NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.6% → 99.8%

Time: 6.8s

Alternatives: 15

Speedup: 3.4×

• 37.5% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -1.5360418905749028e+262
  2. eps = 6.454247965482953e-8

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: 73.6% 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: 99.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \left(x - -1\right) \cdot e^{-x}\\ \mathbf{if}\;eps\_m \leq 0.00068:\\ \;\;\;\;\left(t\_0 + t\_0\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x \cdot eps\_m} + e^{-x \cdot eps\_m}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* (- x -1.0) (exp (- x)))))
   (if (<= eps_m 0.00068)
     (* (+ t_0 t_0) 0.5)
     (* (+ (exp (* x eps_m)) (exp (- (* x eps_m)))) 0.5))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (x - -1.0) * exp(-x);
	double tmp;
	if (eps_m <= 0.00068) {
		tmp = (t_0 + t_0) * 0.5;
	} else {
		tmp = (exp((x * eps_m)) + exp(-(x * eps_m))) * 0.5;
	}
	return tmp;
}

Fortran

eps_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x - (-1.0d0)) * exp(-x)
    if (eps_m <= 0.00068d0) then
        tmp = (t_0 + t_0) * 0.5d0
    else
        tmp = (exp((x * eps_m)) + exp(-(x * eps_m))) * 0.5d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = (x - -1.0) * Math.exp(-x);
	double tmp;
	if (eps_m <= 0.00068) {
		tmp = (t_0 + t_0) * 0.5;
	} else {
		tmp = (Math.exp((x * eps_m)) + Math.exp(-(x * eps_m))) * 0.5;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (x - -1.0) * math.exp(-x)
	tmp = 0
	if eps_m <= 0.00068:
		tmp = (t_0 + t_0) * 0.5
	else:
		tmp = (math.exp((x * eps_m)) + math.exp(-(x * eps_m))) * 0.5
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(x - -1.0) * exp(Float64(-x)))
	tmp = 0.0
	if (eps_m <= 0.00068)
		tmp = Float64(Float64(t_0 + t_0) * 0.5);
	else
		tmp = Float64(Float64(exp(Float64(x * eps_m)) + exp(Float64(-Float64(x * eps_m)))) * 0.5);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (x - -1.0) * exp(-x);
	tmp = 0.0;
	if (eps_m <= 0.00068)
		tmp = (t_0 + t_0) * 0.5;
	else
		tmp = (exp((x * eps_m)) + exp(-(x * eps_m))) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(x - -1.0), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps$95$m, 0.00068], N[(N[(t$95$0 + t$95$0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] + N[Exp[(-N[(x * eps$95$m), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 6.8e-4

   1. Initial program 63.7%

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

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 70.8%

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

   if 6.8e-4 < 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 eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in eps around inf

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

       1. *-commutative N/A

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

       2. lift-*.f64 100.0

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

   11. Applied rewrites 100.0%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.00068:\\ \;\;\;\;\left(\left(x - -1\right) \cdot e^{-x} + \left(x - -1\right) \cdot e^{-x}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} + e^{-x \cdot \varepsilon}\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 0.6× speedup

Math

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

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<=
      (/
       (-
        (* (+ 1.0 (/ 1.0 eps_m)) (exp (* (+ -1.0 eps_m) x)))
        (* (- (/ 1.0 eps_m) 1.0) (exp (* (- -1.0 eps_m) x))))
       2.0)
      0.0)
   (exp (- x))
   (* (+ (exp (* x eps_m)) (exp (- (* x eps_m)))) 0.5)))

C

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

Fortran

eps_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (((((1.0d0 + (1.0d0 / eps_m)) * exp((((-1.0d0) + eps_m) * x))) - (((1.0d0 / eps_m) - 1.0d0) * exp((((-1.0d0) - eps_m) * x)))) / 2.0d0) <= 0.0d0) then
        tmp = exp(-x)
    else
        tmp = (exp((x * eps_m)) + exp(-(x * eps_m))) * 0.5d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[N[(N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(-1.0 + eps$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[N[(N[(-1.0 - eps$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0], N[Exp[(-x)], $MachinePrecision], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] + N[Exp[(-N[(x * eps$95$m), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * 0.5), $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 37.2%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 95.7%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 69.7

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

   8. Applied rewrites 69.7%

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

   9. Taylor expanded in eps around 0

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

       1. lower-exp.f64 N/A

          \[\leadsto e^{\mathsf{neg}\left(x\right)} \]

       2. lift-neg.f64 95.7

          \[\leadsto e^{-x} \]

   11. Applied rewrites 95.7%

       \[\leadsto e^{-x} \]

   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 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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in eps around inf

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

       1. *-commutative N/A

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

       2. lift-*.f64 100.0

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

   11. Applied rewrites 100.0%

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

4. Final simplification 98.2%

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

Alternative 3: 98.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \left(e^{x \cdot \left(-1 + eps\_m\right)} + e^{-\mathsf{fma}\left(x, eps\_m, x\right)}\right) \cdot 0.5 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (* (+ (exp (* x (+ -1.0 eps_m))) (exp (- (fma x eps_m x)))) 0.5))

C

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

Julia

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := N[(N[(N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[(-N[(x * eps$95$m + x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 73.5%

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

3. Taylor expanded in eps around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 98.2%

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

6. Final simplification 98.2%

   \[\leadsto \left(e^{x \cdot \left(-1 + \varepsilon\right)} + e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right) \cdot 0.5 \]
7. Add Preprocessing

