NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.4% → 98.8%

Time: 18.4s

Alternatives: 18

Speedup: 2.0×

• 38.5% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -1.99999999999999997e77
  2. eps = -2e-231

Specification

Math

\[\begin{array}{l} \\ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (-
   (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x))))
   (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x)))))
  2.0))

C

double code(double x, double eps) {
	return (((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-((1.0 + eps) * x)))) / 2.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (((1.0d0 + (1.0d0 / eps)) * exp(-((1.0d0 - eps) * x))) - (((1.0d0 / eps) - 1.0d0) * exp(-((1.0d0 + eps) * x)))) / 2.0d0
end function

Java

public static double code(double x, double eps) {
	return (((1.0 + (1.0 / eps)) * Math.exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * Math.exp(-((1.0 + eps) * x)))) / 2.0;
}

Python

def code(x, eps):
	return (((1.0 + (1.0 / eps)) * math.exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * math.exp(-((1.0 + eps) * x)))) / 2.0

Julia

function code(x, eps)
	return Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(-Float64(Float64(1.0 - eps) * x)))) - Float64(Float64(Float64(1.0 / eps) - 1.0) * exp(Float64(-Float64(Float64(1.0 + eps) * x))))) / 2.0)
end

MATLAB

function tmp = code(x, eps)
	tmp = (((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-((1.0 + eps) * x)))) / 2.0;
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[(1.0 - eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[(-N[(N[(1.0 + eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Initial Program: 73.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (-
   (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x))))
   (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x)))))
  2.0))

C

double code(double x, double eps) {
	return (((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-((1.0 + eps) * x)))) / 2.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (((1.0d0 + (1.0d0 / eps)) * exp(-((1.0d0 - eps) * x))) - (((1.0d0 / eps) - 1.0d0) * exp(-((1.0d0 + eps) * x)))) / 2.0d0
end function

Java

public static double code(double x, double eps) {
	return (((1.0 + (1.0 / eps)) * Math.exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * Math.exp(-((1.0 + eps) * x)))) / 2.0;
}

Python

def code(x, eps):
	return (((1.0 + (1.0 / eps)) * math.exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * math.exp(-((1.0 + eps) * x)))) / 2.0

Julia

function code(x, eps)
	return Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(-Float64(Float64(1.0 - eps) * x)))) - Float64(Float64(Float64(1.0 / eps) - 1.0) * exp(Float64(-Float64(Float64(1.0 + eps) * x))))) / 2.0)
end

MATLAB

function tmp = code(x, eps)
	tmp = (((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-((1.0 + eps) * x)))) / 2.0;
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[(1.0 - eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[(-N[(N[(1.0 + eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Alternative 1: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \frac{e^{x \cdot \left(eps_m + -1\right)} + \frac{1}{e^{\mathsf{fma}\left(eps_m, x, x\right)}}}{2} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (/ (+ (exp (* x (+ eps_m -1.0))) (/ 1.0 (exp (fma eps_m x x)))) 2.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return (exp((x * (eps_m + -1.0))) + (1.0 / exp(fma(eps_m, x, x)))) / 2.0;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + Float64(1.0 / exp(fma(eps_m, x, x)))) / 2.0)
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[Exp[N[(eps$95$m * x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

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1. Add Preprocessing

Alternative 2: 98.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \frac{e^{x \cdot \left(eps_m + -1\right)} + e^{x \cdot \left(-1 - eps_m\right)}}{2} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (/ (+ (exp (* x (+ eps_m -1.0))) (exp (* x (- -1.0 eps_m)))) 2.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return (exp((x * (eps_m + -1.0))) + exp((x * (-1.0 - eps_m)))) / 2.0;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    code = (exp((x * (eps_m + (-1.0d0)))) + exp((x * ((-1.0d0) - eps_m)))) / 2.0d0
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return (Math.exp((x * (eps_m + -1.0))) + Math.exp((x * (-1.0 - eps_m)))) / 2.0;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return (math.exp((x * (eps_m + -1.0))) + math.exp((x * (-1.0 - eps_m)))) / 2.0

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + exp(Float64(x * Float64(-1.0 - eps_m)))) / 2.0)
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = (exp((x * (eps_m + -1.0))) + exp((x * (-1.0 - eps_m)))) / 2.0;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

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1. Add Preprocessing

Alternative 3: 98.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \left(eps_m + -1\right)}\\ \mathbf{if}\;x \leq -3.7 \cdot 10^{-255}:\\ \;\;\;\;\frac{1 + \frac{1}{e^{\mathsf{fma}\left(eps_m, x, x\right)}}}{2}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_0 + \left(1 + x \cdot \left(-1 - eps_m\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (+ eps_m -1.0)))))
   (if (<= x -3.7e-255)
     (/ (+ 1.0 (/ 1.0 (exp (fma eps_m x x)))) 2.0)
     (if (<= x 6.2e-5)
       (/ (+ t_0 (+ 1.0 (* x (- -1.0 eps_m)))) 2.0)
       (/ t_0 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= -3.7e-255) {
		tmp = (1.0 + (1.0 / exp(fma(eps_m, x, x)))) / 2.0;
	} else if (x <= 6.2e-5) {
		tmp = (t_0 + (1.0 + (x * (-1.0 - eps_m)))) / 2.0;
	} else {
		tmp = t_0 / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * Float64(eps_m + -1.0)))
	tmp = 0.0
	if (x <= -3.7e-255)
		tmp = Float64(Float64(1.0 + Float64(1.0 / exp(fma(eps_m, x, x)))) / 2.0);
	elseif (x <= 6.2e-5)
		tmp = Float64(Float64(t_0 + Float64(1.0 + Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	else
		tmp = Float64(t_0 / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -3.7e-255], N[(N[(1.0 + N[(1.0 / N[Exp[N[(eps$95$m * x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 6.2e-5], N[(N[(t$95$0 + N[(1.0 + N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(t$95$0 / 2.0), $MachinePrecision]]]]

