NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.3% → 99.6%

Time: 9.2s

Alternatives: 10

Speedup: 0.5×

• 38.0% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -1.2566123624886292e+100
  2. eps = -6.275306923335652e-68

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0
         (/
          (-
           (* (+ 1.0 (pow eps -1.0)) (exp (* (+ -1.0 eps) x)))
           (* (- (pow eps -1.0) 1.0) (exp (* (- -1.0 eps) x))))
          2.0)))
   (if (<= t_0 0.0) (* (* (exp (- x)) (+ (- (+ 1.0 x) -1.0) x)) 0.5) t_0)))

C

double code(double x, double eps) {
	double t_0 = (((1.0 + pow(eps, -1.0)) * exp(((-1.0 + eps) * x))) - ((pow(eps, -1.0) - 1.0) * exp(((-1.0 - eps) * x)))) / 2.0;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (exp(-x) * (((1.0 + x) - -1.0) + x)) * 0.5;
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (((1.0d0 + (eps ** (-1.0d0))) * exp((((-1.0d0) + eps) * x))) - (((eps ** (-1.0d0)) - 1.0d0) * exp((((-1.0d0) - eps) * x)))) / 2.0d0
    if (t_0 <= 0.0d0) then
        tmp = (exp(-x) * (((1.0d0 + x) - (-1.0d0)) + x)) * 0.5d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = (((1.0 + Math.pow(eps, -1.0)) * Math.exp(((-1.0 + eps) * x))) - ((Math.pow(eps, -1.0) - 1.0) * Math.exp(((-1.0 - eps) * x)))) / 2.0;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (Math.exp(-x) * (((1.0 + x) - -1.0) + x)) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (((1.0 + math.pow(eps, -1.0)) * math.exp(((-1.0 + eps) * x))) - ((math.pow(eps, -1.0) - 1.0) * math.exp(((-1.0 - eps) * x)))) / 2.0
	tmp = 0
	if t_0 <= 0.0:
		tmp = (math.exp(-x) * (((1.0 + x) - -1.0) + x)) * 0.5
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(Float64(Float64(1.0 + (eps ^ -1.0)) * exp(Float64(Float64(-1.0 + eps) * x))) - Float64(Float64((eps ^ -1.0) - 1.0) * exp(Float64(Float64(-1.0 - eps) * x)))) / 2.0)
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(exp(Float64(-x)) * Float64(Float64(Float64(1.0 + x) - -1.0) + x)) * 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (((1.0 + (eps ^ -1.0)) * exp(((-1.0 + eps) * x))) - (((eps ^ -1.0) - 1.0) * exp(((-1.0 - eps) * x)))) / 2.0;
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = (exp(-x) * (((1.0 + x) - -1.0) + x)) * 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

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

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)) < 0.0

   1. Initial program 41.5%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (/.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 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2} \leq 0:\\ \;\;\;\;\left(e^{-x} \cdot \left(\left(\left(1 + x\right) - -1\right) + x\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\varepsilon}^{-1} - 1\\ t_1 := 1 + {\varepsilon}^{-1}\\ \mathbf{if}\;\frac{t\_1 \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - t\_0 \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2} \leq 2:\\ \;\;\;\;\left(e^{-x} \cdot \left(\left(\left(1 + x\right) - -1\right) + x\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 \cdot e^{-\left(-\varepsilon\right) \cdot x} - t\_0}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow eps -1.0) 1.0)) (t_1 (+ 1.0 (pow eps -1.0))))
   (if (<=
        (/
         (- (* t_1 (exp (* (+ -1.0 eps) x))) (* t_0 (exp (* (- -1.0 eps) x))))
         2.0)
        2.0)
     (* (* (exp (- x)) (+ (- (+ 1.0 x) -1.0) x)) 0.5)
     (/ (- (* t_1 (exp (- (* (- eps) x)))) t_0) 2.0))))

