NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.0% → 99.9%

Time: 9.1s

Alternatives: 15

Speedup: 2.3×

• 62.0% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -9.506279990331684e+175
  2. eps = 1.4472403251217463e-122

Specification

Math

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

FPCore

(FPCore (x eps)
  :precision binary64
  (/
 (-
  (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x))))
  (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x)))))
 2))

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 + N[(1 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[(1 - eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1 / eps), $MachinePrecision] - 1), $MachinePrecision] * N[Exp[(-N[(N[(1 + eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2), $MachinePrecision]

Initial Program: 73.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x eps)
  :precision binary64
  (/
 (-
  (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x))))
  (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x)))))
 2))

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 + N[(1 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[(1 - eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1 / eps), $MachinePrecision] - 1), $MachinePrecision] * N[Exp[(-N[(N[(1 + eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|\varepsilon\right| \leq \frac{3512807709348987}{18014398509481984}:\\ \;\;\;\;\frac{\left(x - -1\right) + \left(x - -1\right)}{e^{x}} \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\left(\left(-\left|\varepsilon\right|\right) - 1\right) \cdot x} + e^{\left|\varepsilon\right| \cdot x}\right) \cdot \frac{1}{2}\\ \end{array} \]

FPCore

(FPCore (x eps)
  :precision binary64
  (if (<= (fabs eps) 3512807709348987/18014398509481984)
  (* (/ (+ (- x -1) (- x -1)) (exp x)) 1/2)
  (* (+ (exp (* (- (- (fabs eps)) 1) x)) (exp (* (fabs eps) x))) 1/2)))

C

double code(double x, double eps) {
	double tmp;
	if (fabs(eps) <= 0.195) {
		tmp = (((x - -1.0) + (x - -1.0)) / exp(x)) * 0.5;
	} else {
		tmp = (exp(((-fabs(eps) - 1.0) * x)) + exp((fabs(eps) * x))) * 0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (abs(eps) <= 0.195d0) then
        tmp = (((x - (-1.0d0)) + (x - (-1.0d0))) / exp(x)) * 0.5d0
    else
        tmp = (exp(((-abs(eps) - 1.0d0) * x)) + exp((abs(eps) * x))) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (Math.abs(eps) <= 0.195) {
		tmp = (((x - -1.0) + (x - -1.0)) / Math.exp(x)) * 0.5;
	} else {
		tmp = (Math.exp(((-Math.abs(eps) - 1.0) * x)) + Math.exp((Math.abs(eps) * x))) * 0.5;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if math.fabs(eps) <= 0.195:
		tmp = (((x - -1.0) + (x - -1.0)) / math.exp(x)) * 0.5
	else:
		tmp = (math.exp(((-math.fabs(eps) - 1.0) * x)) + math.exp((math.fabs(eps) * x))) * 0.5
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (abs(eps) <= 0.195)
		tmp = Float64(Float64(Float64(Float64(x - -1.0) + Float64(x - -1.0)) / exp(x)) * 0.5);
	else
		tmp = Float64(Float64(exp(Float64(Float64(Float64(-abs(eps)) - 1.0) * x)) + exp(Float64(abs(eps) * x))) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (abs(eps) <= 0.195)
		tmp = (((x - -1.0) + (x - -1.0)) / exp(x)) * 0.5;
	else
		tmp = (exp(((-abs(eps) - 1.0) * x)) + exp((abs(eps) * x))) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[Abs[eps], $MachinePrecision], 3512807709348987/18014398509481984], N[(N[(N[(N[(x - -1), $MachinePrecision] + N[(x - -1), $MachinePrecision]), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] * 1/2), $MachinePrecision], N[(N[(N[Exp[N[(N[((-N[Abs[eps], $MachinePrecision]) - 1), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(N[Abs[eps], $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1/2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 0.19500000000000001

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 57.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 57.4%

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

   6. Applied rewrites 57.5%

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

   if 0.19500000000000001 < eps

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1%

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.1%

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in eps around inf

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

      1. lower-*.f64 88.6%

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

   9. Applied rewrites 88.6%

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

Alternative 2: 99.1% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x eps)
  :precision binary64
  (* (+ (exp (* (- (- eps) 1) x)) (exp (* (- eps 1) x))) 1/2))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (exp(((-eps - 1.0d0) * x)) + exp(((eps - 1.0d0) * x))) * 0.5d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(N[(N[Exp[N[(N[((-eps) - 1), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(N[(eps - 1), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1/2), $MachinePrecision]

Derivation

1. Initial program 73.0%

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

2. Taylor expanded in eps around inf

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-exp.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-+.f64 99.1%

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

4. Applied rewrites 99.1%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.1%

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

6. Applied rewrites 99.1%

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

Alternative 3: 84.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq \frac{-6032057205060441}{3016028602530220424421062271578867838525126125874252890898307532480811172246863646685486789069132871854112712507200418582406770249989531589552959798883475511096677545853948017425342019529539590198394174553047792145043723038206885734470238620775335376572758801465612196212014773714996912064944617579072807182486470656}:\\ \;\;\;\;\frac{1}{2} \cdot \left(1 - -1 \cdot e^{-x \cdot \left(1 + \left|\varepsilon\right|\right)}\right)\\ \mathbf{elif}\;x \leq 19999999999999999862796380718940425895318388736:\\ \;\;\;\;\left(1 + e^{\left|\varepsilon\right| \cdot x}\right) \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-x} \cdot 2\right) \cdot \frac{1}{2}\\ \end{array} \]

