Jmat.Real.erf

Percentage Accurate: 79.3% → 99.1%

Time: 26.2s

Alternatives: 9

Speedup: 8.3×

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: 99.1% accurate, 4.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 1.25 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{\sqrt[3]{10^{-54}}}\\ \mathbf{elif}\;x_m \leq 0.89:\\ \;\;\;\;{\left({\left(10^{-9} + x_m \cdot 1.128386358070218\right)}^{3}\right)}^{0.3333333333333333}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.25e-24)
   (sqrt (cbrt 1e-54))
   (if (<= x_m 0.89)
     (pow (pow (+ 1e-9 (* x_m 1.128386358070218)) 3.0) 0.3333333333333333)
     1.0)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.25e-24) {
		tmp = sqrt(cbrt(1e-54));
	} else if (x_m <= 0.89) {
		tmp = pow(pow((1e-9 + (x_m * 1.128386358070218)), 3.0), 0.3333333333333333);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.25e-24) {
		tmp = Math.sqrt(Math.cbrt(1e-54));
	} else if (x_m <= 0.89) {
		tmp = Math.pow(Math.pow((1e-9 + (x_m * 1.128386358070218)), 3.0), 0.3333333333333333);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.25e-24)
		tmp = sqrt(cbrt(1e-54));
	elseif (x_m <= 0.89)
		tmp = (Float64(1e-9 + Float64(x_m * 1.128386358070218)) ^ 3.0) ^ 0.3333333333333333;
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.25e-24], N[Sqrt[N[Power[1e-54, 1/3], $MachinePrecision]], $MachinePrecision], If[LessEqual[x$95$m, 0.89], N[Power[N[Power[N[(1e-9 + N[(x$95$m * 1.128386358070218), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision], 1.0]]

