NMSE Section 6.1 mentioned, A

Percentage Accurate: 74.1% → 99.1%

Time: 7.5s

Alternatives: 16

Speedup: 2.4×

• 61.8% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -3.5692506274687863e+140
  2. eps = -2.1018930033968455e-15

Specification

Math

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

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: 74.1% accurate, 1.0× speedup

Math

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

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: 99.1% accurate, 1.2× speedup

Math

\[0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

FPCore

(FPCore (x eps)
  :precision binary64
  (*
 0.5
 (- (exp (- (* x (- 1.0 eps)))) (* -1.0 (exp (- (* x (+ 1.0 eps))))))))

C

double code(double x, double eps) {
	return 0.5 * (exp(-(x * (1.0 - eps))) - (-1.0 * exp(-(x * (1.0 + eps)))));
}

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 = 0.5d0 * (exp(-(x * (1.0d0 - eps))) - ((-1.0d0) * exp(-(x * (1.0d0 + eps)))))
end function

Java

public static double code(double x, double eps) {
	return 0.5 * (Math.exp(-(x * (1.0 - eps))) - (-1.0 * Math.exp(-(x * (1.0 + eps)))));
}

Python

def code(x, eps):
	return 0.5 * (math.exp(-(x * (1.0 - eps))) - (-1.0 * math.exp(-(x * (1.0 + eps)))))

Julia

function code(x, eps)
	return Float64(0.5 * Float64(exp(Float64(-Float64(x * Float64(1.0 - eps)))) - Float64(-1.0 * exp(Float64(-Float64(x * Float64(1.0 + eps)))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.5 * (exp(-(x * (1.0 - eps))) - (-1.0 * exp(-(x * (1.0 + eps)))));
end

Wolfram

code[x_, eps_] := N[(0.5 * N[(N[Exp[(-N[(x * N[(1.0 - eps), $MachinePrecision]), $MachinePrecision])], $MachinePrecision] - N[(-1.0 * N[Exp[(-N[(x * N[(1.0 + eps), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.1%

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

2. Taylor expanded in eps around inf

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-exp.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-+.f64 99.1%

       \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

4. Applied rewrites 99.1%

   \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]
5. Add Preprocessing

Alternative 2: 90.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\left|\varepsilon\right|}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-295}:\\ \;\;\;\;\left(1 + e^{\left(\left(-\left|\varepsilon\right|\right) - 1\right) \cdot x}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 390000:\\ \;\;\;\;0.5 \cdot \left(e^{\left|\varepsilon\right| \cdot x} - -1\right)\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+267}:\\ \;\;\;\;\frac{\left(1 + t\_0\right) \cdot e^{-\left(-1 \cdot \left|\varepsilon\right|\right) \cdot x} - \left(t\_0 - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot 0 + \frac{x}{x \cdot x}\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
  :precision binary64
  (let* ((t_0 (/ 1.0 (fabs eps))))
  (if (<= x -2e-295)
    (* (+ 1.0 (exp (* (- (- (fabs eps)) 1.0) x))) 0.5)
    (if (<= x 390000.0)
      (* 0.5 (- (exp (* (fabs eps) x)) -1.0))
      (if (<= x 6.5e+267)
        (/
         (-
          (* (+ 1.0 t_0) (exp (- (* (* -1.0 (fabs eps)) x))))
          (- t_0 1.0))
         2.0)
        (* x (+ (* 0.5 0.0) (/ x (* x x)))))))))

C

double code(double x, double eps) {
	double t_0 = 1.0 / fabs(eps);
	double tmp;
	if (x <= -2e-295) {
		tmp = (1.0 + exp(((-fabs(eps) - 1.0) * x))) * 0.5;
	} else if (x <= 390000.0) {
		tmp = 0.5 * (exp((fabs(eps) * x)) - -1.0);
	} else if (x <= 6.5e+267) {
		tmp = (((1.0 + t_0) * exp(-((-1.0 * fabs(eps)) * x))) - (t_0 - 1.0)) / 2.0;
	} else {
		tmp = x * ((0.5 * 0.0) + (x / (x * x)));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / abs(eps)
    if (x <= (-2d-295)) then
        tmp = (1.0d0 + exp(((-abs(eps) - 1.0d0) * x))) * 0.5d0
    else if (x <= 390000.0d0) then
        tmp = 0.5d0 * (exp((abs(eps) * x)) - (-1.0d0))
    else if (x <= 6.5d+267) then
        tmp = (((1.0d0 + t_0) * exp(-(((-1.0d0) * abs(eps)) * x))) - (t_0 - 1.0d0)) / 2.0d0
    else
        tmp = x * ((0.5d0 * 0.0d0) + (x / (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = 1.0 / Math.abs(eps);
	double tmp;
	if (x <= -2e-295) {
		tmp = (1.0 + Math.exp(((-Math.abs(eps) - 1.0) * x))) * 0.5;
	} else if (x <= 390000.0) {
		tmp = 0.5 * (Math.exp((Math.abs(eps) * x)) - -1.0);
	} else if (x <= 6.5e+267) {
		tmp = (((1.0 + t_0) * Math.exp(-((-1.0 * Math.abs(eps)) * x))) - (t_0 - 1.0)) / 2.0;
	} else {
		tmp = x * ((0.5 * 0.0) + (x / (x * x)));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = 1.0 / math.fabs(eps)
	tmp = 0
	if x <= -2e-295:
		tmp = (1.0 + math.exp(((-math.fabs(eps) - 1.0) * x))) * 0.5
	elif x <= 390000.0:
		tmp = 0.5 * (math.exp((math.fabs(eps) * x)) - -1.0)
	elif x <= 6.5e+267:
		tmp = (((1.0 + t_0) * math.exp(-((-1.0 * math.fabs(eps)) * x))) - (t_0 - 1.0)) / 2.0
	else:
		tmp = x * ((0.5 * 0.0) + (x / (x * x)))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(1.0 / abs(eps))
	tmp = 0.0
	if (x <= -2e-295)
		tmp = Float64(Float64(1.0 + exp(Float64(Float64(Float64(-abs(eps)) - 1.0) * x))) * 0.5);
	elseif (x <= 390000.0)
		tmp = Float64(0.5 * Float64(exp(Float64(abs(eps) * x)) - -1.0));
	elseif (x <= 6.5e+267)
		tmp = Float64(Float64(Float64(Float64(1.0 + t_0) * exp(Float64(-Float64(Float64(-1.0 * abs(eps)) * x)))) - Float64(t_0 - 1.0)) / 2.0);
	else
		tmp = Float64(x * Float64(Float64(0.5 * 0.0) + Float64(x / Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = 1.0 / abs(eps);
	tmp = 0.0;
	if (x <= -2e-295)
		tmp = (1.0 + exp(((-abs(eps) - 1.0) * x))) * 0.5;
	elseif (x <= 390000.0)
		tmp = 0.5 * (exp((abs(eps) * x)) - -1.0);
	elseif (x <= 6.5e+267)
		tmp = (((1.0 + t_0) * exp(-((-1.0 * abs(eps)) * x))) - (t_0 - 1.0)) / 2.0;
	else
		tmp = x * ((0.5 * 0.0) + (x / (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 / N[Abs[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2e-295], N[(N[(1.0 + N[Exp[N[(N[((-N[Abs[eps], $MachinePrecision]) - 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 390000.0], N[(0.5 * N[(N[Exp[N[(N[Abs[eps], $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.5e+267], N[(N[(N[(N[(1.0 + t$95$0), $MachinePrecision] * N[Exp[(-N[(N[(-1.0 * N[Abs[eps], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(x * N[(N[(0.5 * 0.0), $MachinePrecision] + N[(x / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.0000000000000001e-295

   1. Initial program 74.1%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1%

          \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto 0.5 \cdot \left(1 - \color{blue}{-1} \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 64.0%

      \[\leadsto 0.5 \cdot \left(1 - \color{blue}{-1} \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 64.0%

         \[\leadsto \left(1 - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \cdot \color{blue}{0.5} \]

   9. Applied rewrites 64.0%

      \[\leadsto \left(1 + e^{\left(\left(-\varepsilon\right) - 1\right) \cdot x}\right) \cdot \color{blue}{0.5} \]

   if -2.0000000000000001e-295 < x < 3.9e5

   1. Initial program 74.1%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1%

          \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 63.8%

      \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1\right) \]

   8. Taylor expanded in eps around inf

      \[\leadsto 0.5 \cdot \left(e^{\varepsilon \cdot x} - -1\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 64.0%

         \[\leadsto 0.5 \cdot \left(e^{\varepsilon \cdot x} - -1\right) \]

   10. Applied rewrites 64.0%

       \[\leadsto 0.5 \cdot \left(e^{\varepsilon \cdot x} - -1\right) \]

   if 3.9e5 < x < 6.4999999999999998e267

   1. Initial program 74.1%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 38.7%

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

   4. Applied rewrites 38.7%

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

   5. Taylor expanded in eps around inf

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

      1. lower-*.f64 44.8%

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

   7. Applied rewrites 44.8%

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

   if 6.4999999999999998e267 < x

   1. Initial program 74.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 43.7%

      \[\leadsto \color{blue}{1 + 0.5 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

   7. Applied rewrites 43.6%

      \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{1}{x}\right)} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. rgt-mult-inverse N/A

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

      4. mult-flip-rev N/A

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

      5. frac-times N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 29.8%

         \[\leadsto x \cdot \left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{x}{x \cdot x}\right) \]

   9. Applied rewrites 29.8%

      \[\leadsto x \cdot \left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{x}{x \cdot \color{blue}{x}}\right) \]

   10. Taylor expanded in eps around 0

       \[\leadsto x \cdot \left(0.5 \cdot 0 + \frac{x}{x \cdot x}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 29.9%

       \[\leadsto x \cdot \left(0.5 \cdot 0 + \frac{x}{x \cdot x}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 84.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-295}:\\ \;\;\;\;\left(1 + e^{\left(\left(-\left|\varepsilon\right|\right) - 1\right) \cdot x}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 6:\\ \;\;\;\;0.5 \cdot \left(e^{\left|\varepsilon\right| \cdot x} - -1\right)\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+89}:\\ \;\;\;\;\left(2 \cdot e^{-x}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+267}:\\ \;\;\;\;0.5 \cdot \left(e^{-x \cdot \left(1 - \left|\varepsilon\right|\right)} - -1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot 0 + \frac{x}{x \cdot x}\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
  :precision binary64
  (if (<= x -2e-295)
  (* (+ 1.0 (exp (* (- (- (fabs eps)) 1.0) x))) 0.5)
  (if (<= x 6.0)
    (* 0.5 (- (exp (* (fabs eps) x)) -1.0))
    (if (<= x 1.45e+89)
      (* (* 2.0 (exp (- x))) 0.5)
      (if (<= x 6.5e+267)
        (* 0.5 (- (exp (- (* x (- 1.0 (fabs eps))))) -1.0))
        (* x (+ (* 0.5 0.0) (/ x (* x x)))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2e-295) {
		tmp = (1.0 + exp(((-fabs(eps) - 1.0) * x))) * 0.5;
	} else if (x <= 6.0) {
		tmp = 0.5 * (exp((fabs(eps) * x)) - -1.0);
	} else if (x <= 1.45e+89) {
		tmp = (2.0 * exp(-x)) * 0.5;
	} else if (x <= 6.5e+267) {
		tmp = 0.5 * (exp(-(x * (1.0 - fabs(eps)))) - -1.0);
	} else {
		tmp = x * ((0.5 * 0.0) + (x / (x * x)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (x <= (-2d-295)) then
        tmp = (1.0d0 + exp(((-abs(eps) - 1.0d0) * x))) * 0.5d0
    else if (x <= 6.0d0) then
        tmp = 0.5d0 * (exp((abs(eps) * x)) - (-1.0d0))
    else if (x <= 1.45d+89) then
        tmp = (2.0d0 * exp(-x)) * 0.5d0
    else if (x <= 6.5d+267) then
        tmp = 0.5d0 * (exp(-(x * (1.0d0 - abs(eps)))) - (-1.0d0))
    else
        tmp = x * ((0.5d0 * 0.0d0) + (x / (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2e-295) {
		tmp = (1.0 + Math.exp(((-Math.abs(eps) - 1.0) * x))) * 0.5;
	} else if (x <= 6.0) {
		tmp = 0.5 * (Math.exp((Math.abs(eps) * x)) - -1.0);
	} else if (x <= 1.45e+89) {
		tmp = (2.0 * Math.exp(-x)) * 0.5;
	} else if (x <= 6.5e+267) {
		tmp = 0.5 * (Math.exp(-(x * (1.0 - Math.abs(eps)))) - -1.0);
	} else {
		tmp = x * ((0.5 * 0.0) + (x / (x * x)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2e-295:
		tmp = (1.0 + math.exp(((-math.fabs(eps) - 1.0) * x))) * 0.5
	elif x <= 6.0:
		tmp = 0.5 * (math.exp((math.fabs(eps) * x)) - -1.0)
	elif x <= 1.45e+89:
		tmp = (2.0 * math.exp(-x)) * 0.5
	elif x <= 6.5e+267:
		tmp = 0.5 * (math.exp(-(x * (1.0 - math.fabs(eps)))) - -1.0)
	else:
		tmp = x * ((0.5 * 0.0) + (x / (x * x)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2e-295)
		tmp = Float64(Float64(1.0 + exp(Float64(Float64(Float64(-abs(eps)) - 1.0) * x))) * 0.5);
	elseif (x <= 6.0)
		tmp = Float64(0.5 * Float64(exp(Float64(abs(eps) * x)) - -1.0));
	elseif (x <= 1.45e+89)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) * 0.5);
	elseif (x <= 6.5e+267)
		tmp = Float64(0.5 * Float64(exp(Float64(-Float64(x * Float64(1.0 - abs(eps))))) - -1.0));
	else
		tmp = Float64(x * Float64(Float64(0.5 * 0.0) + Float64(x / Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2e-295)
		tmp = (1.0 + exp(((-abs(eps) - 1.0) * x))) * 0.5;
	elseif (x <= 6.0)
		tmp = 0.5 * (exp((abs(eps) * x)) - -1.0);
	elseif (x <= 1.45e+89)
		tmp = (2.0 * exp(-x)) * 0.5;
	elseif (x <= 6.5e+267)
		tmp = 0.5 * (exp(-(x * (1.0 - abs(eps)))) - -1.0);
	else
		tmp = x * ((0.5 * 0.0) + (x / (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2e-295], N[(N[(1.0 + N[Exp[N[(N[((-N[Abs[eps], $MachinePrecision]) - 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 6.0], N[(0.5 * N[(N[Exp[N[(N[Abs[eps], $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.45e+89], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 6.5e+267], N[(0.5 * N[(N[Exp[(-N[(x * N[(1.0 - N[Abs[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(0.5 * 0.0), $MachinePrecision] + N[(x / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -2.0000000000000001e-295

   1. Initial program 74.1%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1%

          \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto 0.5 \cdot \left(1 - \color{blue}{-1} \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 64.0%

      \[\leadsto 0.5 \cdot \left(1 - \color{blue}{-1} \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 64.0%

         \[\leadsto \left(1 - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \cdot \color{blue}{0.5} \]

   9. Applied rewrites 64.0%

      \[\leadsto \left(1 + e^{\left(\left(-\varepsilon\right) - 1\right) \cdot x}\right) \cdot \color{blue}{0.5} \]

   if -2.0000000000000001e-295 < x < 6

   1. Initial program 74.1%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1%

          \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 63.8%

      \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1\right) \]

   8. Taylor expanded in eps around inf

      \[\leadsto 0.5 \cdot \left(e^{\varepsilon \cdot x} - -1\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 64.0%

         \[\leadsto 0.5 \cdot \left(e^{\varepsilon \cdot x} - -1\right) \]

   10. Applied rewrites 64.0%

       \[\leadsto 0.5 \cdot \left(e^{\varepsilon \cdot x} - -1\right) \]

   if 6 < x < 1.4500000000000001e89

   1. Initial program 74.1%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1%

          \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]

   5. Taylor expanded in eps around 0

      \[\leadsto 0.5 \cdot \left(e^{\mathsf{neg}\left(x\right)} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x\right)}}\right) \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 70.6%

         \[\leadsto 0.5 \cdot \left(e^{-x} - -1 \cdot e^{-x}\right) \]

   7. Applied rewrites 70.6%

      \[\leadsto 0.5 \cdot \left(e^{-x} - \color{blue}{-1 \cdot e^{-x}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 70.6%

         \[\leadsto \left(e^{-x} - -1 \cdot e^{-x}\right) \cdot \color{blue}{0.5} \]