FPCore

(FPCore (x eps)
  :precision binary64
  (if (<=
     x
     -6032057205060441/3016028602530220424421062271578867838525126125874252890898307532480811172246863646685486789069132871854112712507200418582406770249989531589552959798883475511096677545853948017425342019529539590198394174553047792145043723038206885734470238620775335376572758801465612196212014773714996912064944617579072807182486470656)
  (* 1/2 (- 1 (* -1 (exp (- (* x (+ 1 (fabs eps))))))))
  (if (<= x 19999999999999999862796380718940425895318388736)
    (* (+ 1 (exp (* (fabs eps) x))) 1/2)
    (* (* (exp (- x)) 2) 1/2))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2e-300) {
		tmp = 0.5 * (1.0 - (-1.0 * exp(-(x * (1.0 + fabs(eps))))));
	} else if (x <= 2e+46) {
		tmp = (1.0 + exp((fabs(eps) * x))) * 0.5;
	} else {
		tmp = (exp(-x) * 2.0) * 0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2d-300)) then
        tmp = 0.5d0 * (1.0d0 - ((-1.0d0) * exp(-(x * (1.0d0 + abs(eps))))))
    else if (x <= 2d+46) then
        tmp = (1.0d0 + exp((abs(eps) * x))) * 0.5d0
    else
        tmp = (exp(-x) * 2.0d0) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2e-300) {
		tmp = 0.5 * (1.0 - (-1.0 * Math.exp(-(x * (1.0 + Math.abs(eps))))));
	} else if (x <= 2e+46) {
		tmp = (1.0 + Math.exp((Math.abs(eps) * x))) * 0.5;
	} else {
		tmp = (Math.exp(-x) * 2.0) * 0.5;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2e-300:
		tmp = 0.5 * (1.0 - (-1.0 * math.exp(-(x * (1.0 + math.fabs(eps))))))
	elif x <= 2e+46:
		tmp = (1.0 + math.exp((math.fabs(eps) * x))) * 0.5
	else:
		tmp = (math.exp(-x) * 2.0) * 0.5
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2e-300)
		tmp = Float64(0.5 * Float64(1.0 - Float64(-1.0 * exp(Float64(-Float64(x * Float64(1.0 + abs(eps))))))));
	elseif (x <= 2e+46)
		tmp = Float64(Float64(1.0 + exp(Float64(abs(eps) * x))) * 0.5);
	else
		tmp = Float64(Float64(exp(Float64(-x)) * 2.0) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2e-300)
		tmp = 0.5 * (1.0 - (-1.0 * exp(-(x * (1.0 + abs(eps))))));
	elseif (x <= 2e+46)
		tmp = (1.0 + exp((abs(eps) * x))) * 0.5;
	else
		tmp = (exp(-x) * 2.0) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -6032057205060441/3016028602530220424421062271578867838525126125874252890898307532480811172246863646685486789069132871854112712507200418582406770249989531589552959798883475511096677545853948017425342019529539590198394174553047792145043723038206885734470238620775335376572758801465612196212014773714996912064944617579072807182486470656], N[(1/2 * N[(1 - N[(-1 * N[Exp[(-N[(x * N[(1 + N[Abs[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 19999999999999999862796380718940425895318388736], N[(N[(1 + N[Exp[N[(N[Abs[eps], $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1/2), $MachinePrecision], N[(N[(N[Exp[(-x)], $MachinePrecision] * 2), $MachinePrecision] * 1/2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.0000000000000001e-300

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 63.8%

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

   if -2.0000000000000001e-300 < x < 2e46

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1%

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.1%

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in eps around inf

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

      1. lower-*.f64 88.6%

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

   9. Applied rewrites 88.6%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 64.5%

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

   if 2e46 < x

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1%

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.1%

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-*.f64 70.1%

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

   9. Applied rewrites 70.1%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-exp.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. mul-1-neg N/A

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

       6. lower-*.f64 N/A

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

       7. mul-1-neg N/A

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

       8. lift-*.f64 N/A

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

       9. lift-exp.f64 70.1%

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

       10. lift-*.f64 N/A

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

       11. mul-1-neg N/A

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

       12. lower-neg.f64 70.1%

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

   11. Applied rewrites 70.1%

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

Alternative 4: 84.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq \frac{-6032057205060441}{3016028602530220424421062271578867838525126125874252890898307532480811172246863646685486789069132871854112712507200418582406770249989531589552959798883475511096677545853948017425342019529539590198394174553047792145043723038206885734470238620775335376572758801465612196212014773714996912064944617579072807182486470656}:\\ \;\;\;\;\frac{1}{2} \cdot \left(1 - -1 \cdot e^{-x \cdot \left(1 + \left|\varepsilon\right|\right)}\right)\\ \mathbf{elif}\;x \leq 19999999999999999862796380718940425895318388736:\\ \;\;\;\;\left(1 + e^{\left|\varepsilon\right| \cdot x}\right) \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - -1\right) + \left(x - -1\right)}{e^{x}} \cdot \frac{1}{2}\\ \end{array} \]

FPCore

(FPCore (x eps)
  :precision binary64
  (if (<=
     x
     -6032057205060441/3016028602530220424421062271578867838525126125874252890898307532480811172246863646685486789069132871854112712507200418582406770249989531589552959798883475511096677545853948017425342019529539590198394174553047792145043723038206885734470238620775335376572758801465612196212014773714996912064944617579072807182486470656)
  (* 1/2 (- 1 (* -1 (exp (- (* x (+ 1 (fabs eps))))))))
  (if (<= x 19999999999999999862796380718940425895318388736)
    (* (+ 1 (exp (* (fabs eps) x))) 1/2)
    (* (/ (+ (- x -1) (- x -1)) (exp x)) 1/2))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2e-300) {
		tmp = 0.5 * (1.0 - (-1.0 * exp(-(x * (1.0 + fabs(eps))))));
	} else if (x <= 2e+46) {
		tmp = (1.0 + exp((fabs(eps) * x))) * 0.5;
	} else {
		tmp = (((x - -1.0) + (x - -1.0)) / exp(x)) * 0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2d-300)) then
        tmp = 0.5d0 * (1.0d0 - ((-1.0d0) * exp(-(x * (1.0d0 + abs(eps))))))
    else if (x <= 2d+46) then
        tmp = (1.0d0 + exp((abs(eps) * x))) * 0.5d0
    else
        tmp = (((x - (-1.0d0)) + (x - (-1.0d0))) / exp(x)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2e-300) {
		tmp = 0.5 * (1.0 - (-1.0 * Math.exp(-(x * (1.0 + Math.abs(eps))))));
	} else if (x <= 2e+46) {
		tmp = (1.0 + Math.exp((Math.abs(eps) * x))) * 0.5;
	} else {
		tmp = (((x - -1.0) + (x - -1.0)) / Math.exp(x)) * 0.5;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2e-300:
		tmp = 0.5 * (1.0 - (-1.0 * math.exp(-(x * (1.0 + math.fabs(eps))))))
	elif x <= 2e+46:
		tmp = (1.0 + math.exp((math.fabs(eps) * x))) * 0.5
	else:
		tmp = (((x - -1.0) + (x - -1.0)) / math.exp(x)) * 0.5
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2e-300)
		tmp = Float64(0.5 * Float64(1.0 - Float64(-1.0 * exp(Float64(-Float64(x * Float64(1.0 + abs(eps))))))));
	elseif (x <= 2e+46)
		tmp = Float64(Float64(1.0 + exp(Float64(abs(eps) * x))) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(x - -1.0) + Float64(x - -1.0)) / exp(x)) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2e-300)
		tmp = 0.5 * (1.0 - (-1.0 * exp(-(x * (1.0 + abs(eps))))));
	elseif (x <= 2e+46)
		tmp = (1.0 + exp((abs(eps) * x))) * 0.5;
	else
		tmp = (((x - -1.0) + (x - -1.0)) / exp(x)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -6032057205060441/3016028602530220424421062271578867838525126125874252890898307532480811172246863646685486789069132871854112712507200418582406770249989531589552959798883475511096677545853948017425342019529539590198394174553047792145043723038206885734470238620775335376572758801465612196212014773714996912064944617579072807182486470656], N[(1/2 * N[(1 - N[(-1 * N[Exp[(-N[(x * N[(1 + N[Abs[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 19999999999999999862796380718940425895318388736], N[(N[(1 + N[Exp[N[(N[Abs[eps], $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1/2), $MachinePrecision], N[(N[(N[(N[(x - -1), $MachinePrecision] + N[(x - -1), $MachinePrecision]), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] * 1/2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.0000000000000001e-300

