NMSE Section 6.1 mentioned, A

Percentage Accurate: 72.2% → 99.9%

Time: 9.3s

Alternatives: 17

Speedup: 1.6×

• 37.5% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -1.0578645557035018e+103
  2. eps = 4.757752329810648e-129

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, eps)
use fmin_fmax_functions
    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: 72.2% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, eps)
use fmin_fmax_functions
    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, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if eps < 1

   1. Initial program 63.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0

      \[\leadsto \frac{\color{blue}{\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 70.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 + x, e^{-x}, \frac{1 + x}{e^{x}}\right)}}{2} \]

   if 1 < eps

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 98.7%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}}{2} \]

   6. Taylor expanded in eps around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{\varepsilon \cdot x}} - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 98.7%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{x \cdot \varepsilon}} - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]

   9. Taylor expanded in eps around inf

      \[\leadsto \frac{\color{blue}{1} \cdot e^{x \cdot \varepsilon} - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 100.0%

       \[\leadsto \frac{\color{blue}{1} \cdot e^{x \cdot \varepsilon} - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 83.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := 1 + \frac{1}{eps\_m}\\ t_1 := \frac{1}{eps\_m} - 1\\ \mathbf{if}\;x \leq -8 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{e^{-x}}{eps\_m} - -1}{2}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-108}:\\ \;\;\;\;\frac{1 \cdot \mathsf{fma}\left(eps\_m - 1, x, 1\right) - \frac{-1}{e^{\mathsf{fma}\left(x, eps\_m, x\right)}}}{2}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+210}:\\ \;\;\;\;\frac{t\_0 \cdot e^{x \cdot eps\_m} - t\_1}{2}\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{+247}:\\ \;\;\;\;\frac{\left(\frac{1}{eps\_m} - -1\right) - t\_1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot e^{\left(-1 + eps\_m\right) \cdot x} - \frac{-1}{\mathsf{fma}\left(eps\_m - -1, x, 1\right)}}{2}\\ \end{array} \end{array} \]

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 eps_m) 1.0)))
   (if (<= x -8e+36)
     (/ (- (/ (exp (- x)) eps_m) -1.0) 2.0)
     (if (<= x 1.65e-108)
       (/
        (- (* 1.0 (fma (- eps_m 1.0) x 1.0)) (/ -1.0 (exp (fma x eps_m x))))
        2.0)
       (if (<= x 3e+210)
         (/ (- (* t_0 (exp (* x eps_m))) t_1) 2.0)
         (if (<= x 1.22e+247)
           (/ (- (- (/ 1.0 eps_m) -1.0) t_1) 2.0)
           (/
            (-
             (* t_0 (exp (* (+ -1.0 eps_m) x)))
             (/ -1.0 (fma (- eps_m -1.0) x 1.0)))
            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 / eps_m) - 1.0;
	double tmp;
	if (x <= -8e+36) {
		tmp = ((exp(-x) / eps_m) - -1.0) / 2.0;
	} else if (x <= 1.65e-108) {
		tmp = ((1.0 * fma((eps_m - 1.0), x, 1.0)) - (-1.0 / exp(fma(x, eps_m, x)))) / 2.0;
	} else if (x <= 3e+210) {
		tmp = ((t_0 * exp((x * eps_m))) - t_1) / 2.0;
	} else if (x <= 1.22e+247) {
		tmp = (((1.0 / eps_m) - -1.0) - t_1) / 2.0;
	} else {
		tmp = ((t_0 * exp(((-1.0 + eps_m) * x))) - (-1.0 / fma((eps_m - -1.0), x, 1.0))) / 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(Float64(1.0 / eps_m) - 1.0)
	tmp = 0.0
	if (x <= -8e+36)
		tmp = Float64(Float64(Float64(exp(Float64(-x)) / eps_m) - -1.0) / 2.0);
	elseif (x <= 1.65e-108)
		tmp = Float64(Float64(Float64(1.0 * fma(Float64(eps_m - 1.0), x, 1.0)) - Float64(-1.0 / exp(fma(x, eps_m, x)))) / 2.0);
	elseif (x <= 3e+210)
		tmp = Float64(Float64(Float64(t_0 * exp(Float64(x * eps_m))) - t_1) / 2.0);
	elseif (x <= 1.22e+247)
		tmp = Float64(Float64(Float64(Float64(1.0 / eps_m) - -1.0) - t_1) / 2.0);
	else
		tmp = Float64(Float64(Float64(t_0 * exp(Float64(Float64(-1.0 + eps_m) * x))) - Float64(-1.0 / fma(Float64(eps_m - -1.0), x, 1.0))) / 2.0);
	end
	return 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[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -8e+36], N[(N[(N[(N[Exp[(-x)], $MachinePrecision] / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.65e-108], N[(N[(N[(1.0 * N[(N[(eps$95$m - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(-1.0 / N[Exp[N[(x * eps$95$m + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3e+210], N[(N[(N[(t$95$0 * N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.22e+247], N[(N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision] - t$95$1), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(t$95$0 * N[Exp[N[(N[(-1.0 + eps$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(-1.0 / N[(N[(eps$95$m - -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -8.00000000000000034e36

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 49.5%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in eps around 0

      \[\leadsto \frac{\color{blue}{\frac{e^{\mathsf{neg}\left(x\right)}}{\varepsilon}} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 52.2%

      \[\leadsto \frac{\color{blue}{\frac{e^{-x}}{\varepsilon}} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]

   9. Taylor expanded in eps around inf

      \[\leadsto \frac{\frac{e^{-x}}{\varepsilon} - -1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 52.2%

       \[\leadsto \frac{\frac{e^{-x}}{\varepsilon} - -1}{2} \]

   if -8.00000000000000034e36 < x < 1.6500000000000001e-108

   1. Initial program 58.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 56.5%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(1 + x \cdot \left(\varepsilon - 1\right)\right)} - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 42.6%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\mathsf{fma}\left(\varepsilon - 1, x, 1\right)} - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]

   9. Taylor expanded in eps around inf

      \[\leadsto \frac{\color{blue}{1} \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 85.5%

       \[\leadsto \frac{\color{blue}{1} \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]

   if 1.6500000000000001e-108 < x < 3.00000000000000022e210

   1. Initial program 88.9%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 34.7%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in eps around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{\varepsilon \cdot x}} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.6%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{x \cdot \varepsilon}} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]

   if 3.00000000000000022e210 < x < 1.22000000000000006e247

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 8.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 87.1%

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{\varepsilon} + 1\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]

   if 1.22000000000000006e247 < x

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 100.0%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \frac{-1}{1 + \color{blue}{x \cdot \left(1 + \varepsilon\right)}}}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.5%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \frac{-1}{\mathsf{fma}\left(\varepsilon - -1, \color{blue}{x}, 1\right)}}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{e^{-x}}{\varepsilon} - -1}{2}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-108}:\\ \;\;\;\;\frac{1 \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+210}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \varepsilon} - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{+247}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \frac{-1}{\mathsf{fma}\left(\varepsilon - -1, x, 1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := 1 + \frac{1}{eps\_m}\\ t_1 := \frac{1}{eps\_m} - 1\\ \mathbf{if}\;x \leq -8 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{e^{-x}}{eps\_m} - -1}{2}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-108}:\\ \;\;\;\;\frac{1 \cdot \mathsf{fma}\left(eps\_m - 1, x, 1\right) - \frac{-1}{e^{\mathsf{fma}\left(x, eps\_m, x\right)}}}{2}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+210}:\\ \;\;\;\;\frac{t\_0 \cdot e^{x \cdot eps\_m} - t\_1}{2}\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{+247}:\\ \;\;\;\;\frac{\left(\frac{1}{eps\_m} - -1\right) - t\_1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot e^{\left(-1 + eps\_m\right) \cdot x} - t\_1}{2}\\ \end{array} \end{array} \]