Alternative 4: 84.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-241}:\\ \;\;\;\;\left(1 + e^{-x \cdot eps\_m}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 850000:\\ \;\;\;\;\left(e^{x \cdot eps\_m} - \left(\mathsf{fma}\left(x, eps\_m, x\right) - 1\right)\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+71} \lor \neg \left(x \leq 4.5 \cdot 10^{+123} \lor \neg \left(x \leq 2.05 \cdot 10^{+167}\right)\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) \cdot 1 - \frac{\mathsf{fma}\left(-1, eps\_m, 1\right)}{eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x \cdot \left(-1 + eps\_m\right)} - -1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -8.2e-241)
   (* (+ 1.0 (exp (- (* x eps_m)))) 0.5)
   (if (<= x 850000.0)
     (* (- (exp (* x eps_m)) (- (fma x eps_m x) 1.0)) 0.5)
     (if (or (<= x 8.5e+71) (not (or (<= x 4.5e+123) (not (<= x 2.05e+167)))))
       (/ (- (* (+ 1.0 (/ 1.0 eps_m)) 1.0) (/ (fma -1.0 eps_m 1.0) eps_m)) 2.0)
       (* (- (exp (* x (+ -1.0 eps_m))) -1.0) 0.5)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -8.2e-241) {
		tmp = (1.0 + exp(-(x * eps_m))) * 0.5;
	} else if (x <= 850000.0) {
		tmp = (exp((x * eps_m)) - (fma(x, eps_m, x) - 1.0)) * 0.5;
	} else if ((x <= 8.5e+71) || !((x <= 4.5e+123) || !(x <= 2.05e+167))) {
		tmp = (((1.0 + (1.0 / eps_m)) * 1.0) - (fma(-1.0, eps_m, 1.0) / eps_m)) / 2.0;
	} else {
		tmp = (exp((x * (-1.0 + eps_m))) - -1.0) * 0.5;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -8.2e-241)
		tmp = Float64(Float64(1.0 + exp(Float64(-Float64(x * eps_m)))) * 0.5);
	elseif (x <= 850000.0)
		tmp = Float64(Float64(exp(Float64(x * eps_m)) - Float64(fma(x, eps_m, x) - 1.0)) * 0.5);
	elseif ((x <= 8.5e+71) || !((x <= 4.5e+123) || !(x <= 2.05e+167)))
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * 1.0) - Float64(fma(-1.0, eps_m, 1.0) / eps_m)) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 + eps_m))) - -1.0) * 0.5);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -8.2e-241], N[(N[(1.0 + N[Exp[(-N[(x * eps$95$m), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 850000.0], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] - N[(N[(x * eps$95$m + x), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[Or[LessEqual[x, 8.5e+71], N[Not[Or[LessEqual[x, 4.5e+123], N[Not[LessEqual[x, 2.05e+167]], $MachinePrecision]]], $MachinePrecision]], N[(N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] - N[(N[(-1.0 * eps$95$m + 1.0), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.1999999999999997e-241

   1. Initial program 69.6%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 97.6

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

   8. Applied rewrites 97.6%

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

   9. Taylor expanded in eps around inf

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

       1. *-commutative N/A

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

       2. lift-*.f64 98.0

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

   11. Applied rewrites 98.0%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 68.3%

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

   if -8.1999999999999997e-241 < x < 8.5e5

   1. Initial program 56.2%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 97.4

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

   8. Applied rewrites 97.4%

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

   9. Taylor expanded in eps around inf

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

       1. *-commutative N/A

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

       2. lift-*.f64 97.8

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

   11. Applied rewrites 97.8%

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

   12. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. +-commutative N/A

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

       3. distribute-rgt1-in N/A

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

       4. +-commutative N/A

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

       5. *-commutative N/A

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

       6. lower--.f64 N/A

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

       7. lift-fma.f64 86.1

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

   14. Applied rewrites 86.1%

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

   if 8.5e5 < x < 8.4999999999999996e71 or 4.49999999999999983e123 < x < 2.05e167

   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

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

   5. Applied rewrites 19.1%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. inv-pow N/A

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

      3. lower-pow.f64 83.3

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

   8. Applied rewrites 83.3%

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

   9. Taylor expanded in eps around 0

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

       1. lower-/.f64 N/A

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

       2. mul-1-neg N/A

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

       3. +-commutative N/A

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

       4. mul-1-neg N/A

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

       5. lower-fma.f64 83.3

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

   11. Applied rewrites 83.3%

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

   if 8.4999999999999996e71 < x < 4.49999999999999983e123 or 2.05e167 < 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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 40.9%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-241}:\\ \;\;\;\;\left(1 + e^{-x \cdot \varepsilon}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 850000:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - \left(\mathsf{fma}\left(x, \varepsilon, x\right) - 1\right)\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+71} \lor \neg \left(x \leq 4.5 \cdot 10^{+123} \lor \neg \left(x \leq 2.05 \cdot 10^{+167}\right)\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot 1 - \frac{\mathsf{fma}\left(-1, \varepsilon, 1\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x \cdot \left(-1 + \varepsilon\right)} - -1\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-241}:\\ \;\;\;\;\left(1 + e^{-x \cdot eps\_m}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 850000:\\ \;\;\;\;\left(e^{x \cdot eps\_m} - -1\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+71} \lor \neg \left(x \leq 4.5 \cdot 10^{+123} \lor \neg \left(x \leq 2.05 \cdot 10^{+167}\right)\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) \cdot 1 - \frac{\mathsf{fma}\left(-1, eps\_m, 1\right)}{eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x \cdot \left(-1 + eps\_m\right)} - -1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -8.2e-241)
   (* (+ 1.0 (exp (- (* x eps_m)))) 0.5)
   (if (<= x 850000.0)
     (* (- (exp (* x eps_m)) -1.0) 0.5)
     (if (or (<= x 8.5e+71) (not (or (<= x 4.5e+123) (not (<= x 2.05e+167)))))
       (/ (- (* (+ 1.0 (/ 1.0 eps_m)) 1.0) (/ (fma -1.0 eps_m 1.0) eps_m)) 2.0)
       (* (- (exp (* x (+ -1.0 eps_m))) -1.0) 0.5)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -8.2e-241) {
		tmp = (1.0 + exp(-(x * eps_m))) * 0.5;
	} else if (x <= 850000.0) {
		tmp = (exp((x * eps_m)) - -1.0) * 0.5;
	} else if ((x <= 8.5e+71) || !((x <= 4.5e+123) || !(x <= 2.05e+167))) {
		tmp = (((1.0 + (1.0 / eps_m)) * 1.0) - (fma(-1.0, eps_m, 1.0) / eps_m)) / 2.0;
	} else {
		tmp = (exp((x * (-1.0 + eps_m))) - -1.0) * 0.5;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -8.2e-241)
		tmp = Float64(Float64(1.0 + exp(Float64(-Float64(x * eps_m)))) * 0.5);
	elseif (x <= 850000.0)
		tmp = Float64(Float64(exp(Float64(x * eps_m)) - -1.0) * 0.5);
	elseif ((x <= 8.5e+71) || !((x <= 4.5e+123) || !(x <= 2.05e+167)))
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * 1.0) - Float64(fma(-1.0, eps_m, 1.0) / eps_m)) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 + eps_m))) - -1.0) * 0.5);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -8.2e-241], N[(N[(1.0 + N[Exp[(-N[(x * eps$95$m), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 850000.0], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision], If[Or[LessEqual[x, 8.5e+71], N[Not[Or[LessEqual[x, 4.5e+123], N[Not[LessEqual[x, 2.05e+167]], $MachinePrecision]]], $MachinePrecision]], N[(N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] - N[(N[(-1.0 * eps$95$m + 1.0), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.1999999999999997e-241