C

double code(double x, double eps) {
	double t_0 = pow(eps, -1.0) - 1.0;
	double t_1 = 1.0 + pow(eps, -1.0);
	double tmp;
	if ((((t_1 * exp(((-1.0 + eps) * x))) - (t_0 * exp(((-1.0 - eps) * x)))) / 2.0) <= 2.0) {
		tmp = (exp(-x) * (((1.0 + x) - -1.0) + x)) * 0.5;
	} else {
		tmp = ((t_1 * exp(-(-eps * x))) - t_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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (eps ** (-1.0d0)) - 1.0d0
    t_1 = 1.0d0 + (eps ** (-1.0d0))
    if ((((t_1 * exp((((-1.0d0) + eps) * x))) - (t_0 * exp((((-1.0d0) - eps) * x)))) / 2.0d0) <= 2.0d0) then
        tmp = (exp(-x) * (((1.0d0 + x) - (-1.0d0)) + x)) * 0.5d0
    else
        tmp = ((t_1 * exp(-(-eps * x))) - t_0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow(eps, -1.0) - 1.0;
	double t_1 = 1.0 + Math.pow(eps, -1.0);
	double tmp;
	if ((((t_1 * Math.exp(((-1.0 + eps) * x))) - (t_0 * Math.exp(((-1.0 - eps) * x)))) / 2.0) <= 2.0) {
		tmp = (Math.exp(-x) * (((1.0 + x) - -1.0) + x)) * 0.5;
	} else {
		tmp = ((t_1 * Math.exp(-(-eps * x))) - t_0) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow(eps, -1.0) - 1.0
	t_1 = 1.0 + math.pow(eps, -1.0)
	tmp = 0
	if (((t_1 * math.exp(((-1.0 + eps) * x))) - (t_0 * math.exp(((-1.0 - eps) * x)))) / 2.0) <= 2.0:
		tmp = (math.exp(-x) * (((1.0 + x) - -1.0) + x)) * 0.5
	else:
		tmp = ((t_1 * math.exp(-(-eps * x))) - t_0) / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((eps ^ -1.0) - 1.0)
	t_1 = Float64(1.0 + (eps ^ -1.0))
	tmp = 0.0
	if (Float64(Float64(Float64(t_1 * exp(Float64(Float64(-1.0 + eps) * x))) - Float64(t_0 * exp(Float64(Float64(-1.0 - eps) * x)))) / 2.0) <= 2.0)
		tmp = Float64(Float64(exp(Float64(-x)) * Float64(Float64(Float64(1.0 + x) - -1.0) + x)) * 0.5);
	else
		tmp = Float64(Float64(Float64(t_1 * exp(Float64(-Float64(Float64(-eps) * x)))) - t_0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (eps ^ -1.0) - 1.0;
	t_1 = 1.0 + (eps ^ -1.0);
	tmp = 0.0;
	if ((((t_1 * exp(((-1.0 + eps) * x))) - (t_0 * exp(((-1.0 - eps) * x)))) / 2.0) <= 2.0)
		tmp = (exp(-x) * (((1.0 + x) - -1.0) + x)) * 0.5;
	else
		tmp = ((t_1 * exp(-(-eps * x))) - t_0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$1 * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 2.0], N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(N[(N[(1.0 + x), $MachinePrecision] - -1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(t$95$1 * N[Exp[(-N[((-eps) * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] / 2.0), $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)) < 2

   1. Initial program 59.2%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   if 2 < (/.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 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 58.1

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

   5. Applied rewrites 58.1%

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

   6. Taylor expanded in eps around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 58.1

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

   8. Applied rewrites 58.1%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2} \leq 2:\\ \;\;\;\;\left(e^{-x} \cdot \left(\left(\left(1 + x\right) - -1\right) + x\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + {\varepsilon}^{-1}\right) \cdot e^{-\left(-\varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.0% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow eps -1.0) 1.0)))
   (if (<=
        (/
         (-
          (* (+ 1.0 (pow eps -1.0)) (exp (* (+ -1.0 eps) x)))
          (* t_0 (exp (* (- -1.0 eps) x))))
         2.0)
        2.0)
     (* (* (exp (- x)) (+ (- (+ 1.0 x) -1.0) x)) 0.5)
     (/ (- (* (exp (- (* x eps) x)) (- (pow eps -1.0) -1.0)) t_0) 2.0))))

