NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.8% → 99.6%

Time: 10.9s

Alternatives: 16

Speedup: 0.8×

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

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.3× speedup

Math

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

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) (/ (* 0.5 (+ (+ x x) 2.0)) (exp x)) 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 = (0.5 * ((x + x) + 2.0)) / exp(x);
	} 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 = (0.5d0 * ((x + x) + 2.0d0)) / exp(x)
    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 = (0.5 * ((x + x) + 2.0)) / Math.exp(x);
	} 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 = (0.5 * ((x + x) + 2.0)) / math.exp(x)
	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(0.5 * Float64(Float64(x + x) + 2.0)) / exp(x));
	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 = (0.5 * ((x + x) + 2.0)) / exp(x);
	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[(0.5 * N[(N[(x + x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $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 31.4%

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

   3. Taylor expanded in eps around 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%

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

   7. Applied rewrites 100.0%

      \[\leadsto \frac{\left(\left(x + 2\right) + x\right) \cdot 1}{e^{x}} \cdot 0.5 \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

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

   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 98.6%

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

4. Final simplification 99.2%

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

Alternative 2: 63.1% accurate, 0.3× speedup

Math

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

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)) (+ 2.0 x)) 0.5)
     (/
      (- (fma (fma x (pow eps -1.0) x) (- eps 1.0) (+ (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) * (2.0 + x)) * 0.5;
	} else {
		tmp = (fma(fma(x, pow(eps, -1.0), x), (eps - 1.0), (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(2.0 + x)) * 0.5);
	else
		tmp = Float64(Float64(fma(fma(x, (eps ^ -1.0), x), Float64(eps - 1.0), Float64((eps ^ -1.0) + 1.0)) - t_0) / 2.0);
	end
	return 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[(2.0 + x), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(x * N[Power[eps, -1.0], $MachinePrecision] + x), $MachinePrecision] * N[(eps - 1.0), $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 50.0%

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

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.9%

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

   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 98.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 52.0

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

   5. Applied rewrites 52.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 18.7

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

   8. Applied rewrites 18.7%

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

4. Final simplification 67.2%

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

Alternative 3: 78.1% accurate, 0.4× speedup

Math

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

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)
      2.0)
   (/ (* 0.5 (+ (+ x x) 2.0)) (exp x))
   (/ (- (+ (pow eps -1.0) 1.0) (- (exp (- (fma eps x x))))) 2.0)))

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) <= 2.0) {
		tmp = (0.5 * ((x + x) + 2.0)) / exp(x);
	} else {
		tmp = ((pow(eps, -1.0) + 1.0) - -exp(-fma(eps, x, x))) / 2.0;
	}
	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) <= 2.0)
		tmp = Float64(Float64(0.5 * Float64(Float64(x + x) + 2.0)) / exp(x));
	else
		tmp = Float64(Float64(Float64((eps ^ -1.0) + 1.0) - Float64(-exp(Float64(-fma(eps, x, x))))) / 2.0);
	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], 2.0], N[(N[(0.5 * N[(N[(x + x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Power[eps, -1.0], $MachinePrecision] + 1.0), $MachinePrecision] - (-N[Exp[(-N[(eps * x + x), $MachinePrecision])], $MachinePrecision])), $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 50.0%

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

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

   7. Applied rewrites 100.0%

      \[\leadsto \frac{\left(\left(x + 2\right) + x\right) \cdot 1}{e^{x}} \cdot 0.5 \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

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

   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 98.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 51.1

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

   5. Applied rewrites 51.1%

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

   6. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 49.1

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

   8. Applied rewrites 49.1%

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

   10. Applied rewrites 49.1%

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

4. Final simplification 79.9%

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

Alternative 4: 60.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2} \leq 0:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (/
       (-
        (* (+ 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)
   (* (/ 2.0 (fma (fma (fma 0.16666666666666666 x 0.5) x 1.0) x 1.0)) 0.5)
   (fma (- (* 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 = (2.0 / fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0)) * 0.5;
	} else {
		tmp = fma(((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(2.0 / fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0)) * 0.5);
	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[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[(2.0 / N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(0.3333333333333333 * x), $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 31.4%

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

   3. Taylor expanded in eps around 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%

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

   7. Applied rewrites 100.0%

      \[\leadsto \frac{\left(\left(x + 2\right) + x\right) \cdot 1}{e^{x}} \cdot 0.5 \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 98.4%

       \[\leadsto \frac{2}{e^{x}} \cdot 0.5 \]