   1. Initial program 69.6%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 97.6

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

   8. Applied rewrites 97.6%

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

   9. Taylor expanded in eps around inf

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

       1. *-commutative N/A

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

       2. lift-*.f64 98.0

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

   11. Applied rewrites 98.0%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 68.3%

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

   if -8.1999999999999997e-241 < x < 8.5e5

   1. Initial program 56.2%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 97.4

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

   8. Applied rewrites 97.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 85.7%

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

   if 8.5e5 < x < 8.4999999999999996e71 or 4.49999999999999983e123 < x < 2.05e167

   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

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

   5. Applied rewrites 19.1%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. inv-pow N/A

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

      3. lower-pow.f64 83.3

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

   8. Applied rewrites 83.3%

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

   9. Taylor expanded in eps around 0

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

       1. lower-/.f64 N/A

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

       2. mul-1-neg N/A

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

       3. +-commutative N/A

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

       4. mul-1-neg N/A

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

       5. lower-fma.f64 83.3

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

   11. Applied rewrites 83.3%

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

   if 8.4999999999999996e71 < x < 4.49999999999999983e123 or 2.05e167 < 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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 40.9%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-241}:\\ \;\;\;\;\left(1 + e^{-x \cdot \varepsilon}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 850000:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - -1\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+71} \lor \neg \left(x \leq 4.5 \cdot 10^{+123} \lor \neg \left(x \leq 2.05 \cdot 10^{+167}\right)\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot 1 - \frac{\mathsf{fma}\left(-1, \varepsilon, 1\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x \cdot \left(-1 + \varepsilon\right)} - -1\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-241}:\\ \;\;\;\;\left(1 + e^{-x \cdot eps\_m}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 850000 \lor \neg \left(x \leq 8.5 \cdot 10^{+71} \lor \neg \left(x \leq 4.4 \cdot 10^{+123} \lor \neg \left(x \leq 2.05 \cdot 10^{+167}\right)\right)\right):\\ \;\;\;\;\left(e^{x \cdot eps\_m} - -1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) \cdot 1 - \frac{\mathsf{fma}\left(-1, eps\_m, 1\right)}{eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -8.2e-241)
   (* (+ 1.0 (exp (- (* x eps_m)))) 0.5)
   (if (or (<= x 850000.0)
           (not
            (or (<= x 8.5e+71)
                (not (or (<= x 4.4e+123) (not (<= x 2.05e+167)))))))
     (* (- (exp (* x eps_m)) -1.0) 0.5)
     (/
      (- (* (+ 1.0 (/ 1.0 eps_m)) 1.0) (/ (fma -1.0 eps_m 1.0) eps_m))
      2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -8.2e-241) {
		tmp = (1.0 + exp(-(x * eps_m))) * 0.5;
	} else if ((x <= 850000.0) || !((x <= 8.5e+71) || !((x <= 4.4e+123) || !(x <= 2.05e+167)))) {
		tmp = (exp((x * eps_m)) - -1.0) * 0.5;
	} else {
		tmp = (((1.0 + (1.0 / eps_m)) * 1.0) - (fma(-1.0, eps_m, 1.0) / eps_m)) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -8.2e-241)
		tmp = Float64(Float64(1.0 + exp(Float64(-Float64(x * eps_m)))) * 0.5);
	elseif ((x <= 850000.0) || !((x <= 8.5e+71) || !((x <= 4.4e+123) || !(x <= 2.05e+167))))
		tmp = Float64(Float64(exp(Float64(x * eps_m)) - -1.0) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * 1.0) - Float64(fma(-1.0, eps_m, 1.0) / eps_m)) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -8.2e-241], N[(N[(1.0 + N[Exp[(-N[(x * eps$95$m), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[Or[LessEqual[x, 850000.0], N[Not[Or[LessEqual[x, 8.5e+71], N[Not[Or[LessEqual[x, 4.4e+123], N[Not[LessEqual[x, 2.05e+167]], $MachinePrecision]]], $MachinePrecision]]], $MachinePrecision]], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] - N[(N[(-1.0 * eps$95$m + 1.0), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.1999999999999997e-241

   1. Initial program 69.6%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 97.6

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

   8. Applied rewrites 97.6%

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

   9. Taylor expanded in eps around inf

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

       1. *-commutative N/A

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

       2. lift-*.f64 98.0

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

   11. Applied rewrites 98.0%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 68.3%

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

   if -8.1999999999999997e-241 < x < 8.5e5 or 8.4999999999999996e71 < x < 4.39999999999999984e123 or 2.05e167 < x