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1%

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 63.8%

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

   if -2.0000000000000001e-300 < x < 2e46

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1%

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.1%

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in eps around inf

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

      1. lower-*.f64 88.6%

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

   9. Applied rewrites 88.6%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 64.5%

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

   if 2e46 < x

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 57.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 57.4%

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

   6. Applied rewrites 57.5%

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

Alternative 5: 78.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|\varepsilon\right| \leq 94999999999999995027949442561445199872:\\ \;\;\;\;\left(e^{-x} \cdot 2\right) \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + e^{\left|\varepsilon\right| \cdot x}\right) \cdot \frac{1}{2}\\ \end{array} \]

FPCore

(FPCore (x eps)
  :precision binary64
  (if (<= (fabs eps) 94999999999999995027949442561445199872)
  (* (* (exp (- x)) 2) 1/2)
  (* (+ 1 (exp (* (fabs eps) x))) 1/2)))

C

double code(double x, double eps) {
	double tmp;
	if (fabs(eps) <= 9.5e+37) {
		tmp = (exp(-x) * 2.0) * 0.5;
	} else {
		tmp = (1.0 + exp((fabs(eps) * x))) * 0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (abs(eps) <= 9.5d+37) then
        tmp = (exp(-x) * 2.0d0) * 0.5d0
    else
        tmp = (1.0d0 + exp((abs(eps) * x))) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (Math.abs(eps) <= 9.5e+37) {
		tmp = (Math.exp(-x) * 2.0) * 0.5;
	} else {
		tmp = (1.0 + Math.exp((Math.abs(eps) * x))) * 0.5;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if math.fabs(eps) <= 9.5e+37:
		tmp = (math.exp(-x) * 2.0) * 0.5
	else:
		tmp = (1.0 + math.exp((math.fabs(eps) * x))) * 0.5
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (abs(eps) <= 9.5e+37)
		tmp = Float64(Float64(exp(Float64(-x)) * 2.0) * 0.5);
	else
		tmp = Float64(Float64(1.0 + exp(Float64(abs(eps) * x))) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (abs(eps) <= 9.5e+37)
		tmp = (exp(-x) * 2.0) * 0.5;
	else
		tmp = (1.0 + exp((abs(eps) * x))) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[Abs[eps], $MachinePrecision], 94999999999999995027949442561445199872], N[(N[(N[Exp[(-x)], $MachinePrecision] * 2), $MachinePrecision] * 1/2), $MachinePrecision], N[(N[(1 + N[Exp[N[(N[Abs[eps], $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1/2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 9.4999999999999995e37

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1%

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.1%

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-*.f64 70.1%

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

   9. Applied rewrites 70.1%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-exp.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. mul-1-neg N/A

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

       6. lower-*.f64 N/A

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

       7. mul-1-neg N/A

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

       8. lift-*.f64 N/A

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

       9. lift-exp.f64 70.1%

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

       10. lift-*.f64 N/A

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

       11. mul-1-neg N/A

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

       12. lower-neg.f64 70.1%

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

   11. Applied rewrites 70.1%

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

   if 9.4999999999999995e37 < eps

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1%

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.1%

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in eps around inf

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

      1. lower-*.f64 88.6%

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

   9. Applied rewrites 88.6%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 64.5%

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

Alternative 6: 73.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|\varepsilon\right| \leq 4200000000000000259547572063149966815813128127108259927169943245193807723488603782330037791698095292567457170557823028095081566830920435527163929535154945160713535366880771451178801327025282867003392:\\ \;\;\;\;\left(e^{-x} \cdot 2\right) \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\left|\varepsilon\right|}\right) - \left(\frac{1}{\frac{\left|\varepsilon\right|}{\left(\left(\left(\left|\varepsilon\right| - -1\right) \cdot x\right) \cdot \frac{\left|\varepsilon\right| - 1}{\left|\varepsilon\right|}\right) \cdot \left|\varepsilon\right| - -1}} - 1\right)}{2}\\ \end{array} \]

FPCore

(FPCore (x eps)
  :precision binary64
  (if (<=
     (fabs eps)
     4200000000000000259547572063149966815813128127108259927169943245193807723488603782330037791698095292567457170557823028095081566830920435527163929535154945160713535366880771451178801327025282867003392)
  (* (* (exp (- x)) 2) 1/2)
  (/
   (-
    (+ 1 (/ 1 (fabs eps)))
    (-
     (/
      1
      (/
       (fabs eps)
       (-
        (*
         (* (* (- (fabs eps) -1) x) (/ (- (fabs eps) 1) (fabs eps)))
         (fabs eps))
        -1)))
     1))
   2)))