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 eps_m) 1.0)))
   (if (<= x -8e+36)
     (/ (- (/ (exp (- x)) eps_m) -1.0) 2.0)
     (if (<= x 1.65e-108)
       (/
        (- (* 1.0 (fma (- eps_m 1.0) x 1.0)) (/ -1.0 (exp (fma x eps_m x))))
        2.0)
       (if (<= x 3e+210)
         (/ (- (* t_0 (exp (* x eps_m))) t_1) 2.0)
         (if (<= x 1.28e+247)
           (/ (- (- (/ 1.0 eps_m) -1.0) t_1) 2.0)
           (/ (- (* t_0 (exp (* (+ -1.0 eps_m) x))) t_1) 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 / eps_m) - 1.0;
	double tmp;
	if (x <= -8e+36) {
		tmp = ((exp(-x) / eps_m) - -1.0) / 2.0;
	} else if (x <= 1.65e-108) {
		tmp = ((1.0 * fma((eps_m - 1.0), x, 1.0)) - (-1.0 / exp(fma(x, eps_m, x)))) / 2.0;
	} else if (x <= 3e+210) {
		tmp = ((t_0 * exp((x * eps_m))) - t_1) / 2.0;
	} else if (x <= 1.28e+247) {
		tmp = (((1.0 / eps_m) - -1.0) - t_1) / 2.0;
	} else {
		tmp = ((t_0 * exp(((-1.0 + eps_m) * x))) - t_1) / 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(Float64(1.0 / eps_m) - 1.0)
	tmp = 0.0
	if (x <= -8e+36)
		tmp = Float64(Float64(Float64(exp(Float64(-x)) / eps_m) - -1.0) / 2.0);
	elseif (x <= 1.65e-108)
		tmp = Float64(Float64(Float64(1.0 * fma(Float64(eps_m - 1.0), x, 1.0)) - Float64(-1.0 / exp(fma(x, eps_m, x)))) / 2.0);
	elseif (x <= 3e+210)
		tmp = Float64(Float64(Float64(t_0 * exp(Float64(x * eps_m))) - t_1) / 2.0);
	elseif (x <= 1.28e+247)
		tmp = Float64(Float64(Float64(Float64(1.0 / eps_m) - -1.0) - t_1) / 2.0);
	else
		tmp = Float64(Float64(Float64(t_0 * exp(Float64(Float64(-1.0 + eps_m) * x))) - t_1) / 2.0);
	end
	return 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[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -8e+36], N[(N[(N[(N[Exp[(-x)], $MachinePrecision] / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.65e-108], N[(N[(N[(1.0 * N[(N[(eps$95$m - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(-1.0 / N[Exp[N[(x * eps$95$m + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3e+210], N[(N[(N[(t$95$0 * N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.28e+247], N[(N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision] - t$95$1), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(t$95$0 * N[Exp[N[(N[(-1.0 + eps$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -8.00000000000000034e36

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 49.5%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in eps around 0

      \[\leadsto \frac{\color{blue}{\frac{e^{\mathsf{neg}\left(x\right)}}{\varepsilon}} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 52.2%

      \[\leadsto \frac{\color{blue}{\frac{e^{-x}}{\varepsilon}} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]

   9. Taylor expanded in eps around inf

      \[\leadsto \frac{\frac{e^{-x}}{\varepsilon} - -1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 52.2%

       \[\leadsto \frac{\frac{e^{-x}}{\varepsilon} - -1}{2} \]

   if -8.00000000000000034e36 < x < 1.6500000000000001e-108

   1. Initial program 58.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 56.5%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(1 + x \cdot \left(\varepsilon - 1\right)\right)} - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 42.6%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\mathsf{fma}\left(\varepsilon - 1, x, 1\right)} - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]

   9. Taylor expanded in eps around inf

      \[\leadsto \frac{\color{blue}{1} \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 85.5%

       \[\leadsto \frac{\color{blue}{1} \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]

   if 1.6500000000000001e-108 < x < 3.00000000000000022e210

   1. Initial program 88.9%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 34.7%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in eps around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{\varepsilon \cdot x}} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.6%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{x \cdot \varepsilon}} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]

   if 3.00000000000000022e210 < x < 1.28000000000000006e247

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 8.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 87.1%

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{\varepsilon} + 1\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]

   if 1.28000000000000006e247 < x

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 47.0%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{e^{-x}}{\varepsilon} - -1}{2}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-108}:\\ \;\;\;\;\frac{1 \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+210}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \varepsilon} - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{+247}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{1 \cdot \mathsf{fma}\left(eps\_m - 1, x, 1\right) - \frac{-1}{e^{\mathsf{fma}\left(x, eps\_m, x\right)}}}{2}\\ t_1 := \frac{1}{eps\_m} - 1\\ \mathbf{if}\;x \leq -8 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{e^{-x}}{eps\_m} - -1}{2}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-108}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+210}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) \cdot e^{x \cdot eps\_m} - t\_1}{2}\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{+247}:\\ \;\;\;\;\frac{\left(\frac{1}{eps\_m} - -1\right) - t\_1}{2}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0
         (/
          (- (* 1.0 (fma (- eps_m 1.0) x 1.0)) (/ -1.0 (exp (fma x eps_m x))))
          2.0))
        (t_1 (- (/ 1.0 eps_m) 1.0)))
   (if (<= x -8e+36)
     (/ (- (/ (exp (- x)) eps_m) -1.0) 2.0)
     (if (<= x 1.65e-108)
       t_0
       (if (<= x 3e+210)
         (/ (- (* (+ 1.0 (/ 1.0 eps_m)) (exp (* x eps_m))) t_1) 2.0)
         (if (<= x 1.28e+247) (/ (- (- (/ 1.0 eps_m) -1.0) t_1) 2.0) t_0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = ((1.0 * fma((eps_m - 1.0), x, 1.0)) - (-1.0 / exp(fma(x, eps_m, x)))) / 2.0;
	double t_1 = (1.0 / eps_m) - 1.0;
	double tmp;
	if (x <= -8e+36) {
		tmp = ((exp(-x) / eps_m) - -1.0) / 2.0;
	} else if (x <= 1.65e-108) {
		tmp = t_0;
	} else if (x <= 3e+210) {
		tmp = (((1.0 + (1.0 / eps_m)) * exp((x * eps_m))) - t_1) / 2.0;
	} else if (x <= 1.28e+247) {
		tmp = (((1.0 / eps_m) - -1.0) - t_1) / 2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(Float64(1.0 * fma(Float64(eps_m - 1.0), x, 1.0)) - Float64(-1.0 / exp(fma(x, eps_m, x)))) / 2.0)
	t_1 = Float64(Float64(1.0 / eps_m) - 1.0)
	tmp = 0.0
	if (x <= -8e+36)
		tmp = Float64(Float64(Float64(exp(Float64(-x)) / eps_m) - -1.0) / 2.0);
	elseif (x <= 1.65e-108)
		tmp = t_0;
	elseif (x <= 3e+210)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * exp(Float64(x * eps_m))) - t_1) / 2.0);
	elseif (x <= 1.28e+247)
		tmp = Float64(Float64(Float64(Float64(1.0 / eps_m) - -1.0) - t_1) / 2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(N[(1.0 * N[(N[(eps$95$m - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(-1.0 / N[Exp[N[(x * eps$95$m + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -8e+36], N[(N[(N[(N[Exp[(-x)], $MachinePrecision] / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.65e-108], t$95$0, If[LessEqual[x, 3e+210], N[(N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.28e+247], N[(N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision] - t$95$1), $MachinePrecision] / 2.0), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.00000000000000034e36

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 49.5%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in eps around 0

      \[\leadsto \frac{\color{blue}{\frac{e^{\mathsf{neg}\left(x\right)}}{\varepsilon}} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 52.2%

      \[\leadsto \frac{\color{blue}{\frac{e^{-x}}{\varepsilon}} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]

   9. Taylor expanded in eps around inf

      \[\leadsto \frac{\frac{e^{-x}}{\varepsilon} - -1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 52.2%

       \[\leadsto \frac{\frac{e^{-x}}{\varepsilon} - -1}{2} \]

   if -8.00000000000000034e36 < x < 1.6500000000000001e-108 or 1.28000000000000006e247 < x