C

double code(double x, double eps) {
	double t_0 = pow(eps, -1.0) - 1.0;
	double tmp;
	if (((((1.0 + pow(eps, -1.0)) * exp(((-1.0 + eps) * x))) - (t_0 * exp(((-1.0 - eps) * x)))) / 2.0) <= 2.0) {
		tmp = (exp(-x) * (((1.0 + x) - -1.0) + x)) * 0.5;
	} else {
		tmp = ((exp(((x * eps) - x)) * (pow(eps, -1.0) - -1.0)) - t_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) :: t_0
    real(8) :: tmp
    t_0 = (eps ** (-1.0d0)) - 1.0d0
    if (((((1.0d0 + (eps ** (-1.0d0))) * exp((((-1.0d0) + eps) * x))) - (t_0 * exp((((-1.0d0) - eps) * x)))) / 2.0d0) <= 2.0d0) then
        tmp = (exp(-x) * (((1.0d0 + x) - (-1.0d0)) + x)) * 0.5d0
    else
        tmp = ((exp(((x * eps) - x)) * ((eps ** (-1.0d0)) - (-1.0d0))) - t_0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow(eps, -1.0) - 1.0;
	double tmp;
	if (((((1.0 + Math.pow(eps, -1.0)) * Math.exp(((-1.0 + eps) * x))) - (t_0 * Math.exp(((-1.0 - eps) * x)))) / 2.0) <= 2.0) {
		tmp = (Math.exp(-x) * (((1.0 + x) - -1.0) + x)) * 0.5;
	} else {
		tmp = ((Math.exp(((x * eps) - x)) * (Math.pow(eps, -1.0) - -1.0)) - t_0) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow(eps, -1.0) - 1.0
	tmp = 0
	if ((((1.0 + math.pow(eps, -1.0)) * math.exp(((-1.0 + eps) * x))) - (t_0 * math.exp(((-1.0 - eps) * x)))) / 2.0) <= 2.0:
		tmp = (math.exp(-x) * (((1.0 + x) - -1.0) + x)) * 0.5
	else:
		tmp = ((math.exp(((x * eps) - x)) * (math.pow(eps, -1.0) - -1.0)) - t_0) / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((eps ^ -1.0) - 1.0)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(1.0 + (eps ^ -1.0)) * exp(Float64(Float64(-1.0 + eps) * x))) - Float64(t_0 * exp(Float64(Float64(-1.0 - eps) * x)))) / 2.0) <= 2.0)
		tmp = Float64(Float64(exp(Float64(-x)) * Float64(Float64(Float64(1.0 + x) - -1.0) + x)) * 0.5);
	else
		tmp = Float64(Float64(Float64(exp(Float64(Float64(x * eps) - x)) * Float64((eps ^ -1.0) - -1.0)) - t_0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (eps ^ -1.0) - 1.0;
	tmp = 0.0;
	if (((((1.0 + (eps ^ -1.0)) * exp(((-1.0 + eps) * x))) - (t_0 * exp(((-1.0 - eps) * x)))) / 2.0) <= 2.0)
		tmp = (exp(-x) * (((1.0 + x) - -1.0) + x)) * 0.5;
	else
		tmp = ((exp(((x * eps) - x)) * ((eps ^ -1.0) - -1.0)) - t_0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(1.0 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 2.0], N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(N[(N[(1.0 + x), $MachinePrecision] - -1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[Exp[N[(N[(x * eps), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision] * N[(N[Power[eps, -1.0], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] / 2.0), $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)) < 2

   1. Initial program 59.2%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   if 2 < (/.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 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 58.1

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

   5. Applied rewrites 58.1%

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

   6. Taylor expanded in x around 0

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

      1. *-inverses N/A

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

      2. div-add N/A

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

      3. +-commutative N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. mul-1-neg N/A

