot 10^{-6}:\\ \;\;\;\;\frac{{x_m}^{3} \cdot 1.436724444676459 + 10^{-27}}{{\left(x_m \cdot 1.128386358070218\right)}^{2} + \left(10^{-18} - \left(x_m \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + t_1 \cdot \left(e^{x_m \cdot \left(-x_m\right)} \cdot \left(\left(-0.284496736 + t_1 \cdot \left(1.421413741 + \frac{1}{t_2} \cdot \left(-1.453152027 + \frac{1.061405429}{t_2}\right)\right)\right) \cdot \frac{-1}{t_0} - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x_m 0.3275911)))
        (t_1 (/ 1.0 t_0))
        (t_2 (+ 1.0 (* (fabs x_m) 0.3275911))))
   (if (<= x_m 1.42e-6)
     (/
      (+ (* (pow x_m 3.0) 1.436724444676459) 1e-27)
      (+
       (pow (* x_m 1.128386358070218) 2.0)
       (- 1e-18 (* (* x_m 1.128386358070218) 1e-9))))
     (+
      1.0
      (*
       t_1
       (*
        (exp (* x_m (- x_m)))
        (-
         (*
          (+
           -0.284496736
           (*
            t_1
            (+
             1.421413741
             (* (/ 1.0 t_2) (+ -1.453152027 (/ 1.061405429 t_2))))))
          (/ -1.0 t_0))
         0.254829592)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 + (x_m * 0.3275911);
	double t_1 = 1.0 / t_0;
	double t_2 = 1.0 + (fabs(x_m) * 0.3275911);
	double tmp;
	if (x_m <= 1.42e-6) {
		tmp = ((pow(x_m, 3.0) * 1.436724444676459) + 1e-27) / (pow((x_m * 1.128386358070218), 2.0) + (1e-18 - ((x_m * 1.128386358070218) * 1e-9)));
	} else {
		tmp = 1.0 + (t_1 * (exp((x_m * -x_m)) * (((-0.284496736 + (t_1 * (1.421413741 + ((1.0 / t_2) * (-1.453152027 + (1.061405429 / t_2)))))) * (-1.0 / t_0)) - 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) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 + (x_m * 0.3275911d0)
    t_1 = 1.0d0 / t_0
    t_2 = 1.0d0 + (abs(x_m) * 0.3275911d0)
    if (x_m <= 1.42d-6) then
        tmp = (((x_m ** 3.0d0) * 1.436724444676459d0) + 1d-27) / (((x_m * 1.128386358070218d0) ** 2.0d0) + (1d-18 - ((x_m * 1.128386358070218d0) * 1d-9)))
    else
        tmp = 1.0d0 + (t_1 * (exp((x_m * -x_m)) * ((((-0.284496736d0) + (t_1 * (1.421413741d0 + ((1.0d0 / t_2) * ((-1.453152027d0) + (1.061405429d0 / t_2)))))) * ((-1.0d0) / t_0)) - 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 + (x_m * 0.3275911);
	double t_1 = 1.0 / t_0;
	double t_2 = 1.0 + (Math.abs(x_m) * 0.3275911);
	double tmp;
	if (x_m <= 1.42e-6) {
		tmp = ((Math.pow(x_m, 3.0) * 1.436724444676459) + 1e-27) / (Math.pow((x_m * 1.128386358070218), 2.0) + (1e-18 - ((x_m * 1.128386358070218) * 1e-9)));
	} else {
		tmp = 1.0 + (t_1 * (Math.exp((x_m * -x_m)) * (((-0.284496736 + (t_1 * (1.421413741 + ((1.0 / t_2) * (-1.453152027 + (1.061405429 / t_2)))))) * (-1.0 / t_0)) - 0.254829592)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = 1.0 + (x_m * 0.3275911)
	t_1 = 1.0 / t_0
	t_2 = 1.0 + (math.fabs(x_m) * 0.3275911)
	tmp = 0
	if x_m <= 1.42e-6:
		tmp = ((math.pow(x_m, 3.0) * 1.436724444676459) + 1e-27) / (math.pow((x_m * 1.128386358070218), 2.0) + (1e-18 - ((x_m * 1.128386358070218) * 1e-9)))
	else:
		tmp = 1.0 + (t_1 * (math.exp((x_m * -x_m)) * (((-0.284496736 + (t_1 * (1.421413741 + ((1.0 / t_2) * (-1.453152027 + (1.061405429 / t_2)))))) * (-1.0 / t_0)) - 0.254829592)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 + Float64(x_m * 0.3275911))
	t_1 = Float64(1.0 / t_0)
	t_2 = Float64(1.0 + Float64(abs(x_m) * 0.3275911))
	tmp = 0.0
	if (x_m <= 1.42e-6)
		tmp = Float64(Float64(Float64((x_m ^ 3.0) * 1.436724444676459) + 1e-27) / Float64((Float64(x_m * 1.128386358070218) ^ 2.0) + Float64(1e-18 - Float64(Float64(x_m * 1.128386358070218) * 1e-9))));
	else
		tmp = Float64(1.0 + Float64(t_1 * Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(Float64(Float64(-0.284496736 + Float64(t_1 * Float64(1.421413741 + Float64(Float64(1.0 / t_2) * Float64(-1.453152027 + Float64(1.061405429 / t_2)))))) * Float64(-1.0 / t_0)) - 0.254829592))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = 1.0 + (x_m * 0.3275911);
	t_1 = 1.0 / t_0;
	t_2 = 1.0 + (abs(x_m) * 0.3275911);
	tmp = 0.0;
	if (x_m <= 1.42e-6)
		tmp = (((x_m ^ 3.0) * 1.436724444676459) + 1e-27) / (((x_m * 1.128386358070218) ^ 2.0) + (1e-18 - ((x_m * 1.128386358070218) * 1e-9)));
	else
		tmp = 1.0 + (t_1 * (exp((x_m * -x_m)) * (((-0.284496736 + (t_1 * (1.421413741 + ((1.0 / t_2) * (-1.453152027 + (1.061405429 / t_2)))))) * (-1.0 / t_0)) - 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[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 1.42e-6], N[(N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 1.436724444676459), $MachinePrecision] + 1e-27), $MachinePrecision] / N[(N[Power[N[(x$95$m * 1.128386358070218), $MachinePrecision], 2.0], $MachinePrecision] + N[(1e-18 - N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] * 1e-9), $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[(N[(-0.284496736 + N[(t$95$1 * N[(1.421413741 + N[(N[(1.0 / t$95$2), $MachinePrecision] * N[(-1.453152027 + N[(1.061405429 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