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 85.8%

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

   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 98.6%

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

   3. Taylor expanded in eps around 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 31.8%

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

   7. Applied rewrites 31.8%

      \[\leadsto \frac{\left(\left(x + 2\right) + x\right) \cdot 1}{e^{x}} \cdot 0.5 \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 42.4%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2} \leq 0:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.6% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (/
       (-
        (* (+ 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)
   (* (/ 2.0 (fma (fma 0.5 x 1.0) x 1.0)) 0.5)
   (fma (- (* 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 = (2.0 / fma(fma(0.5, x, 1.0), x, 1.0)) * 0.5;
	} else {
		tmp = fma(((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(2.0 / fma(fma(0.5, x, 1.0), x, 1.0)) * 0.5);
	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[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[(2.0 / N[(N[(0.5 * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(0.3333333333333333 * x), $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 31.4%

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

   3. Taylor expanded in eps around 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%

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

   7. Applied rewrites 100.0%

      \[\leadsto \frac{\left(\left(x + 2\right) + x\right) \cdot 1}{e^{x}} \cdot 0.5 \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 98.4%

       \[\leadsto \frac{2}{e^{x}} \cdot 0.5 \]

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 84.0%

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

   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 98.6%

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

   3. Taylor expanded in eps around 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 31.8%

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

   7. Applied rewrites 31.8%

      \[\leadsto \frac{\left(\left(x + 2\right) + x\right) \cdot 1}{e^{x}} \cdot 0.5 \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 42.4%

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

4. Final simplification 60.8%

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

Alternative 6: 77.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\varepsilon}^{-1} - 1\\ t_1 := {\varepsilon}^{-1} + 1\\ \mathbf{if}\;\varepsilon \leq -1.85 \cdot 10^{+170}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x - x} \cdot t\_1 - t\_0}{2}\\ \mathbf{elif}\;\varepsilon \leq -29:\\ \;\;\;\;\frac{t\_1 - t\_0 \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.15 \cdot 10^{+15}:\\ \;\;\;\;\frac{0.5 \cdot \left(\left(x + x\right) + 2\right)}{e^{x}}\\ \mathbf{elif}\;\varepsilon \leq 3.9 \cdot 10^{+208}:\\ \;\;\;\;\frac{t\_1 - \left(-e^{-\mathsf{fma}\left(\varepsilon, x, x\right)}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\varepsilon}^{-1} \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - t\_0}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow eps -1.0) 1.0)) (t_1 (+ (pow eps -1.0) 1.0)))
   (if (<= eps -1.85e+170)
     (/ (- (* (exp (- (* eps x) x)) t_1) t_0) 2.0)
     (if (<= eps -29.0)
       (/ (- t_1 (* t_0 (exp (* (- -1.0 eps) x)))) 2.0)
       (if (<= eps 1.15e+15)
         (/ (* 0.5 (+ (+ x x) 2.0)) (exp x))
         (if (<= eps 3.9e+208)
           (/ (- t_1 (- (exp (- (fma eps x x))))) 2.0)
           (/ (- (* (pow eps -1.0) (exp (* (+ -1.0 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 = pow(eps, -1.0) + 1.0;
	double tmp;
	if (eps <= -1.85e+170) {
		tmp = ((exp(((eps * x) - x)) * t_1) - t_0) / 2.0;
	} else if (eps <= -29.0) {
		tmp = (t_1 - (t_0 * exp(((-1.0 - eps) * x)))) / 2.0;
	} else if (eps <= 1.15e+15) {
		tmp = (0.5 * ((x + x) + 2.0)) / exp(x);
	} else if (eps <= 3.9e+208) {
		tmp = (t_1 - -exp(-fma(eps, x, x))) / 2.0;
	} else {
		tmp = ((pow(eps, -1.0) * exp(((-1.0 + eps) * x))) - t_0) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((eps ^ -1.0) - 1.0)
	t_1 = Float64((eps ^ -1.0) + 1.0)
	tmp = 0.0
	if (eps <= -1.85e+170)
		tmp = Float64(Float64(Float64(exp(Float64(Float64(eps * x) - x)) * t_1) - t_0) / 2.0);
	elseif (eps <= -29.0)
		tmp = Float64(Float64(t_1 - Float64(t_0 * exp(Float64(Float64(-1.0 - eps) * x)))) / 2.0);
	elseif (eps <= 1.15e+15)
		tmp = Float64(Float64(0.5 * Float64(Float64(x + x) + 2.0)) / exp(x));
	elseif (eps <= 3.9e+208)
		tmp = Float64(Float64(t_1 - Float64(-exp(Float64(-fma(eps, x, x))))) / 2.0);
	else
		tmp = Float64(Float64(Float64((eps ^ -1.0) * exp(Float64(Float64(-1.0 + eps) * x))) - t_0) / 2.0);
	end
	return 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[(N[Power[eps, -1.0], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[eps, -1.85e+170], N[(N[(N[(N[Exp[N[(N[(eps * x), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] - t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps, -29.0], N[(N[(t$95$1 - N[(t$95$0 * N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps, 1.15e+15], N[(N[(0.5 * N[(N[(x + x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3.9e+208], N[(N[(t$95$1 - (-N[Exp[(-N[(eps * x + x), $MachinePrecision])], $MachinePrecision])), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[Power[eps, -1.0], $MachinePrecision] * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if eps < -1.84999999999999994e170