   1. Initial program 70.6%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 91.3

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

   8. Applied rewrites 91.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 71.0%

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

   if 8.5e5 < x < 8.4999999999999996e71 or 4.39999999999999984e123 < x < 2.05e167

   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

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

   5. Applied rewrites 19.1%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. inv-pow N/A

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

      3. lower-pow.f64 83.3

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

   8. Applied rewrites 83.3%

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

   9. Taylor expanded in eps around 0

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

       1. lower-/.f64 N/A

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

       2. mul-1-neg N/A

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

       3. +-commutative N/A

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

       4. mul-1-neg N/A

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

       5. lower-fma.f64 83.3

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

   11. Applied rewrites 83.3%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-241}:\\ \;\;\;\;\left(1 + e^{-x \cdot \varepsilon}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 850000 \lor \neg \left(x \leq 8.5 \cdot 10^{+71} \lor \neg \left(x \leq 4.4 \cdot 10^{+123} \lor \neg \left(x \leq 2.05 \cdot 10^{+167}\right)\right)\right):\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - -1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot 1 - \frac{\mathsf{fma}\left(-1, \varepsilon, 1\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 1.05 \cdot 10^{+37}:\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x \cdot eps\_m} - -1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 1.05e+37) (exp (- x)) (* (- (exp (* x eps_m)) -1.0) 0.5)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.05e+37) {
		tmp = exp(-x);
	} else {
		tmp = (exp((x * eps_m)) - -1.0) * 0.5;
	}
	return tmp;
}

Fortran

eps_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 1.05d+37) then
        tmp = exp(-x)
    else
        tmp = (exp((x * eps_m)) - (-1.0d0)) * 0.5d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.05e+37) {
		tmp = Math.exp(-x);
	} else {
		tmp = (Math.exp((x * eps_m)) - -1.0) * 0.5;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 1.05e+37:
		tmp = math.exp(-x)
	else:
		tmp = (math.exp((x * eps_m)) - -1.0) * 0.5
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 1.05e+37)
		tmp = exp(Float64(-x));
	else
		tmp = Float64(Float64(exp(Float64(x * eps_m)) - -1.0) * 0.5);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 1.05e+37)
		tmp = exp(-x);
	else
		tmp = (exp((x * eps_m)) - -1.0) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 1.05e+37], N[Exp[(-x)], $MachinePrecision], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.0500000000000001e37

   1. Initial program 65.6%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 83.4

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

   8. Applied rewrites 83.4%

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

   9. Taylor expanded in eps around 0

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

       1. lower-exp.f64 N/A

          \[\leadsto e^{\mathsf{neg}\left(x\right)} \]

       2. lift-neg.f64 80.1

          \[\leadsto e^{-x} \]

   11. Applied rewrites 80.1%

       \[\leadsto e^{-x} \]

   if 1.0500000000000001e37 < 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 eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 61.4%

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

Alternative 8: 79.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -720:\\ \;\;\;\;e^{-x}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-222}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{eps\_m \cdot eps\_m - 1}{eps\_m - 1}, -1 + eps\_m\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-208}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 750:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, eps\_m - -1, \frac{-1 + eps\_m \cdot eps\_m}{eps\_m - -1}\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+167}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) \cdot 1 - \frac{\mathsf{fma}\left(-1, eps\_m, 1\right)}{eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x - 1, x, 1\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -720.0)
   (exp (- x))
   (if (<= x -5e-222)
     (*
      (fma
       (fma -1.0 (/ (- (* eps_m eps_m) 1.0) (- eps_m 1.0)) (+ -1.0 eps_m))
       x
       2.0)
      0.5)
     (if (<= x 2e-208)
       1.0
       (if (<= x 750.0)
         (*
          (fma
           (fma
            -1.0
            (- eps_m -1.0)
            (/ (+ -1.0 (* eps_m eps_m)) (- eps_m -1.0)))
           x
           2.0)
          0.5)
         (if (<= x 2.05e+167)
           (/
            (- (* (+ 1.0 (/ 1.0 eps_m)) 1.0) (/ (fma -1.0 eps_m 1.0) eps_m))
            2.0)
           (fma (- (* 0.5 x) 1.0) x 1.0)))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -720.0) {
		tmp = exp(-x);
	} else if (x <= -5e-222) {
		tmp = fma(fma(-1.0, (((eps_m * eps_m) - 1.0) / (eps_m - 1.0)), (-1.0 + eps_m)), x, 2.0) * 0.5;
	} else if (x <= 2e-208) {
		tmp = 1.0;
	} else if (x <= 750.0) {
		tmp = fma(fma(-1.0, (eps_m - -1.0), ((-1.0 + (eps_m * eps_m)) / (eps_m - -1.0))), x, 2.0) * 0.5;
	} else if (x <= 2.05e+167) {
		tmp = (((1.0 + (1.0 / eps_m)) * 1.0) - (fma(-1.0, eps_m, 1.0) / eps_m)) / 2.0;
	} else {
		tmp = fma(((0.5 * x) - 1.0), x, 1.0);
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -720.0)
		tmp = exp(Float64(-x));
	elseif (x <= -5e-222)
		tmp = Float64(fma(fma(-1.0, Float64(Float64(Float64(eps_m * eps_m) - 1.0) / Float64(eps_m - 1.0)), Float64(-1.0 + eps_m)), x, 2.0) * 0.5);
	elseif (x <= 2e-208)
		tmp = 1.0;
	elseif (x <= 750.0)
		tmp = Float64(fma(fma(-1.0, Float64(eps_m - -1.0), Float64(Float64(-1.0 + Float64(eps_m * eps_m)) / Float64(eps_m - -1.0))), x, 2.0) * 0.5);
	elseif (x <= 2.05e+167)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * 1.0) - Float64(fma(-1.0, eps_m, 1.0) / eps_m)) / 2.0);
	else
		tmp = fma(Float64(Float64(0.5 * x) - 1.0), x, 1.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -720.0], N[Exp[(-x)], $MachinePrecision], If[LessEqual[x, -5e-222], N[(N[(N[(-1.0 * N[(N[(N[(eps$95$m * eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision] / N[(eps$95$m - 1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 2e-208], 1.0, If[LessEqual[x, 750.0], N[(N[(N[(-1.0 * N[(eps$95$m - -1.0), $MachinePrecision] + N[(N[(-1.0 + N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision] / N[(eps$95$m - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 2.05e+167], N[(N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] - N[(N[(-1.0 * eps$95$m + 1.0), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(0.5 * x), $MachinePrecision] - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < -720

