NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.5% → 99.9%

Time: 12.0s

Alternatives: 10

Speedup: 2.1×

• 38.9% 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: 73.5% 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: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps_m \leq 10^{-13}:\\ \;\;\;\;\frac{\frac{x + 1}{e^{x}} + \left(x + 1\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps_m}\right) \cdot e^{x \cdot \left(eps_m + -1\right)} + e^{x \cdot \left(-1 - eps_m\right)} \cdot \left(1 + \frac{-1}{eps_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 1e-13)
   (/ (+ (/ (+ x 1.0) (exp x)) (* (+ x 1.0) (exp (- x)))) 2.0)
   (/
    (+
     (* (+ 1.0 (/ 1.0 eps_m)) (exp (* x (+ eps_m -1.0))))
     (* (exp (* x (- -1.0 eps_m))) (+ 1.0 (/ -1.0 eps_m))))
    2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1e-13) {
		tmp = (((x + 1.0) / exp(x)) + ((x + 1.0) * exp(-x))) / 2.0;
	} else {
		tmp = (((1.0 + (1.0 / eps_m)) * exp((x * (eps_m + -1.0)))) + (exp((x * (-1.0 - eps_m))) * (1.0 + (-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 <= 1d-13) then
        tmp = (((x + 1.0d0) / exp(x)) + ((x + 1.0d0) * exp(-x))) / 2.0d0
    else
        tmp = (((1.0d0 + (1.0d0 / eps_m)) * exp((x * (eps_m + (-1.0d0))))) + (exp((x * ((-1.0d0) - eps_m))) * (1.0d0 + ((-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 <= 1e-13) {
		tmp = (((x + 1.0) / Math.exp(x)) + ((x + 1.0) * Math.exp(-x))) / 2.0;
	} else {
		tmp = (((1.0 + (1.0 / eps_m)) * Math.exp((x * (eps_m + -1.0)))) + (Math.exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 1e-13:
		tmp = (((x + 1.0) / math.exp(x)) + ((x + 1.0) * math.exp(-x))) / 2.0
	else:
		tmp = (((1.0 + (1.0 / eps_m)) * math.exp((x * (eps_m + -1.0)))) + (math.exp((x * (-1.0 - eps_m))) * (1.0 + (-1.0 / eps_m)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 1e-13)
		tmp = Float64(Float64(Float64(Float64(x + 1.0) / exp(x)) + Float64(Float64(x + 1.0) * exp(Float64(-x)))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * exp(Float64(x * Float64(eps_m + -1.0)))) + Float64(exp(Float64(x * Float64(-1.0 - eps_m))) * Float64(1.0 + 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 <= 1e-13)
		tmp = (((x + 1.0) / exp(x)) + ((x + 1.0) * exp(-x))) / 2.0;
	else
		tmp = (((1.0 + (1.0 / eps_m)) * exp((x * (eps_m + -1.0)))) + (exp((x * (-1.0 - eps_m))) * (1.0 + (-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, 1e-13], N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] + N[(N[(x + 1.0), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