   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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 41.2

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

   5. Applied rewrites 41.2%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 13.5

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

   8. Applied rewrites 13.5%

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

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{e^{\varepsilon \cdot x - x} \cdot \left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   10. 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. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-+.f64 N/A

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

       7. lower-/.f64 72.3

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

   11. Applied rewrites 72.3%

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

   if -1.84999999999999994e170 < eps < -29

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 81.7

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

   5. Applied rewrites 81.7%

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

   if -29 < eps < 1.15e15

   1. Initial program 34.8%

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

   3. Taylor expanded in eps around 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%

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

   7. Applied rewrites 100.0%

      \[\leadsto \frac{\left(\left(x + 2\right) + x\right) \cdot 1}{e^{x}} \cdot 0.5 \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

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

   if 1.15e15 < eps < 3.9000000000000001e208

   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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 76.7

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

   5. Applied rewrites 76.7%

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

   6. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 76.7

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

   8. Applied rewrites 76.7%

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

   10. Applied rewrites 76.7%

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

   if 3.9000000000000001e208 < eps

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 70.5

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

   5. Applied rewrites 70.5%

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

   6. Taylor expanded in eps around 0

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

      1. lower-/.f64 70.5

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

   8. Applied rewrites 70.5%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.85 \cdot 10^{+170}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x - x} \cdot \left({\varepsilon}^{-1} + 1\right) - \left({\varepsilon}^{-1} - 1\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq -29:\\ \;\;\;\;\frac{\left({\varepsilon}^{-1} + 1\right) - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.15 \cdot 10^{+15}:\\ \;\;\;\;\frac{0.5 \cdot \left(\left(x + x\right) + 2\right)}{e^{x}}\\ \mathbf{elif}\;\varepsilon \leq 3.9 \cdot 10^{+208}:\\ \;\;\;\;\frac{\left({\varepsilon}^{-1} + 1\right) - \left(-e^{-\mathsf{fma}\left(\varepsilon, x, x\right)}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\varepsilon}^{-1} \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\varepsilon}^{-1} - 1\\ t_1 := {\varepsilon}^{-1} + 1\\ t_2 := \frac{t\_1 - \left(-e^{-\mathsf{fma}\left(\varepsilon, x, x\right)}\right)}{2}\\ \mathbf{if}\;\varepsilon \leq -1.85 \cdot 10^{+170}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x - x} \cdot t\_1 - t\_0}{2}\\ \mathbf{elif}\;\varepsilon \leq -4 \cdot 10^{+22}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\varepsilon \leq 1.15 \cdot 10^{+15}:\\ \;\;\;\;\frac{0.5 \cdot \left(\left(x + x\right) + 2\right)}{e^{x}}\\ \mathbf{elif}\;\varepsilon \leq 3.9 \cdot 10^{+208}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{{\varepsilon}^{-1} \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - t\_0}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow eps -1.0) 1.0))
        (t_1 (+ (pow eps -1.0) 1.0))
        (t_2 (/ (- t_1 (- (exp (- (fma eps x x))))) 2.0)))
   (if (<= eps -1.85e+170)
     (/ (- (* (exp (- (* eps x) x)) t_1) t_0) 2.0)
     (if (<= eps -4e+22)
       t_2
       (if (<= eps 1.15e+15)
         (/ (* 0.5 (+ (+ x x) 2.0)) (exp x))
         (if (<= eps 3.9e+208)
           t_2
           (/ (- (* (pow eps -1.0) (exp (* (+ -1.0 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 = pow(eps, -1.0) + 1.0;
	double t_2 = (t_1 - -exp(-fma(eps, x, x))) / 2.0;
	double tmp;
	if (eps <= -1.85e+170) {
		tmp = ((exp(((eps * x) - x)) * t_1) - t_0) / 2.0;
	} else if (eps <= -4e+22) {
		tmp = t_2;
	} else if (eps <= 1.15e+15) {
		tmp = (0.5 * ((x + x) + 2.0)) / exp(x);
	} else if (eps <= 3.9e+208) {
		tmp = t_2;
	} else {
		tmp = ((pow(eps, -1.0) * exp(((-1.0 + eps) * x))) - t_0) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((eps ^ -1.0) - 1.0)
	t_1 = Float64((eps ^ -1.0) + 1.0)
	t_2 = Float64(Float64(t_1 - Float64(-exp(Float64(-fma(eps, x, x))))) / 2.0)
	tmp = 0.0
	if (eps <= -1.85e+170)
		tmp = Float64(Float64(Float64(exp(Float64(Float64(eps * x) - x)) * t_1) - t_0) / 2.0);
	elseif (eps <= -4e+22)
		tmp = t_2;
	elseif (eps <= 1.15e+15)
		tmp = Float64(Float64(0.5 * Float64(Float64(x + x) + 2.0)) / exp(x));
	elseif (eps <= 3.9e+208)
		tmp = t_2;
	else
		tmp = Float64(Float64(Float64((eps ^ -1.0) * exp(Float64(Float64(-1.0 + eps) * x))) - t_0) / 2.0);
	end
	return 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[(N[Power[eps, -1.0], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 - (-N[Exp[(-N[(eps * x + x), $MachinePrecision])], $MachinePrecision])), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[eps, -1.85e+170], N[(N[(N[(N[Exp[N[(N[(eps * x), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] - t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps, -4e+22], t$95$2, If[LessEqual[eps, 1.15e+15], N[(N[(0.5 * N[(N[(x + x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3.9e+208], t$95$2, N[(N[(N[(N[Power[eps, -1.0], $MachinePrecision] * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if eps < -1.84999999999999994e170