   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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in eps around 0

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

       1. lower-exp.f64 N/A

          \[\leadsto e^{\mathsf{neg}\left(x\right)} \]

       2. lift-neg.f64 100.0

          \[\leadsto e^{-x} \]

   11. Applied rewrites 100.0%

       \[\leadsto e^{-x} \]

   if -720 < x < -5.00000000000000008e-222

   1. Initial program 53.9%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lift--.f64 65.4

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

   8. Applied rewrites 65.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 71.9

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

   10. Applied rewrites 71.9%

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

   if -5.00000000000000008e-222 < x < 2.0000000000000002e-208

   1. Initial program 66.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 97.6%

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

   if 2.0000000000000002e-208 < x < 750

   1. Initial program 49.6%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lift--.f64 62.7

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

   8. Applied rewrites 62.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   9. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 72.0

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

   10. Applied rewrites 72.0%

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

   if 750 < x < 2.05e167

   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

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

   5. Applied rewrites 23.3%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. inv-pow N/A

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

      3. lower-pow.f64 63.1

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

   8. Applied rewrites 63.1%

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

   9. Taylor expanded in eps around 0

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

       1. lower-/.f64 N/A

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

       2. mul-1-neg N/A

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

       3. +-commutative N/A

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

       4. mul-1-neg N/A

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

       5. lower-fma.f64 63.1

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

   11. Applied rewrites 63.1%

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

   if 2.05e167 < 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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 75.6

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

   8. Applied rewrites 75.6%

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

   9. Taylor expanded in eps around 0

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

       1. lower-exp.f64 N/A

          \[\leadsto e^{\mathsf{neg}\left(x\right)} \]

       2. lift-neg.f64 43.8

          \[\leadsto e^{-x} \]

   11. Applied rewrites 43.8%

       \[\leadsto e^{-x} \]

   12. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower--.f64 N/A

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

       5. lower-*.f64 57.8

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

   14. Applied rewrites 57.8%

       \[\leadsto \mathsf{fma}\left(0.5 \cdot x - 1, x, 1\right) \]
3. Recombined 6 regimes into one program.

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -720:\\ \;\;\;\;e^{-x}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-222}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, -1 + \varepsilon\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-208}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 750:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, \frac{-1 + \varepsilon \cdot \varepsilon}{\varepsilon - -1}\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+167}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot 1 - \frac{\mathsf{fma}\left(-1, \varepsilon, 1\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x - 1, x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right) \cdot x - 1, x, 1\right)\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-222}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{eps\_m \cdot eps\_m - 1}{eps\_m - 1}, -1 + eps\_m\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-208}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 750:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, eps\_m - -1, \frac{-1 + eps\_m \cdot eps\_m}{eps\_m - -1}\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+167}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) \cdot 1 - \frac{\mathsf{fma}\left(-1, eps\_m, 1\right)}{eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x - 1, x, 1\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.35e+122)
   (fma (- (* (fma -0.16666666666666666 x 0.5) x) 1.0) x 1.0)
   (if (<= x -5e-222)
     (*
      (fma
       (fma -1.0 (/ (- (* eps_m eps_m) 1.0) (- eps_m 1.0)) (+ -1.0 eps_m))
       x
       2.0)
      0.5)
     (if (<= x 2e-208)
       1.0
       (if (<= x 750.0)
         (*
          (fma
           (fma
            -1.0
            (- eps_m -1.0)
            (/ (+ -1.0 (* eps_m eps_m)) (- eps_m -1.0)))
           x
           2.0)
          0.5)
         (if (<= x 2.05e+167)
           (/
            (- (* (+ 1.0 (/ 1.0 eps_m)) 1.0) (/ (fma -1.0 eps_m 1.0) eps_m))
            2.0)
           (fma (- (* 0.5 x) 1.0) x 1.0)))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.35e+122) {
		tmp = fma(((fma(-0.16666666666666666, x, 0.5) * x) - 1.0), x, 1.0);
	} else if (x <= -5e-222) {
		tmp = fma(fma(-1.0, (((eps_m * eps_m) - 1.0) / (eps_m - 1.0)), (-1.0 + eps_m)), x, 2.0) * 0.5;
	} else if (x <= 2e-208) {
		tmp = 1.0;
	} else if (x <= 750.0) {
		tmp = fma(fma(-1.0, (eps_m - -1.0), ((-1.0 + (eps_m * eps_m)) / (eps_m - -1.0))), x, 2.0) * 0.5;
	} else if (x <= 2.05e+167) {
		tmp = (((1.0 + (1.0 / eps_m)) * 1.0) - (fma(-1.0, eps_m, 1.0) / eps_m)) / 2.0;
	} else {
		tmp = fma(((0.5 * x) - 1.0), x, 1.0);
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.35e+122)
		tmp = fma(Float64(Float64(fma(-0.16666666666666666, x, 0.5) * x) - 1.0), x, 1.0);
	elseif (x <= -5e-222)
		tmp = Float64(fma(fma(-1.0, Float64(Float64(Float64(eps_m * eps_m) - 1.0) / Float64(eps_m - 1.0)), Float64(-1.0 + eps_m)), x, 2.0) * 0.5);
	elseif (x <= 2e-208)
		tmp = 1.0;
	elseif (x <= 750.0)
		tmp = Float64(fma(fma(-1.0, Float64(eps_m - -1.0), Float64(Float64(-1.0 + Float64(eps_m * eps_m)) / Float64(eps_m - -1.0))), x, 2.0) * 0.5);
	elseif (x <= 2.05e+167)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * 1.0) - Float64(fma(-1.0, eps_m, 1.0) / eps_m)) / 2.0);
	else
		tmp = fma(Float64(Float64(0.5 * x) - 1.0), x, 1.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.35e+122], N[(N[(N[(N[(-0.16666666666666666 * x + 0.5), $MachinePrecision] * x), $MachinePrecision] - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision], If[LessEqual[x, -5e-222], N[(N[(N[(-1.0 * N[(N[(N[(eps$95$m * eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision] / N[(eps$95$m - 1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 2e-208], 1.0, If[LessEqual[x, 750.0], N[(N[(N[(-1.0 * N[(eps$95$m - -1.0), $MachinePrecision] + N[(N[(-1.0 + N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision] / N[(eps$95$m - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 2.05e+167], N[(N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] - N[(N[(-1.0 * eps$95$m + 1.0), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(0.5 * x), $MachinePrecision] - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < -1.3499999999999999e122