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

Alternative 6: 99.4% accurate, 3.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.88:\\ \;\;\;\;\frac{{x_m}^{3} \cdot 1.436724444676459 + 10^{-27}}{{\left(x_m \cdot 1.128386358070218\right)}^{2} + \left(10^{-18} - \left(x_m \cdot 1.128386358070218\right) \cdot 10^{-9}\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.88)
   (/
    (+ (* (pow x_m 3.0) 1.436724444676459) 1e-27)
    (+
     (pow (* x_m 1.128386358070218) 2.0)
     (- 1e-18 (* (* x_m 1.128386358070218) 1e-9))))
   1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.88) {
		tmp = ((pow(x_m, 3.0) * 1.436724444676459) + 1e-27) / (pow((x_m * 1.128386358070218), 2.0) + (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 (x_m <= 0.88d0) then
        tmp = (((x_m ** 3.0d0) * 1.436724444676459d0) + 1d-27) / (((x_m * 1.128386358070218d0) ** 2.0d0) + (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 (x_m <= 0.88) {
		tmp = ((Math.pow(x_m, 3.0) * 1.436724444676459) + 1e-27) / (Math.pow((x_m * 1.128386358070218), 2.0) + (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 x_m <= 0.88:
		tmp = ((math.pow(x_m, 3.0) * 1.436724444676459) + 1e-27) / (math.pow((x_m * 1.128386358070218), 2.0) + (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 (x_m <= 0.88)
		tmp = Float64(Float64(Float64((x_m ^ 3.0) * 1.436724444676459) + 1e-27) / Float64((Float64(x_m * 1.128386358070218) ^ 2.0) + Float64(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 (x_m <= 0.88)
		tmp = (((x_m ^ 3.0) * 1.436724444676459) + 1e-27) / (((x_m * 1.128386358070218) ^ 2.0) + (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[x$95$m, 0.88], N[(N[(N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 1.436724444676459), $MachinePrecision] + 1e-27), $MachinePrecision] / N[(N[Power[N[(x$95$m * 1.128386358070218), $MachinePrecision], 2.0], $MachinePrecision] + N[(1e-18 - N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] * 1e-9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

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

Alternative 7: 99.5% accurate, 4.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.88:\\ \;\;\;\;10^{-9} + \mathsf{fma}\left(x_m, 1.128386358070218, {x_m}^{2} \cdot -0.00011824294398844343\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.88)
   (+
    1e-9
    (fma x_m 1.128386358070218 (* (pow x_m 2.0) -0.00011824294398844343)))
   1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.88) {
		tmp = 1e-9 + fma(x_m, 1.128386358070218, (pow(x_m, 2.0) * -0.00011824294398844343));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.88)
		tmp = Float64(1e-9 + fma(x_m, 1.128386358070218, Float64((x_m ^ 2.0) * -0.00011824294398844343)));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.88], N[(1e-9 + N[(x$95$m * 1.128386358070218 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