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. fp-cancel-sign-sub-inv N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 70.6%

          \[\leadsto \left(2 \cdot e^{-x}\right) \cdot 0.5 \]

   9. Applied rewrites 70.6%

      \[\leadsto \left(2 \cdot e^{-x}\right) \cdot \color{blue}{0.5} \]

   if 1.4500000000000001e89 < x < 6.4999999999999998e267

   1. Initial program 74.1%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1%

          \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 63.8%

      \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1\right) \]

   if 6.4999999999999998e267 < x

   1. Initial program 74.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 43.7%

      \[\leadsto \color{blue}{1 + 0.5 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

   7. Applied rewrites 43.6%

      \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{1}{x}\right)} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. rgt-mult-inverse N/A

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

      4. mult-flip-rev N/A

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

      5. frac-times N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 29.8%

         \[\leadsto x \cdot \left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{x}{x \cdot x}\right) \]

   9. Applied rewrites 29.8%

      \[\leadsto x \cdot \left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{x}{x \cdot \color{blue}{x}}\right) \]

   10. Taylor expanded in eps around 0

       \[\leadsto x \cdot \left(0.5 \cdot 0 + \frac{x}{x \cdot x}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 29.9%

       \[\leadsto x \cdot \left(0.5 \cdot 0 + \frac{x}{x \cdot x}\right) \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 4: 78.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|\varepsilon\right| \leq 3.7 \cdot 10^{+38}:\\ \;\;\;\;\left(2 \cdot e^{-x}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{\left|\varepsilon\right| \cdot x} - -1\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
  :precision binary64
  (if (<= (fabs eps) 3.7e+38)
  (* (* 2.0 (exp (- x))) 0.5)
  (* 0.5 (- (exp (* (fabs eps) x)) -1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (fabs(eps) <= 3.7e+38) {
		tmp = (2.0 * exp(-x)) * 0.5;
	} else {
		tmp = 0.5 * (exp((fabs(eps) * x)) - -1.0);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (abs(eps) <= 3.7d+38) then
        tmp = (2.0d0 * exp(-x)) * 0.5d0
    else
        tmp = 0.5d0 * (exp((abs(eps) * x)) - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (Math.abs(eps) <= 3.7e+38) {
		tmp = (2.0 * Math.exp(-x)) * 0.5;
	} else {
		tmp = 0.5 * (Math.exp((Math.abs(eps) * x)) - -1.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if math.fabs(eps) <= 3.7e+38:
		tmp = (2.0 * math.exp(-x)) * 0.5
	else:
		tmp = 0.5 * (math.exp((math.fabs(eps) * x)) - -1.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (abs(eps) <= 3.7e+38)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) * 0.5);
	else
		tmp = Float64(0.5 * Float64(exp(Float64(abs(eps) * x)) - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (abs(eps) <= 3.7e+38)
		tmp = (2.0 * exp(-x)) * 0.5;
	else
		tmp = 0.5 * (exp((abs(eps) * x)) - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[Abs[eps], $MachinePrecision], 3.7e+38], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[(N[Exp[N[(N[Abs[eps], $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 3.7000000000000001e38

   1. Initial program 74.1%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1%

          \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]

   5. Taylor expanded in eps around 0

      \[\leadsto 0.5 \cdot \left(e^{\mathsf{neg}\left(x\right)} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x\right)}}\right) \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 70.6%

         \[\leadsto 0.5 \cdot \left(e^{-x} - -1 \cdot e^{-x}\right) \]

   7. Applied rewrites 70.6%

      \[\leadsto 0.5 \cdot \left(e^{-x} - \color{blue}{-1 \cdot e^{-x}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 70.6%

         \[\leadsto \left(e^{-x} - -1 \cdot e^{-x}\right) \cdot \color{blue}{0.5} \]

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. fp-cancel-sign-sub-inv N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 70.6%

          \[\leadsto \left(2 \cdot e^{-x}\right) \cdot 0.5 \]

   9. Applied rewrites 70.6%

      \[\leadsto \left(2 \cdot e^{-x}\right) \cdot \color{blue}{0.5} \]

   if 3.7000000000000001e38 < eps

   1. Initial program 74.1%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1%

          \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 63.8%

      \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1\right) \]

   8. Taylor expanded in eps around inf

      \[\leadsto 0.5 \cdot \left(e^{\varepsilon \cdot x} - -1\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 64.0%

         \[\leadsto 0.5 \cdot \left(e^{\varepsilon \cdot x} - -1\right) \]

   10. Applied rewrites 64.0%

       \[\leadsto 0.5 \cdot \left(e^{\varepsilon \cdot x} - -1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 77.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|\varepsilon\right| \leq 3.7 \cdot 10^{+38}:\\ \;\;\;\;\left(2 \cdot e^{-x}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-x \cdot \left(1 - \left|\varepsilon\right|\right)} - -1\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
  :precision binary64
  (if (<= (fabs eps) 3.7e+38)
  (* (* 2.0 (exp (- x))) 0.5)
  (* 0.5 (- (exp (- (* x (- 1.0 (fabs eps))))) -1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (fabs(eps) <= 3.7e+38) {
		tmp = (2.0 * exp(-x)) * 0.5;
	} else {
		tmp = 0.5 * (exp(-(x * (1.0 - fabs(eps)))) - -1.0);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (abs(eps) <= 3.7d+38) then
        tmp = (2.0d0 * exp(-x)) * 0.5d0
    else
        tmp = 0.5d0 * (exp(-(x * (1.0d0 - abs(eps)))) - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (Math.abs(eps) <= 3.7e+38) {
		tmp = (2.0 * Math.exp(-x)) * 0.5;
	} else {
		tmp = 0.5 * (Math.exp(-(x * (1.0 - Math.abs(eps)))) - -1.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if math.fabs(eps) <= 3.7e+38:
		tmp = (2.0 * math.exp(-x)) * 0.5
	else:
		tmp = 0.5 * (math.exp(-(x * (1.0 - math.fabs(eps)))) - -1.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (abs(eps) <= 3.7e+38)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) * 0.5);
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-Float64(x * Float64(1.0 - abs(eps))))) - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (abs(eps) <= 3.7e+38)
		tmp = (2.0 * exp(-x)) * 0.5;
	else
		tmp = 0.5 * (exp(-(x * (1.0 - abs(eps)))) - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[Abs[eps], $MachinePrecision], 3.7e+38], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[(N[Exp[(-N[(x * N[(1.0 - N[Abs[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 3.7000000000000001e38

   1. Initial program 74.1%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1%

          \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]

   5. Taylor expanded in eps around 0

      \[\leadsto 0.5 \cdot \left(e^{\mathsf{neg}\left(x\right)} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x\right)}}\right) \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 70.6%

         \[\leadsto 0.5 \cdot \left(e^{-x} - -1 \cdot e^{-x}\right) \]

   7. Applied rewrites 70.6%

      \[\leadsto 0.5 \cdot \left(e^{-x} - \color{blue}{-1 \cdot e^{-x}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 70.6%

         \[\leadsto \left(e^{-x} - -1 \cdot e^{-x}\right) \cdot \color{blue}{0.5} \]

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. fp-cancel-sign-sub-inv N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 70.6%

          \[\leadsto \left(2 \cdot e^{-x}\right) \cdot 0.5 \]

   9. Applied rewrites 70.6%

      \[\leadsto \left(2 \cdot e^{-x}\right) \cdot \color{blue}{0.5} \]

   if 3.7000000000000001e38 < eps

   1. Initial program 74.1%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1%

          \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 63.8%

      \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 77.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\left|\varepsilon\right|}\\ t_1 := t\_0 - 1\\ t_2 := \left|\varepsilon\right| - 1\\ t_3 := \left(1 + t\_0\right) \cdot t\_2\\ t_4 := 1 + \left|\varepsilon\right|\\ t_5 := -1 \cdot \left(t\_4 \cdot t\_1\right)\\ \mathbf{if}\;x \leq -8200:\\ \;\;\;\;0.5 \cdot \left(e^{-x} - -1\right)\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-190}:\\ \;\;\;\;1 + 0.5 \cdot \left(x \cdot \left(t\_3 - -1 \cdot \left(\frac{\left|\varepsilon\right| \cdot \left|\varepsilon\right| - 1 \cdot 1}{t\_2} \cdot t\_1\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-251}:\\ \;\;\;\;1 + 0.5 \cdot \left(x \cdot \left(t\_3 - -1 \cdot \left(t\_4 \cdot \left(\left(1 - \frac{1}{t\_0}\right) \cdot t\_0\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 850000:\\ \;\;\;\;1 + 0.5 \cdot \left(x \cdot \left(\frac{t\_2 \cdot \left(\left|\varepsilon\right| - -1\right)}{\left|\varepsilon\right|} - t\_5\right)\right)\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+135}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(t\_3 - t\_5\right)\right)\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+267}:\\ \;\;\;\;0.5 \cdot \left(2 + x \cdot \left(x - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot 0 + \frac{x}{x \cdot x}\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
  :precision binary64
  (let* ((t_0 (/ 1.0 (fabs eps)))
       (t_1 (- t_0 1.0))
       (t_2 (- (fabs eps) 1.0))
       (t_3 (* (+ 1.0 t_0) t_2))
       (t_4 (+ 1.0 (fabs eps)))
       (t_5 (* -1.0 (* t_4 t_1))))
  (if (<= x -8200.0)
    (* 0.5 (- (exp (- x)) -1.0))
    (if (<= x -1e-190)
      (+
       1.0
       (*
        0.5
        (*
         x
         (-
          t_3
          (*
           -1.0
           (*
            (/ (- (* (fabs eps) (fabs eps)) (* 1.0 1.0)) t_2)
            t_1))))))
      (if (<= x 2.3e-251)
        (+
         1.0
         (*
          0.5
          (* x (- t_3 (* -1.0 (* t_4 (* (- 1.0 (/ 1.0 t_0)) t_0)))))))
        (if (<= x 850000.0)
          (+
           1.0
           (*
            0.5
            (* x (- (/ (* t_2 (- (fabs eps) -1.0)) (fabs eps)) t_5))))
          (if (<= x 1.05e+135)
            (* x (* 0.5 (- t_3 t_5)))
            (if (<= x 4.4e+267)
              (* 0.5 (+ 2.0 (* x (- x 2.0))))
              (* x (+ (* 0.5 0.0) (/ x (* x x))))))))))))

C

double code(double x, double eps) {
	double t_0 = 1.0 / fabs(eps);
	double t_1 = t_0 - 1.0;
	double t_2 = fabs(eps) - 1.0;
	double t_3 = (1.0 + t_0) * t_2;
	double t_4 = 1.0 + fabs(eps);
	double t_5 = -1.0 * (t_4 * t_1);
	double tmp;
	if (x <= -8200.0) {
		tmp = 0.5 * (exp(-x) - -1.0);
	} else if (x <= -1e-190) {
		tmp = 1.0 + (0.5 * (x * (t_3 - (-1.0 * ((((fabs(eps) * fabs(eps)) - (1.0 * 1.0)) / t_2) * t_1)))));
	} else if (x <= 2.3e-251) {
		tmp = 1.0 + (0.5 * (x * (t_3 - (-1.0 * (t_4 * ((1.0 - (1.0 / t_0)) * t_0))))));
	} else if (x <= 850000.0) {
		tmp = 1.0 + (0.5 * (x * (((t_2 * (fabs(eps) - -1.0)) / fabs(eps)) - t_5)));
	} else if (x <= 1.05e+135) {
		tmp = x * (0.5 * (t_3 - t_5));
	} else if (x <= 4.4e+267) {
		tmp = 0.5 * (2.0 + (x * (x - 2.0)));
	} else {
		tmp = x * ((0.5 * 0.0) + (x / (x * x)));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = 1.0d0 / abs(eps)
    t_1 = t_0 - 1.0d0
    t_2 = abs(eps) - 1.0d0
    t_3 = (1.0d0 + t_0) * t_2
    t_4 = 1.0d0 + abs(eps)
    t_5 = (-1.0d0) * (t_4 * t_1)
    if (x <= (-8200.0d0)) then
        tmp = 0.5d0 * (exp(-x) - (-1.0d0))
    else if (x <= (-1d-190)) then
        tmp = 1.0d0 + (0.5d0 * (x * (t_3 - ((-1.0d0) * ((((abs(eps) * abs(eps)) - (1.0d0 * 1.0d0)) / t_2) * t_1)))))
    else if (x <= 2.3d-251) then
        tmp = 1.0d0 + (0.5d0 * (x * (t_3 - ((-1.0d0) * (t_4 * ((1.0d0 - (1.0d0 / t_0)) * t_0))))))
    else if (x <= 850000.0d0) then
        tmp = 1.0d0 + (0.5d0 * (x * (((t_2 * (abs(eps) - (-1.0d0))) / abs(eps)) - t_5)))
    else if (x <= 1.05d+135) then
        tmp = x * (0.5d0 * (t_3 - t_5))
    else if (x <= 4.4d+267) then
        tmp = 0.5d0 * (2.0d0 + (x * (x - 2.0d0)))
    else
        tmp = x * ((0.5d0 * 0.0d0) + (x / (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = 1.0 / Math.abs(eps);
	double t_1 = t_0 - 1.0;
	double t_2 = Math.abs(eps) - 1.0;
	double t_3 = (1.0 + t_0) * t_2;
	double t_4 = 1.0 + Math.abs(eps);
	double t_5 = -1.0 * (t_4 * t_1);
	double tmp;
	if (x <= -8200.0) {
		tmp = 0.5 * (Math.exp(-x) - -1.0);
	} else if (x <= -1e-190) {
		tmp = 1.0 + (0.5 * (x * (t_3 - (-1.0 * ((((Math.abs(eps) * Math.abs(eps)) - (1.0 * 1.0)) / t_2) * t_1)))));
	} else if (x <= 2.3e-251) {
		tmp = 1.0 + (0.5 * (x * (t_3 - (-1.0 * (t_4 * ((1.0 - (1.0 / t_0)) * t_0))))));
	} else if (x <= 850000.0) {
		tmp = 1.0 + (0.5 * (x * (((t_2 * (Math.abs(eps) - -1.0)) / Math.abs(eps)) - t_5)));
	} else if (x <= 1.05e+135) {
		tmp = x * (0.5 * (t_3 - t_5));
	} else if (x <= 4.4e+267) {
		tmp = 0.5 * (2.0 + (x * (x - 2.0)));
	} else {
		tmp = x * ((0.5 * 0.0) + (x / (x * x)));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = 1.0 / math.fabs(eps)
	t_1 = t_0 - 1.0
	t_2 = math.fabs(eps) - 1.0
	t_3 = (1.0 + t_0) * t_2
	t_4 = 1.0 + math.fabs(eps)
	t_5 = -1.0 * (t_4 * t_1)
	tmp = 0
	if x <= -8200.0:
		tmp = 0.5 * (math.exp(-x) - -1.0)
	elif x <= -1e-190:
		tmp = 1.0 + (0.5 * (x * (t_3 - (-1.0 * ((((math.fabs(eps) * math.fabs(eps)) - (1.0 * 1.0)) / t_2) * t_1)))))
	elif x <= 2.3e-251:
		tmp = 1.0 + (0.5 * (x * (t_3 - (-1.0 * (t_4 * ((1.0 - (1.0 / t_0)) * t_0))))))
	elif x <= 850000.0:
		tmp = 1.0 + (0.5 * (x * (((t_2 * (math.fabs(eps) - -1.0)) / math.fabs(eps)) - t_5)))
	elif x <= 1.05e+135:
		tmp = x * (0.5 * (t_3 - t_5))
	elif x <= 4.4e+267:
		tmp = 0.5 * (2.0 + (x * (x - 2.0)))
	else:
		tmp = x * ((0.5 * 0.0) + (x / (x * x)))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(1.0 / abs(eps))
	t_1 = Float64(t_0 - 1.0)
	t_2 = Float64(abs(eps) - 1.0)
	t_3 = Float64(Float64(1.0 + t_0) * t_2)
	t_4 = Float64(1.0 + abs(eps))
	t_5 = Float64(-1.0 * Float64(t_4 * t_1))
	tmp = 0.0
	if (x <= -8200.0)
		tmp = Float64(0.5 * Float64(exp(Float64(-x)) - -1.0));
	elseif (x <= -1e-190)
		tmp = Float64(1.0 + Float64(0.5 * Float64(x * Float64(t_3 - Float64(-1.0 * Float64(Float64(Float64(Float64(abs(eps) * abs(eps)) - Float64(1.0 * 1.0)) / t_2) * t_1))))));
	elseif (x <= 2.3e-251)
		tmp = Float64(1.0 + Float64(0.5 * Float64(x * Float64(t_3 - Float64(-1.0 * Float64(t_4 * Float64(Float64(1.0 - Float64(1.0 / t_0)) * t_0)))))));
	elseif (x <= 850000.0)
		tmp = Float64(1.0 + Float64(0.5 * Float64(x * Float64(Float64(Float64(t_2 * Float64(abs(eps) - -1.0)) / abs(eps)) - t_5))));
	elseif (x <= 1.05e+135)
		tmp = Float64(x * Float64(0.5 * Float64(t_3 - t_5)));
	elseif (x <= 4.4e+267)
		tmp = Float64(0.5 * Float64(2.0 + Float64(x * Float64(x - 2.0))));
	else
		tmp = Float64(x * Float64(Float64(0.5 * 0.0) + Float64(x / Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = 1.0 / abs(eps);
	t_1 = t_0 - 1.0;
	t_2 = abs(eps) - 1.0;
	t_3 = (1.0 + t_0) * t_2;
	t_4 = 1.0 + abs(eps);
	t_5 = -1.0 * (t_4 * t_1);
	tmp = 0.0;
	if (x <= -8200.0)
		tmp = 0.5 * (exp(-x) - -1.0);
	elseif (x <= -1e-190)
		tmp = 1.0 + (0.5 * (x * (t_3 - (-1.0 * ((((abs(eps) * abs(eps)) - (1.0 * 1.0)) / t_2) * t_1)))));
	elseif (x <= 2.3e-251)
		tmp = 1.0 + (0.5 * (x * (t_3 - (-1.0 * (t_4 * ((1.0 - (1.0 / t_0)) * t_0))))));
	elseif (x <= 850000.0)
		tmp = 1.0 + (0.5 * (x * (((t_2 * (abs(eps) - -1.0)) / abs(eps)) - t_5)));
	elseif (x <= 1.05e+135)
		tmp = x * (0.5 * (t_3 - t_5));
	elseif (x <= 4.4e+267)
		tmp = 0.5 * (2.0 + (x * (x - 2.0)));
	else
		tmp = x * ((0.5 * 0.0) + (x / (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 / N[Abs[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[eps], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(1.0 + t$95$0), $MachinePrecision] * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 + N[Abs[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(-1.0 * N[(t$95$4 * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8200.0], N[(0.5 * N[(N[Exp[(-x)], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1e-190], N[(1.0 + N[(0.5 * N[(x * N[(t$95$3 - N[(-1.0 * N[(N[(N[(N[(N[Abs[eps], $MachinePrecision] * N[Abs[eps], $MachinePrecision]), $MachinePrecision] - N[(1.0 * 1.0), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.3e-251], N[(1.0 + N[(0.5 * N[(x * N[(t$95$3 - N[(-1.0 * N[(t$95$4 * N[(N[(1.0 - N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 850000.0], N[(1.0 + N[(0.5 * N[(x * N[(N[(N[(t$95$2 * N[(N[Abs[eps], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] / N[Abs[eps], $MachinePrecision]), $MachinePrecision] - t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.05e+135], N[(x * N[(0.5 * N[(t$95$3 - t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.4e+267], N[(0.5 * N[(2.0 + N[(x * N[(x - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(0.5 * 0.0), $MachinePrecision] + N[(x / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if x < -8200