C

double code(double x, double eps) {
	double tmp;
	if (fabs(eps) <= 4.2e+198) {
		tmp = (exp(-x) * 2.0) * 0.5;
	} else {
		tmp = ((1.0 + (1.0 / fabs(eps))) - ((1.0 / (fabs(eps) / (((((fabs(eps) - -1.0) * x) * ((fabs(eps) - 1.0) / fabs(eps))) * fabs(eps)) - -1.0))) - 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 (abs(eps) <= 4.2d+198) then
        tmp = (exp(-x) * 2.0d0) * 0.5d0
    else
        tmp = ((1.0d0 + (1.0d0 / abs(eps))) - ((1.0d0 / (abs(eps) / (((((abs(eps) - (-1.0d0)) * x) * ((abs(eps) - 1.0d0) / abs(eps))) * abs(eps)) - (-1.0d0)))) - 1.0d0)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (Math.abs(eps) <= 4.2e+198) {
		tmp = (Math.exp(-x) * 2.0) * 0.5;
	} else {
		tmp = ((1.0 + (1.0 / Math.abs(eps))) - ((1.0 / (Math.abs(eps) / (((((Math.abs(eps) - -1.0) * x) * ((Math.abs(eps) - 1.0) / Math.abs(eps))) * Math.abs(eps)) - -1.0))) - 1.0)) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if math.fabs(eps) <= 4.2e+198:
		tmp = (math.exp(-x) * 2.0) * 0.5
	else:
		tmp = ((1.0 + (1.0 / math.fabs(eps))) - ((1.0 / (math.fabs(eps) / (((((math.fabs(eps) - -1.0) * x) * ((math.fabs(eps) - 1.0) / math.fabs(eps))) * math.fabs(eps)) - -1.0))) - 1.0)) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (abs(eps) <= 4.2e+198)
		tmp = Float64(Float64(exp(Float64(-x)) * 2.0) * 0.5);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / abs(eps))) - Float64(Float64(1.0 / Float64(abs(eps) / Float64(Float64(Float64(Float64(Float64(abs(eps) - -1.0) * x) * Float64(Float64(abs(eps) - 1.0) / abs(eps))) * abs(eps)) - -1.0))) - 1.0)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (abs(eps) <= 4.2e+198)
		tmp = (exp(-x) * 2.0) * 0.5;
	else
		tmp = ((1.0 + (1.0 / abs(eps))) - ((1.0 / (abs(eps) / (((((abs(eps) - -1.0) * x) * ((abs(eps) - 1.0) / abs(eps))) * abs(eps)) - -1.0))) - 1.0)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[Abs[eps], $MachinePrecision], 4200000000000000259547572063149966815813128127108259927169943245193807723488603782330037791698095292567457170557823028095081566830920435527163929535154945160713535366880771451178801327025282867003392], N[(N[(N[Exp[(-x)], $MachinePrecision] * 2), $MachinePrecision] * 1/2), $MachinePrecision], N[(N[(N[(1 + N[(1 / N[Abs[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(1 / N[(N[Abs[eps], $MachinePrecision] / N[(N[(N[(N[(N[(N[Abs[eps], $MachinePrecision] - -1), $MachinePrecision] * x), $MachinePrecision] * N[(N[(N[Abs[eps], $MachinePrecision] - 1), $MachinePrecision] / N[Abs[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[eps], $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision] / 2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 4.2000000000000003e198

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1%

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.1%

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-*.f64 70.1%

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

   9. Applied rewrites 70.1%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-exp.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. mul-1-neg N/A

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

       6. lower-*.f64 N/A

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

       7. mul-1-neg N/A

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

       8. lift-*.f64 N/A

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

       9. lift-exp.f64 70.1%

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

       10. lift-*.f64 N/A

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

       11. mul-1-neg N/A

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

       12. lower-neg.f64 70.1%

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

   11. Applied rewrites 70.1%

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

   if 4.2000000000000003e198 < eps

   1. Initial program 73.0%

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

   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

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

      2. lower-/.f64 37.6%

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

   4. Applied rewrites 37.6%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 23.6%

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

   7. Applied rewrites 23.6%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. div-flip N/A

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

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

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

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

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

      7. add-flip N/A

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

      8. metadata-eval N/A

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

      9. lower--.f64 N/A

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

   9. Applied rewrites 30.3%

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

Alternative 7: 69.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|\varepsilon\right| \leq \frac{3512807709348987}{18014398509481984}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{-4}{\left(\left(\frac{2}{3} \cdot x - 1\right) \cdot x\right) \cdot x - 2}\\ \mathbf{elif}\;\left|\varepsilon\right| \leq 131999999999999999827056903170405592854753811920539301339283833622256165996931172376387097399098462440422467456220713849431020442499736547195457715787159345896667024903899286690296999949407092736:\\ \;\;\;\;\left(2 + x \cdot \left(x - 2\right)\right) \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\left|\varepsilon\right|}\right) - \frac{\left(x \cdot \left|\varepsilon\right| - 1\right) \cdot \left|\varepsilon\right|}{\left|\varepsilon\right|}}{2}\\ \end{array} \]

FPCore

(FPCore (x eps)
  :precision binary64
  (if (<= (fabs eps) 3512807709348987/18014398509481984)
  (* 1/2 (/ -4 (- (* (* (- (* 2/3 x) 1) x) x) 2)))
  (if (<=
       (fabs eps)
       131999999999999999827056903170405592854753811920539301339283833622256165996931172376387097399098462440422467456220713849431020442499736547195457715787159345896667024903899286690296999949407092736)
    (* (+ 2 (* x (- x 2))) 1/2)
    (/
     (-
      (+ 1 (/ 1 (fabs eps)))
      (/ (* (- (* x (fabs eps)) 1) (fabs eps)) (fabs eps)))
     2))))

C

double code(double x, double eps) {
	double tmp;
	if (fabs(eps) <= 0.195) {
		tmp = 0.5 * (-4.0 / (((((0.6666666666666666 * x) - 1.0) * x) * x) - 2.0));
	} else if (fabs(eps) <= 1.32e+194) {
		tmp = (2.0 + (x * (x - 2.0))) * 0.5;
	} else {
		tmp = ((1.0 + (1.0 / fabs(eps))) - ((((x * fabs(eps)) - 1.0) * fabs(eps)) / fabs(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 (abs(eps) <= 0.195d0) then
        tmp = 0.5d0 * ((-4.0d0) / (((((0.6666666666666666d0 * x) - 1.0d0) * x) * x) - 2.0d0))
    else if (abs(eps) <= 1.32d+194) then
        tmp = (2.0d0 + (x * (x - 2.0d0))) * 0.5d0
    else
        tmp = ((1.0d0 + (1.0d0 / abs(eps))) - ((((x * abs(eps)) - 1.0d0) * abs(eps)) / abs(eps))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (Math.abs(eps) <= 0.195) {
		tmp = 0.5 * (-4.0 / (((((0.6666666666666666 * x) - 1.0) * x) * x) - 2.0));
	} else if (Math.abs(eps) <= 1.32e+194) {
		tmp = (2.0 + (x * (x - 2.0))) * 0.5;
	} else {
		tmp = ((1.0 + (1.0 / Math.abs(eps))) - ((((x * Math.abs(eps)) - 1.0) * Math.abs(eps)) / Math.abs(eps))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if math.fabs(eps) <= 0.195:
		tmp = 0.5 * (-4.0 / (((((0.6666666666666666 * x) - 1.0) * x) * x) - 2.0))
	elif math.fabs(eps) <= 1.32e+194:
		tmp = (2.0 + (x * (x - 2.0))) * 0.5
	else:
		tmp = ((1.0 + (1.0 / math.fabs(eps))) - ((((x * math.fabs(eps)) - 1.0) * math.fabs(eps)) / math.fabs(eps))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (abs(eps) <= 0.195)
		tmp = Float64(0.5 * Float64(-4.0 / Float64(Float64(Float64(Float64(Float64(0.6666666666666666 * x) - 1.0) * x) * x) - 2.0)));
	elseif (abs(eps) <= 1.32e+194)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(x - 2.0))) * 0.5);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / abs(eps))) - Float64(Float64(Float64(Float64(x * abs(eps)) - 1.0) * abs(eps)) / abs(eps))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (abs(eps) <= 0.195)
		tmp = 0.5 * (-4.0 / (((((0.6666666666666666 * x) - 1.0) * x) * x) - 2.0));
	elseif (abs(eps) <= 1.32e+194)
		tmp = (2.0 + (x * (x - 2.0))) * 0.5;
	else
		tmp = ((1.0 + (1.0 / abs(eps))) - ((((x * abs(eps)) - 1.0) * abs(eps)) / abs(eps))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[Abs[eps], $MachinePrecision], 3512807709348987/18014398509481984], N[(1/2 * N[(-4 / N[(N[(N[(N[(N[(2/3 * x), $MachinePrecision] - 1), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] - 2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[eps], $MachinePrecision], 131999999999999999827056903170405592854753811920539301339283833622256165996931172376387097399098462440422467456220713849431020442499736547195457715787159345896667024903899286690296999949407092736], N[(N[(2 + N[(x * N[(x - 2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1/2), $MachinePrecision], N[(N[(N[(1 + N[(1 / N[Abs[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(x * N[Abs[eps], $MachinePrecision]), $MachinePrecision] - 1), $MachinePrecision] * N[Abs[eps], $MachinePrecision]), $MachinePrecision] / N[Abs[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 0.19500000000000001

