NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.8% → 98.8%

Time: 7.0s

Alternatives: 10

Speedup: 2.0×

• 38.7% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -8.6563428559952e+79
  2. eps = 1.8267631982512907e-218

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return (exp(-(x * (1.0 - eps))) - (-1.0 * exp(-(x * (1.0 + eps))))) / 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 = (exp(-(x * (1.0d0 - eps))) - ((-1.0d0) * exp(-(x * (1.0d0 + eps))))) / 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(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] / 2.0), $MachinePrecision]

Derivation

1. Initial program 68.9%

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

2. Taylor expanded in eps around inf

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

   1. lower--.f64 N/A

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

   2. lower-exp.f64 N/A

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

   3. lower-neg.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-exp.f64 N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-+.f64 98.5

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

4. Applied rewrites 98.5%

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

Alternative 2: 78.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -29:\\ \;\;\;\;\frac{\left(1 + x \cdot \left(\varepsilon - 1\right)\right) - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.9 \cdot 10^{-14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(e^{-x}, \left(x + 1\right) - -1, \frac{x}{e^{x}}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.3 \cdot 10^{+203}:\\ \;\;\;\;\frac{e^{-\mathsf{fma}\left(\varepsilon, x, x\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -29.0)
   (/ (- (+ 1.0 (* x (- eps 1.0))) (* -1.0 (exp (- (* x (+ 1.0 eps)))))) 2.0)
   (if (<= eps 1.9e-14)
     (/ (fma (exp (- x)) (- (+ x 1.0) -1.0) (/ x (exp x))) 2.0)
     (if (<= eps 2.3e+203)
       (/ (+ (exp (- (fma eps x x))) 1.0) 2.0)
       (/
        (-
         (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x))))
         (- (/ 1.0 eps) 1.0))
        2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -29.0) {
		tmp = ((1.0 + (x * (eps - 1.0))) - (-1.0 * exp(-(x * (1.0 + eps))))) / 2.0;
	} else if (eps <= 1.9e-14) {
		tmp = fma(exp(-x), ((x + 1.0) - -1.0), (x / exp(x))) / 2.0;
	} else if (eps <= 2.3e+203) {
		tmp = (exp(-fma(eps, x, x)) + 1.0) / 2.0;
	} else {
		tmp = (((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - ((1.0 / eps) - 1.0)) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -29.0)
		tmp = Float64(Float64(Float64(1.0 + Float64(x * Float64(eps - 1.0))) - Float64(-1.0 * exp(Float64(-Float64(x * Float64(1.0 + eps)))))) / 2.0);
	elseif (eps <= 1.9e-14)
		tmp = Float64(fma(exp(Float64(-x)), Float64(Float64(x + 1.0) - -1.0), Float64(x / exp(x))) / 2.0);
	elseif (eps <= 2.3e+203)
		tmp = Float64(Float64(exp(Float64(-fma(eps, x, x))) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(-Float64(Float64(1.0 - eps) * x)))) - Float64(Float64(1.0 / eps) - 1.0)) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -29.0], N[(N[(N[(1.0 + N[(x * N[(eps - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[Exp[(-N[(x * N[(1.0 + eps), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps, 1.9e-14], N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(N[(x + 1.0), $MachinePrecision] - -1.0), $MachinePrecision] + N[(x / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps, 2.3e+203], N[(N[(N[Exp[(-N[(eps * x + x), $MachinePrecision])], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], 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[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if eps < -29

   1. Initial program 100.0%

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

   2. Taylor expanded in eps around inf

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

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 72.0%

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

   if -29 < eps < 1.9000000000000001e-14

   1. Initial program 32.6%

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

   2. Taylor expanded in eps around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-exp.f64 N/A

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

      14. lower-neg.f64 98.3

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

   4. Applied rewrites 98.3%

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

   6. Applied rewrites 100.0%

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

   if 1.9000000000000001e-14 < eps < 2.2999999999999999e203

   1. Initial program 100.0%

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

   2. Taylor expanded in eps around inf

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

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 72.7%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 78.1%