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

Alternative 2: 91.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-271}:\\ \;\;\;\;\frac{1 + e^{eps_m \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+41}:\\ \;\;\;\;\frac{1 + e^{x + eps_m \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(eps_m + -1\right) + \left(x + 2\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2e-271)
   (/ (+ 1.0 (exp (* eps_m (- x)))) 2.0)
   (if (<= x 3.6e+41)
     (/ (+ 1.0 (exp (+ x (* eps_m x)))) 2.0)
     (/ (+ (* x (+ eps_m -1.0)) (+ x 2.0)) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -2e-271) {
		tmp = (1.0 + exp((eps_m * -x))) / 2.0;
	} else if (x <= 3.6e+41) {
		tmp = (1.0 + exp((x + (eps_m * x)))) / 2.0;
	} else {
		tmp = ((x * (eps_m + -1.0)) + (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 <= (-2d-271)) then
        tmp = (1.0d0 + exp((eps_m * -x))) / 2.0d0
    else if (x <= 3.6d+41) then
        tmp = (1.0d0 + exp((x + (eps_m * x)))) / 2.0d0
    else
        tmp = ((x * (eps_m + (-1.0d0))) + (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 <= -2e-271) {
		tmp = (1.0 + Math.exp((eps_m * -x))) / 2.0;
	} else if (x <= 3.6e+41) {
		tmp = (1.0 + Math.exp((x + (eps_m * x)))) / 2.0;
	} else {
		tmp = ((x * (eps_m + -1.0)) + (x + 2.0)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -2e-271:
		tmp = (1.0 + math.exp((eps_m * -x))) / 2.0
	elif x <= 3.6e+41:
		tmp = (1.0 + math.exp((x + (eps_m * x)))) / 2.0
	else:
		tmp = ((x * (eps_m + -1.0)) + (x + 2.0)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -2e-271)
		tmp = Float64(Float64(1.0 + exp(Float64(eps_m * Float64(-x)))) / 2.0);
	elseif (x <= 3.6e+41)
		tmp = Float64(Float64(1.0 + exp(Float64(x + Float64(eps_m * x)))) / 2.0);
	else
		tmp = Float64(Float64(Float64(x * Float64(eps_m + -1.0)) + 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 <= -2e-271)
		tmp = (1.0 + exp((eps_m * -x))) / 2.0;
	elseif (x <= 3.6e+41)
		tmp = (1.0 + exp((x + (eps_m * x)))) / 2.0;
	else
		tmp = ((x * (eps_m + -1.0)) + (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, -2e-271], N[(N[(1.0 + N[Exp[N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3.6e+41], N[(N[(1.0 + N[Exp[N[(x + N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision] + N[(x + 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

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

Alternative 3: 85.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := x \cdot \left(-1 + \left(1 - eps_m\right)\right)\\ \mathbf{if}\;x \leq -1 \cdot 10^{-282}:\\ \;\;\;\;\frac{1 + e^{eps_m \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+26}:\\ \;\;\;\;\frac{\frac{4 - t_0 \cdot t_0}{2 + t_0}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(eps_m + -1\right) + \left(x + 2\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* x (+ -1.0 (- 1.0 eps_m)))))
   (if (<= x -1e-282)
     (/ (+ 1.0 (exp (* eps_m (- x)))) 2.0)
     (if (<= x 8e+26)
       (/ (/ (- 4.0 (* t_0 t_0)) (+ 2.0 t_0)) 2.0)
       (/ (+ (* x (+ eps_m -1.0)) (+ x 2.0)) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = x * (-1.0 + (1.0 - eps_m));
	double tmp;
	if (x <= -1e-282) {
		tmp = (1.0 + exp((eps_m * -x))) / 2.0;
	} else if (x <= 8e+26) {
		tmp = ((4.0 - (t_0 * t_0)) / (2.0 + t_0)) / 2.0;
	} else {
		tmp = ((x * (eps_m + -1.0)) + (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) :: t_0
    real(8) :: tmp
    t_0 = x * ((-1.0d0) + (1.0d0 - eps_m))
    if (x <= (-1d-282)) then
        tmp = (1.0d0 + exp((eps_m * -x))) / 2.0d0
    else if (x <= 8d+26) then
        tmp = ((4.0d0 - (t_0 * t_0)) / (2.0d0 + t_0)) / 2.0d0
    else
        tmp = ((x * (eps_m + (-1.0d0))) + (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 t_0 = x * (-1.0 + (1.0 - eps_m));
	double tmp;
	if (x <= -1e-282) {
		tmp = (1.0 + Math.exp((eps_m * -x))) / 2.0;
	} else if (x <= 8e+26) {
		tmp = ((4.0 - (t_0 * t_0)) / (2.0 + t_0)) / 2.0;
	} else {
		tmp = ((x * (eps_m + -1.0)) + (x + 2.0)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = x * (-1.0 + (1.0 - eps_m))
	tmp = 0
	if x <= -1e-282:
		tmp = (1.0 + math.exp((eps_m * -x))) / 2.0
	elif x <= 8e+26:
		tmp = ((4.0 - (t_0 * t_0)) / (2.0 + t_0)) / 2.0
	else:
		tmp = ((x * (eps_m + -1.0)) + (x + 2.0)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(x * Float64(-1.0 + Float64(1.0 - eps_m)))
	tmp = 0.0
	if (x <= -1e-282)
		tmp = Float64(Float64(1.0 + exp(Float64(eps_m * Float64(-x)))) / 2.0);
	elseif (x <= 8e+26)
		tmp = Float64(Float64(Float64(4.0 - Float64(t_0 * t_0)) / Float64(2.0 + t_0)) / 2.0);
	else
		tmp = Float64(Float64(Float64(x * Float64(eps_m + -1.0)) + Float64(x + 2.0)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = x * (-1.0 + (1.0 - eps_m));
	tmp = 0.0;
	if (x <= -1e-282)
		tmp = (1.0 + exp((eps_m * -x))) / 2.0;
	elseif (x <= 8e+26)
		tmp = ((4.0 - (t_0 * t_0)) / (2.0 + t_0)) / 2.0;
	else
		tmp = ((x * (eps_m + -1.0)) + (x + 2.0)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(x * N[(-1.0 + N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1e-282], N[(N[(1.0 + N[Exp[N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 8e+26], N[(N[(N[(4.0 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision] + N[(x + 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