Derivation

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1. Add Preprocessing

Alternative 4: 91.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \left(eps_m + -1\right)}\\ \mathbf{if}\;x \leq -27000:\\ \;\;\;\;\frac{\frac{e^{-x}}{eps_m}}{2}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_0 + \left(1 + x \cdot \left(-1 - eps_m\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (+ eps_m -1.0)))))
   (if (<= x -27000.0)
     (/ (/ (exp (- x)) eps_m) 2.0)
     (if (<= x 6.2e-5)
       (/ (+ t_0 (+ 1.0 (* x (- -1.0 eps_m)))) 2.0)
       (/ t_0 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= -27000.0) {
		tmp = (exp(-x) / eps_m) / 2.0;
	} else if (x <= 6.2e-5) {
		tmp = (t_0 + (1.0 + (x * (-1.0 - eps_m)))) / 2.0;
	} else {
		tmp = t_0 / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((x * (eps_m + (-1.0d0))))
    if (x <= (-27000.0d0)) then
        tmp = (exp(-x) / eps_m) / 2.0d0
    else if (x <= 6.2d-5) then
        tmp = (t_0 + (1.0d0 + (x * ((-1.0d0) - eps_m)))) / 2.0d0
    else
        tmp = t_0 / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= -27000.0) {
		tmp = (Math.exp(-x) / eps_m) / 2.0;
	} else if (x <= 6.2e-5) {
		tmp = (t_0 + (1.0 + (x * (-1.0 - eps_m)))) / 2.0;
	} else {
		tmp = t_0 / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp((x * (eps_m + -1.0)))
	tmp = 0
	if x <= -27000.0:
		tmp = (math.exp(-x) / eps_m) / 2.0
	elif x <= 6.2e-5:
		tmp = (t_0 + (1.0 + (x * (-1.0 - eps_m)))) / 2.0
	else:
		tmp = t_0 / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * Float64(eps_m + -1.0)))
	tmp = 0.0
	if (x <= -27000.0)
		tmp = Float64(Float64(exp(Float64(-x)) / eps_m) / 2.0);
	elseif (x <= 6.2e-5)
		tmp = Float64(Float64(t_0 + Float64(1.0 + Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	else
		tmp = Float64(t_0 / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp((x * (eps_m + -1.0)));
	tmp = 0.0;
	if (x <= -27000.0)
		tmp = (exp(-x) / eps_m) / 2.0;
	elseif (x <= 6.2e-5)
		tmp = (t_0 + (1.0 + (x * (-1.0 - eps_m)))) / 2.0;
	else
		tmp = t_0 / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -27000.0], N[(N[(N[Exp[(-x)], $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 6.2e-5], N[(N[(t$95$0 + N[(1.0 + N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(t$95$0 / 2.0), $MachinePrecision]]]]

Derivation

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1. Add Preprocessing

Alternative 5: 98.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \left(eps_m + -1\right)}\\ \mathbf{if}\;x \leq -5 \cdot 10^{+19}:\\ \;\;\;\;\frac{\frac{e^{-x}}{eps_m}}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-268}:\\ \;\;\;\;\frac{e^{x \cdot \left(-eps_m\right)} + \left(1 - x \cdot \left(1 - eps_m\right)\right)}{2}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 + t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (+ eps_m -1.0)))))
   (if (<= x -5e+19)
     (/ (/ (exp (- x)) eps_m) 2.0)
     (if (<= x 2e-268)
       (/ (+ (exp (* x (- eps_m))) (- 1.0 (* x (- 1.0 eps_m)))) 2.0)
       (if (<= x 6.2e-5) (/ (+ 1.0 t_0) 2.0) (/ t_0 2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= -5e+19) {
		tmp = (exp(-x) / eps_m) / 2.0;
	} else if (x <= 2e-268) {
		tmp = (exp((x * -eps_m)) + (1.0 - (x * (1.0 - eps_m)))) / 2.0;
	} else if (x <= 6.2e-5) {
		tmp = (1.0 + t_0) / 2.0;
	} else {
		tmp = t_0 / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((x * (eps_m + (-1.0d0))))
    if (x <= (-5d+19)) then
        tmp = (exp(-x) / eps_m) / 2.0d0
    else if (x <= 2d-268) then
        tmp = (exp((x * -eps_m)) + (1.0d0 - (x * (1.0d0 - eps_m)))) / 2.0d0
    else if (x <= 6.2d-5) then
        tmp = (1.0d0 + t_0) / 2.0d0
    else
        tmp = t_0 / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= -5e+19) {
		tmp = (Math.exp(-x) / eps_m) / 2.0;
	} else if (x <= 2e-268) {
		tmp = (Math.exp((x * -eps_m)) + (1.0 - (x * (1.0 - eps_m)))) / 2.0;
	} else if (x <= 6.2e-5) {
		tmp = (1.0 + t_0) / 2.0;
	} else {
		tmp = t_0 / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp((x * (eps_m + -1.0)))
	tmp = 0
	if x <= -5e+19:
		tmp = (math.exp(-x) / eps_m) / 2.0
	elif x <= 2e-268:
		tmp = (math.exp((x * -eps_m)) + (1.0 - (x * (1.0 - eps_m)))) / 2.0
	elif x <= 6.2e-5:
		tmp = (1.0 + t_0) / 2.0
	else:
		tmp = t_0 / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * Float64(eps_m + -1.0)))
	tmp = 0.0
	if (x <= -5e+19)
		tmp = Float64(Float64(exp(Float64(-x)) / eps_m) / 2.0);
	elseif (x <= 2e-268)
		tmp = Float64(Float64(exp(Float64(x * Float64(-eps_m))) + Float64(1.0 - Float64(x * Float64(1.0 - eps_m)))) / 2.0);
	elseif (x <= 6.2e-5)
		tmp = Float64(Float64(1.0 + t_0) / 2.0);
	else
		tmp = Float64(t_0 / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp((x * (eps_m + -1.0)));
	tmp = 0.0;
	if (x <= -5e+19)
		tmp = (exp(-x) / eps_m) / 2.0;
	elseif (x <= 2e-268)
		tmp = (exp((x * -eps_m)) + (1.0 - (x * (1.0 - eps_m)))) / 2.0;
	elseif (x <= 6.2e-5)
		tmp = (1.0 + t_0) / 2.0;
	else
		tmp = t_0 / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -5e+19], N[(N[(N[Exp[(-x)], $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2e-268], N[(N[(N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision] + N[(1.0 - N[(x * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 6.2e-5], N[(N[(1.0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(t$95$0 / 2.0), $MachinePrecision]]]]]