   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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 41.2

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

   5. Applied rewrites 41.2%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 13.5

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

   8. Applied rewrites 13.5%

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

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{e^{\varepsilon \cdot x - x} \cdot \left(1 + \frac{1}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   10. 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. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-+.f64 N/A

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

       7. lower-/.f64 72.3

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

   11. Applied rewrites 72.3%

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

   if -1.84999999999999994e170 < eps < -4e22 or 1.15e15 < eps < 3.9000000000000001e208

   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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 79.1

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 79.1

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

   8. Applied rewrites 79.1%

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

   10. Applied rewrites 79.1%

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

   if -4e22 < eps < 1.15e15

   1. Initial program 36.4%

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

   3. Taylor expanded in eps around 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 99.2%

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

   7. Applied rewrites 99.2%

      \[\leadsto \frac{\left(\left(x + 2\right) + x\right) \cdot 1}{e^{x}} \cdot 0.5 \]
   8. Step-by-step derivation

   9. Applied rewrites 99.2%

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

   if 3.9000000000000001e208 < eps

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 70.5

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

   5. Applied rewrites 70.5%

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

   6. Taylor expanded in eps around 0

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

      1. lower-/.f64 70.5

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

   8. Applied rewrites 70.5%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.85 \cdot 10^{+170}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x - x} \cdot \left({\varepsilon}^{-1} + 1\right) - \left({\varepsilon}^{-1} - 1\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq -4 \cdot 10^{+22}:\\ \;\;\;\;\frac{\left({\varepsilon}^{-1} + 1\right) - \left(-e^{-\mathsf{fma}\left(\varepsilon, x, x\right)}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.15 \cdot 10^{+15}:\\ \;\;\;\;\frac{0.5 \cdot \left(\left(x + x\right) + 2\right)}{e^{x}}\\ \mathbf{elif}\;\varepsilon \leq 3.9 \cdot 10^{+208}:\\ \;\;\;\;\frac{\left({\varepsilon}^{-1} + 1\right) - \left(-e^{-\mathsf{fma}\left(\varepsilon, x, x\right)}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\varepsilon}^{-1} \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left({\varepsilon}^{-1} + 1\right) - \left(-e^{-\mathsf{fma}\left(\varepsilon, x, x\right)}\right)}{2}\\ \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{+22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\varepsilon \leq 1.15 \cdot 10^{+15}:\\ \;\;\;\;\frac{0.5 \cdot \left(\left(x + x\right) + 2\right)}{e^{x}}\\ \mathbf{elif}\;\varepsilon \leq 3.9 \cdot 10^{+208}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{{\varepsilon}^{-1} \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (- (+ (pow eps -1.0) 1.0) (- (exp (- (fma eps x x))))) 2.0)))
   (if (<= eps -4e+22)
     t_0
     (if (<= eps 1.15e+15)
       (/ (* 0.5 (+ (+ x x) 2.0)) (exp x))
       (if (<= eps 3.9e+208)
         t_0
         (/
          (-
           (* (pow eps -1.0) (exp (* (+ -1.0 eps) x)))
           (- (pow eps -1.0) 1.0))
          2.0))))))