   1. Initial program 74.1%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1%

          \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 63.8%

      \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1\right) \]

   8. Taylor expanded in eps around 0

      \[\leadsto 0.5 \cdot \left(e^{\mathsf{neg}\left(x\right)} - -1\right) \]
   9. Step-by-step derivation

      1. lower-exp.f64 N/A

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

      2. lower-neg.f64 57.0%

         \[\leadsto 0.5 \cdot \left(e^{-x} - -1\right) \]

   10. Applied rewrites 57.0%

       \[\leadsto 0.5 \cdot \left(e^{-x} - -1\right) \]

   if -8200 < x < -1e-190

   1. Initial program 74.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 43.7%

      \[\leadsto \color{blue}{1 + 0.5 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-/.f64 N/A

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

      6. lower-unsound--.f64 N/A

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

      7. lower-unsound-*.f64 N/A

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

      8. lower-unsound-*.f64 51.6%

         \[\leadsto 1 + 0.5 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\frac{\varepsilon \cdot \varepsilon - 1 \cdot 1}{\varepsilon - 1} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

   6. Applied rewrites 51.6%

      \[\leadsto 1 + 0.5 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\frac{\varepsilon \cdot \varepsilon - 1 \cdot 1}{\varepsilon - 1} \cdot \left(\color{blue}{\frac{1}{\varepsilon}} - 1\right)\right)\right)\right) \]

   if -1e-190 < x < 2.3000000000000002e-251

   1. Initial program 74.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 43.7%

      \[\leadsto \color{blue}{1 + 0.5 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-/.f64 44.3%

         \[\leadsto 1 + 0.5 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\left(1 - \frac{1}{\frac{1}{\varepsilon}}\right) \cdot \frac{1}{\varepsilon}\right)\right)\right)\right) \]

   6. Applied rewrites 44.3%

      \[\leadsto 1 + 0.5 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\left(1 - \frac{1}{\frac{1}{\varepsilon}}\right) \cdot \color{blue}{\frac{1}{\varepsilon}}\right)\right)\right)\right) \]

   if 2.3000000000000002e-251 < x < 8.5e5

   1. Initial program 74.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 43.7%

      \[\leadsto \color{blue}{1 + 0.5 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. add-to-fraction N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. *-lft-identity N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. lower-*.f64 52.3%

          \[\leadsto 1 + 0.5 \cdot \left(x \cdot \left(\frac{\left(\varepsilon - 1\right) \cdot \left(1 + \varepsilon\right)}{\varepsilon} - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. add-flip N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 52.3%

          \[\leadsto 1 + 0.5 \cdot \left(x \cdot \left(\frac{\left(\varepsilon - 1\right) \cdot \left(\varepsilon - -1\right)}{\varepsilon} - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

   6. Applied rewrites 52.3%

      \[\leadsto 1 + 0.5 \cdot \left(x \cdot \left(\frac{\left(\varepsilon - 1\right) \cdot \left(\varepsilon - -1\right)}{\varepsilon} - \color{blue}{-1} \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

   if 8.5e5 < x < 1.05e135

   1. Initial program 74.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 43.7%

      \[\leadsto \color{blue}{1 + 0.5 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

   7. Applied rewrites 43.6%

      \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{1}{x}\right)} \]

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 16.1%

          \[\leadsto x \cdot \left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

   10. Applied rewrites 16.1%

       \[\leadsto x \cdot \left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - \color{blue}{-1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)}\right)\right) \]

   if 1.05e135 < x < 4.4000000000000002e267

   1. Initial program 74.1%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1%

          \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]

   5. Taylor expanded in eps around 0

      \[\leadsto 0.5 \cdot \left(e^{\mathsf{neg}\left(x\right)} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x\right)}}\right) \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 70.6%

         \[\leadsto 0.5 \cdot \left(e^{-x} - -1 \cdot e^{-x}\right) \]

   7. Applied rewrites 70.6%

      \[\leadsto 0.5 \cdot \left(e^{-x} - \color{blue}{-1 \cdot e^{-x}}\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto 0.5 \cdot \left(2 + x \cdot \color{blue}{\left(x - 2\right)}\right) \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 57.3%

         \[\leadsto 0.5 \cdot \left(2 + x \cdot \left(x - 2\right)\right) \]

   10. Applied rewrites 57.3%

       \[\leadsto 0.5 \cdot \left(2 + x \cdot \color{blue}{\left(x - 2\right)}\right) \]

   if 4.4000000000000002e267 < x

   1. Initial program 74.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 43.7%

      \[\leadsto \color{blue}{1 + 0.5 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

   7. Applied rewrites 43.6%

      \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{1}{x}\right)} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. rgt-mult-inverse N/A

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

      4. mult-flip-rev N/A

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

      5. frac-times N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 29.8%

         \[\leadsto x \cdot \left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{x}{x \cdot x}\right) \]

   9. Applied rewrites 29.8%

      \[\leadsto x \cdot \left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{x}{x \cdot \color{blue}{x}}\right) \]

   10. Taylor expanded in eps around 0

       \[\leadsto x \cdot \left(0.5 \cdot 0 + \frac{x}{x \cdot x}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 29.9%

       \[\leadsto x \cdot \left(0.5 \cdot 0 + \frac{x}{x \cdot x}\right) \]
3. Recombined 7 regimes into one program.
4. Add Preprocessing

Alternative 7: 76.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\left|\varepsilon\right|}\\ t_1 := t\_0 - 1\\ t_2 := 1 + \left|\varepsilon\right|\\ t_3 := \left|\varepsilon\right| - 1\\ t_4 := \left(1 + t\_0\right) \cdot t\_3\\ t_5 := -1 \cdot \left(t\_2 \cdot t\_1\right)\\ \mathbf{if}\;x \leq -3 \cdot 10^{+106}:\\ \;\;\;\;0.5 \cdot \left(2 + x \cdot \left(x \cdot \left(1 + -0.3333333333333333 \cdot x\right) - 2\right)\right)\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-190}:\\ \;\;\;\;1 + 0.5 \cdot \left(x \cdot \left(t\_4 - -1 \cdot \left(\frac{\left|\varepsilon\right| \cdot \left|\varepsilon\right| - 1 \cdot 1}{t\_3} \cdot t\_1\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-251}:\\ \;\;\;\;1 + 0.5 \cdot \left(x \cdot \left(t\_4 - -1 \cdot \left(t\_2 \cdot \left(\left(1 - \frac{1}{t\_0}\right) \cdot t\_0\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 850000:\\ \;\;\;\;1 + 0.5 \cdot \left(x \cdot \left(\frac{t\_3 \cdot \left(\left|\varepsilon\right| - -1\right)}{\left|\varepsilon\right|} - t\_5\right)\right)\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+135}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(t\_4 - t\_5\right)\right)\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+222}:\\ \;\;\;\;0.5 \cdot \left(2 + x \cdot \left(x - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot 0 + \frac{x}{x \cdot x}\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
  :precision binary64
  (let* ((t_0 (/ 1.0 (fabs eps)))
       (t_1 (- t_0 1.0))
       (t_2 (+ 1.0 (fabs eps)))
       (t_3 (- (fabs eps) 1.0))
       (t_4 (* (+ 1.0 t_0) t_3))
       (t_5 (* -1.0 (* t_2 t_1))))
  (if (<= x -3e+106)
    (*
     0.5
     (+ 2.0 (* x (- (* x (+ 1.0 (* -0.3333333333333333 x))) 2.0))))
    (if (<= x -1e-190)
      (+
       1.0
       (*
        0.5
        (*
         x
         (-
          t_4
          (*
           -1.0
           (*
            (/ (- (* (fabs eps) (fabs eps)) (* 1.0 1.0)) t_3)
            t_1))))))
      (if (<= x 2.3e-251)
        (+
         1.0
         (*
          0.5
          (* x (- t_4 (* -1.0 (* t_2 (* (- 1.0 (/ 1.0 t_0)) t_0)))))))
        (if (<= x 850000.0)
          (+
           1.0
           (*
            0.5
            (* x (- (/ (* t_3 (- (fabs eps) -1.0)) (fabs eps)) t_5))))
          (if (<= x 1.05e+135)
            (* x (* 0.5 (- t_4 t_5)))
            (if (<= x 5.6e+222)
              (* 0.5 (+ 2.0 (* x (- x 2.0))))
              (* x (+ (* 0.5 0.0) (/ x (* x x))))))))))))