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

   12. Applied rewrites 78.1%

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

   if 2.2999999999999999e203 < eps

   1. Initial program 100.0%

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

   2. 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

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

      2. lower-/.f64 71.7

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

   4. Applied rewrites 71.7%

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

Alternative 3: 76.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -11000000000000:\\ \;\;\;\;\frac{1 - -1 \cdot e^{-1 \cdot x}}{2}\\ \mathbf{elif}\;x \leq 0.32:\\ \;\;\;\;\frac{e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -11000000000000.0)
   (/ (- 1.0 (* -1.0 (exp (* -1.0 x)))) 2.0)
   (if (<= x 0.32)
     (/ (- (exp (- (* x (- 1.0 eps)))) (* -1.0 1.0)) 2.0)
     (/ (- (+ 1.0 (/ 1.0 eps)) (- (/ 1.0 eps) 1.0)) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -11000000000000.0) {
		tmp = (1.0 - (-1.0 * exp((-1.0 * x)))) / 2.0;
	} else if (x <= 0.32) {
		tmp = (exp(-(x * (1.0 - eps))) - (-1.0 * 1.0)) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps)) - ((1.0 / eps) - 1.0)) / 2.0;
	}
	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 (x <= (-11000000000000.0d0)) then
        tmp = (1.0d0 - ((-1.0d0) * exp(((-1.0d0) * x)))) / 2.0d0
    else if (x <= 0.32d0) then
        tmp = (exp(-(x * (1.0d0 - eps))) - ((-1.0d0) * 1.0d0)) / 2.0d0
    else
        tmp = ((1.0d0 + (1.0d0 / eps)) - ((1.0d0 / eps) - 1.0d0)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -11000000000000.0) {
		tmp = (1.0 - (-1.0 * Math.exp((-1.0 * x)))) / 2.0;
	} else if (x <= 0.32) {
		tmp = (Math.exp(-(x * (1.0 - eps))) - (-1.0 * 1.0)) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps)) - ((1.0 / eps) - 1.0)) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -11000000000000.0:
		tmp = (1.0 - (-1.0 * math.exp((-1.0 * x)))) / 2.0
	elif x <= 0.32:
		tmp = (math.exp(-(x * (1.0 - eps))) - (-1.0 * 1.0)) / 2.0
	else:
		tmp = ((1.0 + (1.0 / eps)) - ((1.0 / eps) - 1.0)) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -11000000000000.0)
		tmp = Float64(Float64(1.0 - Float64(-1.0 * exp(Float64(-1.0 * x)))) / 2.0);
	elseif (x <= 0.32)
		tmp = Float64(Float64(exp(Float64(-Float64(x * Float64(1.0 - eps)))) - Float64(-1.0 * 1.0)) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) - Float64(Float64(1.0 / eps) - 1.0)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -11000000000000.0)
		tmp = (1.0 - (-1.0 * exp((-1.0 * x)))) / 2.0;
	elseif (x <= 0.32)
		tmp = (exp(-(x * (1.0 - eps))) - (-1.0 * 1.0)) / 2.0;
	else
		tmp = ((1.0 + (1.0 / eps)) - ((1.0 / eps) - 1.0)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -11000000000000.0], N[(N[(1.0 - N[(-1.0 * N[Exp[N[(-1.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 0.32], N[(N[(N[Exp[(-N[(x * N[(1.0 - eps), $MachinePrecision]), $MachinePrecision])], $MachinePrecision] - N[(-1.0 * 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1e13

   1. Initial program 100.0%

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

   2. Taylor expanded in eps around inf

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

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 59.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 61.8%

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

   11. Taylor expanded in eps around 0

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

   13. Applied rewrites 100.0%

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

   if -1.1e13 < x < 0.320000000000000007

   1. Initial program 50.0%

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

   2. Taylor expanded in eps around inf

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

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 97.6

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

   4. Applied rewrites 97.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 89.6%

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

   if 0.320000000000000007 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 22.3

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

   4. Applied rewrites 22.3%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 57.3

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

   7. Applied rewrites 57.3%

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

Alternative 4: 76.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-7}:\\ \;\;\;\;\frac{1 - -1 \cdot e^{-1 \cdot x}}{2}\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{+16}:\\ \;\;\;\;\frac{e^{-\mathsf{fma}\left(\varepsilon, x, x\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.95e-7)
   (/ (- 1.0 (* -1.0 (exp (* -1.0 x)))) 2.0)
   (if (<= x 1.22e+16)
     (/ (+ (exp (- (fma eps x x))) 1.0) 2.0)
     (/ (- (+ 1.0 (/ 1.0 eps)) (- (/ 1.0 eps) 1.0)) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.95e-7) {
		tmp = (1.0 - (-1.0 * exp((-1.0 * x)))) / 2.0;
	} else if (x <= 1.22e+16) {
		tmp = (exp(-fma(eps, x, x)) + 1.0) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps)) - ((1.0 / eps) - 1.0)) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.95e-7)
		tmp = Float64(Float64(1.0 - Float64(-1.0 * exp(Float64(-1.0 * x)))) / 2.0);
	elseif (x <= 1.22e+16)
		tmp = Float64(Float64(exp(Float64(-fma(eps, x, x))) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) - Float64(Float64(1.0 / eps) - 1.0)) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.95e-7], N[(N[(1.0 - N[(-1.0 * N[Exp[N[(-1.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.22e+16], N[(N[(N[Exp[(-N[(eps * x + x), $MachinePrecision])], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.95000000000000012e-7