Derivation

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1. Add Preprocessing

Alternative 6: 91.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \left(eps_m + -1\right)}\\ \mathbf{if}\;x \leq -59000000:\\ \;\;\;\;\frac{\frac{e^{-x}}{eps_m}}{2}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 + t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (+ eps_m -1.0)))))
   (if (<= x -59000000.0)
     (/ (/ (exp (- x)) eps_m) 2.0)
     (if (<= x 6.2e-5) (/ (+ 1.0 t_0) 2.0) (/ t_0 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= -59000000.0) {
		tmp = (exp(-x) / eps_m) / 2.0;
	} else if (x <= 6.2e-5) {
		tmp = (1.0 + t_0) / 2.0;
	} else {
		tmp = t_0 / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((x * (eps_m + (-1.0d0))))
    if (x <= (-59000000.0d0)) then
        tmp = (exp(-x) / eps_m) / 2.0d0
    else if (x <= 6.2d-5) then
        tmp = (1.0d0 + t_0) / 2.0d0
    else
        tmp = t_0 / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= -59000000.0) {
		tmp = (Math.exp(-x) / eps_m) / 2.0;
	} else if (x <= 6.2e-5) {
		tmp = (1.0 + t_0) / 2.0;
	} else {
		tmp = t_0 / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp((x * (eps_m + -1.0)))
	tmp = 0
	if x <= -59000000.0:
		tmp = (math.exp(-x) / eps_m) / 2.0
	elif x <= 6.2e-5:
		tmp = (1.0 + t_0) / 2.0
	else:
		tmp = t_0 / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * Float64(eps_m + -1.0)))
	tmp = 0.0
	if (x <= -59000000.0)
		tmp = Float64(Float64(exp(Float64(-x)) / eps_m) / 2.0);
	elseif (x <= 6.2e-5)
		tmp = Float64(Float64(1.0 + t_0) / 2.0);
	else
		tmp = Float64(t_0 / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp((x * (eps_m + -1.0)));
	tmp = 0.0;
	if (x <= -59000000.0)
		tmp = (exp(-x) / eps_m) / 2.0;
	elseif (x <= 6.2e-5)
		tmp = (1.0 + t_0) / 2.0;
	else
		tmp = t_0 / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -59000000.0], N[(N[(N[Exp[(-x)], $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 6.2e-5], N[(N[(1.0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(t$95$0 / 2.0), $MachinePrecision]]]]

Derivation

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

Alternative 7: 84.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -59000000:\\ \;\;\;\;\frac{\frac{e^{-x}}{eps_m}}{2}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-99}:\\ \;\;\;\;\frac{\left(x + 1\right) - \left(x + -1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps_m + -1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -59000000.0)
   (/ (/ (exp (- x)) eps_m) 2.0)
   (if (<= x 1.12e-99)
     (/ (- (+ x 1.0) (+ x -1.0)) 2.0)
     (/ (exp (* x (+ eps_m -1.0))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -59000000.0) {
		tmp = (exp(-x) / eps_m) / 2.0;
	} else if (x <= 1.12e-99) {
		tmp = ((x + 1.0) - (x + -1.0)) / 2.0;
	} else {
		tmp = exp((x * (eps_m + -1.0))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-59000000.0d0)) then
        tmp = (exp(-x) / eps_m) / 2.0d0
    else if (x <= 1.12d-99) then
        tmp = ((x + 1.0d0) - (x + (-1.0d0))) / 2.0d0
    else
        tmp = exp((x * (eps_m + (-1.0d0)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -59000000.0) {
		tmp = (Math.exp(-x) / eps_m) / 2.0;
	} else if (x <= 1.12e-99) {
		tmp = ((x + 1.0) - (x + -1.0)) / 2.0;
	} else {
		tmp = Math.exp((x * (eps_m + -1.0))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -59000000.0:
		tmp = (math.exp(-x) / eps_m) / 2.0
	elif x <= 1.12e-99:
		tmp = ((x + 1.0) - (x + -1.0)) / 2.0
	else:
		tmp = math.exp((x * (eps_m + -1.0))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -59000000.0)
		tmp = Float64(Float64(exp(Float64(-x)) / eps_m) / 2.0);
	elseif (x <= 1.12e-99)
		tmp = Float64(Float64(Float64(x + 1.0) - Float64(x + -1.0)) / 2.0);
	else
		tmp = Float64(exp(Float64(x * Float64(eps_m + -1.0))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -59000000.0)
		tmp = (exp(-x) / eps_m) / 2.0;
	elseif (x <= 1.12e-99)
		tmp = ((x + 1.0) - (x + -1.0)) / 2.0;
	else
		tmp = exp((x * (eps_m + -1.0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -59000000.0], N[(N[(N[Exp[(-x)], $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.12e-99], N[(N[(N[(x + 1.0), $MachinePrecision] - N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