Derivation

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

Alternative 2: 98.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{1}{1 + \left|x_m\right| \cdot 0.3275911}\\ t_1 := \frac{1}{1 + x_m \cdot 0.3275911}\\ \mathbf{if}\;\left|x_m\right| \leq 10^{-8}:\\ \;\;\;\;\sqrt[3]{10^{-27} + \mathsf{fma}\left(x_m, 3.385159074210654 \cdot 10^{-18}, \mathsf{fma}\left(3.819767313236556 \cdot 10^{-9}, {x_m}^{2}, {x_m}^{3} \cdot 1.4367244311350222\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + t_1 \cdot \left(e^{x_m \cdot \left(-x_m\right)} \cdot \left(t_1 \cdot \left(t_1 \cdot \left(t_0 \cdot \left(1.453152027 - t_0 \cdot 1.061405429\right) - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* (fabs x_m) 0.3275911))))
        (t_1 (/ 1.0 (+ 1.0 (* x_m 0.3275911)))))
   (if (<= (fabs x_m) 1e-8)
     (cbrt
      (+
       1e-27
       (fma
        x_m
        3.385159074210654e-18
        (fma
         3.819767313236556e-9
         (pow x_m 2.0)
         (* (pow x_m 3.0) 1.4367244311350222)))))
     (+
      1.0
      (*
       t_1
       (*
        (exp (* x_m (- x_m)))
        (-
         (*
          t_1
          (-
           (* t_1 (- (* t_0 (- 1.453152027 (* t_0 1.061405429))) 1.421413741))
           -0.284496736))
         0.254829592)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 / (1.0 + (fabs(x_m) * 0.3275911));
	double t_1 = 1.0 / (1.0 + (x_m * 0.3275911));
	double tmp;
	if (fabs(x_m) <= 1e-8) {
		tmp = cbrt((1e-27 + fma(x_m, 3.385159074210654e-18, fma(3.819767313236556e-9, pow(x_m, 2.0), (pow(x_m, 3.0) * 1.4367244311350222)))));
	} else {
		tmp = 1.0 + (t_1 * (exp((x_m * -x_m)) * ((t_1 * ((t_1 * ((t_0 * (1.453152027 - (t_0 * 1.061405429))) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(abs(x_m) * 0.3275911)))
	t_1 = Float64(1.0 / Float64(1.0 + Float64(x_m * 0.3275911)))
	tmp = 0.0
	if (abs(x_m) <= 1e-8)
		tmp = cbrt(Float64(1e-27 + fma(x_m, 3.385159074210654e-18, fma(3.819767313236556e-9, (x_m ^ 2.0), Float64((x_m ^ 3.0) * 1.4367244311350222)))));
	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(t_0 * Float64(1.453152027 - Float64(t_0 * 1.061405429))) - 1.421413741)) - -0.284496736)) - 0.254829592))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 1e-8], N[Power[N[(1e-27 + N[(x$95$m * 3.385159074210654e-18 + N[(3.819767313236556e-9 * N[Power[x$95$m, 2.0], $MachinePrecision] + N[(N[Power[x$95$m, 3.0], $MachinePrecision] * 1.4367244311350222), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $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[(t$95$0 * N[(1.453152027 - N[(t$95$0 * 1.061405429), $MachinePrecision]), $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 3: 98.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{1}{1 + \left|x_m\right| \cdot 0.3275911}\\ t_1 := \frac{1}{1 + x_m \cdot 0.3275911}\\ \mathbf{if}\;x_m \leq 3.4 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\sqrt[3]{10^{-54}}}\\ \mathbf{else}:\\ \;\;\;\;1 + t_1 \cdot \left(e^{x_m \cdot \left(-x_m\right)} \cdot \left(t_1 \cdot \left(t_1 \cdot \left(t_0 \cdot \left(1.453152027 - t_0 \cdot 1.061405429\right) - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* (fabs x_m) 0.3275911))))
        (t_1 (/ 1.0 (+ 1.0 (* x_m 0.3275911)))))
   (if (<= x_m 3.4e-16)
     (sqrt (cbrt 1e-54))
     (+
      1.0
      (*
       t_1
       (*
        (exp (* x_m (- x_m)))
        (-
         (*
          t_1
          (-
           (* t_1 (- (* t_0 (- 1.453152027 (* t_0 1.061405429))) 1.421413741))
           -0.284496736))
         0.254829592)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = 1.0 / (1.0 + (fabs(x_m) * 0.3275911));
	double t_1 = 1.0 / (1.0 + (x_m * 0.3275911));
	double tmp;
	if (x_m <= 3.4e-16) {
		tmp = sqrt(cbrt(1e-54));
	} else {
		tmp = 1.0 + (t_1 * (exp((x_m * -x_m)) * ((t_1 * ((t_1 * ((t_0 * (1.453152027 - (t_0 * 1.061405429))) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = 1.0 / (1.0 + (Math.abs(x_m) * 0.3275911));
	double t_1 = 1.0 / (1.0 + (x_m * 0.3275911));
	double tmp;
	if (x_m <= 3.4e-16) {
		tmp = Math.sqrt(Math.cbrt(1e-54));
	} else {
		tmp = 1.0 + (t_1 * (Math.exp((x_m * -x_m)) * ((t_1 * ((t_1 * ((t_0 * (1.453152027 - (t_0 * 1.061405429))) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(abs(x_m) * 0.3275911)))
	t_1 = Float64(1.0 / Float64(1.0 + Float64(x_m * 0.3275911)))
	tmp = 0.0
	if (x_m <= 3.4e-16)
		tmp = sqrt(cbrt(1e-54));
	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(t_0 * Float64(1.453152027 - Float64(t_0 * 1.061405429))) - 1.421413741)) - -0.284496736)) - 0.254829592))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(1.0 + N[(x$95$m * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 3.4e-16], N[Sqrt[N[Power[1e-54, 1/3], $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[(t$95$0 * N[(1.453152027 - N[(t$95$0 * 1.061405429), $MachinePrecision]), $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 4: 98.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + x_m \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;x_m \leq 3.4 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\sqrt[3]{10^{-54}}}\\ \mathbf{else}:\\ \;\;\;\;1 + t_1 \cdot \left(e^{x_m \cdot \left(-x_m\right)} \cdot \left(t_1 \cdot \left(t_1 \cdot \left(\left(-1.453152027 + \frac{1.061405429}{1 + \left|x_m\right| \cdot 0.3275911}\right) \cdot \frac{-1}{t_0} - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x_m 0.3275911))) (t_1 (/ 1.0 t_0)))
   (if (<= x_m 3.4e-16)
     (sqrt (cbrt 1e-54))
     (+
      1.0
      (*
       t_1
       (*
        (exp (* x_m (- x_m)))
        (-
         (*
          t_1
          (-
           (*
            t_1
            (-
             (*
              (+ -1.453152027 (/ 1.061405429 (+ 1.0 (* (fabs 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 + (x_m * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (x_m <= 3.4e-16) {
		tmp = sqrt(cbrt(1e-54));
	} else {
		tmp = 1.0 + (t_1 * (exp((x_m * -x_m)) * ((t_1 * ((t_1 * (((-1.453152027 + (1.061405429 / (1.0 + (fabs(x_m) * 0.3275911)))) * (-1.0 / t_0)) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