   1. Initial program 61.3%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 59.7%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(1 + x \cdot \left(\varepsilon - 1\right)\right)} - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 42.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\mathsf{fma}\left(\varepsilon - 1, x, 1\right)} - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]

   9. Taylor expanded in eps around inf

      \[\leadsto \frac{\color{blue}{1} \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 82.5%

       \[\leadsto \frac{\color{blue}{1} \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]

   if 1.6500000000000001e-108 < x < 3.00000000000000022e210

   1. Initial program 88.9%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 34.7%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in eps around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{\varepsilon \cdot x}} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.6%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{x \cdot \varepsilon}} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]

   if 3.00000000000000022e210 < x < 1.28000000000000006e247

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 8.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 87.1%

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{\varepsilon} + 1\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{e^{-x}}{\varepsilon} - -1}{2}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-108}:\\ \;\;\;\;\frac{1 \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+210}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \varepsilon} - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{+247}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{1 \cdot \mathsf{fma}\left(eps\_m - 1, x, 1\right) - \frac{-1}{e^{\mathsf{fma}\left(x, eps\_m, x\right)}}}{2}\\ \mathbf{if}\;x \leq -8 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{e^{-x}}{eps\_m} - -1}{2}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-108}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+210}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) \cdot e^{x \cdot eps\_m} - \frac{-1}{1}}{2}\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{+247}:\\ \;\;\;\;\frac{\left(\frac{1}{eps\_m} - -1\right) - \left(\frac{1}{eps\_m} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0
         (/
          (- (* 1.0 (fma (- eps_m 1.0) x 1.0)) (/ -1.0 (exp (fma x eps_m x))))
          2.0)))
   (if (<= x -8e+36)
     (/ (- (/ (exp (- x)) eps_m) -1.0) 2.0)
     (if (<= x 1.65e-108)
       t_0
       (if (<= x 2.8e+210)
         (/ (- (* (+ 1.0 (/ 1.0 eps_m)) (exp (* x eps_m))) (/ -1.0 1.0)) 2.0)
         (if (<= x 1.28e+247)
           (/ (- (- (/ 1.0 eps_m) -1.0) (- (/ 1.0 eps_m) 1.0)) 2.0)
           t_0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = ((1.0 * fma((eps_m - 1.0), x, 1.0)) - (-1.0 / exp(fma(x, eps_m, x)))) / 2.0;
	double tmp;
	if (x <= -8e+36) {
		tmp = ((exp(-x) / eps_m) - -1.0) / 2.0;
	} else if (x <= 1.65e-108) {
		tmp = t_0;
	} else if (x <= 2.8e+210) {
		tmp = (((1.0 + (1.0 / eps_m)) * exp((x * eps_m))) - (-1.0 / 1.0)) / 2.0;
	} else if (x <= 1.28e+247) {
		tmp = (((1.0 / eps_m) - -1.0) - ((1.0 / eps_m) - 1.0)) / 2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(Float64(1.0 * fma(Float64(eps_m - 1.0), x, 1.0)) - Float64(-1.0 / exp(fma(x, eps_m, x)))) / 2.0)
	tmp = 0.0
	if (x <= -8e+36)
		tmp = Float64(Float64(Float64(exp(Float64(-x)) / eps_m) - -1.0) / 2.0);
	elseif (x <= 1.65e-108)
		tmp = t_0;
	elseif (x <= 2.8e+210)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * exp(Float64(x * eps_m))) - Float64(-1.0 / 1.0)) / 2.0);
	elseif (x <= 1.28e+247)
		tmp = Float64(Float64(Float64(Float64(1.0 / eps_m) - -1.0) - Float64(Float64(1.0 / eps_m) - 1.0)) / 2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(N[(1.0 * N[(N[(eps$95$m - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(-1.0 / N[Exp[N[(x * eps$95$m + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -8e+36], N[(N[(N[(N[Exp[(-x)], $MachinePrecision] / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.65e-108], t$95$0, If[LessEqual[x, 2.8e+210], N[(N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(-1.0 / 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.28e+247], N[(N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision] - N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.00000000000000034e36

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 49.5%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in eps around 0

      \[\leadsto \frac{\color{blue}{\frac{e^{\mathsf{neg}\left(x\right)}}{\varepsilon}} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 52.2%

      \[\leadsto \frac{\color{blue}{\frac{e^{-x}}{\varepsilon}} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]

   9. Taylor expanded in eps around inf

      \[\leadsto \frac{\frac{e^{-x}}{\varepsilon} - -1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 52.2%

       \[\leadsto \frac{\frac{e^{-x}}{\varepsilon} - -1}{2} \]

   if -8.00000000000000034e36 < x < 1.6500000000000001e-108 or 1.28000000000000006e247 < x

   1. Initial program 61.3%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 59.7%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(1 + x \cdot \left(\varepsilon - 1\right)\right)} - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 42.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\mathsf{fma}\left(\varepsilon - 1, x, 1\right)} - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]

   9. Taylor expanded in eps around inf

      \[\leadsto \frac{\color{blue}{1} \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 82.5%

       \[\leadsto \frac{\color{blue}{1} \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]

   if 1.6500000000000001e-108 < x < 2.8000000000000002e210

   1. Initial program 88.9%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 87.4%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}}{2} \]

   6. Taylor expanded in eps around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{\varepsilon \cdot x}} - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 66.5%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{x \cdot \varepsilon}} - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \varepsilon} - \frac{-1}{1}}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 33.2%

       \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \varepsilon} - \frac{-1}{1}}{2} \]

   if 2.8000000000000002e210 < x < 1.28000000000000006e247

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 8.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 87.1%

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{\varepsilon} + 1\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{e^{-x}}{\varepsilon} - -1}{2}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-108}:\\ \;\;\;\;\frac{1 \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+210}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \varepsilon} - \frac{-1}{1}}{2}\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{+247}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{1 \cdot \mathsf{fma}\left(eps\_m - 1, x, 1\right) - \frac{-1}{e^{\mathsf{fma}\left(x, eps\_m, x\right)}}}{2}\\ \mathbf{if}\;x \leq -8 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{e^{-x}}{eps\_m} - -1}{2}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-108}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+210}:\\ \;\;\;\;\frac{1 \cdot e^{x \cdot eps\_m} - \frac{1 - eps\_m}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{+247}:\\ \;\;\;\;\frac{\left(\frac{1}{eps\_m} - -1\right) - \left(\frac{1}{eps\_m} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0
         (/
          (- (* 1.0 (fma (- eps_m 1.0) x 1.0)) (/ -1.0 (exp (fma x eps_m x))))
          2.0)))
   (if (<= x -8e+36)
     (/ (- (/ (exp (- x)) eps_m) -1.0) 2.0)
     (if (<= x 1.65e-108)
       t_0
       (if (<= x 2.8e+210)
         (/ (- (* 1.0 (exp (* x eps_m))) (/ (- 1.0 eps_m) eps_m)) 2.0)
         (if (<= x 1.28e+247)
           (/ (- (- (/ 1.0 eps_m) -1.0) (- (/ 1.0 eps_m) 1.0)) 2.0)
           t_0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = ((1.0 * fma((eps_m - 1.0), x, 1.0)) - (-1.0 / exp(fma(x, eps_m, x)))) / 2.0;
	double tmp;
	if (x <= -8e+36) {
		tmp = ((exp(-x) / eps_m) - -1.0) / 2.0;
	} else if (x <= 1.65e-108) {
		tmp = t_0;
	} else if (x <= 2.8e+210) {
		tmp = ((1.0 * exp((x * eps_m))) - ((1.0 - eps_m) / eps_m)) / 2.0;
	} else if (x <= 1.28e+247) {
		tmp = (((1.0 / eps_m) - -1.0) - ((1.0 / eps_m) - 1.0)) / 2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(Float64(1.0 * fma(Float64(eps_m - 1.0), x, 1.0)) - Float64(-1.0 / exp(fma(x, eps_m, x)))) / 2.0)
	tmp = 0.0
	if (x <= -8e+36)
		tmp = Float64(Float64(Float64(exp(Float64(-x)) / eps_m) - -1.0) / 2.0);
	elseif (x <= 1.65e-108)
		tmp = t_0;
	elseif (x <= 2.8e+210)
		tmp = Float64(Float64(Float64(1.0 * exp(Float64(x * eps_m))) - Float64(Float64(1.0 - eps_m) / eps_m)) / 2.0);
	elseif (x <= 1.28e+247)
		tmp = Float64(Float64(Float64(Float64(1.0 / eps_m) - -1.0) - Float64(Float64(1.0 / eps_m) - 1.0)) / 2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(N[(1.0 * N[(N[(eps$95$m - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(-1.0 / N[Exp[N[(x * eps$95$m + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -8e+36], N[(N[(N[(N[Exp[(-x)], $MachinePrecision] / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.65e-108], t$95$0, If[LessEqual[x, 2.8e+210], N[(N[(N[(1.0 * N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 - eps$95$m), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.28e+247], N[(N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision] - N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.00000000000000034e36