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

Alternative 8: 70.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := 1 + \frac{-1}{eps_m}\\ t_1 := 1 + \frac{1}{eps_m}\\ t_2 := 1 - x \cdot \left(1 - eps_m\right)\\ t_3 := \frac{t_1 \cdot t_2 + t_2 \cdot t_0}{2}\\ t_4 := \frac{\left(x + 1\right) - \left(x + -1\right)}{2}\\ \mathbf{if}\;x \leq -59000000:\\ \;\;\;\;\frac{\frac{e^{-x}}{eps_m}}{2}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+144}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+200}:\\ \;\;\;\;\frac{t_1 + t_0}{2}\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+241}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+277}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ -1.0 eps_m)))
        (t_1 (+ 1.0 (/ 1.0 eps_m)))
        (t_2 (- 1.0 (* x (- 1.0 eps_m))))
        (t_3 (/ (+ (* t_1 t_2) (* t_2 t_0)) 2.0))
        (t_4 (/ (- (+ x 1.0) (+ x -1.0)) 2.0)))
   (if (<= x -59000000.0)
     (/ (/ (exp (- x)) eps_m) 2.0)
     (if (<= x 6.2e-5)
       t_4
       (if (<= x 9e+144)
         t_3
         (if (<= x 3.6e+200)
           (/ (+ t_1 t_0) 2.0)
           (if (<= x 8.2e+241)
             t_3
             (if (<= x 6.2e+277) t_4 (/ (* x eps_m) 2.0)))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = 1.0 + (-1.0 / eps_m);
	double t_1 = 1.0 + (1.0 / eps_m);
	double t_2 = 1.0 - (x * (1.0 - eps_m));
	double t_3 = ((t_1 * t_2) + (t_2 * t_0)) / 2.0;
	double t_4 = ((x + 1.0) - (x + -1.0)) / 2.0;
	double tmp;
	if (x <= -59000000.0) {
		tmp = (exp(-x) / eps_m) / 2.0;
	} else if (x <= 6.2e-5) {
		tmp = t_4;
	} else if (x <= 9e+144) {
		tmp = t_3;
	} else if (x <= 3.6e+200) {
		tmp = (t_1 + t_0) / 2.0;
	} else if (x <= 8.2e+241) {
		tmp = t_3;
	} else if (x <= 6.2e+277) {
		tmp = t_4;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = 1.0d0 + ((-1.0d0) / eps_m)
    t_1 = 1.0d0 + (1.0d0 / eps_m)
    t_2 = 1.0d0 - (x * (1.0d0 - eps_m))
    t_3 = ((t_1 * t_2) + (t_2 * t_0)) / 2.0d0
    t_4 = ((x + 1.0d0) - (x + (-1.0d0))) / 2.0d0
    if (x <= (-59000000.0d0)) then
        tmp = (exp(-x) / eps_m) / 2.0d0
    else if (x <= 6.2d-5) then
        tmp = t_4
    else if (x <= 9d+144) then
        tmp = t_3
    else if (x <= 3.6d+200) then
        tmp = (t_1 + t_0) / 2.0d0
    else if (x <= 8.2d+241) then
        tmp = t_3
    else if (x <= 6.2d+277) then
        tmp = t_4
    else
        tmp = (x * eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = 1.0 + (-1.0 / eps_m);
	double t_1 = 1.0 + (1.0 / eps_m);
	double t_2 = 1.0 - (x * (1.0 - eps_m));
	double t_3 = ((t_1 * t_2) + (t_2 * t_0)) / 2.0;
	double t_4 = ((x + 1.0) - (x + -1.0)) / 2.0;
	double tmp;
	if (x <= -59000000.0) {
		tmp = (Math.exp(-x) / eps_m) / 2.0;
	} else if (x <= 6.2e-5) {
		tmp = t_4;
	} else if (x <= 9e+144) {
		tmp = t_3;
	} else if (x <= 3.6e+200) {
		tmp = (t_1 + t_0) / 2.0;
	} else if (x <= 8.2e+241) {
		tmp = t_3;
	} else if (x <= 6.2e+277) {
		tmp = t_4;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = 1.0 + (-1.0 / eps_m)
	t_1 = 1.0 + (1.0 / eps_m)
	t_2 = 1.0 - (x * (1.0 - eps_m))
	t_3 = ((t_1 * t_2) + (t_2 * t_0)) / 2.0
	t_4 = ((x + 1.0) - (x + -1.0)) / 2.0
	tmp = 0
	if x <= -59000000.0:
		tmp = (math.exp(-x) / eps_m) / 2.0
	elif x <= 6.2e-5:
		tmp = t_4
	elif x <= 9e+144:
		tmp = t_3
	elif x <= 3.6e+200:
		tmp = (t_1 + t_0) / 2.0
	elif x <= 8.2e+241:
		tmp = t_3
	elif x <= 6.2e+277:
		tmp = t_4
	else:
		tmp = (x * eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(1.0 + Float64(-1.0 / eps_m))
	t_1 = Float64(1.0 + Float64(1.0 / eps_m))
	t_2 = Float64(1.0 - Float64(x * Float64(1.0 - eps_m)))
	t_3 = Float64(Float64(Float64(t_1 * t_2) + Float64(t_2 * t_0)) / 2.0)
	t_4 = Float64(Float64(Float64(x + 1.0) - Float64(x + -1.0)) / 2.0)
	tmp = 0.0
	if (x <= -59000000.0)
		tmp = Float64(Float64(exp(Float64(-x)) / eps_m) / 2.0);
	elseif (x <= 6.2e-5)
		tmp = t_4;
	elseif (x <= 9e+144)
		tmp = t_3;
	elseif (x <= 3.6e+200)
		tmp = Float64(Float64(t_1 + t_0) / 2.0);
	elseif (x <= 8.2e+241)
		tmp = t_3;
	elseif (x <= 6.2e+277)
		tmp = t_4;
	else
		tmp = Float64(Float64(x * eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = 1.0 + (-1.0 / eps_m);
	t_1 = 1.0 + (1.0 / eps_m);
	t_2 = 1.0 - (x * (1.0 - eps_m));
	t_3 = ((t_1 * t_2) + (t_2 * t_0)) / 2.0;
	t_4 = ((x + 1.0) - (x + -1.0)) / 2.0;
	tmp = 0.0;
	if (x <= -59000000.0)
		tmp = (exp(-x) / eps_m) / 2.0;
	elseif (x <= 6.2e-5)
		tmp = t_4;
	elseif (x <= 9e+144)
		tmp = t_3;
	elseif (x <= 3.6e+200)
		tmp = (t_1 + t_0) / 2.0;
	elseif (x <= 8.2e+241)
		tmp = t_3;
	elseif (x <= 6.2e+277)
		tmp = t_4;
	else
		tmp = (x * eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(x * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$1 * t$95$2), $MachinePrecision] + N[(t$95$2 * t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(x + 1.0), $MachinePrecision] - N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -59000000.0], N[(N[(N[Exp[(-x)], $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 6.2e-5], t$95$4, If[LessEqual[x, 9e+144], t$95$3, If[LessEqual[x, 3.6e+200], N[(N[(t$95$1 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 8.2e+241], t$95$3, If[LessEqual[x, 6.2e+277], t$95$4, N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]]]]]]]