   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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in eps around 0

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

       1. lower-exp.f64 N/A

          \[\leadsto e^{\mathsf{neg}\left(x\right)} \]

       2. lift-neg.f64 100.0

          \[\leadsto e^{-x} \]

   11. Applied rewrites 100.0%

       \[\leadsto e^{-x} \]

   12. Taylor expanded in x around 0

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

       1. +-commutative N/A

          \[\leadsto x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1\right) + 1 \]

       2. *-commutative N/A

          \[\leadsto \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1\right) \cdot x + 1 \]

       3. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1, x, 1\right) \]

       4. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1, x, 1\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) \cdot x - 1, x, 1\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) \cdot x - 1, x, 1\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{6} \cdot x + \frac{1}{2}\right) \cdot x - 1, x, 1\right) \]

       8. lower-fma.f64 100.0

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

   14. Applied rewrites 100.0%

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

   if -1.3499999999999999e122 < x < -5.00000000000000008e-222

   1. Initial program 60.7%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lift--.f64 56.2

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

   8. Applied rewrites 56.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 65.5

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

   10. Applied rewrites 65.5%

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

   if -5.00000000000000008e-222 < x < 2.0000000000000002e-208

   1. Initial program 66.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 97.6%

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

   if 2.0000000000000002e-208 < x < 750

   1. Initial program 49.6%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lift--.f64 62.7

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

   8. Applied rewrites 62.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   9. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 72.0

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

   10. Applied rewrites 72.0%

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

   if 750 < x < 2.05e167

   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

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

   5. Applied rewrites 23.3%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. inv-pow N/A

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

      3. lower-pow.f64 63.1

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

   8. Applied rewrites 63.1%

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

   9. Taylor expanded in eps around 0

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

       1. lower-/.f64 N/A

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

       2. mul-1-neg N/A

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

       3. +-commutative N/A

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

       4. mul-1-neg N/A

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

       5. lower-fma.f64 63.1

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

   11. Applied rewrites 63.1%

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

   if 2.05e167 < 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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 75.6

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

   8. Applied rewrites 75.6%

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

   9. Taylor expanded in eps around 0

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

       1. lower-exp.f64 N/A

          \[\leadsto e^{\mathsf{neg}\left(x\right)} \]

       2. lift-neg.f64 43.8

          \[\leadsto e^{-x} \]

   11. Applied rewrites 43.8%

       \[\leadsto e^{-x} \]

   12. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower--.f64 N/A

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

       5. lower-*.f64 57.8

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

   14. Applied rewrites 57.8%

       \[\leadsto \mathsf{fma}\left(0.5 \cdot x - 1, x, 1\right) \]
3. Recombined 6 regimes into one program.

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right) \cdot x - 1, x, 1\right)\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-222}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, -1 + \varepsilon\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-208}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 750:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, \frac{-1 + \varepsilon \cdot \varepsilon}{\varepsilon - -1}\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+167}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot 1 - \frac{\mathsf{fma}\left(-1, \varepsilon, 1\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x - 1, x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.8% accurate, 4.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right) \cdot x - 1, x, 1\right)\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-222}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{eps\_m \cdot eps\_m - 1}{eps\_m - 1}, -1 + eps\_m\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-195}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, eps\_m - -1, \frac{-1 + eps\_m \cdot eps\_m}{1}\right), x, 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.35e+122)
   (fma (- (* (fma -0.16666666666666666 x 0.5) x) 1.0) x 1.0)
   (if (<= x -5e-222)
     (*
      (fma
       (fma -1.0 (/ (- (* eps_m eps_m) 1.0) (- eps_m 1.0)) (+ -1.0 eps_m))
       x
       2.0)
      0.5)
     (if (<= x 6.5e-195)
       1.0
       (*
        (fma (fma -1.0 (- eps_m -1.0) (/ (+ -1.0 (* eps_m eps_m)) 1.0)) x 2.0)
        0.5)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.35e+122) {
		tmp = fma(((fma(-0.16666666666666666, x, 0.5) * x) - 1.0), x, 1.0);
	} else if (x <= -5e-222) {
		tmp = fma(fma(-1.0, (((eps_m * eps_m) - 1.0) / (eps_m - 1.0)), (-1.0 + eps_m)), x, 2.0) * 0.5;
	} else if (x <= 6.5e-195) {
		tmp = 1.0;
	} else {
		tmp = fma(fma(-1.0, (eps_m - -1.0), ((-1.0 + (eps_m * eps_m)) / 1.0)), x, 2.0) * 0.5;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.35e+122)
		tmp = fma(Float64(Float64(fma(-0.16666666666666666, x, 0.5) * x) - 1.0), x, 1.0);
	elseif (x <= -5e-222)
		tmp = Float64(fma(fma(-1.0, Float64(Float64(Float64(eps_m * eps_m) - 1.0) / Float64(eps_m - 1.0)), Float64(-1.0 + eps_m)), x, 2.0) * 0.5);
	elseif (x <= 6.5e-195)
		tmp = 1.0;
	else
		tmp = Float64(fma(fma(-1.0, Float64(eps_m - -1.0), Float64(Float64(-1.0 + Float64(eps_m * eps_m)) / 1.0)), x, 2.0) * 0.5);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.35e+122], N[(N[(N[(N[(-0.16666666666666666 * x + 0.5), $MachinePrecision] * x), $MachinePrecision] - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision], If[LessEqual[x, -5e-222], N[(N[(N[(-1.0 * N[(N[(N[(eps$95$m * eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision] / N[(eps$95$m - 1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 6.5e-195], 1.0, N[(N[(N[(-1.0 * N[(eps$95$m - -1.0), $MachinePrecision] + N[(N[(-1.0 + N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.3499999999999999e122