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 49.5%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in eps around 0

      \[\leadsto \frac{\color{blue}{\frac{e^{\mathsf{neg}\left(x\right)}}{\varepsilon}} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 52.2%

      \[\leadsto \frac{\color{blue}{\frac{e^{-x}}{\varepsilon}} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]

   9. Taylor expanded in eps around inf

      \[\leadsto \frac{\frac{e^{-x}}{\varepsilon} - -1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 52.2%

       \[\leadsto \frac{\frac{e^{-x}}{\varepsilon} - -1}{2} \]

   if -8.00000000000000034e36 < x < 1.6500000000000001e-108 or 1.28000000000000006e247 < x

   1. Initial program 61.3%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 59.7%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(1 + x \cdot \left(\varepsilon - 1\right)\right)} - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 42.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\mathsf{fma}\left(\varepsilon - 1, x, 1\right)} - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]

   9. Taylor expanded in eps around inf

      \[\leadsto \frac{\color{blue}{1} \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 82.5%

       \[\leadsto \frac{\color{blue}{1} \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]

   if 1.6500000000000001e-108 < x < 2.8000000000000002e210

   1. Initial program 88.9%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 87.4%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}}{2} \]

   6. Taylor expanded in eps around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{\varepsilon \cdot x}} - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 66.5%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{x \cdot \varepsilon}} - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]

   9. Taylor expanded in eps around inf

      \[\leadsto \frac{\color{blue}{1} \cdot e^{x \cdot \varepsilon} - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 77.6%

       \[\leadsto \frac{\color{blue}{1} \cdot e^{x \cdot \varepsilon} - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]

   12. Taylor expanded in x around 0

       \[\leadsto \frac{1 \cdot e^{x \cdot \varepsilon} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   13. Step-by-step derivation

   14. Applied rewrites 33.5%

       \[\leadsto \frac{1 \cdot e^{x \cdot \varepsilon} - \color{blue}{\frac{1 - \varepsilon}{\varepsilon}}}{2} \]

   if 2.8000000000000002e210 < x < 1.28000000000000006e247

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 8.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 87.1%

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{\varepsilon} + 1\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{e^{-x}}{\varepsilon} - -1}{2}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-108}:\\ \;\;\;\;\frac{1 \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+210}:\\ \;\;\;\;\frac{1 \cdot e^{x \cdot \varepsilon} - \frac{1 - \varepsilon}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{+247}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{1}{eps\_m} - 1\\ t_1 := \frac{1}{eps\_m} - -1\\ \mathbf{if}\;x \leq -2.95 \cdot 10^{-93}:\\ \;\;\;\;\frac{t\_1 - \frac{-1}{e^{\mathsf{fma}\left(x, eps\_m, x\right)}}}{2}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-108}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+210}:\\ \;\;\;\;\frac{1 \cdot e^{x \cdot eps\_m} - \frac{1 - eps\_m}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{+247}:\\ \;\;\;\;\frac{t\_1 - t\_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) \cdot \mathsf{fma}\left(eps\_m - 1, x, 1\right) - t\_0}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 eps_m) 1.0)) (t_1 (- (/ 1.0 eps_m) -1.0)))
   (if (<= x -2.95e-93)
     (/ (- t_1 (/ -1.0 (exp (fma x eps_m x)))) 2.0)
     (if (<= x 1.65e-108)
       1.0
       (if (<= x 2.8e+210)
         (/ (- (* 1.0 (exp (* x eps_m))) (/ (- 1.0 eps_m) eps_m)) 2.0)
         (if (<= x 1.28e+247)
           (/ (- t_1 t_0) 2.0)
           (/
            (- (* (+ 1.0 (/ 1.0 eps_m)) (fma (- eps_m 1.0) x 1.0)) t_0)
            2.0)))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (1.0 / eps_m) - 1.0;
	double t_1 = (1.0 / eps_m) - -1.0;
	double tmp;
	if (x <= -2.95e-93) {
		tmp = (t_1 - (-1.0 / exp(fma(x, eps_m, x)))) / 2.0;
	} else if (x <= 1.65e-108) {
		tmp = 1.0;
	} else if (x <= 2.8e+210) {
		tmp = ((1.0 * exp((x * eps_m))) - ((1.0 - eps_m) / eps_m)) / 2.0;
	} else if (x <= 1.28e+247) {
		tmp = (t_1 - t_0) / 2.0;
	} else {
		tmp = (((1.0 + (1.0 / eps_m)) * fma((eps_m - 1.0), x, 1.0)) - t_0) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(1.0 / eps_m) - 1.0)
	t_1 = Float64(Float64(1.0 / eps_m) - -1.0)
	tmp = 0.0
	if (x <= -2.95e-93)
		tmp = Float64(Float64(t_1 - Float64(-1.0 / exp(fma(x, eps_m, x)))) / 2.0);
	elseif (x <= 1.65e-108)
		tmp = 1.0;
	elseif (x <= 2.8e+210)
		tmp = Float64(Float64(Float64(1.0 * exp(Float64(x * eps_m))) - Float64(Float64(1.0 - eps_m) / eps_m)) / 2.0);
	elseif (x <= 1.28e+247)
		tmp = Float64(Float64(t_1 - t_0) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * fma(Float64(eps_m - 1.0), x, 1.0)) - 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[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[x, -2.95e-93], N[(N[(t$95$1 - N[(-1.0 / N[Exp[N[(x * eps$95$m + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.65e-108], 1.0, If[LessEqual[x, 2.8e+210], N[(N[(N[(1.0 * N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 - eps$95$m), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.28e+247], N[(N[(t$95$1 - t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(eps$95$m - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -2.95e-93

   1. Initial program 85.9%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 83.0%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 51.3%

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{\varepsilon} + 1\right)} - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]

   if -2.95e-93 < x < 1.6500000000000001e-108

   1. Initial program 52.8%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 52.1%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 78.1%

      \[\leadsto \color{blue}{1} \]

   if 1.6500000000000001e-108 < x < 2.8000000000000002e210

   1. Initial program 88.9%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 87.4%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}}{2} \]

   6. Taylor expanded in eps around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{\varepsilon \cdot x}} - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 66.5%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{x \cdot \varepsilon}} - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]

   9. Taylor expanded in eps around inf

      \[\leadsto \frac{\color{blue}{1} \cdot e^{x \cdot \varepsilon} - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 77.6%

       \[\leadsto \frac{\color{blue}{1} \cdot e^{x \cdot \varepsilon} - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]

   12. Taylor expanded in x around 0

       \[\leadsto \frac{1 \cdot e^{x \cdot \varepsilon} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   13. Step-by-step derivation

   14. Applied rewrites 33.5%

       \[\leadsto \frac{1 \cdot e^{x \cdot \varepsilon} - \color{blue}{\frac{1 - \varepsilon}{\varepsilon}}}{2} \]

   if 2.8000000000000002e210 < x < 1.28000000000000006e247

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 8.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 87.1%