Derivation

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

Alternative 9: 63.2% accurate, 6.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := 1 - x \cdot \left(1 - eps_m\right)\\ \mathbf{if}\;eps_m \leq 1.76 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(x + 1\right) - \left(x + -1\right)}{2}\\ \mathbf{elif}\;eps_m \leq 4.8 \cdot 10^{+205}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps_m}\right) \cdot t_0 + t_0 \cdot \left(1 + \frac{-1}{eps_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(1 + x \cdot \left(-1 - eps_m\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (* x (- 1.0 eps_m)))))
   (if (<= eps_m 1.76e-6)
     (/ (- (+ x 1.0) (+ x -1.0)) 2.0)
     (if (<= eps_m 4.8e+205)
       (/ (+ (* (+ 1.0 (/ 1.0 eps_m)) t_0) (* t_0 (+ 1.0 (/ -1.0 eps_m)))) 2.0)
       (/ (+ 1.0 (+ 1.0 (* x (- -1.0 eps_m)))) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = 1.0 - (x * (1.0 - eps_m));
	double tmp;
	if (eps_m <= 1.76e-6) {
		tmp = ((x + 1.0) - (x + -1.0)) / 2.0;
	} else if (eps_m <= 4.8e+205) {
		tmp = (((1.0 + (1.0 / eps_m)) * t_0) + (t_0 * (1.0 + (-1.0 / eps_m)))) / 2.0;
	} else {
		tmp = (1.0 + (1.0 + (x * (-1.0 - eps_m)))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (x * (1.0d0 - eps_m))
    if (eps_m <= 1.76d-6) then
        tmp = ((x + 1.0d0) - (x + (-1.0d0))) / 2.0d0
    else if (eps_m <= 4.8d+205) then
        tmp = (((1.0d0 + (1.0d0 / eps_m)) * t_0) + (t_0 * (1.0d0 + ((-1.0d0) / eps_m)))) / 2.0d0
    else
        tmp = (1.0d0 + (1.0d0 + (x * ((-1.0d0) - eps_m)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = 1.0 - (x * (1.0 - eps_m));
	double tmp;
	if (eps_m <= 1.76e-6) {
		tmp = ((x + 1.0) - (x + -1.0)) / 2.0;
	} else if (eps_m <= 4.8e+205) {
		tmp = (((1.0 + (1.0 / eps_m)) * t_0) + (t_0 * (1.0 + (-1.0 / eps_m)))) / 2.0;
	} else {
		tmp = (1.0 + (1.0 + (x * (-1.0 - eps_m)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = 1.0 - (x * (1.0 - eps_m))
	tmp = 0
	if eps_m <= 1.76e-6:
		tmp = ((x + 1.0) - (x + -1.0)) / 2.0
	elif eps_m <= 4.8e+205:
		tmp = (((1.0 + (1.0 / eps_m)) * t_0) + (t_0 * (1.0 + (-1.0 / eps_m)))) / 2.0
	else:
		tmp = (1.0 + (1.0 + (x * (-1.0 - eps_m)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(1.0 - Float64(x * Float64(1.0 - eps_m)))
	tmp = 0.0
	if (eps_m <= 1.76e-6)
		tmp = Float64(Float64(Float64(x + 1.0) - Float64(x + -1.0)) / 2.0);
	elseif (eps_m <= 4.8e+205)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * t_0) + Float64(t_0 * Float64(1.0 + Float64(-1.0 / eps_m)))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(1.0 + Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = 1.0 - (x * (1.0 - eps_m));
	tmp = 0.0;
	if (eps_m <= 1.76e-6)
		tmp = ((x + 1.0) - (x + -1.0)) / 2.0;
	elseif (eps_m <= 4.8e+205)
		tmp = (((1.0 + (1.0 / eps_m)) * t_0) + (t_0 * (1.0 + (-1.0 / eps_m)))) / 2.0;
	else
		tmp = (1.0 + (1.0 + (x * (-1.0 - eps_m)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(1.0 - N[(x * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps$95$m, 1.76e-6], N[(N[(N[(x + 1.0), $MachinePrecision] - N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps$95$m, 4.8e+205], N[(N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(t$95$0 * N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(1.0 + N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