   1. Initial program 93.1%

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

   2. Taylor expanded in eps around inf

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

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 93.8

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

   4. Applied rewrites 93.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 53.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 55.6%

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

   11. Taylor expanded in eps around 0

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

   13. Applied rewrites 91.6%

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

   if -1.95000000000000012e-7 < x < 1.22e16

   1. Initial program 51.3%

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

   2. Taylor expanded in eps around inf

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

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 86.4%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 86.2%

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

   12. Applied rewrites 86.2%

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

   if 1.22e16 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 19.9

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

   4. Applied rewrites 19.9%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 58.5

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

   7. Applied rewrites 58.5%

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

Alternative 5: 69.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.22 \cdot 10^{+16}:\\ \;\;\;\;\frac{e^{-\mathsf{fma}\left(\varepsilon, x, x\right)} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 1.22e+16)
   (/ (+ (exp (- (fma eps x x))) 1.0) 2.0)
   (/ (- (+ 1.0 (/ 1.0 eps)) (- (/ 1.0 eps) 1.0)) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 1.22e+16) {
		tmp = (exp(-fma(eps, x, x)) + 1.0) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps)) - ((1.0 / eps) - 1.0)) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 1.22e+16)
		tmp = Float64(Float64(exp(Float64(-fma(eps, x, x))) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) - Float64(Float64(1.0 / eps) - 1.0)) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 1.22e+16], N[(N[(N[Exp[(-N[(eps * x + x), $MachinePrecision])], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.22e16

   1. Initial program 60.3%

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

   2. Taylor expanded in eps around inf

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

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 98.1

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

   4. Applied rewrites 98.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 79.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 79.6%

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

   12. Applied rewrites 79.6%

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

   if 1.22e16 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 19.9

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

   4. Applied rewrites 19.9%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 58.5

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

   7. Applied rewrites 58.5%

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

Alternative 6: 58.7% accurate, 5.6× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 2.0)
   (/ (- 1.0 (* -1.0 (+ 1.0 (* -1.0 (* x (+ 1.0 eps)))))) 2.0)
   (/ (- (+ 1.0 (/ 1.0 eps)) (- (/ 1.0 eps) 1.0)) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 2.0) {
		tmp = (1.0 - (-1.0 * (1.0 + (-1.0 * (x * (1.0 + eps)))))) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps)) - ((1.0 / eps) - 1.0)) / 2.0;
	}
	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 (x <= 2.0d0) then
        tmp = (1.0d0 - ((-1.0d0) * (1.0d0 + ((-1.0d0) * (x * (1.0d0 + eps)))))) / 2.0d0
    else
        tmp = ((1.0d0 + (1.0d0 / eps)) - ((1.0d0 / eps) - 1.0d0)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 2.0) {
		tmp = (1.0 - (-1.0 * (1.0 + (-1.0 * (x * (1.0 + eps)))))) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps)) - ((1.0 / eps) - 1.0)) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 2.0:
		tmp = (1.0 - (-1.0 * (1.0 + (-1.0 * (x * (1.0 + eps)))))) / 2.0
	else:
		tmp = ((1.0 + (1.0 / eps)) - ((1.0 / eps) - 1.0)) / 2.0
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 59.8%

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

   2. Taylor expanded in eps around inf

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

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 98.1

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

   4. Applied rewrites 98.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 79.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 79.9%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 66.0%

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

   if 2 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 21.0

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

   4. Applied rewrites 21.0%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 58.2

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

   7. Applied rewrites 58.2%

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

Alternative 7: 58.7% accurate, 5.9× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 2.0)
   (/ (- 1.0 (* -1.0 (+ 1.0 (* -1.0 (* x (+ 1.0 eps)))))) 2.0)
   (/ (- (/ 1.0 eps) (- (/ 1.0 eps) 1.0)) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 2.0) {
		tmp = (1.0 - (-1.0 * (1.0 + (-1.0 * (x * (1.0 + eps)))))) / 2.0;
	} else {
		tmp = ((1.0 / eps) - ((1.0 / eps) - 1.0)) / 2.0;
	}
	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 (x <= 2.0d0) then
        tmp = (1.0d0 - ((-1.0d0) * (1.0d0 + ((-1.0d0) * (x * (1.0d0 + eps)))))) / 2.0d0
    else
        tmp = ((1.0d0 / eps) - ((1.0d0 / eps) - 1.0d0)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 2.0) {
		tmp = (1.0 - (-1.0 * (1.0 + (-1.0 * (x * (1.0 + eps)))))) / 2.0;
	} else {
		tmp = ((1.0 / eps) - ((1.0 / eps) - 1.0)) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 2.0:
		tmp = (1.0 - (-1.0 * (1.0 + (-1.0 * (x * (1.0 + eps)))))) / 2.0
	else:
		tmp = ((1.0 / eps) - ((1.0 / eps) - 1.0)) / 2.0
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 59.8%