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 57.4%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 52.6%

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

   7. Applied rewrites 52.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

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

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

   9. Applied rewrites 45.2%

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

   10. Taylor expanded in x around 0

       \[\leadsto \frac{1}{2} \cdot \frac{-4}{\left(\left(\frac{2}{3} \cdot x - 1\right) \cdot x\right) \cdot x - 2} \]
   11. Step-by-step derivation

   12. Applied rewrites 52.7%

       \[\leadsto \frac{1}{2} \cdot \frac{-4}{\left(\left(\frac{2}{3} \cdot x - 1\right) \cdot x\right) \cdot x - 2} \]

   if 0.19500000000000001 < eps < 1.32e194

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1%

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.1%

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-*.f64 70.1%

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

   9. Applied rewrites 70.1%

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

   10. Taylor expanded in x around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 56.7%

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

   12. Applied rewrites 56.7%

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

   if 1.32e194 < eps

   1. Initial program 73.0%

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

   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

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

      2. lower-/.f64 37.6%

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

   4. Applied rewrites 37.6%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 23.6%

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

   7. Applied rewrites 23.6%

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

   8. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 21.9%

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

   10. Applied rewrites 21.9%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift--.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. sub-to-fraction N/A

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

       6. associate-*l/ N/A

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

       7. lower-/.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lift-*.f64 N/A

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

       11. lower--.f64 28.6%

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

       12. lift-*.f64 N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 28.6%

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

   12. Applied rewrites 28.6%

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

Alternative 8: 69.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|\varepsilon\right| \leq \frac{3512807709348987}{18014398509481984}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{-4}{\left(\left(\frac{2}{3} \cdot x - 1\right) \cdot x\right) \cdot x - 2}\\ \mathbf{elif}\;\left|\varepsilon\right| \leq 131999999999999999827056903170405592854753811920539301339283833622256165996931172376387097399098462440422467456220713849431020442499736547195457715787159345896667024903899286690296999949407092736:\\ \;\;\;\;\left(2 + x \cdot \left(x - 2\right)\right) \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\left|\varepsilon\right|}\right) - \left(\frac{1}{\frac{\left|\varepsilon\right|}{\left(\left(\left(\left|\varepsilon\right| - -1\right) \cdot x\right) \cdot \frac{\left|\varepsilon\right| - 1}{\left|\varepsilon\right|}\right) \cdot \left|\varepsilon\right| - -1}} - 1\right)}{2}\\ \end{array} \]

FPCore

(FPCore (x eps)
  :precision binary64
  (if (<= (fabs eps) 3512807709348987/18014398509481984)
  (* 1/2 (/ -4 (- (* (* (- (* 2/3 x) 1) x) x) 2)))
  (if (<=
       (fabs eps)
       131999999999999999827056903170405592854753811920539301339283833622256165996931172376387097399098462440422467456220713849431020442499736547195457715787159345896667024903899286690296999949407092736)
    (* (+ 2 (* x (- x 2))) 1/2)
    (/
     (-
      (+ 1 (/ 1 (fabs eps)))
      (-
       (/
        1
        (/
         (fabs eps)
         (-
          (*
           (* (* (- (fabs eps) -1) x) (/ (- (fabs eps) 1) (fabs eps)))
           (fabs eps))
          -1)))
       1))
     2))))