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

Alternative 10: 63.3% accurate, 7.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps_m \leq 510000000000:\\ \;\;\;\;\frac{\left(x + 1\right) - \left(x + -1\right)}{2}\\ \mathbf{elif}\;eps_m \leq 4.5 \cdot 10^{+205}:\\ \;\;\;\;\frac{\left(x \cdot \left(eps_m + -1\right) + \left(x + 1\right)\right) + \left(-1 + \frac{1}{eps_m}\right) \cdot \left(-1 - \left(x + x \cdot eps_m\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(1 + x \cdot \left(-1 - eps_m\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 510000000000.0)
   (/ (- (+ x 1.0) (+ x -1.0)) 2.0)
   (if (<= eps_m 4.5e+205)
     (/
      (+
       (+ (* x (+ eps_m -1.0)) (+ x 1.0))
       (* (+ -1.0 (/ 1.0 eps_m)) (- -1.0 (+ x (* x eps_m)))))
      2.0)
     (/ (+ 1.0 (+ 1.0 (* x (- -1.0 eps_m)))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 510000000000.0) {
		tmp = ((x + 1.0) - (x + -1.0)) / 2.0;
	} else if (eps_m <= 4.5e+205) {
		tmp = (((x * (eps_m + -1.0)) + (x + 1.0)) + ((-1.0 + (1.0 / eps_m)) * (-1.0 - (x + (x * eps_m))))) / 2.0;
	} else {
		tmp = (1.0 + (1.0 + (x * (-1.0 - eps_m)))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 510000000000.0d0) then
        tmp = ((x + 1.0d0) - (x + (-1.0d0))) / 2.0d0
    else if (eps_m <= 4.5d+205) then
        tmp = (((x * (eps_m + (-1.0d0))) + (x + 1.0d0)) + (((-1.0d0) + (1.0d0 / eps_m)) * ((-1.0d0) - (x + (x * eps_m))))) / 2.0d0
    else
        tmp = (1.0d0 + (1.0d0 + (x * ((-1.0d0) - eps_m)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 510000000000.0) {
		tmp = ((x + 1.0) - (x + -1.0)) / 2.0;
	} else if (eps_m <= 4.5e+205) {
		tmp = (((x * (eps_m + -1.0)) + (x + 1.0)) + ((-1.0 + (1.0 / eps_m)) * (-1.0 - (x + (x * eps_m))))) / 2.0;
	} else {
		tmp = (1.0 + (1.0 + (x * (-1.0 - eps_m)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 510000000000.0:
		tmp = ((x + 1.0) - (x + -1.0)) / 2.0
	elif eps_m <= 4.5e+205:
		tmp = (((x * (eps_m + -1.0)) + (x + 1.0)) + ((-1.0 + (1.0 / eps_m)) * (-1.0 - (x + (x * eps_m))))) / 2.0
	else:
		tmp = (1.0 + (1.0 + (x * (-1.0 - eps_m)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 510000000000.0)
		tmp = Float64(Float64(Float64(x + 1.0) - Float64(x + -1.0)) / 2.0);
	elseif (eps_m <= 4.5e+205)
		tmp = Float64(Float64(Float64(Float64(x * Float64(eps_m + -1.0)) + Float64(x + 1.0)) + Float64(Float64(-1.0 + Float64(1.0 / eps_m)) * Float64(-1.0 - Float64(x + Float64(x * eps_m))))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(1.0 + Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 510000000000.0)
		tmp = ((x + 1.0) - (x + -1.0)) / 2.0;
	elseif (eps_m <= 4.5e+205)
		tmp = (((x * (eps_m + -1.0)) + (x + 1.0)) + ((-1.0 + (1.0 / eps_m)) * (-1.0 - (x + (x * eps_m))))) / 2.0;
	else
		tmp = (1.0 + (1.0 + (x * (-1.0 - eps_m)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 510000000000.0], N[(N[(N[(x + 1.0), $MachinePrecision] - N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps$95$m, 4.5e+205], N[(N[(N[(N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision] + N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(-1.0 - N[(x + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(1.0 + N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