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{\varepsilon} + 1\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]

   if 1.28000000000000006e247 < x

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 47.0%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(1 + x \cdot \left(\varepsilon - 1\right)\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\mathsf{fma}\left(\varepsilon - 1, x, 1\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.95 \cdot 10^{-93}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-108}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+210}:\\ \;\;\;\;\frac{1 \cdot e^{x \cdot \varepsilon} - \frac{1 - \varepsilon}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{+247}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{1}{eps\_m} - 1\\ \mathbf{if}\;x \leq -700:\\ \;\;\;\;\frac{\frac{e^{-x}}{eps\_m} - -1}{2}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-108}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+210}:\\ \;\;\;\;\frac{1 \cdot e^{x \cdot eps\_m} - \frac{1 - eps\_m}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{+247}:\\ \;\;\;\;\frac{\left(\frac{1}{eps\_m} - -1\right) - t\_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) \cdot \mathsf{fma}\left(eps\_m - 1, x, 1\right) - t\_0}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 eps_m) 1.0)))
   (if (<= x -700.0)
     (/ (- (/ (exp (- x)) eps_m) -1.0) 2.0)
     (if (<= x 1.65e-108)
       1.0
       (if (<= x 2.8e+210)
         (/ (- (* 1.0 (exp (* x eps_m))) (/ (- 1.0 eps_m) eps_m)) 2.0)
         (if (<= x 1.28e+247)
           (/ (- (- (/ 1.0 eps_m) -1.0) t_0) 2.0)
           (/
            (- (* (+ 1.0 (/ 1.0 eps_m)) (fma (- eps_m 1.0) x 1.0)) t_0)
            2.0)))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (1.0 / eps_m) - 1.0;
	double tmp;
	if (x <= -700.0) {
		tmp = ((exp(-x) / eps_m) - -1.0) / 2.0;
	} else if (x <= 1.65e-108) {
		tmp = 1.0;
	} else if (x <= 2.8e+210) {
		tmp = ((1.0 * exp((x * eps_m))) - ((1.0 - eps_m) / eps_m)) / 2.0;
	} else if (x <= 1.28e+247) {
		tmp = (((1.0 / eps_m) - -1.0) - t_0) / 2.0;
	} else {
		tmp = (((1.0 + (1.0 / eps_m)) * fma((eps_m - 1.0), x, 1.0)) - t_0) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(1.0 / eps_m) - 1.0)
	tmp = 0.0
	if (x <= -700.0)
		tmp = Float64(Float64(Float64(exp(Float64(-x)) / eps_m) - -1.0) / 2.0);
	elseif (x <= 1.65e-108)
		tmp = 1.0;
	elseif (x <= 2.8e+210)
		tmp = Float64(Float64(Float64(1.0 * exp(Float64(x * eps_m))) - Float64(Float64(1.0 - eps_m) / eps_m)) / 2.0);
	elseif (x <= 1.28e+247)
		tmp = Float64(Float64(Float64(Float64(1.0 / eps_m) - -1.0) - t_0) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * fma(Float64(eps_m - 1.0), x, 1.0)) - 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[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -700.0], N[(N[(N[(N[Exp[(-x)], $MachinePrecision] / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.65e-108], 1.0, If[LessEqual[x, 2.8e+210], N[(N[(N[(1.0 * N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 - eps$95$m), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.28e+247], N[(N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision] - t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(eps$95$m - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -700

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 48.5%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in eps around 0

      \[\leadsto \frac{\color{blue}{\frac{e^{\mathsf{neg}\left(x\right)}}{\varepsilon}} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 53.1%

      \[\leadsto \frac{\color{blue}{\frac{e^{-x}}{\varepsilon}} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]

   9. Taylor expanded in eps around inf

      \[\leadsto \frac{\frac{e^{-x}}{\varepsilon} - -1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 53.1%

       \[\leadsto \frac{\frac{e^{-x}}{\varepsilon} - -1}{2} \]

   if -700 < x < 1.6500000000000001e-108

   1. Initial program 55.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 53.4%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 72.4%

      \[\leadsto \color{blue}{1} \]

   if 1.6500000000000001e-108 < x < 2.8000000000000002e210

   1. Initial program 88.9%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 87.4%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}}{2} \]

   6. Taylor expanded in eps around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{\varepsilon \cdot x}} - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 66.5%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{x \cdot \varepsilon}} - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]

   9. Taylor expanded in eps around inf

      \[\leadsto \frac{\color{blue}{1} \cdot e^{x \cdot \varepsilon} - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 77.6%

       \[\leadsto \frac{\color{blue}{1} \cdot e^{x \cdot \varepsilon} - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2} \]

   12. Taylor expanded in x around 0

       \[\leadsto \frac{1 \cdot e^{x \cdot \varepsilon} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   13. Step-by-step derivation

   14. Applied rewrites 33.5%

       \[\leadsto \frac{1 \cdot e^{x \cdot \varepsilon} - \color{blue}{\frac{1 - \varepsilon}{\varepsilon}}}{2} \]

   if 2.8000000000000002e210 < x < 1.28000000000000006e247

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 8.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 87.1%

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{\varepsilon} + 1\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]

   if 1.28000000000000006e247 < x

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 47.0%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(1 + x \cdot \left(\varepsilon - 1\right)\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\mathsf{fma}\left(\varepsilon - 1, x, 1\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -700:\\ \;\;\;\;\frac{\frac{e^{-x}}{\varepsilon} - -1}{2}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{-108}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+210}:\\ \;\;\;\;\frac{1 \cdot e^{x \cdot \varepsilon} - \frac{1 - \varepsilon}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{+247}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{1}{eps\_m} - 1\\ \mathbf{if}\;x \leq -700:\\ \;\;\;\;\frac{\frac{e^{-x}}{eps\_m} - -1}{2}\\ \mathbf{elif}\;x \leq 0.47:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{+247}:\\ \;\;\;\;\frac{\left(\frac{1}{eps\_m} - -1\right) - t\_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) \cdot \mathsf{fma}\left(eps\_m - 1, x, 1\right) - t\_0}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 eps_m) 1.0)))
   (if (<= x -700.0)
     (/ (- (/ (exp (- x)) eps_m) -1.0) 2.0)
     (if (<= x 0.47)
       1.0
       (if (<= x 1.28e+247)
         (/ (- (- (/ 1.0 eps_m) -1.0) t_0) 2.0)
         (/
          (- (* (+ 1.0 (/ 1.0 eps_m)) (fma (- eps_m 1.0) x 1.0)) t_0)
          2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (1.0 / eps_m) - 1.0;
	double tmp;
	if (x <= -700.0) {
		tmp = ((exp(-x) / eps_m) - -1.0) / 2.0;
	} else if (x <= 0.47) {
		tmp = 1.0;
	} else if (x <= 1.28e+247) {
		tmp = (((1.0 / eps_m) - -1.0) - t_0) / 2.0;
	} else {
		tmp = (((1.0 + (1.0 / eps_m)) * fma((eps_m - 1.0), x, 1.0)) - t_0) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(1.0 / eps_m) - 1.0)
	tmp = 0.0
	if (x <= -700.0)
		tmp = Float64(Float64(Float64(exp(Float64(-x)) / eps_m) - -1.0) / 2.0);
	elseif (x <= 0.47)
		tmp = 1.0;
	elseif (x <= 1.28e+247)
		tmp = Float64(Float64(Float64(Float64(1.0 / eps_m) - -1.0) - t_0) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * fma(Float64(eps_m - 1.0), x, 1.0)) - 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[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -700.0], N[(N[(N[(N[Exp[(-x)], $MachinePrecision] / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 0.47], 1.0, If[LessEqual[x, 1.28e+247], N[(N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision] - t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(eps$95$m - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -700

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 48.5%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in eps around 0

      \[\leadsto \frac{\color{blue}{\frac{e^{\mathsf{neg}\left(x\right)}}{\varepsilon}} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 53.1%

      \[\leadsto \frac{\color{blue}{\frac{e^{-x}}{\varepsilon}} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]

   9. Taylor expanded in eps around inf

      \[\leadsto \frac{\frac{e^{-x}}{\varepsilon} - -1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 53.1%

       \[\leadsto \frac{\frac{e^{-x}}{\varepsilon} - -1}{2} \]

   if -700 < x < 0.46999999999999997

   1. Initial program 56.9%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 54.7%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 68.9%

      \[\leadsto \color{blue}{1} \]

   if 0.46999999999999997 < x < 1.28000000000000006e247

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 30.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.0%