C

double code(double x, double eps) {
	double t_0 = ((pow(eps, -1.0) + 1.0) - -exp(-fma(eps, x, x))) / 2.0;
	double tmp;
	if (eps <= -4e+22) {
		tmp = t_0;
	} else if (eps <= 1.15e+15) {
		tmp = (0.5 * ((x + x) + 2.0)) / exp(x);
	} else if (eps <= 3.9e+208) {
		tmp = t_0;
	} else {
		tmp = ((pow(eps, -1.0) * exp(((-1.0 + eps) * x))) - (pow(eps, -1.0) - 1.0)) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(Float64((eps ^ -1.0) + 1.0) - Float64(-exp(Float64(-fma(eps, x, x))))) / 2.0)
	tmp = 0.0
	if (eps <= -4e+22)
		tmp = t_0;
	elseif (eps <= 1.15e+15)
		tmp = Float64(Float64(0.5 * Float64(Float64(x + x) + 2.0)) / exp(x));
	elseif (eps <= 3.9e+208)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64((eps ^ -1.0) * exp(Float64(Float64(-1.0 + eps) * x))) - Float64((eps ^ -1.0) - 1.0)) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(N[Power[eps, -1.0], $MachinePrecision] + 1.0), $MachinePrecision] - (-N[Exp[(-N[(eps * x + x), $MachinePrecision])], $MachinePrecision])), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[eps, -4e+22], t$95$0, If[LessEqual[eps, 1.15e+15], N[(N[(0.5 * N[(N[(x + x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 3.9e+208], t$95$0, N[(N[(N[(N[Power[eps, -1.0], $MachinePrecision] * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -4e22 or 1.15e15 < eps < 3.9000000000000001e208

   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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 69.3

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

   5. Applied rewrites 69.3%

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

   6. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 69.3

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

   8. Applied rewrites 69.3%

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

   10. Applied rewrites 69.3%

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

   if -4e22 < eps < 1.15e15

   1. Initial program 36.4%

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

   3. Taylor expanded in eps around 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 99.2%

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

   7. Applied rewrites 99.2%

      \[\leadsto \frac{\left(\left(x + 2\right) + x\right) \cdot 1}{e^{x}} \cdot 0.5 \]
   8. Step-by-step derivation

   9. Applied rewrites 99.2%

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

   if 3.9000000000000001e208 < eps

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 70.5

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

   5. Applied rewrites 70.5%

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

   6. Taylor expanded in eps around 0

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

      1. lower-/.f64 70.5

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

   8. Applied rewrites 70.5%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4 \cdot 10^{+22}:\\ \;\;\;\;\frac{\left({\varepsilon}^{-1} + 1\right) - \left(-e^{-\mathsf{fma}\left(\varepsilon, x, x\right)}\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.15 \cdot 10^{+15}:\\ \;\;\;\;\frac{0.5 \cdot \left(\left(x + x\right) + 2\right)}{e^{x}}\\ \mathbf{elif}\;\varepsilon \leq 3.9 \cdot 10^{+208}:\\ \;\;\;\;\frac{\left({\varepsilon}^{-1} + 1\right) - \left(-e^{-\mathsf{fma}\left(\varepsilon, x, x\right)}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\varepsilon}^{-1} \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\varepsilon}^{-1} + 1\\ \mathbf{if}\;x \leq -150:\\ \;\;\;\;\frac{t\_0 + \left(\frac{-1}{\varepsilon} - -1\right) \cdot \mathsf{fma}\left(-1 - \varepsilon, x, 1\right)}{2}\\ \mathbf{elif}\;x \leq 0.32:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.3333333333333333\right) \cdot x - 0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - \left({\varepsilon}^{-1} - 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (pow eps -1.0) 1.0)))
   (if (<= x -150.0)
     (/ (+ t_0 (* (- (/ -1.0 eps) -1.0) (fma (- -1.0 eps) x 1.0))) 2.0)
     (if (<= x 0.32)
       (fma (- (* (fma -0.125 x 0.3333333333333333) x) 0.5) (* x x) 1.0)
       (/ (- t_0 (- (pow eps -1.0) 1.0)) 2.0)))))