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

Alternative 11: 63.4% accurate, 9.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps_m \leq 1.76 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(x + 1\right) - \left(x + -1\right)}{2}\\ \mathbf{elif}\;eps_m \leq 2.7 \cdot 10^{+205}:\\ \;\;\;\;\frac{\left(1 - x \cdot \left(1 - eps_m\right)\right) + \frac{1}{1 + x \cdot \left(eps_m + 1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(1 + x \cdot \left(-1 - eps_m\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 1.76e-6)
   (/ (- (+ x 1.0) (+ x -1.0)) 2.0)
   (if (<= eps_m 2.7e+205)
     (/
      (+ (- 1.0 (* x (- 1.0 eps_m))) (/ 1.0 (+ 1.0 (* x (+ eps_m 1.0)))))
      2.0)
     (/ (+ 1.0 (+ 1.0 (* x (- -1.0 eps_m)))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.76e-6) {
		tmp = ((x + 1.0) - (x + -1.0)) / 2.0;
	} else if (eps_m <= 2.7e+205) {
		tmp = ((1.0 - (x * (1.0 - eps_m))) + (1.0 / (1.0 + (x * (eps_m + 1.0))))) / 2.0;
	} else {
		tmp = (1.0 + (1.0 + (x * (-1.0 - eps_m)))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 1.76d-6) then
        tmp = ((x + 1.0d0) - (x + (-1.0d0))) / 2.0d0
    else if (eps_m <= 2.7d+205) then
        tmp = ((1.0d0 - (x * (1.0d0 - eps_m))) + (1.0d0 / (1.0d0 + (x * (eps_m + 1.0d0))))) / 2.0d0
    else
        tmp = (1.0d0 + (1.0d0 + (x * ((-1.0d0) - eps_m)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.76e-6) {
		tmp = ((x + 1.0) - (x + -1.0)) / 2.0;
	} else if (eps_m <= 2.7e+205) {
		tmp = ((1.0 - (x * (1.0 - eps_m))) + (1.0 / (1.0 + (x * (eps_m + 1.0))))) / 2.0;
	} else {
		tmp = (1.0 + (1.0 + (x * (-1.0 - eps_m)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 1.76e-6:
		tmp = ((x + 1.0) - (x + -1.0)) / 2.0
	elif eps_m <= 2.7e+205:
		tmp = ((1.0 - (x * (1.0 - eps_m))) + (1.0 / (1.0 + (x * (eps_m + 1.0))))) / 2.0
	else:
		tmp = (1.0 + (1.0 + (x * (-1.0 - eps_m)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 1.76e-6)
		tmp = Float64(Float64(Float64(x + 1.0) - Float64(x + -1.0)) / 2.0);
	elseif (eps_m <= 2.7e+205)
		tmp = Float64(Float64(Float64(1.0 - Float64(x * Float64(1.0 - eps_m))) + Float64(1.0 / Float64(1.0 + Float64(x * Float64(eps_m + 1.0))))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(1.0 + Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 1.76e-6)
		tmp = ((x + 1.0) - (x + -1.0)) / 2.0;
	elseif (eps_m <= 2.7e+205)
		tmp = ((1.0 - (x * (1.0 - eps_m))) + (1.0 / (1.0 + (x * (eps_m + 1.0))))) / 2.0;
	else
		tmp = (1.0 + (1.0 + (x * (-1.0 - eps_m)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 1.76e-6], N[(N[(N[(x + 1.0), $MachinePrecision] - N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps$95$m, 2.7e+205], N[(N[(N[(1.0 - N[(x * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(1.0 + N[(x * N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(1.0 + N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

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

Alternative 12: 58.6% accurate, 15.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{x \cdot \left(\left(eps_m + 1\right) \cdot \left(-1 + \frac{1}{eps_m}\right)\right)}{2}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{eps_m \cdot \left(x \cdot 2\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.0)
   (/ (* x (* (+ eps_m 1.0) (+ -1.0 (/ 1.0 eps_m)))) 2.0)
   (if (<= x 6.2e-5) 1.0 (/ (* eps_m (* x 2.0)) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = (x * ((eps_m + 1.0) * (-1.0 + (1.0 / eps_m)))) / 2.0;
	} else if (x <= 6.2e-5) {
		tmp = 1.0;
	} else {
		tmp = (eps_m * (x * 2.0)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (x * ((eps_m + 1.0d0) * ((-1.0d0) + (1.0d0 / eps_m)))) / 2.0d0
    else if (x <= 6.2d-5) then
        tmp = 1.0d0
    else
        tmp = (eps_m * (x * 2.0d0)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = (x * ((eps_m + 1.0) * (-1.0 + (1.0 / eps_m)))) / 2.0;
	} else if (x <= 6.2e-5) {
		tmp = 1.0;
	} else {
		tmp = (eps_m * (x * 2.0)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.0:
		tmp = (x * ((eps_m + 1.0) * (-1.0 + (1.0 / eps_m)))) / 2.0
	elif x <= 6.2e-5:
		tmp = 1.0
	else:
		tmp = (eps_m * (x * 2.0)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(x * Float64(Float64(eps_m + 1.0) * Float64(-1.0 + Float64(1.0 / eps_m)))) / 2.0);
	elseif (x <= 6.2e-5)
		tmp = 1.0;
	else
		tmp = Float64(Float64(eps_m * Float64(x * 2.0)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (x * ((eps_m + 1.0) * (-1.0 + (1.0 / eps_m)))) / 2.0;
	elseif (x <= 6.2e-5)
		tmp = 1.0;
	else
		tmp = (eps_m * (x * 2.0)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.0], N[(N[(x * N[(N[(eps$95$m + 1.0), $MachinePrecision] * N[(-1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 6.2e-5], 1.0, N[(N[(eps$95$m * N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

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

Alternative 13: 57.9% accurate, 17.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 + \left(1 + x \cdot \left(-1 - eps_m\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{eps_m \cdot \left(x \cdot 2\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 6.2e-5)
   (/ (+ 1.0 (+ 1.0 (* x (- -1.0 eps_m)))) 2.0)
   (/ (* eps_m (* x 2.0)) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 6.2e-5) {
		tmp = (1.0 + (1.0 + (x * (-1.0 - eps_m)))) / 2.0;
	} else {
		tmp = (eps_m * (x * 2.0)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 6.2d-5) then
        tmp = (1.0d0 + (1.0d0 + (x * ((-1.0d0) - eps_m)))) / 2.0d0
    else
        tmp = (eps_m * (x * 2.0d0)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 6.2e-5) {
		tmp = (1.0 + (1.0 + (x * (-1.0 - eps_m)))) / 2.0;
	} else {
		tmp = (eps_m * (x * 2.0)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 6.2e-5:
		tmp = (1.0 + (1.0 + (x * (-1.0 - eps_m)))) / 2.0
	else:
		tmp = (eps_m * (x * 2.0)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 6.2e-5)
		tmp = Float64(Float64(1.0 + Float64(1.0 + Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	else
		tmp = Float64(Float64(eps_m * Float64(x * 2.0)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 6.2e-5)
		tmp = (1.0 + (1.0 + (x * (-1.0 - eps_m)))) / 2.0;
	else
		tmp = (eps_m * (x * 2.0)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 6.2e-5], N[(N[(1.0 + N[(1.0 + N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(eps$95$m * N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