C

double code(double x, double eps) {
	double t_0 = 1.0 / fabs(eps);
	double t_1 = t_0 - 1.0;
	double t_2 = 1.0 + fabs(eps);
	double t_3 = fabs(eps) - 1.0;
	double t_4 = (1.0 + t_0) * t_3;
	double t_5 = -1.0 * (t_2 * t_1);
	double tmp;
	if (x <= -3e+106) {
		tmp = 0.5 * (2.0 + (x * ((x * (1.0 + (-0.3333333333333333 * x))) - 2.0)));
	} else if (x <= -1e-190) {
		tmp = 1.0 + (0.5 * (x * (t_4 - (-1.0 * ((((fabs(eps) * fabs(eps)) - (1.0 * 1.0)) / t_3) * t_1)))));
	} else if (x <= 2.3e-251) {
		tmp = 1.0 + (0.5 * (x * (t_4 - (-1.0 * (t_2 * ((1.0 - (1.0 / t_0)) * t_0))))));
	} else if (x <= 850000.0) {
		tmp = 1.0 + (0.5 * (x * (((t_3 * (fabs(eps) - -1.0)) / fabs(eps)) - t_5)));
	} else if (x <= 1.05e+135) {
		tmp = x * (0.5 * (t_4 - t_5));
	} else if (x <= 5.6e+222) {
		tmp = 0.5 * (2.0 + (x * (x - 2.0)));
	} else {
		tmp = x * ((0.5 * 0.0) + (x / (x * x)));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = 1.0d0 / abs(eps)
    t_1 = t_0 - 1.0d0
    t_2 = 1.0d0 + abs(eps)
    t_3 = abs(eps) - 1.0d0
    t_4 = (1.0d0 + t_0) * t_3
    t_5 = (-1.0d0) * (t_2 * t_1)
    if (x <= (-3d+106)) then
        tmp = 0.5d0 * (2.0d0 + (x * ((x * (1.0d0 + ((-0.3333333333333333d0) * x))) - 2.0d0)))
    else if (x <= (-1d-190)) then
        tmp = 1.0d0 + (0.5d0 * (x * (t_4 - ((-1.0d0) * ((((abs(eps) * abs(eps)) - (1.0d0 * 1.0d0)) / t_3) * t_1)))))
    else if (x <= 2.3d-251) then
        tmp = 1.0d0 + (0.5d0 * (x * (t_4 - ((-1.0d0) * (t_2 * ((1.0d0 - (1.0d0 / t_0)) * t_0))))))
    else if (x <= 850000.0d0) then
        tmp = 1.0d0 + (0.5d0 * (x * (((t_3 * (abs(eps) - (-1.0d0))) / abs(eps)) - t_5)))
    else if (x <= 1.05d+135) then
        tmp = x * (0.5d0 * (t_4 - t_5))
    else if (x <= 5.6d+222) then
        tmp = 0.5d0 * (2.0d0 + (x * (x - 2.0d0)))
    else
        tmp = x * ((0.5d0 * 0.0d0) + (x / (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = 1.0 / Math.abs(eps);
	double t_1 = t_0 - 1.0;
	double t_2 = 1.0 + Math.abs(eps);
	double t_3 = Math.abs(eps) - 1.0;
	double t_4 = (1.0 + t_0) * t_3;
	double t_5 = -1.0 * (t_2 * t_1);
	double tmp;
	if (x <= -3e+106) {
		tmp = 0.5 * (2.0 + (x * ((x * (1.0 + (-0.3333333333333333 * x))) - 2.0)));
	} else if (x <= -1e-190) {
		tmp = 1.0 + (0.5 * (x * (t_4 - (-1.0 * ((((Math.abs(eps) * Math.abs(eps)) - (1.0 * 1.0)) / t_3) * t_1)))));
	} else if (x <= 2.3e-251) {
		tmp = 1.0 + (0.5 * (x * (t_4 - (-1.0 * (t_2 * ((1.0 - (1.0 / t_0)) * t_0))))));
	} else if (x <= 850000.0) {
		tmp = 1.0 + (0.5 * (x * (((t_3 * (Math.abs(eps) - -1.0)) / Math.abs(eps)) - t_5)));
	} else if (x <= 1.05e+135) {
		tmp = x * (0.5 * (t_4 - t_5));
	} else if (x <= 5.6e+222) {
		tmp = 0.5 * (2.0 + (x * (x - 2.0)));
	} else {
		tmp = x * ((0.5 * 0.0) + (x / (x * x)));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = 1.0 / math.fabs(eps)
	t_1 = t_0 - 1.0
	t_2 = 1.0 + math.fabs(eps)
	t_3 = math.fabs(eps) - 1.0
	t_4 = (1.0 + t_0) * t_3
	t_5 = -1.0 * (t_2 * t_1)
	tmp = 0
	if x <= -3e+106:
		tmp = 0.5 * (2.0 + (x * ((x * (1.0 + (-0.3333333333333333 * x))) - 2.0)))
	elif x <= -1e-190:
		tmp = 1.0 + (0.5 * (x * (t_4 - (-1.0 * ((((math.fabs(eps) * math.fabs(eps)) - (1.0 * 1.0)) / t_3) * t_1)))))
	elif x <= 2.3e-251:
		tmp = 1.0 + (0.5 * (x * (t_4 - (-1.0 * (t_2 * ((1.0 - (1.0 / t_0)) * t_0))))))
	elif x <= 850000.0:
		tmp = 1.0 + (0.5 * (x * (((t_3 * (math.fabs(eps) - -1.0)) / math.fabs(eps)) - t_5)))
	elif x <= 1.05e+135:
		tmp = x * (0.5 * (t_4 - t_5))
	elif x <= 5.6e+222:
		tmp = 0.5 * (2.0 + (x * (x - 2.0)))
	else:
		tmp = x * ((0.5 * 0.0) + (x / (x * x)))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(1.0 / abs(eps))
	t_1 = Float64(t_0 - 1.0)
	t_2 = Float64(1.0 + abs(eps))
	t_3 = Float64(abs(eps) - 1.0)
	t_4 = Float64(Float64(1.0 + t_0) * t_3)
	t_5 = Float64(-1.0 * Float64(t_2 * t_1))
	tmp = 0.0
	if (x <= -3e+106)
		tmp = Float64(0.5 * Float64(2.0 + Float64(x * Float64(Float64(x * Float64(1.0 + Float64(-0.3333333333333333 * x))) - 2.0))));
	elseif (x <= -1e-190)
		tmp = Float64(1.0 + Float64(0.5 * Float64(x * Float64(t_4 - Float64(-1.0 * Float64(Float64(Float64(Float64(abs(eps) * abs(eps)) - Float64(1.0 * 1.0)) / t_3) * t_1))))));
	elseif (x <= 2.3e-251)
		tmp = Float64(1.0 + Float64(0.5 * Float64(x * Float64(t_4 - Float64(-1.0 * Float64(t_2 * Float64(Float64(1.0 - Float64(1.0 / t_0)) * t_0)))))));
	elseif (x <= 850000.0)
		tmp = Float64(1.0 + Float64(0.5 * Float64(x * Float64(Float64(Float64(t_3 * Float64(abs(eps) - -1.0)) / abs(eps)) - t_5))));
	elseif (x <= 1.05e+135)
		tmp = Float64(x * Float64(0.5 * Float64(t_4 - t_5)));
	elseif (x <= 5.6e+222)
		tmp = Float64(0.5 * Float64(2.0 + Float64(x * Float64(x - 2.0))));
	else
		tmp = Float64(x * Float64(Float64(0.5 * 0.0) + Float64(x / Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = 1.0 / abs(eps);
	t_1 = t_0 - 1.0;
	t_2 = 1.0 + abs(eps);
	t_3 = abs(eps) - 1.0;
	t_4 = (1.0 + t_0) * t_3;
	t_5 = -1.0 * (t_2 * t_1);
	tmp = 0.0;
	if (x <= -3e+106)
		tmp = 0.5 * (2.0 + (x * ((x * (1.0 + (-0.3333333333333333 * x))) - 2.0)));
	elseif (x <= -1e-190)
		tmp = 1.0 + (0.5 * (x * (t_4 - (-1.0 * ((((abs(eps) * abs(eps)) - (1.0 * 1.0)) / t_3) * t_1)))));
	elseif (x <= 2.3e-251)
		tmp = 1.0 + (0.5 * (x * (t_4 - (-1.0 * (t_2 * ((1.0 - (1.0 / t_0)) * t_0))))));
	elseif (x <= 850000.0)
		tmp = 1.0 + (0.5 * (x * (((t_3 * (abs(eps) - -1.0)) / abs(eps)) - t_5)));
	elseif (x <= 1.05e+135)
		tmp = x * (0.5 * (t_4 - t_5));
	elseif (x <= 5.6e+222)
		tmp = 0.5 * (2.0 + (x * (x - 2.0)));
	else
		tmp = x * ((0.5 * 0.0) + (x / (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 / N[Abs[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[Abs[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[eps], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(1.0 + t$95$0), $MachinePrecision] * t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(-1.0 * N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3e+106], N[(0.5 * N[(2.0 + N[(x * N[(N[(x * N[(1.0 + N[(-0.3333333333333333 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1e-190], N[(1.0 + N[(0.5 * N[(x * N[(t$95$4 - N[(-1.0 * N[(N[(N[(N[(N[Abs[eps], $MachinePrecision] * N[Abs[eps], $MachinePrecision]), $MachinePrecision] - N[(1.0 * 1.0), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.3e-251], N[(1.0 + N[(0.5 * N[(x * N[(t$95$4 - N[(-1.0 * N[(t$95$2 * N[(N[(1.0 - N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 850000.0], N[(1.0 + N[(0.5 * N[(x * N[(N[(N[(t$95$3 * N[(N[Abs[eps], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] / N[Abs[eps], $MachinePrecision]), $MachinePrecision] - t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.05e+135], N[(x * N[(0.5 * N[(t$95$4 - t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.6e+222], N[(0.5 * N[(2.0 + N[(x * N[(x - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(0.5 * 0.0), $MachinePrecision] + N[(x / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if x < -3.0000000000000001e106

   1. Initial program 74.1%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1%

          \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]

   5. Taylor expanded in eps around 0

      \[\leadsto 0.5 \cdot \left(e^{\mathsf{neg}\left(x\right)} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x\right)}}\right) \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 70.6%

         \[\leadsto 0.5 \cdot \left(e^{-x} - -1 \cdot e^{-x}\right) \]

   7. Applied rewrites 70.6%

      \[\leadsto 0.5 \cdot \left(e^{-x} - \color{blue}{-1 \cdot e^{-x}}\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto 0.5 \cdot \left(2 + x \cdot \color{blue}{\left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2\right)}\right) \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + x \cdot \left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - \color{blue}{2}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + x \cdot \left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2\right)\right) \]

      3. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + x \cdot \left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + x \cdot \left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + x \cdot \left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2\right)\right) \]

      6. lower-*.f64 52.1%

         \[\leadsto 0.5 \cdot \left(2 + x \cdot \left(x \cdot \left(1 + -0.3333333333333333 \cdot x\right) - 2\right)\right) \]

   10. Applied rewrites 52.1%

       \[\leadsto 0.5 \cdot \left(2 + x \cdot \color{blue}{\left(x \cdot \left(1 + -0.3333333333333333 \cdot x\right) - 2\right)}\right) \]

   if -3.0000000000000001e106 < x < -1e-190

   1. Initial program 74.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 43.7%

      \[\leadsto \color{blue}{1 + 0.5 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-/.f64 N/A

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

      6. lower-unsound--.f64 N/A

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

      7. lower-unsound-*.f64 N/A

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

      8. lower-unsound-*.f64 51.6%

         \[\leadsto 1 + 0.5 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\frac{\varepsilon \cdot \varepsilon - 1 \cdot 1}{\varepsilon - 1} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

   6. Applied rewrites 51.6%

      \[\leadsto 1 + 0.5 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\frac{\varepsilon \cdot \varepsilon - 1 \cdot 1}{\varepsilon - 1} \cdot \left(\color{blue}{\frac{1}{\varepsilon}} - 1\right)\right)\right)\right) \]

   if -1e-190 < x < 2.3000000000000002e-251

   1. Initial program 74.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 43.7%

      \[\leadsto \color{blue}{1 + 0.5 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-/.f64 44.3%

         \[\leadsto 1 + 0.5 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\left(1 - \frac{1}{\frac{1}{\varepsilon}}\right) \cdot \frac{1}{\varepsilon}\right)\right)\right)\right) \]

   6. Applied rewrites 44.3%

      \[\leadsto 1 + 0.5 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\left(1 - \frac{1}{\frac{1}{\varepsilon}}\right) \cdot \color{blue}{\frac{1}{\varepsilon}}\right)\right)\right)\right) \]

   if 2.3000000000000002e-251 < x < 8.5e5

   1. Initial program 74.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 43.7%

      \[\leadsto \color{blue}{1 + 0.5 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. add-to-fraction N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. *-lft-identity N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. lower-*.f64 52.3%

          \[\leadsto 1 + 0.5 \cdot \left(x \cdot \left(\frac{\left(\varepsilon - 1\right) \cdot \left(1 + \varepsilon\right)}{\varepsilon} - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. add-flip N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 52.3%

          \[\leadsto 1 + 0.5 \cdot \left(x \cdot \left(\frac{\left(\varepsilon - 1\right) \cdot \left(\varepsilon - -1\right)}{\varepsilon} - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

   6. Applied rewrites 52.3%

      \[\leadsto 1 + 0.5 \cdot \left(x \cdot \left(\frac{\left(\varepsilon - 1\right) \cdot \left(\varepsilon - -1\right)}{\varepsilon} - \color{blue}{-1} \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

   if 8.5e5 < x < 1.05e135

   1. Initial program 74.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 43.7%

      \[\leadsto \color{blue}{1 + 0.5 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

   7. Applied rewrites 43.6%

      \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{1}{x}\right)} \]

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 16.1%

          \[\leadsto x \cdot \left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

   10. Applied rewrites 16.1%

       \[\leadsto x \cdot \left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - \color{blue}{-1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)}\right)\right) \]

   if 1.05e135 < x < 5.6000000000000003e222

   1. Initial program 74.1%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1%

          \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]

   5. Taylor expanded in eps around 0

      \[\leadsto 0.5 \cdot \left(e^{\mathsf{neg}\left(x\right)} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x\right)}}\right) \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 70.6%

         \[\leadsto 0.5 \cdot \left(e^{-x} - -1 \cdot e^{-x}\right) \]

   7. Applied rewrites 70.6%

      \[\leadsto 0.5 \cdot \left(e^{-x} - \color{blue}{-1 \cdot e^{-x}}\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto 0.5 \cdot \left(2 + x \cdot \color{blue}{\left(x - 2\right)}\right) \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 57.3%

         \[\leadsto 0.5 \cdot \left(2 + x \cdot \left(x - 2\right)\right) \]

   10. Applied rewrites 57.3%

       \[\leadsto 0.5 \cdot \left(2 + x \cdot \color{blue}{\left(x - 2\right)}\right) \]

   if 5.6000000000000003e222 < x

   1. Initial program 74.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 43.7%

      \[\leadsto \color{blue}{1 + 0.5 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

   7. Applied rewrites 43.6%

      \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{1}{x}\right)} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. rgt-mult-inverse N/A

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

      4. mult-flip-rev N/A

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

      5. frac-times N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 29.8%

         \[\leadsto x \cdot \left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{x}{x \cdot x}\right) \]

   9. Applied rewrites 29.8%

      \[\leadsto x \cdot \left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{x}{x \cdot \color{blue}{x}}\right) \]

   10. Taylor expanded in eps around 0

       \[\leadsto x \cdot \left(0.5 \cdot 0 + \frac{x}{x \cdot x}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 29.9%

       \[\leadsto x \cdot \left(0.5 \cdot 0 + \frac{x}{x \cdot x}\right) \]
3. Recombined 7 regimes into one program.
4. Add Preprocessing

Alternative 8: 76.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\left|\varepsilon\right|}\\ t_1 := t\_0 - 1\\ t_2 := -1 \cdot \left(\left(1 + \left|\varepsilon\right|\right) \cdot t\_1\right)\\ t_3 := \left|\varepsilon\right| - 1\\ t_4 := \left(1 + t\_0\right) \cdot t\_3\\ \mathbf{if}\;x \leq -3 \cdot 10^{+106}:\\ \;\;\;\;0.5 \cdot \left(2 + x \cdot \left(x \cdot \left(1 + -0.3333333333333333 \cdot x\right) - 2\right)\right)\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-190}:\\ \;\;\;\;1 + 0.5 \cdot \left(x \cdot \left(t\_4 - -1 \cdot \left(\frac{\left|\varepsilon\right| \cdot \left|\varepsilon\right| - 1 \cdot 1}{t\_3} \cdot t\_1\right)\right)\right)\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-251}:\\ \;\;\;\;2 \cdot 0.5\\ \mathbf{elif}\;x \leq 850000:\\ \;\;\;\;1 + 0.5 \cdot \left(x \cdot \left(\frac{t\_3 \cdot \left(\left|\varepsilon\right| - -1\right)}{\left|\varepsilon\right|} - t\_2\right)\right)\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+135}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(t\_4 - t\_2\right)\right)\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+222}:\\ \;\;\;\;0.5 \cdot \left(2 + x \cdot \left(x - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot 0 + \frac{x}{x \cdot x}\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
  :precision binary64
  (let* ((t_0 (/ 1.0 (fabs eps)))
       (t_1 (- t_0 1.0))
       (t_2 (* -1.0 (* (+ 1.0 (fabs eps)) t_1)))
       (t_3 (- (fabs eps) 1.0))
       (t_4 (* (+ 1.0 t_0) t_3)))
  (if (<= x -3e+106)
    (*
     0.5
     (+ 2.0 (* x (- (* x (+ 1.0 (* -0.3333333333333333 x))) 2.0))))
    (if (<= x -1e-190)
      (+
       1.0
       (*
        0.5
        (*
         x
         (-
          t_4
          (*
           -1.0
           (*
            (/ (- (* (fabs eps) (fabs eps)) (* 1.0 1.0)) t_3)
            t_1))))))
      (if (<= x 2.3e-251)
        (* 2.0 0.5)
        (if (<= x 850000.0)
          (+
           1.0
           (*
            0.5
            (* x (- (/ (* t_3 (- (fabs eps) -1.0)) (fabs eps)) t_2))))
          (if (<= x 1.05e+135)
            (* x (* 0.5 (- t_4 t_2)))
            (if (<= x 5.6e+222)
              (* 0.5 (+ 2.0 (* x (- x 2.0))))
              (* x (+ (* 0.5 0.0) (/ x (* x x))))))))))))