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

      8. div-sub N/A

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

      9. mul-1-neg N/A

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

      10. *-lft-identity N/A

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

      11. times-frac N/A

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

      12. metadata-eval N/A

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

      13. *-inverses N/A

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

      14. metadata-eval N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 3.1

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

   8. Applied rewrites 3.1%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 3.1%

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

   12. Taylor expanded in x around inf

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

       1. lower-*.f64 N/A

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

       2. lower-exp.f64 N/A

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

       3. lower--.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. *-inverses N/A

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

       7. div-add N/A

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

       8. +-commutative N/A

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

       9. *-lft-identity N/A

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

       10. metadata-eval N/A

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

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

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

       12. mul-1-neg N/A

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

       13. div-sub N/A

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

       14. mul-1-neg N/A

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

       15. *-lft-identity N/A

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

       16. times-frac N/A

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

       17. metadata-eval N/A

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

       18. *-inverses N/A

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

       19. metadata-eval N/A

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

       20. lower--.f64 N/A

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

       21. lower-/.f64 58.1

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

   14. Applied rewrites 58.1%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2} \leq 2:\\ \;\;\;\;\left(e^{-x} \cdot \left(\left(\left(1 + x\right) - -1\right) + x\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon - x} \cdot \left({\varepsilon}^{-1} - -1\right) - \left({\varepsilon}^{-1} - 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.8% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (- (pow eps -1.0) 1.0) (exp (* (- -1.0 eps) x)))))
   (if (<=
        (/ (- (* (+ 1.0 (pow eps -1.0)) (exp (* (+ -1.0 eps) x))) t_0) 2.0)
        1.0)
     (* (* (exp (- x)) (+ (- (+ 1.0 x) -1.0) x)) 0.5)
     (/ (- (- (* eps x) -1.0) t_0) 2.0))))

C

double code(double x, double eps) {
	double t_0 = (pow(eps, -1.0) - 1.0) * exp(((-1.0 - eps) * x));
	double tmp;
	if (((((1.0 + pow(eps, -1.0)) * exp(((-1.0 + eps) * x))) - t_0) / 2.0) <= 1.0) {
		tmp = (exp(-x) * (((1.0 + x) - -1.0) + x)) * 0.5;
	} else {
		tmp = (((eps * x) - -1.0) - t_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) :: t_0
    real(8) :: tmp
    t_0 = ((eps ** (-1.0d0)) - 1.0d0) * exp((((-1.0d0) - eps) * x))
    if (((((1.0d0 + (eps ** (-1.0d0))) * exp((((-1.0d0) + eps) * x))) - t_0) / 2.0d0) <= 1.0d0) then
        tmp = (exp(-x) * (((1.0d0 + x) - (-1.0d0)) + x)) * 0.5d0
    else
        tmp = (((eps * x) - (-1.0d0)) - t_0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = (Math.pow(eps, -1.0) - 1.0) * Math.exp(((-1.0 - eps) * x));
	double tmp;
	if (((((1.0 + Math.pow(eps, -1.0)) * Math.exp(((-1.0 + eps) * x))) - t_0) / 2.0) <= 1.0) {
		tmp = (Math.exp(-x) * (((1.0 + x) - -1.0) + x)) * 0.5;
	} else {
		tmp = (((eps * x) - -1.0) - t_0) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (math.pow(eps, -1.0) - 1.0) * math.exp(((-1.0 - eps) * x))
	tmp = 0
	if ((((1.0 + math.pow(eps, -1.0)) * math.exp(((-1.0 + eps) * x))) - t_0) / 2.0) <= 1.0:
		tmp = (math.exp(-x) * (((1.0 + x) - -1.0) + x)) * 0.5
	else:
		tmp = (((eps * x) - -1.0) - t_0) / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64((eps ^ -1.0) - 1.0) * exp(Float64(Float64(-1.0 - eps) * x)))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(1.0 + (eps ^ -1.0)) * exp(Float64(Float64(-1.0 + eps) * x))) - t_0) / 2.0) <= 1.0)
		tmp = Float64(Float64(exp(Float64(-x)) * Float64(Float64(Float64(1.0 + x) - -1.0) + x)) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(eps * x) - -1.0) - t_0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((eps ^ -1.0) - 1.0) * exp(((-1.0 - eps) * x));
	tmp = 0.0;
	if (((((1.0 + (eps ^ -1.0)) * exp(((-1.0 + eps) * x))) - t_0) / 2.0) <= 1.0)
		tmp = (exp(-x) * (((1.0 + x) - -1.0) + x)) * 0.5;
	else
		tmp = (((eps * x) - -1.0) - t_0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(1.0 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], 1.0], N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(N[(N[(1.0 + x), $MachinePrecision] - -1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(eps * x), $MachinePrecision] - -1.0), $MachinePrecision] - t$95$0), $MachinePrecision] / 2.0), $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)) < 1