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

Alternative 14: 63.0% accurate, 17.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps_m \leq 4.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{\left(x + 1\right) - \left(x + -1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(1 + x \cdot \left(-1 - eps_m\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 4.5e+17)
   (/ (- (+ x 1.0) (+ x -1.0)) 2.0)
   (/ (+ 1.0 (+ 1.0 (* x (- -1.0 eps_m)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 4.5e+17) {
		tmp = ((x + 1.0) - (x + -1.0)) / 2.0;
	} else {
		tmp = (1.0 + (1.0 + (x * (-1.0 - eps_m)))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 4.5d+17) then
        tmp = ((x + 1.0d0) - (x + (-1.0d0))) / 2.0d0
    else
        tmp = (1.0d0 + (1.0d0 + (x * ((-1.0d0) - eps_m)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 4.5e+17) {
		tmp = ((x + 1.0) - (x + -1.0)) / 2.0;
	} else {
		tmp = (1.0 + (1.0 + (x * (-1.0 - eps_m)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 4.5e+17:
		tmp = ((x + 1.0) - (x + -1.0)) / 2.0
	else:
		tmp = (1.0 + (1.0 + (x * (-1.0 - eps_m)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 4.5e+17)
		tmp = Float64(Float64(Float64(x + 1.0) - Float64(x + -1.0)) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(1.0 + Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 4.5e+17)
		tmp = ((x + 1.0) - (x + -1.0)) / 2.0;
	else
		tmp = (1.0 + (1.0 + (x * (-1.0 - eps_m)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 4.5e+17], N[(N[(N[(x + 1.0), $MachinePrecision] - N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(1.0 + N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

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1. Add Preprocessing

Alternative 15: 58.5% accurate, 20.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;-\frac{x \cdot eps_m}{2}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{eps_m \cdot \left(x \cdot 2\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.0)
   (- (/ (* x eps_m) 2.0))
   (if (<= x 6.2e-5) 1.0 (/ (* eps_m (* x 2.0)) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = -((x * eps_m) / 2.0);
	} else if (x <= 6.2e-5) {
		tmp = 1.0;
	} else {
		tmp = (eps_m * (x * 2.0)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = -((x * eps_m) / 2.0d0)
    else if (x <= 6.2d-5) then
        tmp = 1.0d0
    else
        tmp = (eps_m * (x * 2.0d0)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = -((x * eps_m) / 2.0);
	} else if (x <= 6.2e-5) {
		tmp = 1.0;
	} else {
		tmp = (eps_m * (x * 2.0)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.0:
		tmp = -((x * eps_m) / 2.0)
	elif x <= 6.2e-5:
		tmp = 1.0
	else:
		tmp = (eps_m * (x * 2.0)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(-Float64(Float64(x * eps_m) / 2.0));
	elseif (x <= 6.2e-5)
		tmp = 1.0;
	else
		tmp = Float64(Float64(eps_m * Float64(x * 2.0)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = -((x * eps_m) / 2.0);
	elseif (x <= 6.2e-5)
		tmp = 1.0;
	else
		tmp = (eps_m * (x * 2.0)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.0], (-N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]), If[LessEqual[x, 6.2e-5], 1.0, N[(N[(eps$95$m * N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

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

Alternative 16: 58.5% accurate, 24.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{x \cdot eps_m}{2}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;-t_0\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (/ (* x eps_m) 2.0)))
   (if (<= x -1.0) (- t_0) (if (<= x 6.2e-5) 1.0 t_0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (x * eps_m) / 2.0;
	double tmp;
	if (x <= -1.0) {
		tmp = -t_0;
	} else if (x <= 6.2e-5) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * eps_m) / 2.0d0
    if (x <= (-1.0d0)) then
        tmp = -t_0
    else if (x <= 6.2d-5) then
        tmp = 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = (x * eps_m) / 2.0;
	double tmp;
	if (x <= -1.0) {
		tmp = -t_0;
	} else if (x <= 6.2e-5) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (x * eps_m) / 2.0
	tmp = 0
	if x <= -1.0:
		tmp = -t_0
	elif x <= 6.2e-5:
		tmp = 1.0
	else:
		tmp = t_0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(x * eps_m) / 2.0)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(-t_0);
	elseif (x <= 6.2e-5)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (x * eps_m) / 2.0;
	tmp = 0.0;
	if (x <= -1.0)
		tmp = -t_0;
	elseif (x <= 6.2e-5)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -1.0], (-t$95$0), If[LessEqual[x, 6.2e-5], 1.0, t$95$0]]]

Derivation

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

Alternative 17: 51.3% accurate, 32.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 6.2 \cdot 10^{-5}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 6.2e-5) 1.0 (/ (* x eps_m) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 6.2e-5) {
		tmp = 1.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 6.2d-5) then
        tmp = 1.0d0
    else
        tmp = (x * eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 6.2e-5) {
		tmp = 1.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 6.2e-5:
		tmp = 1.0
	else:
		tmp = (x * eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 6.2e-5)
		tmp = 1.0;
	else
		tmp = Float64(Float64(x * eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 6.2e-5)
		tmp = 1.0;
	else
		tmp = (x * eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 6.2e-5], 1.0, N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

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

Alternative 18: 43.9% accurate, 227.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ 1 \end{array} \]

FPCore

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

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return 1.0;
}

Fortran

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

Java

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

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return 1.0

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return 1.0
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = 1.0;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := 1.0

Derivation

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

Reproduce

herbie shell --seed 2024008 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  :precision binary64
  (/ (- (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x)))) (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x))))) 2.0))