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{\varepsilon} + 1\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]

   if 1.28000000000000006e247 < x

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 47.0%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(1 + x \cdot \left(\varepsilon - 1\right)\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\mathsf{fma}\left(\varepsilon - 1, x, 1\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -700:\\ \;\;\;\;\frac{\frac{e^{-x}}{\varepsilon} - -1}{2}\\ \mathbf{elif}\;x \leq 0.47:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{+247}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := -1 + x \cdot eps\_m\\ t_1 := \frac{1}{eps\_m} - 1\\ t_2 := \frac{1}{eps\_m} - -1\\ \mathbf{if}\;x \leq -2 \cdot 10^{+104}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right)}{eps\_m} \cdot x - \frac{1}{eps\_m}, x, \frac{1}{eps\_m}\right) - t\_1}{2}\\ \mathbf{elif}\;x \leq -0.0146:\\ \;\;\;\;\frac{t\_2 + \left(\frac{-1}{eps\_m} - -1\right) \cdot \frac{t\_0 \cdot t\_0 - x \cdot x}{\left(1 - x \cdot eps\_m\right) + x}}{2}\\ \mathbf{elif}\;x \leq 0.47:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{+247}:\\ \;\;\;\;\frac{t\_2 - t\_1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) \cdot \mathsf{fma}\left(eps\_m - 1, x, 1\right) - t\_1}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (+ -1.0 (* x eps_m)))
        (t_1 (- (/ 1.0 eps_m) 1.0))
        (t_2 (- (/ 1.0 eps_m) -1.0)))
   (if (<= x -2e+104)
     (/
      (-
       (fma
        (- (* (/ (fma -0.16666666666666666 x 0.5) eps_m) x) (/ 1.0 eps_m))
        x
        (/ 1.0 eps_m))
       t_1)
      2.0)
     (if (<= x -0.0146)
       (/
        (+
         t_2
         (*
          (- (/ -1.0 eps_m) -1.0)
          (/ (- (* t_0 t_0) (* x x)) (+ (- 1.0 (* x eps_m)) x))))
        2.0)
       (if (<= x 0.47)
         1.0
         (if (<= x 1.28e+247)
           (/ (- t_2 t_1) 2.0)
           (/
            (- (* (+ 1.0 (/ 1.0 eps_m)) (fma (- eps_m 1.0) x 1.0)) t_1)
            2.0)))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = -1.0 + (x * eps_m);
	double t_1 = (1.0 / eps_m) - 1.0;
	double t_2 = (1.0 / eps_m) - -1.0;
	double tmp;
	if (x <= -2e+104) {
		tmp = (fma((((fma(-0.16666666666666666, x, 0.5) / eps_m) * x) - (1.0 / eps_m)), x, (1.0 / eps_m)) - t_1) / 2.0;
	} else if (x <= -0.0146) {
		tmp = (t_2 + (((-1.0 / eps_m) - -1.0) * (((t_0 * t_0) - (x * x)) / ((1.0 - (x * eps_m)) + x)))) / 2.0;
	} else if (x <= 0.47) {
		tmp = 1.0;
	} else if (x <= 1.28e+247) {
		tmp = (t_2 - t_1) / 2.0;
	} else {
		tmp = (((1.0 + (1.0 / eps_m)) * fma((eps_m - 1.0), x, 1.0)) - t_1) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(-1.0 + Float64(x * eps_m))
	t_1 = Float64(Float64(1.0 / eps_m) - 1.0)
	t_2 = Float64(Float64(1.0 / eps_m) - -1.0)
	tmp = 0.0
	if (x <= -2e+104)
		tmp = Float64(Float64(fma(Float64(Float64(Float64(fma(-0.16666666666666666, x, 0.5) / eps_m) * x) - Float64(1.0 / eps_m)), x, Float64(1.0 / eps_m)) - t_1) / 2.0);
	elseif (x <= -0.0146)
		tmp = Float64(Float64(t_2 + Float64(Float64(Float64(-1.0 / eps_m) - -1.0) * Float64(Float64(Float64(t_0 * t_0) - Float64(x * x)) / Float64(Float64(1.0 - Float64(x * eps_m)) + x)))) / 2.0);
	elseif (x <= 0.47)
		tmp = 1.0;
	elseif (x <= 1.28e+247)
		tmp = Float64(Float64(t_2 - t_1) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * fma(Float64(eps_m - 1.0), x, 1.0)) - t_1) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(-1.0 + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[x, -2e+104], N[(N[(N[(N[(N[(N[(N[(-0.16666666666666666 * x + 0.5), $MachinePrecision] / eps$95$m), $MachinePrecision] * x), $MachinePrecision] - N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * x + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -0.0146], N[(N[(t$95$2 + N[(N[(N[(-1.0 / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision] * N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 0.47], 1.0, If[LessEqual[x, 1.28e+247], N[(N[(t$95$2 - t$95$1), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(eps$95$m - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -2e104

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 39.5%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in eps around 0

      \[\leadsto \frac{\color{blue}{\frac{e^{\mathsf{neg}\left(x\right)}}{\varepsilon}} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 62.5%

      \[\leadsto \frac{\color{blue}{\frac{e^{-x}}{\varepsilon}} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\left(x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{\varepsilon} + \frac{1}{2} \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right) + \color{blue}{\frac{1}{\varepsilon}}\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 51.0%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right)}{\varepsilon} \cdot x - \frac{1}{\varepsilon}, \color{blue}{x}, \frac{1}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]

   if -2e104 < x < -0.0146000000000000001

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 52.5%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\left(1 - \mathsf{fma}\left(x, \varepsilon, x\right)\right)}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(1 - \mathsf{fma}\left(x, \varepsilon, x\right)\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 8.2%

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{\varepsilon} + 1\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(1 - \mathsf{fma}\left(x, \varepsilon, x\right)\right)}{2} \]
   9. Step-by-step derivation

   10. Applied rewrites 28.5%

       \[\leadsto \frac{\left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{\left(1 - x \cdot \varepsilon\right) \cdot \left(1 - x \cdot \varepsilon\right) - x \cdot x}{\color{blue}{\left(1 - x \cdot \varepsilon\right) + x}}}{2} \]

   if -0.0146000000000000001 < x < 0.46999999999999997

   1. Initial program 56.3%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 54.1%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 69.8%

      \[\leadsto \color{blue}{1} \]

   if 0.46999999999999997 < x < 1.28000000000000006e247

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 30.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.0%

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{\varepsilon} + 1\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]