C

double code(double x, double eps) {
	double t_0 = 1.0 / fabs(eps);
	double t_1 = t_0 - 1.0;
	double t_2 = -1.0 * ((1.0 + fabs(eps)) * t_1);
	double t_3 = fabs(eps) - 1.0;
	double t_4 = (1.0 + t_0) * t_3;
	double tmp;
	if (x <= -3e+106) {
		tmp = 0.5 * (2.0 + (x * ((x * (1.0 + (-0.3333333333333333 * x))) - 2.0)));
	} else if (x <= -1e-190) {
		tmp = 1.0 + (0.5 * (x * (t_4 - (-1.0 * ((((fabs(eps) * fabs(eps)) - (1.0 * 1.0)) / t_3) * t_1)))));
	} else if (x <= 2.3e-251) {
		tmp = 2.0 * 0.5;
	} else if (x <= 850000.0) {
		tmp = 1.0 + (0.5 * (x * (((t_3 * (fabs(eps) - -1.0)) / fabs(eps)) - t_2)));
	} else if (x <= 1.05e+135) {
		tmp = x * (0.5 * (t_4 - t_2));
	} else if (x <= 5.6e+222) {
		tmp = 0.5 * (2.0 + (x * (x - 2.0)));
	} else {
		tmp = x * ((0.5 * 0.0) + (x / (x * x)));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = 1.0d0 / abs(eps)
    t_1 = t_0 - 1.0d0
    t_2 = (-1.0d0) * ((1.0d0 + abs(eps)) * t_1)
    t_3 = abs(eps) - 1.0d0
    t_4 = (1.0d0 + t_0) * t_3
    if (x <= (-3d+106)) then
        tmp = 0.5d0 * (2.0d0 + (x * ((x * (1.0d0 + ((-0.3333333333333333d0) * x))) - 2.0d0)))
    else if (x <= (-1d-190)) then
        tmp = 1.0d0 + (0.5d0 * (x * (t_4 - ((-1.0d0) * ((((abs(eps) * abs(eps)) - (1.0d0 * 1.0d0)) / t_3) * t_1)))))
    else if (x <= 2.3d-251) then
        tmp = 2.0d0 * 0.5d0
    else if (x <= 850000.0d0) then
        tmp = 1.0d0 + (0.5d0 * (x * (((t_3 * (abs(eps) - (-1.0d0))) / abs(eps)) - t_2)))
    else if (x <= 1.05d+135) then
        tmp = x * (0.5d0 * (t_4 - t_2))
    else if (x <= 5.6d+222) then
        tmp = 0.5d0 * (2.0d0 + (x * (x - 2.0d0)))
    else
        tmp = x * ((0.5d0 * 0.0d0) + (x / (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = 1.0 / Math.abs(eps);
	double t_1 = t_0 - 1.0;
	double t_2 = -1.0 * ((1.0 + Math.abs(eps)) * t_1);
	double t_3 = Math.abs(eps) - 1.0;
	double t_4 = (1.0 + t_0) * t_3;
	double tmp;
	if (x <= -3e+106) {
		tmp = 0.5 * (2.0 + (x * ((x * (1.0 + (-0.3333333333333333 * x))) - 2.0)));
	} else if (x <= -1e-190) {
		tmp = 1.0 + (0.5 * (x * (t_4 - (-1.0 * ((((Math.abs(eps) * Math.abs(eps)) - (1.0 * 1.0)) / t_3) * t_1)))));
	} else if (x <= 2.3e-251) {
		tmp = 2.0 * 0.5;
	} else if (x <= 850000.0) {
		tmp = 1.0 + (0.5 * (x * (((t_3 * (Math.abs(eps) - -1.0)) / Math.abs(eps)) - t_2)));
	} else if (x <= 1.05e+135) {
		tmp = x * (0.5 * (t_4 - t_2));
	} else if (x <= 5.6e+222) {
		tmp = 0.5 * (2.0 + (x * (x - 2.0)));
	} else {
		tmp = x * ((0.5 * 0.0) + (x / (x * x)));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = 1.0 / math.fabs(eps)
	t_1 = t_0 - 1.0
	t_2 = -1.0 * ((1.0 + math.fabs(eps)) * t_1)
	t_3 = math.fabs(eps) - 1.0
	t_4 = (1.0 + t_0) * t_3
	tmp = 0
	if x <= -3e+106:
		tmp = 0.5 * (2.0 + (x * ((x * (1.0 + (-0.3333333333333333 * x))) - 2.0)))
	elif x <= -1e-190:
		tmp = 1.0 + (0.5 * (x * (t_4 - (-1.0 * ((((math.fabs(eps) * math.fabs(eps)) - (1.0 * 1.0)) / t_3) * t_1)))))
	elif x <= 2.3e-251:
		tmp = 2.0 * 0.5
	elif x <= 850000.0:
		tmp = 1.0 + (0.5 * (x * (((t_3 * (math.fabs(eps) - -1.0)) / math.fabs(eps)) - t_2)))
	elif x <= 1.05e+135:
		tmp = x * (0.5 * (t_4 - t_2))
	elif x <= 5.6e+222:
		tmp = 0.5 * (2.0 + (x * (x - 2.0)))
	else:
		tmp = x * ((0.5 * 0.0) + (x / (x * x)))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(1.0 / abs(eps))
	t_1 = Float64(t_0 - 1.0)
	t_2 = Float64(-1.0 * Float64(Float64(1.0 + abs(eps)) * t_1))
	t_3 = Float64(abs(eps) - 1.0)
	t_4 = Float64(Float64(1.0 + t_0) * t_3)
	tmp = 0.0
	if (x <= -3e+106)
		tmp = Float64(0.5 * Float64(2.0 + Float64(x * Float64(Float64(x * Float64(1.0 + Float64(-0.3333333333333333 * x))) - 2.0))));
	elseif (x <= -1e-190)
		tmp = Float64(1.0 + Float64(0.5 * Float64(x * Float64(t_4 - Float64(-1.0 * Float64(Float64(Float64(Float64(abs(eps) * abs(eps)) - Float64(1.0 * 1.0)) / t_3) * t_1))))));
	elseif (x <= 2.3e-251)
		tmp = Float64(2.0 * 0.5);
	elseif (x <= 850000.0)
		tmp = Float64(1.0 + Float64(0.5 * Float64(x * Float64(Float64(Float64(t_3 * Float64(abs(eps) - -1.0)) / abs(eps)) - t_2))));
	elseif (x <= 1.05e+135)
		tmp = Float64(x * Float64(0.5 * Float64(t_4 - t_2)));
	elseif (x <= 5.6e+222)
		tmp = Float64(0.5 * Float64(2.0 + Float64(x * Float64(x - 2.0))));
	else
		tmp = Float64(x * Float64(Float64(0.5 * 0.0) + Float64(x / Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = 1.0 / abs(eps);
	t_1 = t_0 - 1.0;
	t_2 = -1.0 * ((1.0 + abs(eps)) * t_1);
	t_3 = abs(eps) - 1.0;
	t_4 = (1.0 + t_0) * t_3;
	tmp = 0.0;
	if (x <= -3e+106)
		tmp = 0.5 * (2.0 + (x * ((x * (1.0 + (-0.3333333333333333 * x))) - 2.0)));
	elseif (x <= -1e-190)
		tmp = 1.0 + (0.5 * (x * (t_4 - (-1.0 * ((((abs(eps) * abs(eps)) - (1.0 * 1.0)) / t_3) * t_1)))));
	elseif (x <= 2.3e-251)
		tmp = 2.0 * 0.5;
	elseif (x <= 850000.0)
		tmp = 1.0 + (0.5 * (x * (((t_3 * (abs(eps) - -1.0)) / abs(eps)) - t_2)));
	elseif (x <= 1.05e+135)
		tmp = x * (0.5 * (t_4 - t_2));
	elseif (x <= 5.6e+222)
		tmp = 0.5 * (2.0 + (x * (x - 2.0)));
	else
		tmp = x * ((0.5 * 0.0) + (x / (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 / N[Abs[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 * N[(N[(1.0 + N[Abs[eps], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[eps], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(1.0 + t$95$0), $MachinePrecision] * t$95$3), $MachinePrecision]}, If[LessEqual[x, -3e+106], N[(0.5 * N[(2.0 + N[(x * N[(N[(x * N[(1.0 + N[(-0.3333333333333333 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1e-190], N[(1.0 + N[(0.5 * N[(x * N[(t$95$4 - N[(-1.0 * N[(N[(N[(N[(N[Abs[eps], $MachinePrecision] * N[Abs[eps], $MachinePrecision]), $MachinePrecision] - N[(1.0 * 1.0), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.3e-251], N[(2.0 * 0.5), $MachinePrecision], If[LessEqual[x, 850000.0], N[(1.0 + N[(0.5 * N[(x * N[(N[(N[(t$95$3 * N[(N[Abs[eps], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] / N[Abs[eps], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.05e+135], N[(x * N[(0.5 * N[(t$95$4 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.6e+222], N[(0.5 * N[(2.0 + N[(x * N[(x - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(0.5 * 0.0), $MachinePrecision] + N[(x / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if x < -3.0000000000000001e106

   1. Initial program 74.1%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1%

          \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]

   5. Taylor expanded in eps around 0

      \[\leadsto 0.5 \cdot \left(e^{\mathsf{neg}\left(x\right)} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x\right)}}\right) \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 70.6%

         \[\leadsto 0.5 \cdot \left(e^{-x} - -1 \cdot e^{-x}\right) \]

   7. Applied rewrites 70.6%

      \[\leadsto 0.5 \cdot \left(e^{-x} - \color{blue}{-1 \cdot e^{-x}}\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto 0.5 \cdot \left(2 + x \cdot \color{blue}{\left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2\right)}\right) \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + x \cdot \left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - \color{blue}{2}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + x \cdot \left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2\right)\right) \]

      3. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + x \cdot \left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + x \cdot \left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + x \cdot \left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2\right)\right) \]

      6. lower-*.f64 52.1%

         \[\leadsto 0.5 \cdot \left(2 + x \cdot \left(x \cdot \left(1 + -0.3333333333333333 \cdot x\right) - 2\right)\right) \]

   10. Applied rewrites 52.1%

       \[\leadsto 0.5 \cdot \left(2 + x \cdot \color{blue}{\left(x \cdot \left(1 + -0.3333333333333333 \cdot x\right) - 2\right)}\right) \]

   if -3.0000000000000001e106 < x < -1e-190

   1. Initial program 74.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 43.7%

      \[\leadsto \color{blue}{1 + 0.5 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. lower-unsound--.f64 N/A

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

      5. lower-unsound-/.f64 N/A

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

      6. lower-unsound--.f64 N/A

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

      7. lower-unsound-*.f64 N/A

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

      8. lower-unsound-*.f64 51.6%

         \[\leadsto 1 + 0.5 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\frac{\varepsilon \cdot \varepsilon - 1 \cdot 1}{\varepsilon - 1} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

   6. Applied rewrites 51.6%

      \[\leadsto 1 + 0.5 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\frac{\varepsilon \cdot \varepsilon - 1 \cdot 1}{\varepsilon - 1} \cdot \left(\color{blue}{\frac{1}{\varepsilon}} - 1\right)\right)\right)\right) \]

   if -1e-190 < x < 2.3000000000000002e-251

   1. Initial program 74.1%

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

   2. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{2}}{2} \]
   3. Step-by-step derivation

   4. Applied rewrites 43.7%

      \[\leadsto \frac{\color{blue}{2}}{2} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{2}} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{2}} \]

      3. metadata-eval N/A

         \[\leadsto 2 \cdot \color{blue}{\frac{1}{2}} \]

      4. lower-*.f64 43.7%

         \[\leadsto \color{blue}{2 \cdot 0.5} \]

   6. Applied rewrites 43.7%

      \[\leadsto \color{blue}{2 \cdot 0.5} \]

   if 2.3000000000000002e-251 < x < 8.5e5

   1. Initial program 74.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 43.7%

      \[\leadsto \color{blue}{1 + 0.5 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. add-to-fraction N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. *-lft-identity N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. lower-*.f64 52.3%

          \[\leadsto 1 + 0.5 \cdot \left(x \cdot \left(\frac{\left(\varepsilon - 1\right) \cdot \left(1 + \varepsilon\right)}{\varepsilon} - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. add-flip N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 52.3%

          \[\leadsto 1 + 0.5 \cdot \left(x \cdot \left(\frac{\left(\varepsilon - 1\right) \cdot \left(\varepsilon - -1\right)}{\varepsilon} - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

   6. Applied rewrites 52.3%

      \[\leadsto 1 + 0.5 \cdot \left(x \cdot \left(\frac{\left(\varepsilon - 1\right) \cdot \left(\varepsilon - -1\right)}{\varepsilon} - \color{blue}{-1} \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

   if 8.5e5 < x < 1.05e135

   1. Initial program 74.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 43.7%

      \[\leadsto \color{blue}{1 + 0.5 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

   7. Applied rewrites 43.6%

      \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{1}{x}\right)} \]

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 16.1%

          \[\leadsto x \cdot \left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

   10. Applied rewrites 16.1%

       \[\leadsto x \cdot \left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - \color{blue}{-1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)}\right)\right) \]

   if 1.05e135 < x < 5.6000000000000003e222

   1. Initial program 74.1%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1%

          \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]

   5. Taylor expanded in eps around 0

      \[\leadsto 0.5 \cdot \left(e^{\mathsf{neg}\left(x\right)} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x\right)}}\right) \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 70.6%

         \[\leadsto 0.5 \cdot \left(e^{-x} - -1 \cdot e^{-x}\right) \]

   7. Applied rewrites 70.6%

      \[\leadsto 0.5 \cdot \left(e^{-x} - \color{blue}{-1 \cdot e^{-x}}\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto 0.5 \cdot \left(2 + x \cdot \color{blue}{\left(x - 2\right)}\right) \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 57.3%

         \[\leadsto 0.5 \cdot \left(2 + x \cdot \left(x - 2\right)\right) \]

   10. Applied rewrites 57.3%

       \[\leadsto 0.5 \cdot \left(2 + x \cdot \color{blue}{\left(x - 2\right)}\right) \]

   if 5.6000000000000003e222 < x

   1. Initial program 74.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 43.7%

      \[\leadsto \color{blue}{1 + 0.5 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

   7. Applied rewrites 43.6%

      \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{1}{x}\right)} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. rgt-mult-inverse N/A

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

      4. mult-flip-rev N/A

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

      5. frac-times N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 29.8%

         \[\leadsto x \cdot \left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{x}{x \cdot x}\right) \]

   9. Applied rewrites 29.8%

      \[\leadsto x \cdot \left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{x}{x \cdot \color{blue}{x}}\right) \]

   10. Taylor expanded in eps around 0

       \[\leadsto x \cdot \left(0.5 \cdot 0 + \frac{x}{x \cdot x}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 29.9%

       \[\leadsto x \cdot \left(0.5 \cdot 0 + \frac{x}{x \cdot x}\right) \]
3. Recombined 7 regimes into one program.
4. Add Preprocessing

Alternative 9: 72.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|\varepsilon\right| \leq 8.2 \cdot 10^{+38}:\\ \;\;\;\;\left(2 \cdot e^{-x}\right) \cdot 0.5\\ \mathbf{elif}\;\left|\varepsilon\right| \leq 3.6 \cdot 10^{+272}:\\ \;\;\;\;0.5 \cdot \left(2 + x \cdot \left(x - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 0.5 \cdot \left(x \cdot \left(\frac{\left(\left|\varepsilon\right| - 1\right) \cdot \left(\left|\varepsilon\right| - -1\right)}{\left|\varepsilon\right|} - -1 \cdot \left(\left(1 + \left|\varepsilon\right|\right) \cdot \left(\frac{1}{\left|\varepsilon\right|} - 1\right)\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
  :precision binary64
  (if (<= (fabs eps) 8.2e+38)
  (* (* 2.0 (exp (- x))) 0.5)
  (if (<= (fabs eps) 3.6e+272)
    (* 0.5 (+ 2.0 (* x (- x 2.0))))
    (+
     1.0
     (*
      0.5
      (*
       x
       (-
        (/ (* (- (fabs eps) 1.0) (- (fabs eps) -1.0)) (fabs eps))
        (*
         -1.0
         (* (+ 1.0 (fabs eps)) (- (/ 1.0 (fabs eps)) 1.0))))))))))

C

double code(double x, double eps) {
	double tmp;
	if (fabs(eps) <= 8.2e+38) {
		tmp = (2.0 * exp(-x)) * 0.5;
	} else if (fabs(eps) <= 3.6e+272) {
		tmp = 0.5 * (2.0 + (x * (x - 2.0)));
	} else {
		tmp = 1.0 + (0.5 * (x * ((((fabs(eps) - 1.0) * (fabs(eps) - -1.0)) / fabs(eps)) - (-1.0 * ((1.0 + fabs(eps)) * ((1.0 / fabs(eps)) - 1.0))))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (abs(eps) <= 8.2d+38) then
        tmp = (2.0d0 * exp(-x)) * 0.5d0
    else if (abs(eps) <= 3.6d+272) then
        tmp = 0.5d0 * (2.0d0 + (x * (x - 2.0d0)))
    else
        tmp = 1.0d0 + (0.5d0 * (x * ((((abs(eps) - 1.0d0) * (abs(eps) - (-1.0d0))) / abs(eps)) - ((-1.0d0) * ((1.0d0 + abs(eps)) * ((1.0d0 / abs(eps)) - 1.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (Math.abs(eps) <= 8.2e+38) {
		tmp = (2.0 * Math.exp(-x)) * 0.5;
	} else if (Math.abs(eps) <= 3.6e+272) {
		tmp = 0.5 * (2.0 + (x * (x - 2.0)));
	} else {
		tmp = 1.0 + (0.5 * (x * ((((Math.abs(eps) - 1.0) * (Math.abs(eps) - -1.0)) / Math.abs(eps)) - (-1.0 * ((1.0 + Math.abs(eps)) * ((1.0 / Math.abs(eps)) - 1.0))))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if math.fabs(eps) <= 8.2e+38:
		tmp = (2.0 * math.exp(-x)) * 0.5
	elif math.fabs(eps) <= 3.6e+272:
		tmp = 0.5 * (2.0 + (x * (x - 2.0)))
	else:
		tmp = 1.0 + (0.5 * (x * ((((math.fabs(eps) - 1.0) * (math.fabs(eps) - -1.0)) / math.fabs(eps)) - (-1.0 * ((1.0 + math.fabs(eps)) * ((1.0 / math.fabs(eps)) - 1.0))))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (abs(eps) <= 8.2e+38)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) * 0.5);
	elseif (abs(eps) <= 3.6e+272)
		tmp = Float64(0.5 * Float64(2.0 + Float64(x * Float64(x - 2.0))));
	else
		tmp = Float64(1.0 + Float64(0.5 * Float64(x * Float64(Float64(Float64(Float64(abs(eps) - 1.0) * Float64(abs(eps) - -1.0)) / abs(eps)) - Float64(-1.0 * Float64(Float64(1.0 + abs(eps)) * Float64(Float64(1.0 / abs(eps)) - 1.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (abs(eps) <= 8.2e+38)
		tmp = (2.0 * exp(-x)) * 0.5;
	elseif (abs(eps) <= 3.6e+272)
		tmp = 0.5 * (2.0 + (x * (x - 2.0)));
	else
		tmp = 1.0 + (0.5 * (x * ((((abs(eps) - 1.0) * (abs(eps) - -1.0)) / abs(eps)) - (-1.0 * ((1.0 + abs(eps)) * ((1.0 / abs(eps)) - 1.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[Abs[eps], $MachinePrecision], 8.2e+38], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[N[Abs[eps], $MachinePrecision], 3.6e+272], N[(0.5 * N[(2.0 + N[(x * N[(x - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(0.5 * N[(x * N[(N[(N[(N[(N[Abs[eps], $MachinePrecision] - 1.0), $MachinePrecision] * N[(N[Abs[eps], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] / N[Abs[eps], $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(N[(1.0 + N[Abs[eps], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[Abs[eps], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 8.2000000000000007e38