   1. Initial program 59.0%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   if 1 < (/.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 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-lft-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 47.2

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

   5. Applied rewrites 47.2%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 47.2%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2} \leq 1:\\ \;\;\;\;\left(e^{-x} \cdot \left(\left(\left(1 + x\right) - -1\right) + x\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\varepsilon \cdot x - -1\right) - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2} \leq 0:\\ \;\;\;\;\left(e^{-x} \cdot \left(\left(\left(1 + x\right) - -1\right) + x\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left|0.3333333333333333 \cdot x\right| - 0.5, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (/
       (-
        (* (+ 1.0 (pow eps -1.0)) (exp (* (+ -1.0 eps) x)))
        (* (- (pow eps -1.0) 1.0) (exp (* (- -1.0 eps) x))))
       2.0)
      0.0)
   (* (* (exp (- x)) (+ (- (+ 1.0 x) -1.0) x)) 0.5)
   (fma (- (fabs (* 0.3333333333333333 x)) 0.5) (* x x) 1.0)))

C

double code(double x, double eps) {
	double tmp;
	if (((((1.0 + pow(eps, -1.0)) * exp(((-1.0 + eps) * x))) - ((pow(eps, -1.0) - 1.0) * exp(((-1.0 - eps) * x)))) / 2.0) <= 0.0) {
		tmp = (exp(-x) * (((1.0 + x) - -1.0) + x)) * 0.5;
	} else {
		tmp = fma((fabs((0.3333333333333333 * x)) - 0.5), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(1.0 + (eps ^ -1.0)) * exp(Float64(Float64(-1.0 + eps) * x))) - Float64(Float64((eps ^ -1.0) - 1.0) * exp(Float64(Float64(-1.0 - eps) * x)))) / 2.0) <= 0.0)
		tmp = Float64(Float64(exp(Float64(-x)) * Float64(Float64(Float64(1.0 + x) - -1.0) + x)) * 0.5);
	else
		tmp = fma(Float64(abs(Float64(0.3333333333333333 * x)) - 0.5), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[(N[(N[(1.0 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0], N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(N[(N[(1.0 + x), $MachinePrecision] - -1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Abs[N[(0.3333333333333333 * x), $MachinePrecision]], $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $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)) < 0.0

   1. Initial program 41.5%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   if 0.0 < (/.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 100.0%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 29.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 44.3%

      \[\leadsto \mathsf{fma}\left(0.6666666666666666 \cdot x - 1, x \cdot x, 2\right) \cdot 0.5 \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 44.3%

       \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, \color{blue}{x \cdot x}, 1\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 61.7%

       \[\leadsto \mathsf{fma}\left(\left|0.3333333333333333 \cdot x\right| - 0.5, x \cdot x, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2} \leq 0:\\ \;\;\;\;\left(e^{-x} \cdot \left(\left(\left(1 + x\right) - -1\right) + x\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left|0.3333333333333333 \cdot x\right| - 0.5, x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 40000000000:\\ \;\;\;\;\mathsf{fma}\left(\left|0.3333333333333333 \cdot x\right| - 0.5, x \cdot x, 1\right)\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+212}:\\ \;\;\;\;\frac{\left({\varepsilon}^{-1} - -1\right) - \left({\varepsilon}^{-1} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 40000000000.0)
   (fma (- (fabs (* 0.3333333333333333 x)) 0.5) (* x x) 1.0)
   (if (<= x 3.6e+212)
     (/ (- (- (pow eps -1.0) -1.0) (- (pow eps -1.0) 1.0)) 2.0)
     (fma (- (* 0.3333333333333333 x) 0.5) (* x x) 1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 40000000000.0) {
		tmp = fma((fabs((0.3333333333333333 * x)) - 0.5), (x * x), 1.0);
	} else if (x <= 3.6e+212) {
		tmp = ((pow(eps, -1.0) - -1.0) - (pow(eps, -1.0) - 1.0)) / 2.0;
	} else {
		tmp = fma(((0.3333333333333333 * x) - 0.5), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 40000000000.0)
		tmp = fma(Float64(abs(Float64(0.3333333333333333 * x)) - 0.5), Float64(x * x), 1.0);
	elseif (x <= 3.6e+212)
		tmp = Float64(Float64(Float64((eps ^ -1.0) - -1.0) - Float64((eps ^ -1.0) - 1.0)) / 2.0);
	else
		tmp = fma(Float64(Float64(0.3333333333333333 * x) - 0.5), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 40000000000.0], N[(N[(N[Abs[N[(0.3333333333333333 * x), $MachinePrecision]], $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 3.6e+212], N[(N[(N[(N[Power[eps, -1.0], $MachinePrecision] - -1.0), $MachinePrecision] - N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(0.3333333333333333 * x), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 4e10