   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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in eps around 0

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

       1. lower-exp.f64 N/A

          \[\leadsto e^{\mathsf{neg}\left(x\right)} \]

       2. lift-neg.f64 100.0

          \[\leadsto e^{-x} \]

   11. Applied rewrites 100.0%

       \[\leadsto e^{-x} \]

   12. Taylor expanded in x around 0

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

       1. +-commutative N/A

          \[\leadsto x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1\right) + 1 \]

       2. *-commutative N/A

          \[\leadsto \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1\right) \cdot x + 1 \]

       3. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1, x, 1\right) \]

       4. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1, x, 1\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) \cdot x - 1, x, 1\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) \cdot x - 1, x, 1\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{6} \cdot x + \frac{1}{2}\right) \cdot x - 1, x, 1\right) \]

       8. lower-fma.f64 100.0

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

   14. Applied rewrites 100.0%

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

   if -1.3499999999999999e122 < x < -5.00000000000000008e-222

   1. Initial program 60.7%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lift--.f64 56.2

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

   8. Applied rewrites 56.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 65.5

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

   10. Applied rewrites 65.5%

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

   if -5.00000000000000008e-222 < x < 6.50000000000000004e-195

   1. Initial program 66.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 97.7%

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

   if 6.50000000000000004e-195 < x

   1. Initial program 79.5%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lift--.f64 26.4

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

   8. Applied rewrites 26.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   9. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 39.3

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

   10. Applied rewrites 39.3%

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

   11. Taylor expanded in eps around 0

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

   13. Applied rewrites 54.8%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right) \cdot x - 1, x, 1\right)\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-222}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, -1 + \varepsilon\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-195}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, \frac{-1 + \varepsilon \cdot \varepsilon}{1}\right), x, 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.9% accurate, 5.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{-195}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right) \cdot x - 1, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, eps\_m - -1, \frac{-1 + eps\_m \cdot eps\_m}{1}\right), x, 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 6.5e-195)
   (fma (- (* (fma -0.16666666666666666 x 0.5) x) 1.0) x 1.0)
   (*
    (fma (fma -1.0 (- eps_m -1.0) (/ (+ -1.0 (* eps_m eps_m)) 1.0)) x 2.0)
    0.5)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 6.5e-195) {
		tmp = fma(((fma(-0.16666666666666666, x, 0.5) * x) - 1.0), x, 1.0);
	} else {
		tmp = fma(fma(-1.0, (eps_m - -1.0), ((-1.0 + (eps_m * eps_m)) / 1.0)), x, 2.0) * 0.5;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 6.5e-195)
		tmp = fma(Float64(Float64(fma(-0.16666666666666666, x, 0.5) * x) - 1.0), x, 1.0);
	else
		tmp = Float64(fma(fma(-1.0, Float64(eps_m - -1.0), Float64(Float64(-1.0 + Float64(eps_m * eps_m)) / 1.0)), x, 2.0) * 0.5);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 6.5e-195], N[(N[(N[(N[(-0.16666666666666666 * x + 0.5), $MachinePrecision] * x), $MachinePrecision] - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision], N[(N[(N[(-1.0 * N[(eps$95$m - -1.0), $MachinePrecision] + N[(N[(-1.0 + N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.50000000000000004e-195

   1. Initial program 68.4%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 98.2

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

   8. Applied rewrites 98.2%

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

   9. Taylor expanded in eps around 0

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

       1. lower-exp.f64 N/A

          \[\leadsto e^{\mathsf{neg}\left(x\right)} \]

       2. lift-neg.f64 83.5

          \[\leadsto e^{-x} \]

   11. Applied rewrites 83.5%

       \[\leadsto e^{-x} \]

   12. Taylor expanded in x around 0

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

       1. +-commutative N/A

          \[\leadsto x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1\right) + 1 \]

       2. *-commutative N/A

          \[\leadsto \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1\right) \cdot x + 1 \]

       3. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1, x, 1\right) \]

       4. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1, x, 1\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) \cdot x - 1, x, 1\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) \cdot x - 1, x, 1\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{6} \cdot x + \frac{1}{2}\right) \cdot x - 1, x, 1\right) \]

       8. lower-fma.f64 77.4

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

   14. Applied rewrites 77.4%

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

   if 6.50000000000000004e-195 < x

   1. Initial program 79.5%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lift--.f64 26.4

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

   8. Applied rewrites 26.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   9. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 39.3