   1. Initial program 74.1%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1%

          \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]

   5. Taylor expanded in eps around 0

      \[\leadsto 0.5 \cdot \left(e^{\mathsf{neg}\left(x\right)} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x\right)}}\right) \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 70.6%

         \[\leadsto 0.5 \cdot \left(e^{-x} - -1 \cdot e^{-x}\right) \]

   7. Applied rewrites 70.6%

      \[\leadsto 0.5 \cdot \left(e^{-x} - \color{blue}{-1 \cdot e^{-x}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 70.6%

         \[\leadsto \left(e^{-x} - -1 \cdot e^{-x}\right) \cdot \color{blue}{0.5} \]

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. fp-cancel-sign-sub-inv N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 70.6%

          \[\leadsto \left(2 \cdot e^{-x}\right) \cdot 0.5 \]

   9. Applied rewrites 70.6%

      \[\leadsto \left(2 \cdot e^{-x}\right) \cdot \color{blue}{0.5} \]

   if 8.2000000000000007e38 < eps < 3.5999999999999998e272

   1. Initial program 74.1%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1%

          \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]

   5. Taylor expanded in eps around 0

      \[\leadsto 0.5 \cdot \left(e^{\mathsf{neg}\left(x\right)} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x\right)}}\right) \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 70.6%

         \[\leadsto 0.5 \cdot \left(e^{-x} - -1 \cdot e^{-x}\right) \]

   7. Applied rewrites 70.6%

      \[\leadsto 0.5 \cdot \left(e^{-x} - \color{blue}{-1 \cdot e^{-x}}\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto 0.5 \cdot \left(2 + x \cdot \color{blue}{\left(x - 2\right)}\right) \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 57.3%

         \[\leadsto 0.5 \cdot \left(2 + x \cdot \left(x - 2\right)\right) \]

   10. Applied rewrites 57.3%

       \[\leadsto 0.5 \cdot \left(2 + x \cdot \color{blue}{\left(x - 2\right)}\right) \]

   if 3.5999999999999998e272 < eps

   1. Initial program 74.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 43.7%

      \[\leadsto \color{blue}{1 + 0.5 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. add-to-fraction N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. *-lft-identity N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. lower-*.f64 52.3%

          \[\leadsto 1 + 0.5 \cdot \left(x \cdot \left(\frac{\left(\varepsilon - 1\right) \cdot \left(1 + \varepsilon\right)}{\varepsilon} - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. add-flip N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 52.3%

          \[\leadsto 1 + 0.5 \cdot \left(x \cdot \left(\frac{\left(\varepsilon - 1\right) \cdot \left(\varepsilon - -1\right)}{\varepsilon} - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

   6. Applied rewrites 52.3%

      \[\leadsto 1 + 0.5 \cdot \left(x \cdot \left(\frac{\left(\varepsilon - 1\right) \cdot \left(\varepsilon - -1\right)}{\varepsilon} - \color{blue}{-1} \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 69.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\left|\varepsilon\right|}\\ t_1 := \left|\varepsilon\right| - 1\\ t_2 := -1 \cdot \left(\left(1 + \left|\varepsilon\right|\right) \cdot \left(t\_0 - 1\right)\right)\\ \mathbf{if}\;x \leq 2.3 \cdot 10^{-251}:\\ \;\;\;\;0.5 \cdot \left(2 + x \cdot \left(x \cdot \left(1 + -0.3333333333333333 \cdot x\right) - 2\right)\right)\\ \mathbf{elif}\;x \leq 850000:\\ \;\;\;\;1 + 0.5 \cdot \left(x \cdot \left(\frac{t\_1 \cdot \left(\left|\varepsilon\right| - -1\right)}{\left|\varepsilon\right|} - t\_2\right)\right)\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+135}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(\left(1 + t\_0\right) \cdot t\_1 - t\_2\right)\right)\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+267}:\\ \;\;\;\;0.5 \cdot \left(2 + x \cdot \left(x - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot 0 + \frac{x}{x \cdot x}\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
  :precision binary64
  (let* ((t_0 (/ 1.0 (fabs eps)))
       (t_1 (- (fabs eps) 1.0))
       (t_2 (* -1.0 (* (+ 1.0 (fabs eps)) (- t_0 1.0)))))
  (if (<= x 2.3e-251)
    (*
     0.5
     (+ 2.0 (* x (- (* x (+ 1.0 (* -0.3333333333333333 x))) 2.0))))
    (if (<= x 850000.0)
      (+
       1.0
       (*
        0.5
        (* x (- (/ (* t_1 (- (fabs eps) -1.0)) (fabs eps)) t_2))))
      (if (<= x 1.05e+135)
        (* x (* 0.5 (- (* (+ 1.0 t_0) t_1) t_2)))
        (if (<= x 4.4e+267)
          (* 0.5 (+ 2.0 (* x (- x 2.0))))
          (* x (+ (* 0.5 0.0) (/ x (* x x))))))))))

C

double code(double x, double eps) {
	double t_0 = 1.0 / fabs(eps);
	double t_1 = fabs(eps) - 1.0;
	double t_2 = -1.0 * ((1.0 + fabs(eps)) * (t_0 - 1.0));
	double tmp;
	if (x <= 2.3e-251) {
		tmp = 0.5 * (2.0 + (x * ((x * (1.0 + (-0.3333333333333333 * x))) - 2.0)));
	} else if (x <= 850000.0) {
		tmp = 1.0 + (0.5 * (x * (((t_1 * (fabs(eps) - -1.0)) / fabs(eps)) - t_2)));
	} else if (x <= 1.05e+135) {
		tmp = x * (0.5 * (((1.0 + t_0) * t_1) - t_2));
	} else if (x <= 4.4e+267) {
		tmp = 0.5 * (2.0 + (x * (x - 2.0)));
	} else {
		tmp = x * ((0.5 * 0.0) + (x / (x * x)));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 / abs(eps)
    t_1 = abs(eps) - 1.0d0
    t_2 = (-1.0d0) * ((1.0d0 + abs(eps)) * (t_0 - 1.0d0))
    if (x <= 2.3d-251) then
        tmp = 0.5d0 * (2.0d0 + (x * ((x * (1.0d0 + ((-0.3333333333333333d0) * x))) - 2.0d0)))
    else if (x <= 850000.0d0) then
        tmp = 1.0d0 + (0.5d0 * (x * (((t_1 * (abs(eps) - (-1.0d0))) / abs(eps)) - t_2)))
    else if (x <= 1.05d+135) then
        tmp = x * (0.5d0 * (((1.0d0 + t_0) * t_1) - t_2))
    else if (x <= 4.4d+267) then
        tmp = 0.5d0 * (2.0d0 + (x * (x - 2.0d0)))
    else
        tmp = x * ((0.5d0 * 0.0d0) + (x / (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = 1.0 / Math.abs(eps);
	double t_1 = Math.abs(eps) - 1.0;
	double t_2 = -1.0 * ((1.0 + Math.abs(eps)) * (t_0 - 1.0));
	double tmp;
	if (x <= 2.3e-251) {
		tmp = 0.5 * (2.0 + (x * ((x * (1.0 + (-0.3333333333333333 * x))) - 2.0)));
	} else if (x <= 850000.0) {
		tmp = 1.0 + (0.5 * (x * (((t_1 * (Math.abs(eps) - -1.0)) / Math.abs(eps)) - t_2)));
	} else if (x <= 1.05e+135) {
		tmp = x * (0.5 * (((1.0 + t_0) * t_1) - t_2));
	} else if (x <= 4.4e+267) {
		tmp = 0.5 * (2.0 + (x * (x - 2.0)));
	} else {
		tmp = x * ((0.5 * 0.0) + (x / (x * x)));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = 1.0 / math.fabs(eps)
	t_1 = math.fabs(eps) - 1.0
	t_2 = -1.0 * ((1.0 + math.fabs(eps)) * (t_0 - 1.0))
	tmp = 0
	if x <= 2.3e-251:
		tmp = 0.5 * (2.0 + (x * ((x * (1.0 + (-0.3333333333333333 * x))) - 2.0)))
	elif x <= 850000.0:
		tmp = 1.0 + (0.5 * (x * (((t_1 * (math.fabs(eps) - -1.0)) / math.fabs(eps)) - t_2)))
	elif x <= 1.05e+135:
		tmp = x * (0.5 * (((1.0 + t_0) * t_1) - t_2))
	elif x <= 4.4e+267:
		tmp = 0.5 * (2.0 + (x * (x - 2.0)))
	else:
		tmp = x * ((0.5 * 0.0) + (x / (x * x)))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(1.0 / abs(eps))
	t_1 = Float64(abs(eps) - 1.0)
	t_2 = Float64(-1.0 * Float64(Float64(1.0 + abs(eps)) * Float64(t_0 - 1.0)))
	tmp = 0.0
	if (x <= 2.3e-251)
		tmp = Float64(0.5 * Float64(2.0 + Float64(x * Float64(Float64(x * Float64(1.0 + Float64(-0.3333333333333333 * x))) - 2.0))));
	elseif (x <= 850000.0)
		tmp = Float64(1.0 + Float64(0.5 * Float64(x * Float64(Float64(Float64(t_1 * Float64(abs(eps) - -1.0)) / abs(eps)) - t_2))));
	elseif (x <= 1.05e+135)
		tmp = Float64(x * Float64(0.5 * Float64(Float64(Float64(1.0 + t_0) * t_1) - t_2)));
	elseif (x <= 4.4e+267)
		tmp = Float64(0.5 * Float64(2.0 + Float64(x * Float64(x - 2.0))));
	else
		tmp = Float64(x * Float64(Float64(0.5 * 0.0) + Float64(x / Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = 1.0 / abs(eps);
	t_1 = abs(eps) - 1.0;
	t_2 = -1.0 * ((1.0 + abs(eps)) * (t_0 - 1.0));
	tmp = 0.0;
	if (x <= 2.3e-251)
		tmp = 0.5 * (2.0 + (x * ((x * (1.0 + (-0.3333333333333333 * x))) - 2.0)));
	elseif (x <= 850000.0)
		tmp = 1.0 + (0.5 * (x * (((t_1 * (abs(eps) - -1.0)) / abs(eps)) - t_2)));
	elseif (x <= 1.05e+135)
		tmp = x * (0.5 * (((1.0 + t_0) * t_1) - t_2));
	elseif (x <= 4.4e+267)
		tmp = 0.5 * (2.0 + (x * (x - 2.0)));
	else
		tmp = x * ((0.5 * 0.0) + (x / (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 / N[Abs[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[eps], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 * N[(N[(1.0 + N[Abs[eps], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.3e-251], N[(0.5 * N[(2.0 + N[(x * N[(N[(x * N[(1.0 + N[(-0.3333333333333333 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 850000.0], N[(1.0 + N[(0.5 * N[(x * N[(N[(N[(t$95$1 * N[(N[Abs[eps], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] / N[Abs[eps], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.05e+135], N[(x * N[(0.5 * N[(N[(N[(1.0 + t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.4e+267], N[(0.5 * N[(2.0 + N[(x * N[(x - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(0.5 * 0.0), $MachinePrecision] + N[(x / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 2.3000000000000002e-251

   1. Initial program 74.1%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1%

          \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]

   5. Taylor expanded in eps around 0

      \[\leadsto 0.5 \cdot \left(e^{\mathsf{neg}\left(x\right)} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x\right)}}\right) \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 70.6%

         \[\leadsto 0.5 \cdot \left(e^{-x} - -1 \cdot e^{-x}\right) \]

   7. Applied rewrites 70.6%

      \[\leadsto 0.5 \cdot \left(e^{-x} - \color{blue}{-1 \cdot e^{-x}}\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto 0.5 \cdot \left(2 + x \cdot \color{blue}{\left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2\right)}\right) \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + x \cdot \left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - \color{blue}{2}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + x \cdot \left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2\right)\right) \]

      3. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + x \cdot \left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + x \cdot \left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + x \cdot \left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2\right)\right) \]

      6. lower-*.f64 52.1%

         \[\leadsto 0.5 \cdot \left(2 + x \cdot \left(x \cdot \left(1 + -0.3333333333333333 \cdot x\right) - 2\right)\right) \]

   10. Applied rewrites 52.1%

       \[\leadsto 0.5 \cdot \left(2 + x \cdot \color{blue}{\left(x \cdot \left(1 + -0.3333333333333333 \cdot x\right) - 2\right)}\right) \]

   if 2.3000000000000002e-251 < x < 8.5e5

   1. Initial program 74.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 43.7%

      \[\leadsto \color{blue}{1 + 0.5 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. add-to-fraction N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. *-lft-identity N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. lower-*.f64 52.3%

          \[\leadsto 1 + 0.5 \cdot \left(x \cdot \left(\frac{\left(\varepsilon - 1\right) \cdot \left(1 + \varepsilon\right)}{\varepsilon} - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. add-flip N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 52.3%

          \[\leadsto 1 + 0.5 \cdot \left(x \cdot \left(\frac{\left(\varepsilon - 1\right) \cdot \left(\varepsilon - -1\right)}{\varepsilon} - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

   6. Applied rewrites 52.3%

      \[\leadsto 1 + 0.5 \cdot \left(x \cdot \left(\frac{\left(\varepsilon - 1\right) \cdot \left(\varepsilon - -1\right)}{\varepsilon} - \color{blue}{-1} \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

   if 8.5e5 < x < 1.05e135

   1. Initial program 74.1%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

   4. Applied rewrites 43.7%

      \[\leadsto \color{blue}{1 + 0.5 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

   7. Applied rewrites 43.6%

      \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{1}{x}\right)} \]

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 16.1%

          \[\leadsto x \cdot \left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

   10. Applied rewrites 16.1%

       \[\leadsto x \cdot \left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - \color{blue}{-1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)}\right)\right) \]

   if 1.05e135 < x < 4.4000000000000002e267

   1. Initial program 74.1%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1%

          \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]

   5. Taylor expanded in eps around 0

      \[\leadsto 0.5 \cdot \left(e^{\mathsf{neg}\left(x\right)} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x\right)}}\right) \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x} - -1 \cdot e^{\mathsf{neg}\left(x\right)}\right) \]

      6. lower-neg.f64 70.6%

         \[\leadsto 0.5 \cdot \left(e^{-x} - -1 \cdot e^{-x}\right) \]

   7. Applied rewrites 70.6%

      \[\leadsto 0.5 \cdot \left(e^{-x} - \color{blue}{-1 \cdot e^{-x}}\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto 0.5 \cdot \left(2 + x \cdot \color{blue}{\left(x - 2\right)}\right) \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + x \cdot \left(x - \color{blue}{2}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + x \cdot \left(x - 2\right)\right) \]

      3. lower--.f64 57.3%

         \[\leadsto 0.5 \cdot \left(2 + x \cdot \left(x - 2\right)\right) \]

   10. Applied rewrites 57.3%

       \[\leadsto 0.5 \cdot \left(2 + x \cdot \color{blue}{\left(x - 2\right)}\right) \]

   if 4.4000000000000002e267 < x

   1. Initial program 74.1%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + \color{blue}{\frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \color{blue}{\left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \color{blue}{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)}\right) \]

      4. lower--.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - \color{blue}{-1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - \color{blue}{-1} \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

      8. lower--.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \color{blue}{\left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)}\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}\right)\right)\right) \]