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

Alternative 4: 78.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := x \cdot \left(-1 + \left(1 - eps_m\right)\right)\\ \mathbf{if}\;x \leq 4.8 \cdot 10^{-143}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{+26}:\\ \;\;\;\;\frac{\frac{4 - t_0 \cdot t_0}{2 + t_0}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(eps_m + -1\right) + \left(x + 2\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* x (+ -1.0 (- 1.0 eps_m)))))
   (if (<= x 4.8e-143)
     (/ (+ 1.0 (exp (- x))) 2.0)
     (if (<= x 6.6e+26)
       (/ (/ (- 4.0 (* t_0 t_0)) (+ 2.0 t_0)) 2.0)
       (/ (+ (* x (+ eps_m -1.0)) (+ x 2.0)) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = x * (-1.0 + (1.0 - eps_m));
	double tmp;
	if (x <= 4.8e-143) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 6.6e+26) {
		tmp = ((4.0 - (t_0 * t_0)) / (2.0 + t_0)) / 2.0;
	} else {
		tmp = ((x * (eps_m + -1.0)) + (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) :: t_0
    real(8) :: tmp
    t_0 = x * ((-1.0d0) + (1.0d0 - eps_m))
    if (x <= 4.8d-143) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if (x <= 6.6d+26) then
        tmp = ((4.0d0 - (t_0 * t_0)) / (2.0d0 + t_0)) / 2.0d0
    else
        tmp = ((x * (eps_m + (-1.0d0))) + (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 t_0 = x * (-1.0 + (1.0 - eps_m));
	double tmp;
	if (x <= 4.8e-143) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 6.6e+26) {
		tmp = ((4.0 - (t_0 * t_0)) / (2.0 + t_0)) / 2.0;
	} else {
		tmp = ((x * (eps_m + -1.0)) + (x + 2.0)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = x * (-1.0 + (1.0 - eps_m))
	tmp = 0
	if x <= 4.8e-143:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 6.6e+26:
		tmp = ((4.0 - (t_0 * t_0)) / (2.0 + t_0)) / 2.0
	else:
		tmp = ((x * (eps_m + -1.0)) + (x + 2.0)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(x * Float64(-1.0 + Float64(1.0 - eps_m)))
	tmp = 0.0
	if (x <= 4.8e-143)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 6.6e+26)
		tmp = Float64(Float64(Float64(4.0 - Float64(t_0 * t_0)) / Float64(2.0 + t_0)) / 2.0);
	else
		tmp = Float64(Float64(Float64(x * Float64(eps_m + -1.0)) + Float64(x + 2.0)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = x * (-1.0 + (1.0 - eps_m));
	tmp = 0.0;
	if (x <= 4.8e-143)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif (x <= 6.6e+26)
		tmp = ((4.0 - (t_0 * t_0)) / (2.0 + t_0)) / 2.0;
	else
		tmp = ((x * (eps_m + -1.0)) + (x + 2.0)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(x * N[(-1.0 + N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 4.8e-143], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 6.6e+26], N[(N[(N[(4.0 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision] + N[(x + 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