C

double code(double x, double eps) {
	double t_0 = pow(eps, -1.0) + 1.0;
	double tmp;
	if (x <= -150.0) {
		tmp = (t_0 + (((-1.0 / eps) - -1.0) * fma((-1.0 - eps), x, 1.0))) / 2.0;
	} else if (x <= 0.32) {
		tmp = fma(((fma(-0.125, x, 0.3333333333333333) * x) - 0.5), (x * x), 1.0);
	} else {
		tmp = (t_0 - (pow(eps, -1.0) - 1.0)) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((eps ^ -1.0) + 1.0)
	tmp = 0.0
	if (x <= -150.0)
		tmp = Float64(Float64(t_0 + Float64(Float64(Float64(-1.0 / eps) - -1.0) * fma(Float64(-1.0 - eps), x, 1.0))) / 2.0);
	elseif (x <= 0.32)
		tmp = fma(Float64(Float64(fma(-0.125, x, 0.3333333333333333) * x) - 0.5), Float64(x * x), 1.0);
	else
		tmp = Float64(Float64(t_0 - Float64((eps ^ -1.0) - 1.0)) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[eps, -1.0], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -150.0], N[(N[(t$95$0 + N[(N[(N[(-1.0 / eps), $MachinePrecision] - -1.0), $MachinePrecision] * N[(N[(-1.0 - eps), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 0.32], N[(N[(N[(N[(-0.125 * x + 0.3333333333333333), $MachinePrecision] * x), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(t$95$0 - N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -150

   1. Initial program 95.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 62.2

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

   5. Applied rewrites 62.2%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. distribute-lft-in N/A

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

      8. metadata-eval N/A

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

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

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. lower--.f64 24.6

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

   8. Applied rewrites 24.6%

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

   if -150 < x < 0.320000000000000007

   1. Initial program 50.3%

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

   3. Taylor expanded in eps around 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 78.9%

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

   7. Applied rewrites 78.9%

      \[\leadsto \frac{\left(\left(x + 2\right) + x\right) \cdot 1}{e^{x}} \cdot 0.5 \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 78.9%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.3333333333333333\right) \cdot x - 0.5, \color{blue}{x \cdot x}, 1\right) \]

   if 0.320000000000000007 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 22.3

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

   5. Applied rewrites 22.3%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 57.3

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

   8. Applied rewrites 57.3%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -150:\\ \;\;\;\;\frac{\left({\varepsilon}^{-1} + 1\right) + \left(\frac{-1}{\varepsilon} - -1\right) \cdot \mathsf{fma}\left(-1 - \varepsilon, x, 1\right)}{2}\\ \mathbf{elif}\;x \leq 0.32:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.3333333333333333\right) \cdot x - 0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left({\varepsilon}^{-1} + 1\right) - \left({\varepsilon}^{-1} - 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\varepsilon}^{-1} + 1\\ \mathbf{if}\;x \leq -11000000000000:\\ \;\;\;\;\frac{t\_0 - \left(\mathsf{fma}\left(\varepsilon, x, x\right) - 1\right)}{2}\\ \mathbf{elif}\;x \leq 0.32:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.3333333333333333\right) \cdot x - 0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - \left({\varepsilon}^{-1} - 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (pow eps -1.0) 1.0)))
   (if (<= x -11000000000000.0)
     (/ (- t_0 (- (fma eps x x) 1.0)) 2.0)
     (if (<= x 0.32)
       (fma (- (* (fma -0.125 x 0.3333333333333333) x) 0.5) (* x x) 1.0)
       (/ (- t_0 (- (pow eps -1.0) 1.0)) 2.0)))))

C

double code(double x, double eps) {
	double t_0 = pow(eps, -1.0) + 1.0;
	double tmp;
	if (x <= -11000000000000.0) {
		tmp = (t_0 - (fma(eps, x, x) - 1.0)) / 2.0;
	} else if (x <= 0.32) {
		tmp = fma(((fma(-0.125, x, 0.3333333333333333) * x) - 0.5), (x * x), 1.0);
	} else {
		tmp = (t_0 - (pow(eps, -1.0) - 1.0)) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((eps ^ -1.0) + 1.0)
	tmp = 0.0
	if (x <= -11000000000000.0)
		tmp = Float64(Float64(t_0 - Float64(fma(eps, x, x) - 1.0)) / 2.0);
	elseif (x <= 0.32)
		tmp = fma(Float64(Float64(fma(-0.125, x, 0.3333333333333333) * x) - 0.5), Float64(x * x), 1.0);
	else
		tmp = Float64(Float64(t_0 - Float64((eps ^ -1.0) - 1.0)) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[eps, -1.0], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -11000000000000.0], N[(N[(t$95$0 - N[(N[(eps * x + x), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 0.32], N[(N[(N[(N[(-0.125 * x + 0.3333333333333333), $MachinePrecision] * x), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(t$95$0 - N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1e13