   4. Applied rewrites 43.7%

      \[\leadsto \color{blue}{1 + 0.5 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{1}{x}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \color{blue}{\frac{1}{x}}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{1}{\color{blue}{x}}\right) \]

   7. Applied rewrites 43.6%

      \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{1}{x}\right)} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{1}{x}\right) \]

      2. mult-flip N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + 1 \cdot \frac{1}{\color{blue}{x}}\right) \]

      3. rgt-mult-inverse N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \left(x \cdot \frac{1}{x}\right) \cdot \frac{1}{x}\right) \]

      4. mult-flip-rev N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{x}{x} \cdot \frac{1}{x}\right) \]

      5. frac-times N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{x \cdot 1}{x \cdot \color{blue}{x}}\right) \]

      6. *-rgt-identity N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{x}{x \cdot x}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{x}{x \cdot \color{blue}{x}}\right) \]

      8. lower-*.f64 29.8%

         \[\leadsto x \cdot \left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{x}{x \cdot x}\right) \]

   9. Applied rewrites 29.8%

      \[\leadsto x \cdot \left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{x}{x \cdot \color{blue}{x}}\right) \]

   10. Taylor expanded in eps around 0

       \[\leadsto x \cdot \left(0.5 \cdot 0 + \frac{x}{x \cdot x}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 29.9%

       \[\leadsto x \cdot \left(0.5 \cdot 0 + \frac{x}{x \cdot x}\right) \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 11: 65.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 0.01:\\ \;\;\;\;0.5 \cdot \left(2 + x \cdot \left(x \cdot \left(1 + -0.3333333333333333 \cdot x\right) - 2\right)\right)\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+135}:\\ \;\;\;\;x \cdot \left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+267}:\\ \;\;\;\;0.5 \cdot \left(2 + x \cdot \left(x - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot 0 + \frac{x}{x \cdot x}\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
  :precision binary64
  (if (<= x 0.01)
  (*
   0.5
   (+ 2.0 (* x (- (* x (+ 1.0 (* -0.3333333333333333 x))) 2.0))))
  (if (<= x 1.05e+135)
    (*
     x
     (*
      0.5
      (-
       (* (+ 1.0 (/ 1.0 eps)) (- eps 1.0))
       (* -1.0 (* (+ 1.0 eps) (- (/ 1.0 eps) 1.0))))))
    (if (<= x 4.4e+267)
      (* 0.5 (+ 2.0 (* x (- x 2.0))))
      (* x (+ (* 0.5 0.0) (/ x (* x x))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 0.01) {
		tmp = 0.5 * (2.0 + (x * ((x * (1.0 + (-0.3333333333333333 * x))) - 2.0)));
	} else if (x <= 1.05e+135) {
		tmp = x * (0.5 * (((1.0 + (1.0 / eps)) * (eps - 1.0)) - (-1.0 * ((1.0 + eps) * ((1.0 / eps) - 1.0)))));
	} else if (x <= 4.4e+267) {
		tmp = 0.5 * (2.0 + (x * (x - 2.0)));
	} else {
		tmp = x * ((0.5 * 0.0) + (x / (x * x)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (x <= 0.01d0) then
        tmp = 0.5d0 * (2.0d0 + (x * ((x * (1.0d0 + ((-0.3333333333333333d0) * x))) - 2.0d0)))
    else if (x <= 1.05d+135) then
        tmp = x * (0.5d0 * (((1.0d0 + (1.0d0 / eps)) * (eps - 1.0d0)) - ((-1.0d0) * ((1.0d0 + eps) * ((1.0d0 / eps) - 1.0d0)))))
    else if (x <= 4.4d+267) then
        tmp = 0.5d0 * (2.0d0 + (x * (x - 2.0d0)))
    else
        tmp = x * ((0.5d0 * 0.0d0) + (x / (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 0.01) {
		tmp = 0.5 * (2.0 + (x * ((x * (1.0 + (-0.3333333333333333 * x))) - 2.0)));
	} else if (x <= 1.05e+135) {
		tmp = x * (0.5 * (((1.0 + (1.0 / eps)) * (eps - 1.0)) - (-1.0 * ((1.0 + eps) * ((1.0 / eps) - 1.0)))));
	} else if (x <= 4.4e+267) {
		tmp = 0.5 * (2.0 + (x * (x - 2.0)));
	} else {
		tmp = x * ((0.5 * 0.0) + (x / (x * x)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 0.01:
		tmp = 0.5 * (2.0 + (x * ((x * (1.0 + (-0.3333333333333333 * x))) - 2.0)))
	elif x <= 1.05e+135:
		tmp = x * (0.5 * (((1.0 + (1.0 / eps)) * (eps - 1.0)) - (-1.0 * ((1.0 + eps) * ((1.0 / eps) - 1.0)))))
	elif x <= 4.4e+267:
		tmp = 0.5 * (2.0 + (x * (x - 2.0)))
	else:
		tmp = x * ((0.5 * 0.0) + (x / (x * x)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 0.01)
		tmp = Float64(0.5 * Float64(2.0 + Float64(x * Float64(Float64(x * Float64(1.0 + Float64(-0.3333333333333333 * x))) - 2.0))));
	elseif (x <= 1.05e+135)
		tmp = Float64(x * Float64(0.5 * Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * Float64(eps - 1.0)) - Float64(-1.0 * Float64(Float64(1.0 + eps) * Float64(Float64(1.0 / eps) - 1.0))))));
	elseif (x <= 4.4e+267)
		tmp = Float64(0.5 * Float64(2.0 + Float64(x * Float64(x - 2.0))));
	else
		tmp = Float64(x * Float64(Float64(0.5 * 0.0) + Float64(x / Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 0.01)
		tmp = 0.5 * (2.0 + (x * ((x * (1.0 + (-0.3333333333333333 * x))) - 2.0)));
	elseif (x <= 1.05e+135)
		tmp = x * (0.5 * (((1.0 + (1.0 / eps)) * (eps - 1.0)) - (-1.0 * ((1.0 + eps) * ((1.0 / eps) - 1.0)))));
	elseif (x <= 4.4e+267)
		tmp = 0.5 * (2.0 + (x * (x - 2.0)));
	else
		tmp = x * ((0.5 * 0.0) + (x / (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 0.01], N[(0.5 * N[(2.0 + N[(x * N[(N[(x * N[(1.0 + N[(-0.3333333333333333 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.05e+135], N[(x * N[(0.5 * N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[(eps - 1.0), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(N[(1.0 + eps), $MachinePrecision] * N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.4e+267], N[(0.5 * N[(2.0 + N[(x * N[(x - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(0.5 * 0.0), $MachinePrecision] + N[(x / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 0.01

   1. Initial program 74.1%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1} \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      6. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot \color{blue}{e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

      11. lower-+.f64 99.1%

          \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]

   5. Taylor expanded in eps around 0

      \[\leadsto 0.5 \cdot \left(e^{\mathsf{neg}\left(x\right)} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x\right)}}\right) \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x\right)} - -1 \cdot \color{blue}{e^{\mathsf{neg}\left(x\right)}}\right) \]

      2. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x\right)} - -1 \cdot e^{\color{blue}{\mathsf{neg}\left(x\right)}}\right) \]

      3. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x} - -1 \cdot e^{\mathsf{neg}\left(\color{blue}{x}\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x} - -1 \cdot e^{\mathsf{neg}\left(x\right)}\right) \]

      5. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x} - -1 \cdot e^{\mathsf{neg}\left(x\right)}\right) \]

      6. lower-neg.f64 70.6%

         \[\leadsto 0.5 \cdot \left(e^{-x} - -1 \cdot e^{-x}\right) \]

   7. Applied rewrites 70.6%

      \[\leadsto 0.5 \cdot \left(e^{-x} - \color{blue}{-1 \cdot e^{-x}}\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto 0.5 \cdot \left(2 + x \cdot \color{blue}{\left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2\right)}\right) \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + x \cdot \left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - \color{blue}{2}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + x \cdot \left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2\right)\right) \]

      3. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + x \cdot \left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + x \cdot \left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + x \cdot \left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2\right)\right) \]

      6. lower-*.f64 52.1%

         \[\leadsto 0.5 \cdot \left(2 + x \cdot \left(x \cdot \left(1 + -0.3333333333333333 \cdot x\right) - 2\right)\right) \]

   10. Applied rewrites 52.1%

       \[\leadsto 0.5 \cdot \left(2 + x \cdot \color{blue}{\left(x \cdot \left(1 + -0.3333333333333333 \cdot x\right) - 2\right)}\right) \]

   if 0.01 < x < 1.05e135

   1. Initial program 74.1%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + \color{blue}{\frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \color{blue}{\left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \color{blue}{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)}\right) \]

      4. lower--.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - \color{blue}{-1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - \color{blue}{-1} \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

      8. lower--.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \color{blue}{\left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)}\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}\right)\right)\right) \]

   4. Applied rewrites 43.7%

      \[\leadsto \color{blue}{1 + 0.5 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{1}{x}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \color{blue}{\frac{1}{x}}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{1}{\color{blue}{x}}\right) \]

   7. Applied rewrites 43.6%

      \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{1}{x}\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - \color{blue}{-1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \color{blue}{\left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)}\right)\right) \]

      2. lower--.f64 N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}\right)\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\color{blue}{\frac{1}{\varepsilon}} - 1\right)\right)\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{\color{blue}{1}}{\varepsilon} - 1\right)\right)\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

      6. lower--.f64 N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\color{blue}{\varepsilon}} - 1\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - \color{blue}{1}\right)\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

      9. lower-+.f64 N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

      11. lower-/.f64 16.1%

          \[\leadsto x \cdot \left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

   10. Applied rewrites 16.1%

       \[\leadsto x \cdot \left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - \color{blue}{-1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)}\right)\right) \]

   if 1.05e135 < x < 4.4000000000000002e267

   1. Initial program 74.1%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1} \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      6. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot \color{blue}{e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

      11. lower-+.f64 99.1%

          \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]

   5. Taylor expanded in eps around 0

      \[\leadsto 0.5 \cdot \left(e^{\mathsf{neg}\left(x\right)} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x\right)}}\right) \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x\right)} - -1 \cdot \color{blue}{e^{\mathsf{neg}\left(x\right)}}\right) \]

      2. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x\right)} - -1 \cdot e^{\color{blue}{\mathsf{neg}\left(x\right)}}\right) \]

      3. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x} - -1 \cdot e^{\mathsf{neg}\left(\color{blue}{x}\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x} - -1 \cdot e^{\mathsf{neg}\left(x\right)}\right) \]

      5. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x} - -1 \cdot e^{\mathsf{neg}\left(x\right)}\right) \]

      6. lower-neg.f64 70.6%

         \[\leadsto 0.5 \cdot \left(e^{-x} - -1 \cdot e^{-x}\right) \]

   7. Applied rewrites 70.6%

      \[\leadsto 0.5 \cdot \left(e^{-x} - \color{blue}{-1 \cdot e^{-x}}\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto 0.5 \cdot \left(2 + x \cdot \color{blue}{\left(x - 2\right)}\right) \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + x \cdot \left(x - \color{blue}{2}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + x \cdot \left(x - 2\right)\right) \]

      3. lower--.f64 57.3%

         \[\leadsto 0.5 \cdot \left(2 + x \cdot \left(x - 2\right)\right) \]

   10. Applied rewrites 57.3%

       \[\leadsto 0.5 \cdot \left(2 + x \cdot \color{blue}{\left(x - 2\right)}\right) \]

   if 4.4000000000000002e267 < x

   1. Initial program 74.1%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + \color{blue}{\frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \color{blue}{\left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \color{blue}{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)}\right) \]

      4. lower--.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - \color{blue}{-1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - \color{blue}{-1} \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

      8. lower--.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \color{blue}{\left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)}\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}\right)\right)\right) \]

   4. Applied rewrites 43.7%

      \[\leadsto \color{blue}{1 + 0.5 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{1}{x}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \color{blue}{\frac{1}{x}}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{1}{\color{blue}{x}}\right) \]

   7. Applied rewrites 43.6%

      \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{1}{x}\right)} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{1}{x}\right) \]

      2. mult-flip N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + 1 \cdot \frac{1}{\color{blue}{x}}\right) \]

      3. rgt-mult-inverse N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \left(x \cdot \frac{1}{x}\right) \cdot \frac{1}{x}\right) \]

      4. mult-flip-rev N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{x}{x} \cdot \frac{1}{x}\right) \]

      5. frac-times N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{x \cdot 1}{x \cdot \color{blue}{x}}\right) \]

      6. *-rgt-identity N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{x}{x \cdot x}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{x}{x \cdot \color{blue}{x}}\right) \]

      8. lower-*.f64 29.8%

         \[\leadsto x \cdot \left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{x}{x \cdot x}\right) \]

   9. Applied rewrites 29.8%

      \[\leadsto x \cdot \left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{x}{x \cdot \color{blue}{x}}\right) \]

   10. Taylor expanded in eps around 0

       \[\leadsto x \cdot \left(0.5 \cdot 0 + \frac{x}{x \cdot x}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 29.9%

       \[\leadsto x \cdot \left(0.5 \cdot 0 + \frac{x}{x \cdot x}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 12: 59.6% accurate, 6.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -11500:\\ \;\;\;\;0.5 \cdot \left(2 + x \cdot \left(x \cdot \left(1 + -0.3333333333333333 \cdot x\right) - 2\right)\right)\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+222}:\\ \;\;\;\;0.5 \cdot \left(2 + x \cdot \left(x - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot 0 + \frac{x}{x \cdot x}\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
  :precision binary64
  (if (<= x -11500.0)
  (*
   0.5
   (+ 2.0 (* x (- (* x (+ 1.0 (* -0.3333333333333333 x))) 2.0))))
  (if (<= x 5.6e+222)
    (* 0.5 (+ 2.0 (* x (- x 2.0))))
    (* x (+ (* 0.5 0.0) (/ x (* x x)))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -11500.0) {
		tmp = 0.5 * (2.0 + (x * ((x * (1.0 + (-0.3333333333333333 * x))) - 2.0)));
	} else if (x <= 5.6e+222) {
		tmp = 0.5 * (2.0 + (x * (x - 2.0)));
	} else {
		tmp = x * ((0.5 * 0.0) + (x / (x * x)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (x <= (-11500.0d0)) then
        tmp = 0.5d0 * (2.0d0 + (x * ((x * (1.0d0 + ((-0.3333333333333333d0) * x))) - 2.0d0)))
    else if (x <= 5.6d+222) then
        tmp = 0.5d0 * (2.0d0 + (x * (x - 2.0d0)))
    else
        tmp = x * ((0.5d0 * 0.0d0) + (x / (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -11500.0) {
		tmp = 0.5 * (2.0 + (x * ((x * (1.0 + (-0.3333333333333333 * x))) - 2.0)));
	} else if (x <= 5.6e+222) {
		tmp = 0.5 * (2.0 + (x * (x - 2.0)));
	} else {
		tmp = x * ((0.5 * 0.0) + (x / (x * x)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -11500.0:
		tmp = 0.5 * (2.0 + (x * ((x * (1.0 + (-0.3333333333333333 * x))) - 2.0)))
	elif x <= 5.6e+222:
		tmp = 0.5 * (2.0 + (x * (x - 2.0)))
	else:
		tmp = x * ((0.5 * 0.0) + (x / (x * x)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -11500.0)
		tmp = Float64(0.5 * Float64(2.0 + Float64(x * Float64(Float64(x * Float64(1.0 + Float64(-0.3333333333333333 * x))) - 2.0))));
	elseif (x <= 5.6e+222)
		tmp = Float64(0.5 * Float64(2.0 + Float64(x * Float64(x - 2.0))));
	else
		tmp = Float64(x * Float64(Float64(0.5 * 0.0) + Float64(x / Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -11500.0)
		tmp = 0.5 * (2.0 + (x * ((x * (1.0 + (-0.3333333333333333 * x))) - 2.0)));
	elseif (x <= 5.6e+222)
		tmp = 0.5 * (2.0 + (x * (x - 2.0)));
	else
		tmp = x * ((0.5 * 0.0) + (x / (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -11500.0], N[(0.5 * N[(2.0 + N[(x * N[(N[(x * N[(1.0 + N[(-0.3333333333333333 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.6e+222], N[(0.5 * N[(2.0 + N[(x * N[(x - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(0.5 * 0.0), $MachinePrecision] + N[(x / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -11500

   1. Initial program 74.1%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1} \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      6. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot \color{blue}{e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

      11. lower-+.f64 99.1%

          \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]