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

   2. Taylor expanded in eps around inf

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

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 98.1

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

   4. Applied rewrites 98.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 79.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 79.9%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 66.0%

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

   if 2 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 21.0

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

   4. Applied rewrites 21.0%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 58.2

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

   7. Applied rewrites 58.2%

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

   8. Taylor expanded in eps around 0

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

   10. Applied rewrites 58.2%

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

Alternative 8: 49.5% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.6 \cdot 10^{+122}:\\ \;\;\;\;\frac{\left(1 + \varepsilon \cdot x\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - -1 \cdot \left(1 + -1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -1.6e+122)
   (/ (- (+ 1.0 (* eps x)) (- (/ 1.0 eps) 1.0)) 2.0)
   (/ (- 1.0 (* -1.0 (+ 1.0 (* -1.0 (* x (+ 1.0 eps)))))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -1.6e+122) {
		tmp = ((1.0 + (eps * x)) - ((1.0 / eps) - 1.0)) / 2.0;
	} else {
		tmp = (1.0 - (-1.0 * (1.0 + (-1.0 * (x * (1.0 + eps)))))) / 2.0;
	}
	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 (eps <= (-1.6d+122)) then
        tmp = ((1.0d0 + (eps * x)) - ((1.0d0 / eps) - 1.0d0)) / 2.0d0
    else
        tmp = (1.0d0 - ((-1.0d0) * (1.0d0 + ((-1.0d0) * (x * (1.0d0 + eps)))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -1.6e+122) {
		tmp = ((1.0 + (eps * x)) - ((1.0 / eps) - 1.0)) / 2.0;
	} else {
		tmp = (1.0 - (-1.0 * (1.0 + (-1.0 * (x * (1.0 + eps)))))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -1.6e+122:
		tmp = ((1.0 + (eps * x)) - ((1.0 / eps) - 1.0)) / 2.0
	else:
		tmp = (1.0 - (-1.0 * (1.0 + (-1.0 * (x * (1.0 + eps)))))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -1.6e+122)
		tmp = Float64(Float64(Float64(1.0 + Float64(eps * x)) - Float64(Float64(1.0 / eps) - 1.0)) / 2.0);
	else
		tmp = Float64(Float64(1.0 - Float64(-1.0 * Float64(1.0 + Float64(-1.0 * Float64(x * Float64(1.0 + eps)))))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -1.6e+122)
		tmp = ((1.0 + (eps * x)) - ((1.0 / eps) - 1.0)) / 2.0;
	else
		tmp = (1.0 - (-1.0 * (1.0 + (-1.0 * (x * (1.0 + eps)))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -1.6e+122], N[(N[(N[(1.0 + N[(eps * x), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 - N[(-1.0 * N[(1.0 + N[(-1.0 * N[(x * N[(1.0 + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.60000000000000006e122

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 51.0

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 24.9

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

   7. Applied rewrites 24.9%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 43.8

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

   10. Applied rewrites 43.8%

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

   11. Taylor expanded in eps around inf

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

   13. Applied rewrites 43.8%

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

   if -1.60000000000000006e122 < eps

   1. Initial program 63.2%

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

   2. Taylor expanded in eps around inf

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

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 98.2

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 69.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 69.6%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 57.2%

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

Alternative 9: 49.3% accurate, 7.6× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return (1.0 - (-1.0 * (1.0 + (-1.0 * (x * (1.0 + eps)))))) / 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) * (1.0d0 + ((-1.0d0) * (x * (1.0d0 + eps)))))) / 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.9%

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

2. Taylor expanded in eps around inf

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

   1. lower--.f64 N/A

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

   2. lower-exp.f64 N/A

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

   3. lower-neg.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-exp.f64 N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-+.f64 98.5

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 68.7%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 66.7%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 52.7%

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

Alternative 10: 43.6% accurate, 22.8× speedup

Math

\[\begin{array}{l} \\ \frac{2}{2} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (/ 2.0 2.0))

C

double code(double x, double eps) {
	return 2.0 / 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 = 2.0d0 / 2.0d0
end function

Java

public static double code(double x, double eps) {
	return 2.0 / 2.0;
}

Python

def code(x, eps):
	return 2.0 / 2.0

Julia

function code(x, eps)
	return Float64(2.0 / 2.0)
end

MATLAB

function tmp = code(x, eps)
	tmp = 2.0 / 2.0;
end

Wolfram

code[x_, eps_] := N[(2.0 / 2.0), $MachinePrecision]

Derivation

1. Initial program 68.9%

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

2. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{2}}{2} \]
3. Step-by-step derivation

4. Applied rewrites 48.8%

   \[\leadsto \frac{\color{blue}{2}}{2} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024363 -o localize:costs -o setup:simplify -o generate:simplify
(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))