   1. Initial program 67.9%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 57.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 57.7%

      \[\leadsto \mathsf{fma}\left(0.6666666666666666 \cdot x - 1, x \cdot x, 2\right) \cdot 0.5 \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 57.7%

       \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, \color{blue}{x \cdot x}, 1\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 72.4%

       \[\leadsto \mathsf{fma}\left(\left|0.3333333333333333 \cdot x\right| - 0.5, x \cdot x, 1\right) \]

   if 4e10 < x < 3.6e212

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 22.3

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

   5. Applied rewrites 22.3%

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

   6. Taylor expanded in x around 0

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

      1. *-inverses N/A

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

      2. div-add N/A

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

      3. +-commutative N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. mul-1-neg N/A

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

      8. div-sub N/A

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

      9. mul-1-neg N/A

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

      10. *-lft-identity N/A

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

      11. times-frac N/A

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

      12. metadata-eval N/A

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

      13. *-inverses N/A

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

      14. metadata-eval N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 66.4

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

   8. Applied rewrites 66.4%

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

   if 3.6e212 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 35.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 65.8%

      \[\leadsto \mathsf{fma}\left(0.6666666666666666 \cdot x - 1, x \cdot x, 2\right) \cdot 0.5 \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 65.8%

       \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, \color{blue}{x \cdot x}, 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 40000000000:\\ \;\;\;\;\mathsf{fma}\left(\left|0.3333333333333333 \cdot x\right| - 0.5, x \cdot x, 1\right)\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+212}:\\ \;\;\;\;\frac{\left({\varepsilon}^{-1} - -1\right) - \left({\varepsilon}^{-1} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 40000000000:\\ \;\;\;\;\mathsf{fma}\left(\left|0.3333333333333333 \cdot x\right| - 0.5, x \cdot x, 1\right)\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+212}:\\ \;\;\;\;\frac{{\varepsilon}^{-1} - \left({\varepsilon}^{-1} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 40000000000.0)
   (fma (- (fabs (* 0.3333333333333333 x)) 0.5) (* x x) 1.0)
   (if (<= x 3.6e+212)
     (/ (- (pow eps -1.0) (- (pow eps -1.0) 1.0)) 2.0)
     (fma (- (* 0.3333333333333333 x) 0.5) (* x x) 1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 40000000000.0) {
		tmp = fma((fabs((0.3333333333333333 * x)) - 0.5), (x * x), 1.0);
	} else if (x <= 3.6e+212) {
		tmp = (pow(eps, -1.0) - (pow(eps, -1.0) - 1.0)) / 2.0;
	} else {
		tmp = fma(((0.3333333333333333 * x) - 0.5), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 40000000000.0)
		tmp = fma(Float64(abs(Float64(0.3333333333333333 * x)) - 0.5), Float64(x * x), 1.0);
	elseif (x <= 3.6e+212)
		tmp = Float64(Float64((eps ^ -1.0) - Float64((eps ^ -1.0) - 1.0)) / 2.0);
	else
		tmp = fma(Float64(Float64(0.3333333333333333 * x) - 0.5), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 40000000000.0], N[(N[(N[Abs[N[(0.3333333333333333 * x), $MachinePrecision]], $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 3.6e+212], N[(N[(N[Power[eps, -1.0], $MachinePrecision] - N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(0.3333333333333333 * x), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 4e10

   1. Initial program 67.9%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 57.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 57.7%

      \[\leadsto \mathsf{fma}\left(0.6666666666666666 \cdot x - 1, x \cdot x, 2\right) \cdot 0.5 \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 57.7%