   5. Taylor expanded in eps around 0

      \[\leadsto 0.5 \cdot \left(e^{\mathsf{neg}\left(x\right)} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x\right)}}\right) \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x\right)} - -1 \cdot \color{blue}{e^{\mathsf{neg}\left(x\right)}}\right) \]

      2. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x\right)} - -1 \cdot e^{\color{blue}{\mathsf{neg}\left(x\right)}}\right) \]

      3. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x} - -1 \cdot e^{\mathsf{neg}\left(\color{blue}{x}\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x} - -1 \cdot e^{\mathsf{neg}\left(x\right)}\right) \]

      5. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x} - -1 \cdot e^{\mathsf{neg}\left(x\right)}\right) \]

      6. lower-neg.f64 70.6%

         \[\leadsto 0.5 \cdot \left(e^{-x} - -1 \cdot e^{-x}\right) \]

   7. Applied rewrites 70.6%

      \[\leadsto 0.5 \cdot \left(e^{-x} - \color{blue}{-1 \cdot e^{-x}}\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto 0.5 \cdot \left(2 + x \cdot \color{blue}{\left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2\right)}\right) \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + x \cdot \left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - \color{blue}{2}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + x \cdot \left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2\right)\right) \]

      3. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + x \cdot \left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + x \cdot \left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + x \cdot \left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2\right)\right) \]

      6. lower-*.f64 52.1%

         \[\leadsto 0.5 \cdot \left(2 + x \cdot \left(x \cdot \left(1 + -0.3333333333333333 \cdot x\right) - 2\right)\right) \]

   10. Applied rewrites 52.1%

       \[\leadsto 0.5 \cdot \left(2 + x \cdot \color{blue}{\left(x \cdot \left(1 + -0.3333333333333333 \cdot x\right) - 2\right)}\right) \]

   if -11500 < x < 5.6000000000000003e222

   1. Initial program 74.1%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1} \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      6. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot \color{blue}{e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

      11. lower-+.f64 99.1%

          \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]

   5. Taylor expanded in eps around 0

      \[\leadsto 0.5 \cdot \left(e^{\mathsf{neg}\left(x\right)} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x\right)}}\right) \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x\right)} - -1 \cdot \color{blue}{e^{\mathsf{neg}\left(x\right)}}\right) \]

      2. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x\right)} - -1 \cdot e^{\color{blue}{\mathsf{neg}\left(x\right)}}\right) \]

      3. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x} - -1 \cdot e^{\mathsf{neg}\left(\color{blue}{x}\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x} - -1 \cdot e^{\mathsf{neg}\left(x\right)}\right) \]

      5. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x} - -1 \cdot e^{\mathsf{neg}\left(x\right)}\right) \]

      6. lower-neg.f64 70.6%

         \[\leadsto 0.5 \cdot \left(e^{-x} - -1 \cdot e^{-x}\right) \]

   7. Applied rewrites 70.6%

      \[\leadsto 0.5 \cdot \left(e^{-x} - \color{blue}{-1 \cdot e^{-x}}\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto 0.5 \cdot \left(2 + x \cdot \color{blue}{\left(x - 2\right)}\right) \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + x \cdot \left(x - \color{blue}{2}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + x \cdot \left(x - 2\right)\right) \]

      3. lower--.f64 57.3%

         \[\leadsto 0.5 \cdot \left(2 + x \cdot \left(x - 2\right)\right) \]

   10. Applied rewrites 57.3%

       \[\leadsto 0.5 \cdot \left(2 + x \cdot \color{blue}{\left(x - 2\right)}\right) \]

   if 5.6000000000000003e222 < x

   1. Initial program 74.1%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + \color{blue}{\frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \color{blue}{\left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \color{blue}{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)}\right) \]

      4. lower--.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - \color{blue}{-1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - \color{blue}{-1} \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

      8. lower--.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \color{blue}{\left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)}\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}\right)\right)\right) \]

   4. Applied rewrites 43.7%

      \[\leadsto \color{blue}{1 + 0.5 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{1}{x}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \color{blue}{\frac{1}{x}}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{1}{\color{blue}{x}}\right) \]

   7. Applied rewrites 43.6%

      \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{1}{x}\right)} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{1}{x}\right) \]

      2. mult-flip N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + 1 \cdot \frac{1}{\color{blue}{x}}\right) \]

      3. rgt-mult-inverse N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \left(x \cdot \frac{1}{x}\right) \cdot \frac{1}{x}\right) \]

      4. mult-flip-rev N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{x}{x} \cdot \frac{1}{x}\right) \]

      5. frac-times N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{x \cdot 1}{x \cdot \color{blue}{x}}\right) \]

      6. *-rgt-identity N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{x}{x \cdot x}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{x}{x \cdot \color{blue}{x}}\right) \]

      8. lower-*.f64 29.8%

         \[\leadsto x \cdot \left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{x}{x \cdot x}\right) \]

   9. Applied rewrites 29.8%

      \[\leadsto x \cdot \left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{x}{x \cdot \color{blue}{x}}\right) \]

   10. Taylor expanded in eps around 0

       \[\leadsto x \cdot \left(0.5 \cdot 0 + \frac{x}{x \cdot x}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 29.9%

       \[\leadsto x \cdot \left(0.5 \cdot 0 + \frac{x}{x \cdot x}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 57.3% accurate, 7.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 4.4 \cdot 10^{+267}:\\ \;\;\;\;0.5 \cdot \left(2 + x \cdot \left(x - 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 \cdot 0 + \frac{x}{x \cdot x}\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
  :precision binary64
  (if (<= x 4.4e+267)
  (* 0.5 (+ 2.0 (* x (- x 2.0))))
  (* x (+ (* 0.5 0.0) (/ x (* x x))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 4.4e+267) {
		tmp = 0.5 * (2.0 + (x * (x - 2.0)));
	} else {
		tmp = x * ((0.5 * 0.0) + (x / (x * x)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (x <= 4.4d+267) then
        tmp = 0.5d0 * (2.0d0 + (x * (x - 2.0d0)))
    else
        tmp = x * ((0.5d0 * 0.0d0) + (x / (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 4.4e+267) {
		tmp = 0.5 * (2.0 + (x * (x - 2.0)));
	} else {
		tmp = x * ((0.5 * 0.0) + (x / (x * x)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 4.4e+267:
		tmp = 0.5 * (2.0 + (x * (x - 2.0)))
	else:
		tmp = x * ((0.5 * 0.0) + (x / (x * x)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 4.4e+267)
		tmp = Float64(0.5 * Float64(2.0 + Float64(x * Float64(x - 2.0))));
	else
		tmp = Float64(x * Float64(Float64(0.5 * 0.0) + Float64(x / Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 4.4e+267)
		tmp = 0.5 * (2.0 + (x * (x - 2.0)));
	else
		tmp = x * ((0.5 * 0.0) + (x / (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 4.4e+267], N[(0.5 * N[(2.0 + N[(x * N[(x - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(0.5 * 0.0), $MachinePrecision] + N[(x / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.4000000000000002e267

   1. Initial program 74.1%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1} \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      6. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot \color{blue}{e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

      11. lower-+.f64 99.1%

          \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]

   5. Taylor expanded in eps around 0

      \[\leadsto 0.5 \cdot \left(e^{\mathsf{neg}\left(x\right)} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x\right)}}\right) \]
   6. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x\right)} - -1 \cdot \color{blue}{e^{\mathsf{neg}\left(x\right)}}\right) \]

      2. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x\right)} - -1 \cdot e^{\color{blue}{\mathsf{neg}\left(x\right)}}\right) \]

      3. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x} - -1 \cdot e^{\mathsf{neg}\left(\color{blue}{x}\right)}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x} - -1 \cdot e^{\mathsf{neg}\left(x\right)}\right) \]

      5. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x} - -1 \cdot e^{\mathsf{neg}\left(x\right)}\right) \]

      6. lower-neg.f64 70.6%

         \[\leadsto 0.5 \cdot \left(e^{-x} - -1 \cdot e^{-x}\right) \]

   7. Applied rewrites 70.6%

      \[\leadsto 0.5 \cdot \left(e^{-x} - \color{blue}{-1 \cdot e^{-x}}\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto 0.5 \cdot \left(2 + x \cdot \color{blue}{\left(x - 2\right)}\right) \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + x \cdot \left(x - \color{blue}{2}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(2 + x \cdot \left(x - 2\right)\right) \]

      3. lower--.f64 57.3%

         \[\leadsto 0.5 \cdot \left(2 + x \cdot \left(x - 2\right)\right) \]

   10. Applied rewrites 57.3%

       \[\leadsto 0.5 \cdot \left(2 + x \cdot \color{blue}{\left(x - 2\right)}\right) \]

   if 4.4000000000000002e267 < x

   1. Initial program 74.1%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + \color{blue}{\frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \color{blue}{\left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \color{blue}{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)}\right) \]

      4. lower--.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - \color{blue}{-1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - \color{blue}{-1} \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

      6. lower-+.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

      8. lower--.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \color{blue}{\left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)}\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto 1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}\right)\right)\right) \]

   4. Applied rewrites 43.7%

      \[\leadsto \color{blue}{1 + 0.5 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{1}{x}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \color{blue}{\frac{1}{x}}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{1}{\color{blue}{x}}\right) \]

   7. Applied rewrites 43.6%

      \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{1}{x}\right)} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{1}{x}\right) \]

      2. mult-flip N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + 1 \cdot \frac{1}{\color{blue}{x}}\right) \]

      3. rgt-mult-inverse N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \left(x \cdot \frac{1}{x}\right) \cdot \frac{1}{x}\right) \]

      4. mult-flip-rev N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{x}{x} \cdot \frac{1}{x}\right) \]

      5. frac-times N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{x \cdot 1}{x \cdot \color{blue}{x}}\right) \]

      6. *-rgt-identity N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{x}{x \cdot x}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{x}{x \cdot \color{blue}{x}}\right) \]

      8. lower-*.f64 29.8%

         \[\leadsto x \cdot \left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{x}{x \cdot x}\right) \]

   9. Applied rewrites 29.8%

      \[\leadsto x \cdot \left(0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{x}{x \cdot \color{blue}{x}}\right) \]

   10. Taylor expanded in eps around 0

       \[\leadsto x \cdot \left(0.5 \cdot 0 + \frac{x}{x \cdot x}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 29.9%

       \[\leadsto x \cdot \left(0.5 \cdot 0 + \frac{x}{x \cdot x}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 57.3% accurate, 16.1× speedup

Math

\[0.5 \cdot \left(2 + x \cdot \left(x - 2\right)\right) \]

FPCore

(FPCore (x eps)
  :precision binary64
  (* 0.5 (+ 2.0 (* x (- x 2.0)))))

C

double code(double x, double eps) {
	return 0.5 * (2.0 + (x * (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 = 0.5d0 * (2.0d0 + (x * (x - 2.0d0)))
end function

Java

public static double code(double x, double eps) {
	return 0.5 * (2.0 + (x * (x - 2.0)));
}

Python

def code(x, eps):
	return 0.5 * (2.0 + (x * (x - 2.0)))

Julia

function code(x, eps)
	return Float64(0.5 * Float64(2.0 + Float64(x * Float64(x - 2.0))))
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.5 * (2.0 + (x * (x - 2.0)));
end

Wolfram

code[x_, eps_] := N[(0.5 * N[(2.0 + N[(x * N[(x - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.1%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

2. Taylor expanded in eps around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

   3. lower-exp.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1} \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

   4. lower-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

   6. lower--.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot \color{blue}{e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

   8. lower-exp.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

   9. lower-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   11. lower-+.f64 99.1%

       \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

4. Applied rewrites 99.1%

   \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]

5. Taylor expanded in eps around 0

   \[\leadsto 0.5 \cdot \left(e^{\mathsf{neg}\left(x\right)} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x\right)}}\right) \]
6. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x\right)} - -1 \cdot \color{blue}{e^{\mathsf{neg}\left(x\right)}}\right) \]

   2. lower-exp.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x\right)} - -1 \cdot e^{\color{blue}{\mathsf{neg}\left(x\right)}}\right) \]

   3. lower-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{-x} - -1 \cdot e^{\mathsf{neg}\left(\color{blue}{x}\right)}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{-x} - -1 \cdot e^{\mathsf{neg}\left(x\right)}\right) \]

   5. lower-exp.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{-x} - -1 \cdot e^{\mathsf{neg}\left(x\right)}\right) \]

   6. lower-neg.f64 70.6%

      \[\leadsto 0.5 \cdot \left(e^{-x} - -1 \cdot e^{-x}\right) \]

7. Applied rewrites 70.6%

   \[\leadsto 0.5 \cdot \left(e^{-x} - \color{blue}{-1 \cdot e^{-x}}\right) \]

8. Taylor expanded in x around 0

   \[\leadsto 0.5 \cdot \left(2 + x \cdot \color{blue}{\left(x - 2\right)}\right) \]
9. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(2 + x \cdot \left(x - \color{blue}{2}\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(2 + x \cdot \left(x - 2\right)\right) \]

   3. lower--.f64 57.3%

      \[\leadsto 0.5 \cdot \left(2 + x \cdot \left(x - 2\right)\right) \]

10. Applied rewrites 57.3%

    \[\leadsto 0.5 \cdot \left(2 + x \cdot \color{blue}{\left(x - 2\right)}\right) \]
11. Add Preprocessing

Alternative 15: 43.7% accurate, 45.5× speedup

Math

\[2 \cdot 0.5 \]

FPCore

(FPCore (x eps)
  :precision binary64
  (* 2.0 0.5))

C

double code(double x, double eps) {
	return 2.0 * 0.5;
}

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 = 2.0d0 * 0.5d0
end function

Java

public static double code(double x, double eps) {
	return 2.0 * 0.5;
}

Python

def code(x, eps):
	return 2.0 * 0.5

Julia

function code(x, eps)
	return Float64(2.0 * 0.5)
end

MATLAB

function tmp = code(x, eps)
	tmp = 2.0 * 0.5;
end

Wolfram

code[x_, eps_] := N[(2.0 * 0.5), $MachinePrecision]

Derivation

1. Initial program 74.1%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

2. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{2}}{2} \]
3. Step-by-step derivation

4. Applied rewrites 43.7%

   \[\leadsto \frac{\color{blue}{2}}{2} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2}{2}} \]

   2. mult-flip N/A

      \[\leadsto \color{blue}{2 \cdot \frac{1}{2}} \]

   3. metadata-eval N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{1}{2}} \]

   4. lower-*.f64 43.7%

      \[\leadsto \color{blue}{2 \cdot 0.5} \]

6. Applied rewrites 43.7%

   \[\leadsto \color{blue}{2 \cdot 0.5} \]
7. Add Preprocessing

Alternative 16: 43.2% accurate, 68.3× speedup

Math

\[1 - x \]

FPCore

(FPCore (x eps)
  :precision binary64
  (- 1.0 x))

C

double code(double x, double eps) {
	return 1.0 - x;
}

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 - x
end function

Java

public static double code(double x, double eps) {
	return 1.0 - x;
}

Python

def code(x, eps):
	return 1.0 - x

Julia

function code(x, eps)
	return Float64(1.0 - x)
end

MATLAB

function tmp = code(x, eps)
	tmp = 1.0 - x;
end

Wolfram

code[x_, eps_] := N[(1.0 - x), $MachinePrecision]

Derivation

1. Initial program 74.1%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

2. Taylor expanded in eps around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

   3. lower-exp.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1} \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

   4. lower-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

   6. lower--.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot \color{blue}{e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

   8. lower-exp.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

   9. lower-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   11. lower-+.f64 99.1%

       \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

4. Applied rewrites 99.1%

   \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]

5. Taylor expanded in x around 0

   \[\leadsto 1 + \color{blue}{-1 \cdot x} \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto 1 + -1 \cdot \color{blue}{x} \]

   2. lower-*.f64 43.2%

      \[\leadsto 1 + -1 \cdot x \]

7. Applied rewrites 43.2%

   \[\leadsto 1 + \color{blue}{-1 \cdot x} \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto 1 + -1 \cdot \color{blue}{x} \]

   2. lift-*.f64 N/A

      \[\leadsto 1 + -1 \cdot x \]

   3. mul-1-neg N/A

      \[\leadsto 1 + \left(\mathsf{neg}\left(x\right)\right) \]

   4. sub-flip-reverse N/A

      \[\leadsto 1 - x \]

   5. lower--.f64 43.2%

      \[\leadsto 1 - x \]

9. Applied rewrites 43.2%

   \[\leadsto 1 - x \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025258 
(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))