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 61.8

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

   5. Applied rewrites 61.8%

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

   6. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 61.8

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

   8. Applied rewrites 61.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 26.2%

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

   if -1.1e13 < x < 0.320000000000000007

   1. Initial program 50.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. 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 78.6%

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

   7. Applied rewrites 78.7%

      \[\leadsto \frac{\left(\left(x + 2\right) + x\right) \cdot 1}{e^{x}} \cdot 0.5 \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 77.4%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.3333333333333333\right) \cdot x - 0.5, \color{blue}{x \cdot x}, 1\right) \]

   if 0.320000000000000007 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 22.3

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

   5. Applied rewrites 22.3%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 57.3

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

   8. Applied rewrites 57.3%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -11000000000000:\\ \;\;\;\;\frac{\left({\varepsilon}^{-1} + 1\right) - \left(\mathsf{fma}\left(\varepsilon, x, x\right) - 1\right)}{2}\\ \mathbf{elif}\;x \leq 0.32:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.3333333333333333\right) \cdot x - 0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left({\varepsilon}^{-1} + 1\right) - \left({\varepsilon}^{-1} - 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -11000000000000:\\ \;\;\;\;\frac{\left({\varepsilon}^{-1} + 1\right) - \left(\mathsf{fma}\left(\varepsilon, x, x\right) - 1\right)}{2}\\ \mathbf{elif}\;x \leq 1.8:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.3333333333333333\right) \cdot x - 0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\varepsilon}^{-1} - \left({\varepsilon}^{-1} - 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -11000000000000.0)
   (/ (- (+ (pow eps -1.0) 1.0) (- (fma eps x x) 1.0)) 2.0)
   (if (<= x 1.8)
     (fma (- (* (fma -0.125 x 0.3333333333333333) x) 0.5) (* x x) 1.0)
     (/ (- (pow eps -1.0) (- (pow eps -1.0) 1.0)) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -11000000000000.0) {
		tmp = ((pow(eps, -1.0) + 1.0) - (fma(eps, x, x) - 1.0)) / 2.0;
	} else if (x <= 1.8) {
		tmp = fma(((fma(-0.125, x, 0.3333333333333333) * x) - 0.5), (x * x), 1.0);
	} else {
		tmp = (pow(eps, -1.0) - (pow(eps, -1.0) - 1.0)) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -11000000000000.0)
		tmp = Float64(Float64(Float64((eps ^ -1.0) + 1.0) - Float64(fma(eps, x, x) - 1.0)) / 2.0);
	elseif (x <= 1.8)
		tmp = fma(Float64(Float64(fma(-0.125, x, 0.3333333333333333) * x) - 0.5), Float64(x * x), 1.0);
	else
		tmp = Float64(Float64((eps ^ -1.0) - Float64((eps ^ -1.0) - 1.0)) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -11000000000000.0], N[(N[(N[(N[Power[eps, -1.0], $MachinePrecision] + 1.0), $MachinePrecision] - N[(N[(eps * x + x), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.8], N[(N[(N[(N[(-0.125 * x + 0.3333333333333333), $MachinePrecision] * x), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[Power[eps, -1.0], $MachinePrecision] - N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1e13

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 61.8

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

   5. Applied rewrites 61.8%

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

   6. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. +-commutative N/A

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

      9. distribute-lft-in N/A

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

      10. *-rgt-identity N/A

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

      11. lower-fma.f64 61.8

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

   8. Applied rewrites 61.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 26.2%

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

   if -1.1e13 < x < 1.80000000000000004

   1. Initial program 50.3%

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

   3. Taylor expanded in eps around 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 78.2%

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

   7. Applied rewrites 78.2%

      \[\leadsto \frac{\left(\left(x + 2\right) + x\right) \cdot 1}{e^{x}} \cdot 0.5 \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 76.9%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.3333333333333333\right) \cdot x - 0.5, \color{blue}{x \cdot x}, 1\right) \]

   if 1.80000000000000004 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 21.0