C

double code(double x, double eps) {
	double tmp;
	if (fabs(eps) <= 0.195) {
		tmp = 0.5 * (-4.0 / (((((0.6666666666666666 * x) - 1.0) * x) * x) - 2.0));
	} else if (fabs(eps) <= 1.32e+194) {
		tmp = (2.0 + (x * (x - 2.0))) * 0.5;
	} else {
		tmp = ((1.0 + (1.0 / fabs(eps))) - ((1.0 / (fabs(eps) / (((((fabs(eps) - -1.0) * x) * ((fabs(eps) - 1.0) / fabs(eps))) * fabs(eps)) - -1.0))) - 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 (abs(eps) <= 0.195d0) then
        tmp = 0.5d0 * ((-4.0d0) / (((((0.6666666666666666d0 * x) - 1.0d0) * x) * x) - 2.0d0))
    else if (abs(eps) <= 1.32d+194) then
        tmp = (2.0d0 + (x * (x - 2.0d0))) * 0.5d0
    else
        tmp = ((1.0d0 + (1.0d0 / abs(eps))) - ((1.0d0 / (abs(eps) / (((((abs(eps) - (-1.0d0)) * x) * ((abs(eps) - 1.0d0) / abs(eps))) * abs(eps)) - (-1.0d0)))) - 1.0d0)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (Math.abs(eps) <= 0.195) {
		tmp = 0.5 * (-4.0 / (((((0.6666666666666666 * x) - 1.0) * x) * x) - 2.0));
	} else if (Math.abs(eps) <= 1.32e+194) {
		tmp = (2.0 + (x * (x - 2.0))) * 0.5;
	} else {
		tmp = ((1.0 + (1.0 / Math.abs(eps))) - ((1.0 / (Math.abs(eps) / (((((Math.abs(eps) - -1.0) * x) * ((Math.abs(eps) - 1.0) / Math.abs(eps))) * Math.abs(eps)) - -1.0))) - 1.0)) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if math.fabs(eps) <= 0.195:
		tmp = 0.5 * (-4.0 / (((((0.6666666666666666 * x) - 1.0) * x) * x) - 2.0))
	elif math.fabs(eps) <= 1.32e+194:
		tmp = (2.0 + (x * (x - 2.0))) * 0.5
	else:
		tmp = ((1.0 + (1.0 / math.fabs(eps))) - ((1.0 / (math.fabs(eps) / (((((math.fabs(eps) - -1.0) * x) * ((math.fabs(eps) - 1.0) / math.fabs(eps))) * math.fabs(eps)) - -1.0))) - 1.0)) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (abs(eps) <= 0.195)
		tmp = Float64(0.5 * Float64(-4.0 / Float64(Float64(Float64(Float64(Float64(0.6666666666666666 * x) - 1.0) * x) * x) - 2.0)));
	elseif (abs(eps) <= 1.32e+194)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(x - 2.0))) * 0.5);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / abs(eps))) - Float64(Float64(1.0 / Float64(abs(eps) / Float64(Float64(Float64(Float64(Float64(abs(eps) - -1.0) * x) * Float64(Float64(abs(eps) - 1.0) / abs(eps))) * abs(eps)) - -1.0))) - 1.0)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (abs(eps) <= 0.195)
		tmp = 0.5 * (-4.0 / (((((0.6666666666666666 * x) - 1.0) * x) * x) - 2.0));
	elseif (abs(eps) <= 1.32e+194)
		tmp = (2.0 + (x * (x - 2.0))) * 0.5;
	else
		tmp = ((1.0 + (1.0 / abs(eps))) - ((1.0 / (abs(eps) / (((((abs(eps) - -1.0) * x) * ((abs(eps) - 1.0) / abs(eps))) * abs(eps)) - -1.0))) - 1.0)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[Abs[eps], $MachinePrecision], 3512807709348987/18014398509481984], N[(1/2 * N[(-4 / N[(N[(N[(N[(N[(2/3 * x), $MachinePrecision] - 1), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] - 2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[eps], $MachinePrecision], 131999999999999999827056903170405592854753811920539301339283833622256165996931172376387097399098462440422467456220713849431020442499736547195457715787159345896667024903899286690296999949407092736], N[(N[(2 + N[(x * N[(x - 2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1/2), $MachinePrecision], N[(N[(N[(1 + N[(1 / N[Abs[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(1 / N[(N[Abs[eps], $MachinePrecision] / N[(N[(N[(N[(N[(N[Abs[eps], $MachinePrecision] - -1), $MachinePrecision] * x), $MachinePrecision] * N[(N[(N[Abs[eps], $MachinePrecision] - 1), $MachinePrecision] / N[Abs[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[eps], $MachinePrecision]), $MachinePrecision] - -1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision] / 2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 0.19500000000000001

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 57.4%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 52.6%

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

   7. Applied rewrites 52.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

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

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

   9. Applied rewrites 45.2%

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

   10. Taylor expanded in x around 0

       \[\leadsto \frac{1}{2} \cdot \frac{-4}{\left(\left(\frac{2}{3} \cdot x - 1\right) \cdot x\right) \cdot x - 2} \]
   11. Step-by-step derivation

   12. Applied rewrites 52.7%

       \[\leadsto \frac{1}{2} \cdot \frac{-4}{\left(\left(\frac{2}{3} \cdot x - 1\right) \cdot x\right) \cdot x - 2} \]

   if 0.19500000000000001 < eps < 1.32e194

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1%

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.1%

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-*.f64 70.1%

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

   9. Applied rewrites 70.1%

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

   10. Taylor expanded in x around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 56.7%

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

   12. Applied rewrites 56.7%

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

   if 1.32e194 < eps

   1. Initial program 73.0%

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

   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

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

      2. lower-/.f64 37.6%

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

   4. Applied rewrites 37.6%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 23.6%

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

   7. Applied rewrites 23.6%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. div-flip N/A

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

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

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

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

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

      7. add-flip N/A

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

      8. metadata-eval N/A

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

      9. lower--.f64 N/A

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

   9. Applied rewrites 30.3%

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

Alternative 9: 66.4% accurate, 5.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq \frac{6313494200355439}{2475880078570760549798248448}:\\ \;\;\;\;\left(2 + x \cdot \left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2\right)\right) \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \]

FPCore

(FPCore (x eps)
  :precision binary64
  (if (<= x 6313494200355439/2475880078570760549798248448)
  (* (+ 2 (* x (- (* x (+ 1 (* -1/3 x))) 2))) 1/2)
  (/ (- (+ 1 (/ 1 eps)) (- (/ 1 eps) 1)) 2)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 2.55e-12) {
		tmp = (2.0 + (x * ((x * (1.0 + (-0.3333333333333333 * x))) - 2.0))) * 0.5;
	} 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 <= 2.55d-12) then
        tmp = (2.0d0 + (x * ((x * (1.0d0 + ((-0.3333333333333333d0) * x))) - 2.0d0))) * 0.5d0
    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 <= 2.55e-12) {
		tmp = (2.0 + (x * ((x * (1.0 + (-0.3333333333333333 * x))) - 2.0))) * 0.5;
	} 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 <= 2.55e-12:
		tmp = (2.0 + (x * ((x * (1.0 + (-0.3333333333333333 * x))) - 2.0))) * 0.5
	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 <= 2.55e-12)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(x * Float64(1.0 + Float64(-0.3333333333333333 * x))) - 2.0))) * 0.5);
	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 <= 2.55e-12)
		tmp = (2.0 + (x * ((x * (1.0 + (-0.3333333333333333 * x))) - 2.0))) * 0.5;
	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, 6313494200355439/2475880078570760549798248448], N[(N[(2 + N[(x * N[(N[(x * N[(1 + N[(-1/3 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1/2), $MachinePrecision], N[(N[(N[(1 + N[(1 / eps), $MachinePrecision]), $MachinePrecision] - N[(N[(1 / eps), $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision] / 2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.5499999999999998e-12

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1%

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.1%

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-*.f64 70.1%

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

   9. Applied rewrites 70.1%

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

   10. Taylor expanded in x around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-*.f64 51.6%

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

   12. Applied rewrites 51.6%

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

   if 2.5499999999999998e-12 < x

   1. Initial program 73.0%

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

   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

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

      2. lower-/.f64 37.6%

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

   4. Applied rewrites 37.6%

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

   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

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

      2. lower-/.f64 30.7%

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

   7. Applied rewrites 30.7%

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

Alternative 10: 65.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \leq 2000:\\ \;\;\;\;\frac{1}{2} \cdot \frac{-4}{\left(\left(\frac{2}{3} \cdot x - 1\right) \cdot x\right) \cdot x - 2}\\ \mathbf{else}:\\ \;\;\;\;\left(2 + x \cdot \left(x - 2\right)\right) \cdot \frac{1}{2}\\ \end{array} \]

FPCore

(FPCore (x eps)
  :precision binary64
  (if (<=
     (/
      (-
       (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x))))
       (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x)))))
      2)
     2000)
  (* 1/2 (/ -4 (- (* (* (- (* 2/3 x) 1) x) x) 2)))
  (* (+ 2 (* x (- x 2))) 1/2)))