   if 1.28000000000000006e247 < x

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 47.0%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(1 + x \cdot \left(\varepsilon - 1\right)\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\mathsf{fma}\left(\varepsilon - 1, x, 1\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+104}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right)}{\varepsilon} \cdot x - \frac{1}{\varepsilon}, x, \frac{1}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \mathbf{elif}\;x \leq -0.0146:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) + \left(\frac{-1}{\varepsilon} - -1\right) \cdot \frac{\left(-1 + x \cdot \varepsilon\right) \cdot \left(-1 + x \cdot \varepsilon\right) - x \cdot x}{\left(1 - x \cdot \varepsilon\right) + x}}{2}\\ \mathbf{elif}\;x \leq 0.47:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{+247}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{1}{eps\_m} - -1\\ t_1 := \frac{1}{eps\_m} - 1\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{t\_0 + \left(\frac{-1}{eps\_m} - -1\right) \cdot \left(\left(-1 - eps\_m\right) \cdot x\right)}{2}\\ \mathbf{elif}\;x \leq 0.47:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{+247}:\\ \;\;\;\;\frac{t\_0 - t\_1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) \cdot \mathsf{fma}\left(eps\_m - 1, x, 1\right) - t\_1}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 eps_m) -1.0)) (t_1 (- (/ 1.0 eps_m) 1.0)))
   (if (<= x -1.0)
     (/ (+ t_0 (* (- (/ -1.0 eps_m) -1.0) (* (- -1.0 eps_m) x))) 2.0)
     (if (<= x 0.47)
       1.0
       (if (<= x 1.28e+247)
         (/ (- t_0 t_1) 2.0)
         (/
          (- (* (+ 1.0 (/ 1.0 eps_m)) (fma (- eps_m 1.0) x 1.0)) t_1)
          2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (1.0 / eps_m) - -1.0;
	double t_1 = (1.0 / eps_m) - 1.0;
	double tmp;
	if (x <= -1.0) {
		tmp = (t_0 + (((-1.0 / eps_m) - -1.0) * ((-1.0 - eps_m) * x))) / 2.0;
	} else if (x <= 0.47) {
		tmp = 1.0;
	} else if (x <= 1.28e+247) {
		tmp = (t_0 - t_1) / 2.0;
	} else {
		tmp = (((1.0 + (1.0 / eps_m)) * fma((eps_m - 1.0), x, 1.0)) - t_1) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(1.0 / eps_m) - -1.0)
	t_1 = Float64(Float64(1.0 / eps_m) - 1.0)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(t_0 + Float64(Float64(Float64(-1.0 / eps_m) - -1.0) * Float64(Float64(-1.0 - eps_m) * x))) / 2.0);
	elseif (x <= 0.47)
		tmp = 1.0;
	elseif (x <= 1.28e+247)
		tmp = Float64(Float64(t_0 - t_1) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * fma(Float64(eps_m - 1.0), x, 1.0)) - t_1) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(1.0 / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -1.0], N[(N[(t$95$0 + N[(N[(N[(-1.0 / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision] * N[(N[(-1.0 - eps$95$m), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 0.47], 1.0, If[LessEqual[x, 1.28e+247], N[(N[(t$95$0 - t$95$1), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(eps$95$m - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 51.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\left(1 - \mathsf{fma}\left(x, \varepsilon, x\right)\right)}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(1 - \mathsf{fma}\left(x, \varepsilon, x\right)\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 25.5%

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{\varepsilon} + 1\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(1 - \mathsf{fma}\left(x, \varepsilon, x\right)\right)}{2} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(-1 \cdot \color{blue}{\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 25.5%

       \[\leadsto \frac{\left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(\left(-1 - \varepsilon\right) \cdot \color{blue}{x}\right)}{2} \]

   if -1 < x < 0.46999999999999997

   1. Initial program 56.3%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 54.1%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 69.8%

      \[\leadsto \color{blue}{1} \]

   if 0.46999999999999997 < x < 1.28000000000000006e247

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 30.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.0%

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{\varepsilon} + 1\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]

   if 1.28000000000000006e247 < x

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 47.0%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(1 + x \cdot \left(\varepsilon - 1\right)\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\mathsf{fma}\left(\varepsilon - 1, x, 1\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) + \left(\frac{-1}{\varepsilon} - -1\right) \cdot \left(\left(-1 - \varepsilon\right) \cdot x\right)}{2}\\ \mathbf{elif}\;x \leq 0.47:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{+247}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.5% accurate, 4.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{1}{eps\_m} - -1\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{t\_0 + \left(\frac{-1}{eps\_m} - -1\right) \cdot \left(\left(-1 - eps\_m\right) \cdot x\right)}{2}\\ \mathbf{elif}\;x \leq 0.47:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{+247}:\\ \;\;\;\;\frac{t\_0 - \left(\frac{1}{eps\_m} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) \cdot \mathsf{fma}\left(eps\_m - 1, x, 1\right) - -1}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 eps_m) -1.0)))
   (if (<= x -1.0)
     (/ (+ t_0 (* (- (/ -1.0 eps_m) -1.0) (* (- -1.0 eps_m) x))) 2.0)
     (if (<= x 0.47)
       1.0
       (if (<= x 1.28e+247)
         (/ (- t_0 (- (/ 1.0 eps_m) 1.0)) 2.0)
         (/
          (- (* (+ 1.0 (/ 1.0 eps_m)) (fma (- eps_m 1.0) x 1.0)) -1.0)
          2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (1.0 / eps_m) - -1.0;
	double tmp;
	if (x <= -1.0) {
		tmp = (t_0 + (((-1.0 / eps_m) - -1.0) * ((-1.0 - eps_m) * x))) / 2.0;
	} else if (x <= 0.47) {
		tmp = 1.0;
	} else if (x <= 1.28e+247) {
		tmp = (t_0 - ((1.0 / eps_m) - 1.0)) / 2.0;
	} else {
		tmp = (((1.0 + (1.0 / eps_m)) * fma((eps_m - 1.0), x, 1.0)) - -1.0) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(1.0 / eps_m) - -1.0)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(t_0 + Float64(Float64(Float64(-1.0 / eps_m) - -1.0) * Float64(Float64(-1.0 - eps_m) * x))) / 2.0);
	elseif (x <= 0.47)
		tmp = 1.0;
	elseif (x <= 1.28e+247)
		tmp = Float64(Float64(t_0 - Float64(Float64(1.0 / eps_m) - 1.0)) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * fma(Float64(eps_m - 1.0), x, 1.0)) - -1.0) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(1.0 / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[x, -1.0], N[(N[(t$95$0 + N[(N[(N[(-1.0 / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision] * N[(N[(-1.0 - eps$95$m), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 0.47], 1.0, If[LessEqual[x, 1.28e+247], N[(N[(t$95$0 - N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(eps$95$m - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 51.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\left(1 - \mathsf{fma}\left(x, \varepsilon, x\right)\right)}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(1 - \mathsf{fma}\left(x, \varepsilon, x\right)\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 25.5%

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{\varepsilon} + 1\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(1 - \mathsf{fma}\left(x, \varepsilon, x\right)\right)}{2} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(-1 \cdot \color{blue}{\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 25.5%

       \[\leadsto \frac{\left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(\left(-1 - \varepsilon\right) \cdot \color{blue}{x}\right)}{2} \]

   if -1 < x < 0.46999999999999997

   1. Initial program 56.3%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 54.1%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 69.8%

      \[\leadsto \color{blue}{1} \]

   if 0.46999999999999997 < x < 1.28000000000000006e247

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 30.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.0%

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{\varepsilon} + 1\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]

   if 1.28000000000000006e247 < x

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 47.0%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(1 + x \cdot \left(\varepsilon - 1\right)\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\mathsf{fma}\left(\varepsilon - 1, x, 1\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]

   9. Taylor expanded in eps around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - -1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 45.9%

       \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - -1}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) + \left(\frac{-1}{\varepsilon} - -1\right) \cdot \left(\left(-1 - \varepsilon\right) \cdot x\right)}{2}\\ \mathbf{elif}\;x \leq 0.47:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{+247}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - -1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.5% accurate, 4.5× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{1}{eps\_m} - -1\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{t\_0 + \left(\frac{-1}{eps\_m} - -1\right) \cdot \left(\left(-x\right) \cdot eps\_m\right)}{2}\\ \mathbf{elif}\;x \leq 0.47:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{+247}:\\ \;\;\;\;\frac{t\_0 - \left(\frac{1}{eps\_m} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) \cdot \mathsf{fma}\left(eps\_m - 1, x, 1\right) - -1}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 eps_m) -1.0)))
   (if (<= x -1.0)
     (/ (+ t_0 (* (- (/ -1.0 eps_m) -1.0) (* (- x) eps_m))) 2.0)
     (if (<= x 0.47)
       1.0
       (if (<= x 1.28e+247)
         (/ (- t_0 (- (/ 1.0 eps_m) 1.0)) 2.0)
         (/
          (- (* (+ 1.0 (/ 1.0 eps_m)) (fma (- eps_m 1.0) x 1.0)) -1.0)
          2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (1.0 / eps_m) - -1.0;
	double tmp;
	if (x <= -1.0) {
		tmp = (t_0 + (((-1.0 / eps_m) - -1.0) * (-x * eps_m))) / 2.0;
	} else if (x <= 0.47) {
		tmp = 1.0;
	} else if (x <= 1.28e+247) {
		tmp = (t_0 - ((1.0 / eps_m) - 1.0)) / 2.0;
	} else {
		tmp = (((1.0 + (1.0 / eps_m)) * fma((eps_m - 1.0), x, 1.0)) - -1.0) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(1.0 / eps_m) - -1.0)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(t_0 + Float64(Float64(Float64(-1.0 / eps_m) - -1.0) * Float64(Float64(-x) * eps_m))) / 2.0);
	elseif (x <= 0.47)
		tmp = 1.0;
	elseif (x <= 1.28e+247)
		tmp = Float64(Float64(t_0 - Float64(Float64(1.0 / eps_m) - 1.0)) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * fma(Float64(eps_m - 1.0), x, 1.0)) - -1.0) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(1.0 / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[x, -1.0], N[(N[(t$95$0 + N[(N[(N[(-1.0 / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision] * N[((-x) * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 0.47], 1.0, If[LessEqual[x, 1.28e+247], N[(N[(t$95$0 - N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(eps$95$m - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 51.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\left(1 - \mathsf{fma}\left(x, \varepsilon, x\right)\right)}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(1 - \mathsf{fma}\left(x, \varepsilon, x\right)\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 25.5%