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

Alternative 8: 99.5% accurate, 7.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.88:\\ \;\;\;\;x_m \cdot 1.128386358070218 + \left(10^{-9} + {x_m}^{2} \cdot -0.00011824294398844343\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.88)
   (+
    (* x_m 1.128386358070218)
    (+ 1e-9 (* (pow x_m 2.0) -0.00011824294398844343)))
   1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.88) {
		tmp = (x_m * 1.128386358070218) + (1e-9 + (pow(x_m, 2.0) * -0.00011824294398844343));
	} 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 <= 0.88d0) then
        tmp = (x_m * 1.128386358070218d0) + (1d-9 + ((x_m ** 2.0d0) * (-0.00011824294398844343d0)))
    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 <= 0.88) {
		tmp = (x_m * 1.128386358070218) + (1e-9 + (Math.pow(x_m, 2.0) * -0.00011824294398844343));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.88:
		tmp = (x_m * 1.128386358070218) + (1e-9 + (math.pow(x_m, 2.0) * -0.00011824294398844343))
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.88)
		tmp = Float64(Float64(x_m * 1.128386358070218) + Float64(1e-9 + Float64((x_m ^ 2.0) * -0.00011824294398844343)));
	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 <= 0.88)
		tmp = (x_m * 1.128386358070218) + (1e-9 + ((x_m ^ 2.0) * -0.00011824294398844343));
	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, 0.88], N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] + N[(1e-9 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * -0.00011824294398844343), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

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

Alternative 9: 99.4% accurate, 37.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 10^{-9} - x_m \cdot 1.128386358070218\\ \mathbf{if}\;x_m \leq 0.88:\\ \;\;\;\;\frac{10^{-18}}{t_0} - \frac{\left(x_m \cdot 1.128386358070218\right) \cdot \left(x_m \cdot 1.128386358070218\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (- 1e-9 (* x_m 1.128386358070218))))
   (if (<= x_m 0.88)
     (-
      (/ 1e-18 t_0)
      (/ (* (* x_m 1.128386358070218) (* x_m 1.128386358070218)) t_0))
     1.0)))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1e-9 - (x_m * 1.128386358070218);
	double tmp;
	if (x_m <= 0.88) {
		tmp = (1e-18 / t_0) - (((x_m * 1.128386358070218) * (x_m * 1.128386358070218)) / t_0);
	} 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) :: t_0
    real(8) :: tmp
    t_0 = 1d-9 - (x_m * 1.128386358070218d0)
    if (x_m <= 0.88d0) then
        tmp = (1d-18 / t_0) - (((x_m * 1.128386358070218d0) * (x_m * 1.128386358070218d0)) / t_0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = 1e-9 - (x_m * 1.128386358070218);
	double tmp;
	if (x_m <= 0.88) {
		tmp = (1e-18 / t_0) - (((x_m * 1.128386358070218) * (x_m * 1.128386358070218)) / t_0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = 1e-9 - (x_m * 1.128386358070218)
	tmp = 0
	if x_m <= 0.88:
		tmp = (1e-18 / t_0) - (((x_m * 1.128386358070218) * (x_m * 1.128386358070218)) / t_0)
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1e-9 - Float64(x_m * 1.128386358070218))
	tmp = 0.0
	if (x_m <= 0.88)
		tmp = Float64(Float64(1e-18 / t_0) - Float64(Float64(Float64(x_m * 1.128386358070218) * Float64(x_m * 1.128386358070218)) / t_0));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = 1e-9 - (x_m * 1.128386358070218);
	tmp = 0.0;
	if (x_m <= 0.88)
		tmp = (1e-18 / t_0) - (((x_m * 1.128386358070218) * (x_m * 1.128386358070218)) / t_0);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1e-9 - N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 0.88], N[(N[(1e-18 / t$95$0), $MachinePrecision] - N[(N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] * N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

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

Alternative 10: 99.4% accurate, 121.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.88:\\ \;\;\;\;x_m \cdot 1.128386358070218 + 10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.88) (+ (* x_m 1.128386358070218) 1e-9) 1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.88) {
		tmp = (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 (x_m <= 0.88d0) then
        tmp = (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 (x_m <= 0.88) {
		tmp = (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 x_m <= 0.88:
		tmp = (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 (x_m <= 0.88)
		tmp = 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 (x_m <= 0.88)
		tmp = (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[x$95$m, 0.88], N[(N[(x$95$m * 1.128386358070218), $MachinePrecision] + 1e-9), $MachinePrecision], 1.0]

Derivation

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

Alternative 11: 97.8% accurate, 279.5× 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: 52.3% 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 2023350 
(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)))))))