C

double code(double x, double eps) {
	double tmp;
	if (((((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-((1.0 + eps) * x)))) / 2.0) <= 2000.0) {
		tmp = 0.5 * (-4.0 / (((((0.6666666666666666 * x) - 1.0) * x) * x) - 2.0));
	} else {
		tmp = (2.0 + (x * (x - 2.0))) * 0.5;
	}
	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 (((((1.0d0 + (1.0d0 / eps)) * exp(-((1.0d0 - eps) * x))) - (((1.0d0 / eps) - 1.0d0) * exp(-((1.0d0 + eps) * x)))) / 2.0d0) <= 2000.0d0) then
        tmp = 0.5d0 * ((-4.0d0) / (((((0.6666666666666666d0 * x) - 1.0d0) * x) * x) - 2.0d0))
    else
        tmp = (2.0d0 + (x * (x - 2.0d0))) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (((((1.0 + (1.0 / eps)) * Math.exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * Math.exp(-((1.0 + eps) * x)))) / 2.0) <= 2000.0) {
		tmp = 0.5 * (-4.0 / (((((0.6666666666666666 * x) - 1.0) * x) * x) - 2.0));
	} else {
		tmp = (2.0 + (x * (x - 2.0))) * 0.5;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if ((((1.0 + (1.0 / eps)) * math.exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * math.exp(-((1.0 + eps) * x)))) / 2.0) <= 2000.0:
		tmp = 0.5 * (-4.0 / (((((0.6666666666666666 * x) - 1.0) * x) * x) - 2.0))
	else:
		tmp = (2.0 + (x * (x - 2.0))) * 0.5
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (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) <= 2000.0)
		tmp = Float64(0.5 * Float64(-4.0 / Float64(Float64(Float64(Float64(Float64(0.6666666666666666 * x) - 1.0) * x) * x) - 2.0)));
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(x - 2.0))) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (((((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-((1.0 + eps) * x)))) / 2.0) <= 2000.0)
		tmp = 0.5 * (-4.0 / (((((0.6666666666666666 * x) - 1.0) * x) * x) - 2.0));
	else
		tmp = (2.0 + (x * (x - 2.0))) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[(N[(N[(1 + N[(1 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[(1 - eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1 / eps), $MachinePrecision] - 1), $MachinePrecision] * N[Exp[(-N[(N[(1 + eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2), $MachinePrecision], 2000], N[(1/2 * N[(-4 / N[(N[(N[(N[(N[(2/3 * x), $MachinePrecision] - 1), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] - 2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2 + N[(x * N[(x - 2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1/2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) #s(literal 2 binary64)) < 2e3

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 57.4%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 52.6%

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

   7. Applied rewrites 52.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

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

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

   9. Applied rewrites 45.2%

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

   10. Taylor expanded in x around 0

       \[\leadsto \frac{1}{2} \cdot \frac{-4}{\left(\left(\frac{2}{3} \cdot x - 1\right) \cdot x\right) \cdot x - 2} \]
   11. Step-by-step derivation

   12. Applied rewrites 52.7%

       \[\leadsto \frac{1}{2} \cdot \frac{-4}{\left(\left(\frac{2}{3} \cdot x - 1\right) \cdot x\right) \cdot x - 2} \]

   if 2e3 < (/.f64 (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) #s(literal 2 binary64))

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1%

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.1%

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-*.f64 70.1%

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

   9. Applied rewrites 70.1%

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

   10. Taylor expanded in x around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 56.7%

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

   12. Applied rewrites 56.7%

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

Alternative 11: 61.7% accurate, 7.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq \frac{-7385903388887613}{9007199254740992}:\\ \;\;\;\;\left(2 + x \cdot \left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2\right)\right) \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(2 - \left(1 - \frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
  :precision binary64
  (if (<= x -7385903388887613/9007199254740992)
  (* (+ 2 (* x (- (* x (+ 1 (* -1/3 x))) 2))) 1/2)
  (* 1/2 (- 2 (* (- 1 (* 2/3 x)) (* x x))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -0.82) {
		tmp = (2.0 + (x * ((x * (1.0 + (-0.3333333333333333 * x))) - 2.0))) * 0.5;
	} else {
		tmp = 0.5 * (2.0 - ((1.0 - (0.6666666666666666 * x)) * (x * x)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-0.82d0)) then
        tmp = (2.0d0 + (x * ((x * (1.0d0 + ((-0.3333333333333333d0) * x))) - 2.0d0))) * 0.5d0
    else
        tmp = 0.5d0 * (2.0d0 - ((1.0d0 - (0.6666666666666666d0 * x)) * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -0.82) {
		tmp = (2.0 + (x * ((x * (1.0 + (-0.3333333333333333 * x))) - 2.0))) * 0.5;
	} else {
		tmp = 0.5 * (2.0 - ((1.0 - (0.6666666666666666 * x)) * (x * x)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -0.82:
		tmp = (2.0 + (x * ((x * (1.0 + (-0.3333333333333333 * x))) - 2.0))) * 0.5
	else:
		tmp = 0.5 * (2.0 - ((1.0 - (0.6666666666666666 * x)) * (x * x)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -0.82)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(x * Float64(1.0 + Float64(-0.3333333333333333 * x))) - 2.0))) * 0.5);
	else
		tmp = Float64(0.5 * Float64(2.0 - Float64(Float64(1.0 - Float64(0.6666666666666666 * x)) * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -0.82)
		tmp = (2.0 + (x * ((x * (1.0 + (-0.3333333333333333 * x))) - 2.0))) * 0.5;
	else
		tmp = 0.5 * (2.0 - ((1.0 - (0.6666666666666666 * x)) * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -7385903388887613/9007199254740992], N[(N[(2 + N[(x * N[(N[(x * N[(1 + N[(-1/3 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1/2), $MachinePrecision], N[(1/2 * N[(2 - N[(N[(1 - N[(2/3 * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.81999999999999995

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1%

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.1%

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-*.f64 70.1%

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

   9. Applied rewrites 70.1%

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

   10. Taylor expanded in x around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-*.f64 51.6%

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

   12. Applied rewrites 51.6%

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

   if -0.81999999999999995 < x

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 57.4%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 52.6%

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

   7. Applied rewrites 52.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lower--.f64 N/A

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

      5. distribute-lft-neg-out N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 52.6%