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1. Add Preprocessing

Alternative 5: 71.4% accurate, 6.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := x \cdot \left(-1 + \left(1 - eps_m\right)\right)\\ \mathbf{if}\;x \leq -2 \cdot 10^{-271}:\\ \;\;\;\;\frac{2 - eps_m \cdot x}{2}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+27}:\\ \;\;\;\;\frac{\frac{4 - t_0 \cdot t_0}{2 + t_0}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(eps_m + -1\right) + \left(x + 2\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* x (+ -1.0 (- 1.0 eps_m)))))
   (if (<= x -2e-271)
     (/ (- 2.0 (* eps_m x)) 2.0)
     (if (<= x 3.2e+27)
       (/ (/ (- 4.0 (* t_0 t_0)) (+ 2.0 t_0)) 2.0)
       (/ (+ (* x (+ eps_m -1.0)) (+ x 2.0)) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = x * (-1.0 + (1.0 - eps_m));
	double tmp;
	if (x <= -2e-271) {
		tmp = (2.0 - (eps_m * x)) / 2.0;
	} else if (x <= 3.2e+27) {
		tmp = ((4.0 - (t_0 * t_0)) / (2.0 + t_0)) / 2.0;
	} else {
		tmp = ((x * (eps_m + -1.0)) + (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) :: t_0
    real(8) :: tmp
    t_0 = x * ((-1.0d0) + (1.0d0 - eps_m))
    if (x <= (-2d-271)) then
        tmp = (2.0d0 - (eps_m * x)) / 2.0d0
    else if (x <= 3.2d+27) then
        tmp = ((4.0d0 - (t_0 * t_0)) / (2.0d0 + t_0)) / 2.0d0
    else
        tmp = ((x * (eps_m + (-1.0d0))) + (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 t_0 = x * (-1.0 + (1.0 - eps_m));
	double tmp;
	if (x <= -2e-271) {
		tmp = (2.0 - (eps_m * x)) / 2.0;
	} else if (x <= 3.2e+27) {
		tmp = ((4.0 - (t_0 * t_0)) / (2.0 + t_0)) / 2.0;
	} else {
		tmp = ((x * (eps_m + -1.0)) + (x + 2.0)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = x * (-1.0 + (1.0 - eps_m))
	tmp = 0
	if x <= -2e-271:
		tmp = (2.0 - (eps_m * x)) / 2.0
	elif x <= 3.2e+27:
		tmp = ((4.0 - (t_0 * t_0)) / (2.0 + t_0)) / 2.0
	else:
		tmp = ((x * (eps_m + -1.0)) + (x + 2.0)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(x * Float64(-1.0 + Float64(1.0 - eps_m)))
	tmp = 0.0
	if (x <= -2e-271)
		tmp = Float64(Float64(2.0 - Float64(eps_m * x)) / 2.0);
	elseif (x <= 3.2e+27)
		tmp = Float64(Float64(Float64(4.0 - Float64(t_0 * t_0)) / Float64(2.0 + t_0)) / 2.0);
	else
		tmp = Float64(Float64(Float64(x * Float64(eps_m + -1.0)) + Float64(x + 2.0)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = x * (-1.0 + (1.0 - eps_m));
	tmp = 0.0;
	if (x <= -2e-271)
		tmp = (2.0 - (eps_m * x)) / 2.0;
	elseif (x <= 3.2e+27)
		tmp = ((4.0 - (t_0 * t_0)) / (2.0 + t_0)) / 2.0;
	else
		tmp = ((x * (eps_m + -1.0)) + (x + 2.0)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(x * N[(-1.0 + N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2e-271], N[(N[(2.0 - N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3.2e+27], N[(N[(N[(4.0 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision] + N[(x + 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