       \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, \color{blue}{x \cdot x}, 1\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 72.4%

       \[\leadsto \mathsf{fma}\left(\left|0.3333333333333333 \cdot x\right| - 0.5, x \cdot x, 1\right) \]

   if 4e10 < x < 3.6e212

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 22.3

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

   5. Applied rewrites 22.3%

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

   6. Taylor expanded in x around 0

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

      1. *-inverses N/A

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

      2. div-add N/A

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

      3. +-commutative N/A

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

      4. *-lft-identity N/A

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

      5. metadata-eval N/A

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

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

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

      7. mul-1-neg N/A

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

      8. div-sub N/A

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

      9. mul-1-neg N/A

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

      10. *-lft-identity N/A

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

      11. times-frac N/A

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

      12. metadata-eval N/A

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

      13. *-inverses N/A

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

      14. metadata-eval N/A

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

      15. lower--.f64 N/A

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

      16. lower-/.f64 66.4

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

   8. Applied rewrites 66.4%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 66.4%

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

   if 3.6e212 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 35.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 65.8%

      \[\leadsto \mathsf{fma}\left(0.6666666666666666 \cdot x - 1, x \cdot x, 2\right) \cdot 0.5 \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 65.8%

       \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, \color{blue}{x \cdot x}, 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 40000000000:\\ \;\;\;\;\mathsf{fma}\left(\left|0.3333333333333333 \cdot x\right| - 0.5, x \cdot x, 1\right)\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{+212}:\\ \;\;\;\;\frac{{\varepsilon}^{-1} - \left({\varepsilon}^{-1} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.1% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left|0.3333333333333333 \cdot x\right| - 0.5, x \cdot x, 1\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma (- (fabs (* 0.3333333333333333 x)) 0.5) (* x x) 1.0))

C

double code(double x, double eps) {
	return fma((fabs((0.3333333333333333 * x)) - 0.5), (x * x), 1.0);
}

Julia

function code(x, eps)
	return fma(Float64(abs(Float64(0.3333333333333333 * x)) - 0.5), Float64(x * x), 1.0)
end

Wolfram

code[x_, eps_] := N[(N[(N[Abs[N[(0.3333333333333333 * x), $MachinePrecision]], $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 76.9%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 57.3%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 51.0%

   \[\leadsto \mathsf{fma}\left(0.6666666666666666 \cdot x - 1, x \cdot x, 2\right) \cdot 0.5 \]

9. Taylor expanded in x around 0

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

11. Applied rewrites 51.0%

    \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, \color{blue}{x \cdot x}, 1\right) \]
12. Step-by-step derivation

13. Applied rewrites 61.5%

    \[\leadsto \mathsf{fma}\left(\left|0.3333333333333333 \cdot x\right| - 0.5, x \cdot x, 1\right) \]
14. Add Preprocessing

Alternative 9: 52.8% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, x \cdot x, 1\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma (- (* 0.3333333333333333 x) 0.5) (* x x) 1.0))

C

double code(double x, double eps) {
	return fma(((0.3333333333333333 * x) - 0.5), (x * x), 1.0);
}

Julia

function code(x, eps)
	return fma(Float64(Float64(0.3333333333333333 * x) - 0.5), Float64(x * x), 1.0)
end

Wolfram

code[x_, eps_] := N[(N[(N[(0.3333333333333333 * x), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 76.9%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 57.3%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 51.0%

   \[\leadsto \mathsf{fma}\left(0.6666666666666666 \cdot x - 1, x \cdot x, 2\right) \cdot 0.5 \]

9. Taylor expanded in x around 0

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

11. Applied rewrites 51.0%

    \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, \color{blue}{x \cdot x}, 1\right) \]
12. Add Preprocessing

Alternative 10: 43.9% accurate, 273.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 1.0)

C

double code(double x, double eps) {
	return 1.0;
}

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
end function

Java

public static double code(double x, double eps) {
	return 1.0;
}

Python

def code(x, eps):
	return 1.0

Julia

function code(x, eps)
	return 1.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 1.0;
end

Wolfram

code[x_, eps_] := 1.0

Derivation

1. Initial program 76.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 42.9%

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

Reproduce

herbie shell --seed 2024359 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  :precision binary64
  (/ (- (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x)))) (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x))))) 2.0))