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 tmp;
	if (x_m <= 3.4e-16) {
		tmp = Math.sqrt(Math.cbrt(1e-54));
	} else {
		tmp = 1.0 + (t_1 * (Math.exp((x_m * -x_m)) * ((t_1 * ((t_1 * (((-1.453152027 + (1.061405429 / (1.0 + (Math.abs(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(x_m * 0.3275911))
	t_1 = Float64(1.0 / t_0)
	tmp = 0.0
	if (x_m <= 3.4e-16)
		tmp = sqrt(cbrt(1e-54));
	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(abs(x_m) * 0.3275911)))) * Float64(-1.0 / t_0)) - 1.421413741)) - -0.284496736)) - 0.254829592))));
	end
	return 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]}, If[LessEqual[x$95$m, 3.4e-16], N[Sqrt[N[Power[1e-54, 1/3], $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[(N[Abs[x$95$m], $MachinePrecision] * 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 5: 98.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := 1 + x_m \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;x_m \leq 3.4 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\sqrt[3]{10^{-54}}}\\ \mathbf{else}:\\ \;\;\;\;1 + t_1 \cdot \left(e^{x_m \cdot \left(-x_m\right)} \cdot \left(t_1 \cdot \left(\left(1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{t_0}}{t_0}\right) \cdot \frac{-1}{1 + \left|x_m\right| \cdot 0.3275911} - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x_m 0.3275911))) (t_1 (/ 1.0 t_0)))
   (if (<= x_m 3.4e-16)
     (sqrt (cbrt 1e-54))
     (+
      1.0
      (*
       t_1
       (*
        (exp (* x_m (- x_m)))
        (-
         (*
          t_1
          (-
           (*
            (+ 1.421413741 (/ (+ -1.453152027 (/ 1.061405429 t_0)) t_0))
            (/ -1.0 (+ 1.0 (* (fabs x_m) 0.3275911))))
           -0.284496736))
         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 tmp;
	if (x_m <= 3.4e-16) {
		tmp = sqrt(cbrt(1e-54));
	} else {
		tmp = 1.0 + (t_1 * (exp((x_m * -x_m)) * ((t_1 * (((1.421413741 + ((-1.453152027 + (1.061405429 / t_0)) / t_0)) * (-1.0 / (1.0 + (fabs(x_m) * 0.3275911)))) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

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 tmp;
	if (x_m <= 3.4e-16) {
		tmp = Math.sqrt(Math.cbrt(1e-54));
	} else {
		tmp = 1.0 + (t_1 * (Math.exp((x_m * -x_m)) * ((t_1 * (((1.421413741 + ((-1.453152027 + (1.061405429 / t_0)) / t_0)) * (-1.0 / (1.0 + (Math.abs(x_m) * 0.3275911)))) - -0.284496736)) - 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)
	tmp = 0.0
	if (x_m <= 3.4e-16)
		tmp = sqrt(cbrt(1e-54));
	else
		tmp = Float64(1.0 + Float64(t_1 * Float64(exp(Float64(x_m * Float64(-x_m))) * Float64(Float64(t_1 * Float64(Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / t_0)) / t_0)) * Float64(-1.0 / Float64(1.0 + Float64(abs(x_m) * 0.3275911)))) - -0.284496736)) - 0.254829592))));
	end
	return 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]}, If[LessEqual[x$95$m, 3.4e-16], N[Sqrt[N[Power[1e-54, 1/3], $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[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(1.0 + N[(N[Abs[x$95$m], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -0.284496736), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

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

Alternative 6: 97.6% accurate, 4.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 2.8 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\sqrt[3]{10^{-54}}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 2.8e-5) (sqrt (cbrt 1e-54)) 1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.8e-5) {
		tmp = sqrt(cbrt(1e-54));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 2.8e-5) {
		tmp = Math.sqrt(Math.cbrt(1e-54));
	} 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 = sqrt(cbrt(1e-54));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.8e-5], N[Sqrt[N[Power[1e-54, 1/3], $MachinePrecision]], $MachinePrecision], 1.0]

Derivation

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

Alternative 7: 97.4% accurate, 8.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.00029:\\ \;\;\;\;\sqrt[3]{10^{-27} + x_m \cdot 3.385159074210654 \cdot 10^{-18}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.00029) (cbrt (+ 1e-27 (* x_m 3.385159074210654e-18))) 1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.00029) {
		tmp = cbrt((1e-27 + (x_m * 3.385159074210654e-18)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.00029) {
		tmp = Math.cbrt((1e-27 + (x_m * 3.385159074210654e-18)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.00029)
		tmp = cbrt(Float64(1e-27 + Float64(x_m * 3.385159074210654e-18)));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.00029], N[Power[N[(1e-27 + N[(x$95$m * 3.385159074210654e-18), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 1.0]

Derivation

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

Alternative 8: 96.9% accurate, 8.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 2.8 \cdot 10^{-5}:\\ \;\;\;\;\sqrt[3]{10^{-27}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 2.8e-5) (cbrt 1e-27) 1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.8e-5) {
		tmp = cbrt(1e-27);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 2.8e-5) {
		tmp = Math.cbrt(1e-27);
	} 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 = cbrt(1e-27);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.8e-5], N[Power[1e-27, 1/3], $MachinePrecision], 1.0]

Derivation

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

Alternative 9: 55.7% accurate, 856.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 1 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 1.0)

C

x_m = fabs(x);
double code(double x_m) {
	return 1.0;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = 1.0d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 1.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 1.0

Julia

x_m = abs(x)
function code(x_m)
	return 1.0
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 1.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 1.0

Derivation

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

Reproduce

herbie shell --seed 2024010 
(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)))))))