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

   5. Applied rewrites 21.0%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 58.2

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

   8. Applied rewrites 58.2%

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

   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 58.2%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -11000000000000:\\ \;\;\;\;\frac{\left({\varepsilon}^{-1} + 1\right) - \left(\mathsf{fma}\left(\varepsilon, x, x\right) - 1\right)}{2}\\ \mathbf{elif}\;x \leq 1.8:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.3333333333333333\right) \cdot x - 0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\varepsilon}^{-1} - \left({\varepsilon}^{-1} - 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{2}{e^{x}} \cdot 0.5 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* (/ 2.0 (exp x)) 0.5))

C

double code(double x, double eps) {
	return (2.0 / exp(x)) * 0.5;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (2.0d0 / exp(x)) * 0.5d0
end function

Java

public static double code(double x, double eps) {
	return (2.0 / Math.exp(x)) * 0.5;
}

Python

def code(x, eps):
	return (2.0 / math.exp(x)) * 0.5

Julia

function code(x, eps)
	return Float64(Float64(2.0 / exp(x)) * 0.5)
end

MATLAB

function tmp = code(x, eps)
	tmp = (2.0 / exp(x)) * 0.5;
end

Wolfram

code[x_, eps_] := N[(N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 68.9%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
2. 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 61.9%

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

7. Applied rewrites 61.9%

   \[\leadsto \frac{\left(\left(x + 2\right) + x\right) \cdot 1}{e^{x}} \cdot 0.5 \]

8. Taylor expanded in x around 0

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

10. Applied rewrites 75.7%

    \[\leadsto \frac{2}{e^{x}} \cdot 0.5 \]
11. Add Preprocessing

Alternative 13: 52.2% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+207}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 2e+207)
   (fma (- (* 0.3333333333333333 x) 0.5) (* x x) 1.0)
   (/ x (fma (fma 0.5 x 1.0) x 1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 2e+207) {
		tmp = fma(((0.3333333333333333 * x) - 0.5), (x * x), 1.0);
	} else {
		tmp = x / fma(fma(0.5, x, 1.0), x, 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 2e+207)
		tmp = fma(Float64(Float64(0.3333333333333333 * x) - 0.5), Float64(x * x), 1.0);
	else
		tmp = Float64(x / fma(fma(0.5, x, 1.0), x, 1.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 2e+207], N[(N[(N[(0.3333333333333333 * x), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(x / N[(N[(0.5 * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.0000000000000001e207

   1. Initial program 66.7%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 61.3%

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

   7. Applied rewrites 61.3%

      \[\leadsto \frac{\left(\left(x + 2\right) + x\right) \cdot 1}{e^{x}} \cdot 0.5 \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 56.8%

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

   if 2.0000000000000001e207 < 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 71.0%

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

   7. Applied rewrites 71.0%

      \[\leadsto \frac{\left(\left(x + 2\right) + x\right) \cdot 1}{e^{x}} \cdot 0.5 \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{x}{\color{blue}{e^{x}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 71.0%

       \[\leadsto \frac{x}{\color{blue}{e^{x}}} \]

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 71.0%

       \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 52.5% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+55}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x - 1, x, 1\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 5e+55) 1.0 (* (fma (- (* 0.5 x) 1.0) x 1.0) x)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 5e+55) {
		tmp = 1.0;
	} else {
		tmp = fma(((0.5 * x) - 1.0), x, 1.0) * x;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 5e+55)
		tmp = 1.0;
	else
		tmp = Float64(fma(Float64(Float64(0.5 * x) - 1.0), x, 1.0) * x);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 5e+55], 1.0, N[(N[(N[(N[(0.5 * x), $MachinePrecision] - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.00000000000000046e55

   1. Initial program 62.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 58.8%

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

   if 5.00000000000000046e55 < 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 52.9%

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

   7. Applied rewrites 52.9%

      \[\leadsto \frac{\left(\left(x + 2\right) + x\right) \cdot 1}{e^{x}} \cdot 0.5 \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{x}{\color{blue}{e^{x}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 52.9%

       \[\leadsto \frac{x}{\color{blue}{e^{x}}} \]

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 38.5%

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

Alternative 15: 52.3% 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 68.9%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
2. 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 61.9%

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

7. Applied rewrites 61.9%

   \[\leadsto \frac{\left(\left(x + 2\right) + x\right) \cdot 1}{e^{x}} \cdot 0.5 \]

8. Taylor expanded in x around 0

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

10. Applied rewrites 55.0%

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

Alternative 16: 43.6% 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 68.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 48.8%

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

Reproduce

herbie shell --seed 2024363 
(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))