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1. Add Preprocessing

Alternative 6: 69.8% accurate, 17.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-282}:\\ \;\;\;\;\frac{2 - eps_m \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(eps_m + -1\right) + \left(x + 2\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1e-282)
   (/ (- 2.0 (* eps_m x)) 2.0)
   (/ (+ (* x (+ eps_m -1.0)) (+ x 2.0)) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1e-282) {
		tmp = (2.0 - (eps_m * x)) / 2.0;
	} else {
		tmp = ((x * (eps_m + -1.0)) + (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 <= (-1d-282)) then
        tmp = (2.0d0 - (eps_m * x)) / 2.0d0
    else
        tmp = ((x * (eps_m + (-1.0d0))) + (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 <= -1e-282) {
		tmp = (2.0 - (eps_m * x)) / 2.0;
	} else {
		tmp = ((x * (eps_m + -1.0)) + (x + 2.0)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1e-282:
		tmp = (2.0 - (eps_m * x)) / 2.0
	else:
		tmp = ((x * (eps_m + -1.0)) + (x + 2.0)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1e-282)
		tmp = Float64(Float64(2.0 - Float64(eps_m * x)) / 2.0);
	else
		tmp = Float64(Float64(Float64(x * Float64(eps_m + -1.0)) + 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 <= -1e-282)
		tmp = (2.0 - (eps_m * x)) / 2.0;
	else
		tmp = ((x * (eps_m + -1.0)) + (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, -1e-282], N[(N[(2.0 - N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision] + N[(x + 2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

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1. Add Preprocessing

Alternative 7: 57.6% accurate, 25.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-283}:\\ \;\;\;\;\frac{2 - eps_m \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + eps_m \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2e-283) (/ (- 2.0 (* eps_m x)) 2.0) (/ (+ 2.0 (* eps_m x)) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -2e-283) {
		tmp = (2.0 - (eps_m * x)) / 2.0;
	} else {
		tmp = (2.0 + (eps_m * x)) / 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 <= (-2d-283)) then
        tmp = (2.0d0 - (eps_m * x)) / 2.0d0
    else
        tmp = (2.0d0 + (eps_m * x)) / 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 <= -2e-283) {
		tmp = (2.0 - (eps_m * x)) / 2.0;
	} else {
		tmp = (2.0 + (eps_m * x)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -2e-283:
		tmp = (2.0 - (eps_m * x)) / 2.0
	else:
		tmp = (2.0 + (eps_m * x)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -2e-283)
		tmp = Float64(Float64(2.0 - Float64(eps_m * x)) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(eps_m * x)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -2e-283)
		tmp = (2.0 - (eps_m * x)) / 2.0;
	else
		tmp = (2.0 + (eps_m * x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -2e-283], N[(N[(2.0 - N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

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

Alternative 8: 50.8% accurate, 32.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{eps_m \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 1.7) 1.0 (/ (* eps_m x) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.7) {
		tmp = 1.0;
	} else {
		tmp = (eps_m * x) / 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 <= 1.7d0) then
        tmp = 1.0d0
    else
        tmp = (eps_m * x) / 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 <= 1.7) {
		tmp = 1.0;
	} else {
		tmp = (eps_m * x) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 1.7:
		tmp = 1.0
	else:
		tmp = (eps_m * x) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 1.7)
		tmp = 1.0;
	else
		tmp = Float64(Float64(eps_m * x) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 1.7)
		tmp = 1.0;
	else
		tmp = (eps_m * x) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 1.7], 1.0, N[(N[(eps$95$m * x), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

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

Alternative 9: 49.9% accurate, 32.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \frac{2 + eps_m \cdot x}{2} \end{array} \]

FPCore

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

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return (2.0 + (eps_m * x)) / 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 = (2.0d0 + (eps_m * x)) / 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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

Alternative 10: 43.7% 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 2023350 
(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))