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{\varepsilon} + 1\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(1 - \mathsf{fma}\left(x, \varepsilon, x\right)\right)}{2} \]

   9. Taylor expanded in eps around inf

      \[\leadsto \frac{\left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(-1 \cdot \color{blue}{\left(\varepsilon \cdot x\right)}\right)}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 25.5%

       \[\leadsto \frac{\left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(\left(-x\right) \cdot \color{blue}{\varepsilon}\right)}{2} \]

   if -1 < x < 0.46999999999999997

   1. Initial program 56.3%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 54.1%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 69.8%

      \[\leadsto \color{blue}{1} \]

   if 0.46999999999999997 < x < 1.28000000000000006e247

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 30.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.0%

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{\varepsilon} + 1\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]

   if 1.28000000000000006e247 < x

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 47.0%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(1 + x \cdot \left(\varepsilon - 1\right)\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\mathsf{fma}\left(\varepsilon - 1, x, 1\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]

   9. Taylor expanded in eps around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - -1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 45.9%

       \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - -1}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) + \left(\frac{-1}{\varepsilon} - -1\right) \cdot \left(\left(-x\right) \cdot \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 0.47:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{+247}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - -1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.4% accurate, 5.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 0.47:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{+247}:\\ \;\;\;\;\frac{\left(\frac{1}{eps\_m} - -1\right) - \left(\frac{1}{eps\_m} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) \cdot \mathsf{fma}\left(eps\_m - 1, x, 1\right) - -1}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 0.47)
   1.0
   (if (<= x 1.28e+247)
     (/ (- (- (/ 1.0 eps_m) -1.0) (- (/ 1.0 eps_m) 1.0)) 2.0)
     (/ (- (* (+ 1.0 (/ 1.0 eps_m)) (fma (- eps_m 1.0) x 1.0)) -1.0) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 0.47) {
		tmp = 1.0;
	} else if (x <= 1.28e+247) {
		tmp = (((1.0 / eps_m) - -1.0) - ((1.0 / eps_m) - 1.0)) / 2.0;
	} else {
		tmp = (((1.0 + (1.0 / eps_m)) * fma((eps_m - 1.0), x, 1.0)) - -1.0) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 0.47)
		tmp = 1.0;
	elseif (x <= 1.28e+247)
		tmp = Float64(Float64(Float64(Float64(1.0 / eps_m) - -1.0) - Float64(Float64(1.0 / eps_m) - 1.0)) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * fma(Float64(eps_m - 1.0), x, 1.0)) - -1.0) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 0.47], 1.0, If[LessEqual[x, 1.28e+247], N[(N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision] - N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(eps$95$m - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.46999999999999997

   1. Initial program 64.4%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 62.6%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 57.4%

      \[\leadsto \color{blue}{1} \]

   if 0.46999999999999997 < x < 1.28000000000000006e247

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 30.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.0%

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{\varepsilon} + 1\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]

   if 1.28000000000000006e247 < x

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 47.0%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\left(1 + x \cdot \left(\varepsilon - 1\right)\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.9%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\mathsf{fma}\left(\varepsilon - 1, x, 1\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]

   9. Taylor expanded in eps around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - -1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 45.9%

       \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - -1}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.47:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.28 \cdot 10^{+247}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - -1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 57.9% accurate, 5.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 0.47:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{eps\_m} - -1\right) - \left(\frac{1}{eps\_m} - 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 0.47) 1.0 (/ (- (- (/ 1.0 eps_m) -1.0) (- (/ 1.0 eps_m) 1.0)) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 0.47) {
		tmp = 1.0;
	} else {
		tmp = (((1.0 / eps_m) - -1.0) - ((1.0 / eps_m) - 1.0)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, eps_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 0.47d0) then
        tmp = 1.0d0
    else
        tmp = (((1.0d0 / eps_m) - (-1.0d0)) - ((1.0d0 / 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 <= 0.47) {
		tmp = 1.0;
	} else {
		tmp = (((1.0 / eps_m) - -1.0) - ((1.0 / eps_m) - 1.0)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 0.47:
		tmp = 1.0
	else:
		tmp = (((1.0 / eps_m) - -1.0) - ((1.0 / eps_m) - 1.0)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 0.47)
		tmp = 1.0;
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / eps_m) - -1.0) - Float64(Float64(1.0 / 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 <= 0.47)
		tmp = 1.0;
	else
		tmp = (((1.0 / eps_m) - -1.0) - ((1.0 / 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, 0.47], 1.0, N[(N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision] - N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.46999999999999997

   1. Initial program 64.4%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 62.6%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 57.4%

      \[\leadsto \color{blue}{1} \]

   if 0.46999999999999997 < x

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 33.3%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 46.9%

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{\varepsilon} + 1\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.47:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 57.7% accurate, 5.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 180000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{eps\_m} - \left(\frac{1}{eps\_m} - 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 180000000.0) 1.0 (/ (- (/ 1.0 eps_m) (- (/ 1.0 eps_m) 1.0)) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 180000000.0) {
		tmp = 1.0;
	} else {
		tmp = ((1.0 / eps_m) - ((1.0 / eps_m) - 1.0)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, eps_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 180000000.0d0) then
        tmp = 1.0d0
    else
        tmp = ((1.0d0 / eps_m) - ((1.0d0 / 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 <= 180000000.0) {
		tmp = 1.0;
	} else {
		tmp = ((1.0 / eps_m) - ((1.0 / eps_m) - 1.0)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 180000000.0:
		tmp = 1.0
	else:
		tmp = ((1.0 / eps_m) - ((1.0 / eps_m) - 1.0)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 180000000.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(Float64(1.0 / eps_m) - Float64(Float64(1.0 / 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 <= 180000000.0)
		tmp = 1.0;
	else
		tmp = ((1.0 / eps_m) - ((1.0 / 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, 180000000.0], 1.0, N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] - N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8e8

   1. Initial program 64.8%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 63.0%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}}{2} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.8%

      \[\leadsto \color{blue}{1} \]

   if 1.8e8 < x

   1. Initial program 100.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
   4. Step-by-step derivation

   5. Applied rewrites 32.8%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]

   6. Taylor expanded in eps around 0

      \[\leadsto \frac{\color{blue}{\frac{e^{\mathsf{neg}\left(x\right)}}{\varepsilon}} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 2.8%

      \[\leadsto \frac{\color{blue}{\frac{e^{-x}}{\varepsilon}} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{1}{\color{blue}{\varepsilon}} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 48.2%

       \[\leadsto \frac{\frac{1}{\color{blue}{\varepsilon}} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 44.5% accurate, 273.0× 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 =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, eps_m)
use fmin_fmax_functions
    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

1. Initial program 74.6%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
2. Add Preprocessing

3. Taylor expanded in eps around inf

   \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]
4. Step-by-step derivation

5. Applied rewrites 73.3%

   \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}}{2} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Applied rewrites 41.9%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025026 
(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))