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 52.6%

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

   9. Applied rewrites 52.6%

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

Alternative 12: 59.4% accurate, 8.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq \frac{-7385903388887613}{9007199254740992}:\\ \;\;\;\;\left(2 + x \cdot \left(x - 2\right)\right) \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(2 - \left(1 - \frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
  :precision binary64
  (if (<= x -7385903388887613/9007199254740992)
  (* (+ 2 (* x (- x 2))) 1/2)
  (* 1/2 (- 2 (* (- 1 (* 2/3 x)) (* x x))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -0.82) {
		tmp = (2.0 + (x * (x - 2.0))) * 0.5;
	} else {
		tmp = 0.5 * (2.0 - ((1.0 - (0.6666666666666666 * x)) * (x * x)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-0.82d0)) then
        tmp = (2.0d0 + (x * (x - 2.0d0))) * 0.5d0
    else
        tmp = 0.5d0 * (2.0d0 - ((1.0d0 - (0.6666666666666666d0 * x)) * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -0.82) {
		tmp = (2.0 + (x * (x - 2.0))) * 0.5;
	} else {
		tmp = 0.5 * (2.0 - ((1.0 - (0.6666666666666666 * x)) * (x * x)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -0.82:
		tmp = (2.0 + (x * (x - 2.0))) * 0.5
	else:
		tmp = 0.5 * (2.0 - ((1.0 - (0.6666666666666666 * x)) * (x * x)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -0.82)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(x - 2.0))) * 0.5);
	else
		tmp = Float64(0.5 * Float64(2.0 - Float64(Float64(1.0 - Float64(0.6666666666666666 * x)) * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -0.82)
		tmp = (2.0 + (x * (x - 2.0))) * 0.5;
	else
		tmp = 0.5 * (2.0 - ((1.0 - (0.6666666666666666 * x)) * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -7385903388887613/9007199254740992], N[(N[(2 + N[(x * N[(x - 2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1/2), $MachinePrecision], N[(1/2 * N[(2 - N[(N[(1 - N[(2/3 * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.81999999999999995

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1%

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.1%

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-*.f64 70.1%

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

   9. Applied rewrites 70.1%

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

   10. Taylor expanded in x around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 56.7%

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

   12. Applied rewrites 56.7%

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

   if -0.81999999999999995 < x

   1. Initial program 73.0%

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

   2. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 57.4%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 52.6%

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

   7. Applied rewrites 52.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lower--.f64 N/A

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

      5. distribute-lft-neg-out N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 52.6%

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 52.6%

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

   9. Applied rewrites 52.6%

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

Alternative 13: 56.7% accurate, 16.1× speedup

Math

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

FPCore

(FPCore (x eps)
  :precision binary64
  (* (+ 2 (* x (- x 2))) 1/2))

C

double code(double x, double eps) {
	return (2.0 + (x * (x - 2.0))) * 0.5;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (2.0d0 + (x * (x - 2.0d0))) * 0.5d0
end function

Java

public static double code(double x, double eps) {
	return (2.0 + (x * (x - 2.0))) * 0.5;
}

Python

def code(x, eps):
	return (2.0 + (x * (x - 2.0))) * 0.5

Julia

function code(x, eps)
	return Float64(Float64(2.0 + Float64(x * Float64(x - 2.0))) * 0.5)
end

MATLAB

function tmp = code(x, eps)
	tmp = (2.0 + (x * (x - 2.0))) * 0.5;
end

Wolfram

code[x_, eps_] := N[(N[(2 + N[(x * N[(x - 2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1/2), $MachinePrecision]

Derivation

1. Initial program 73.0%

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

2. Taylor expanded in eps around inf

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-exp.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-+.f64 99.1%

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

4. Applied rewrites 99.1%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.1%

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

6. Applied rewrites 99.1%

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

7. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. lower-exp.f64 N/A

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

   3. lower-*.f64 70.1%

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

9. Applied rewrites 70.1%

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

10. Taylor expanded in x around 0

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

    1. lower-+.f64 N/A

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

    2. lower-*.f64 N/A

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

    3. lower--.f64 56.7%

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

12. Applied rewrites 56.7%

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

Alternative 14: 43.8% accurate, 45.5× speedup

Math

\[2 \cdot \frac{1}{2} \]

FPCore

(FPCore (x eps)
  :precision binary64
  (* 2 1/2))

C

double code(double x, double eps) {
	return 2.0 * 0.5;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 2.0d0 * 0.5d0
end function

Java

public static double code(double x, double eps) {
	return 2.0 * 0.5;
}

Python

def code(x, eps):
	return 2.0 * 0.5

Julia

function code(x, eps)
	return Float64(2.0 * 0.5)
end

MATLAB

function tmp = code(x, eps)
	tmp = 2.0 * 0.5;
end

Wolfram

code[x_, eps_] := N[(2 * 1/2), $MachinePrecision]

Derivation

1. Initial program 73.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 43.8%

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

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2}{2}} \]

   2. mult-flip N/A

      \[\leadsto \color{blue}{2 \cdot \frac{1}{2}} \]

   3. metadata-eval N/A

      \[\leadsto 2 \cdot \color{blue}{\frac{1}{2}} \]

   4. lower-*.f64 43.8%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{2}} \]

6. Applied rewrites 43.8%

   \[\leadsto \color{blue}{2 \cdot \frac{1}{2}} \]
7. Add Preprocessing

Alternative 15: 43.3% accurate, 68.3× speedup

Math

\[1 - x \]

FPCore

(FPCore (x eps)
  :precision binary64
  (- 1 x))

C

double code(double x, double eps) {
	return 1.0 - x;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 1.0d0 - x
end function

Java

public static double code(double x, double eps) {
	return 1.0 - x;
}

Python

def code(x, eps):
	return 1.0 - x

Julia

function code(x, eps)
	return Float64(1.0 - x)
end

MATLAB

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

Wolfram

code[x_, eps_] := N[(1 - x), $MachinePrecision]

Derivation

1. Initial program 73.0%

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

2. Taylor expanded in eps around inf

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-exp.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-+.f64 99.1%

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

4. Applied rewrites 99.1%

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

5. Taylor expanded in x around 0

   \[\leadsto 1 + \color{blue}{-1 \cdot x} \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto 1 + -1 \cdot \color{blue}{x} \]

   2. lower-*.f64 43.3%

      \[\leadsto 1 + -1 \cdot x \]

7. Applied rewrites 43.3%

   \[\leadsto 1 + \color{blue}{-1 \cdot x} \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto 1 + -1 \cdot \color{blue}{x} \]

   2. lift-*.f64 N/A

      \[\leadsto 1 + -1 \cdot x \]

   3. mul-1-neg N/A

      \[\leadsto 1 + \left(\mathsf{neg}\left(x\right)\right) \]

   4. sub-flip-reverse N/A

      \[\leadsto 1 - x \]

   5. lower--.f64 43.3%

      \[\leadsto 1 - x \]

9. Applied rewrites 43.3%

   \[\leadsto 1 - x \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  :precision binary64
  (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2))