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

   10. Applied rewrites 39.3%

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

   11. Taylor expanded in eps around 0

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

   13. Applied rewrites 54.8%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{-195}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right) \cdot x - 1, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, \frac{-1 + \varepsilon \cdot \varepsilon}{1}\right), x, 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.8% accurate, 10.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 6.2 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right) \cdot x - 1, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, 1, -1 + eps\_m\right), x, 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 6.2e-30)
   (fma (- (* (fma -0.16666666666666666 x 0.5) x) 1.0) x 1.0)
   (* (fma (fma -1.0 1.0 (+ -1.0 eps_m)) x 2.0) 0.5)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 6.2e-30) {
		tmp = fma(((fma(-0.16666666666666666, x, 0.5) * x) - 1.0), x, 1.0);
	} else {
		tmp = fma(fma(-1.0, 1.0, (-1.0 + eps_m)), x, 2.0) * 0.5;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 6.2e-30)
		tmp = fma(Float64(Float64(fma(-0.16666666666666666, x, 0.5) * x) - 1.0), x, 1.0);
	else
		tmp = Float64(fma(fma(-1.0, 1.0, Float64(-1.0 + eps_m)), x, 2.0) * 0.5);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 6.2e-30], N[(N[(N[(N[(-0.16666666666666666 * x + 0.5), $MachinePrecision] * x), $MachinePrecision] - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision], N[(N[(N[(-1.0 * 1.0 + N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.19999999999999982e-30

   1. Initial program 63.9%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 98.6

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

   8. Applied rewrites 98.6%

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

   9. Taylor expanded in eps around 0

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

       1. lower-exp.f64 N/A

          \[\leadsto e^{\mathsf{neg}\left(x\right)} \]

       2. lift-neg.f64 80.6

          \[\leadsto e^{-x} \]

   11. Applied rewrites 80.6%

       \[\leadsto e^{-x} \]

   12. Taylor expanded in x around 0

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

       1. +-commutative N/A

          \[\leadsto x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1\right) + 1 \]

       2. *-commutative N/A

          \[\leadsto \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1\right) \cdot x + 1 \]

       3. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1, x, 1\right) \]

       4. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1, x, 1\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) \cdot x - 1, x, 1\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) \cdot x - 1, x, 1\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{6} \cdot x + \frac{1}{2}\right) \cdot x - 1, x, 1\right) \]

       8. lower-fma.f64 75.8

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

   14. Applied rewrites 75.8%

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

   if 6.19999999999999982e-30 < x

   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 eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lift--.f64 4.8

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

   8. Applied rewrites 4.8%

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

   9. Taylor expanded in eps around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 19.2%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.2 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right) \cdot x - 1, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, 1, -1 + \varepsilon\right), x, 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.0% accurate, 10.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -155:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right) \cdot x - 1, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x - 1, x, 1\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -155.0)
   (fma (- (* (fma -0.16666666666666666 x 0.5) x) 1.0) x 1.0)
   (fma (- (* 0.5 x) 1.0) x 1.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -155.0) {
		tmp = fma(((fma(-0.16666666666666666, x, 0.5) * x) - 1.0), x, 1.0);
	} else {
		tmp = fma(((0.5 * x) - 1.0), x, 1.0);
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -155.0)
		tmp = fma(Float64(Float64(fma(-0.16666666666666666, x, 0.5) * x) - 1.0), x, 1.0);
	else
		tmp = fma(Float64(Float64(0.5 * x) - 1.0), x, 1.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -155.0], N[(N[(N[(N[(-0.16666666666666666 * x + 0.5), $MachinePrecision] * x), $MachinePrecision] - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision], N[(N[(N[(0.5 * x), $MachinePrecision] - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -155

   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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in eps around 0

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

       1. lower-exp.f64 N/A

          \[\leadsto e^{\mathsf{neg}\left(x\right)} \]

       2. lift-neg.f64 100.0

          \[\leadsto e^{-x} \]

   11. Applied rewrites 100.0%

       \[\leadsto e^{-x} \]

   12. Taylor expanded in x around 0

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

       1. +-commutative N/A

          \[\leadsto x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1\right) + 1 \]

       2. *-commutative N/A

          \[\leadsto \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1\right) \cdot x + 1 \]

       3. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1, x, 1\right) \]

       4. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1, x, 1\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) \cdot x - 1, x, 1\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) \cdot x - 1, x, 1\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{6} \cdot x + \frac{1}{2}\right) \cdot x - 1, x, 1\right) \]

       8. lower-fma.f64 73.9

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

   14. Applied rewrites 73.9%

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

   if -155 < x

   1. Initial program 69.7%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 85.4

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

   8. Applied rewrites 85.4%

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

   9. Taylor expanded in eps around 0

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

       1. lower-exp.f64 N/A

          \[\leadsto e^{\mathsf{neg}\left(x\right)} \]

       2. lift-neg.f64 67.4

          \[\leadsto e^{-x} \]

   11. Applied rewrites 67.4%

       \[\leadsto e^{-x} \]

   12. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower--.f64 N/A

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

       5. lower-*.f64 58.6

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

   14. Applied rewrites 58.6%

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

Alternative 14: 57.8% accurate, 18.2× speedup

Math

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

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (fma (- (* 0.5 x) 1.0) x 1.0))

C

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

Julia

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := N[(N[(N[(0.5 * x), $MachinePrecision] - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]

Derivation

1. Initial program 73.5%

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

3. Taylor expanded in eps around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 98.2%

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

6. Taylor expanded in eps around inf

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

   1. *-commutative N/A

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

   2. distribute-rgt-neg-in N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 87.2

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

8. Applied rewrites 87.2%

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

9. Taylor expanded in eps around 0

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

    1. lower-exp.f64 N/A

       \[\leadsto e^{\mathsf{neg}\left(x\right)} \]

    2. lift-neg.f64 71.5

       \[\leadsto e^{-x} \]

11. Applied rewrites 71.5%

    \[\leadsto e^{-x} \]

12. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. lower--.f64 N/A

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

    5. lower-*.f64 58.2

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

14. Applied rewrites 58.2%

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

Alternative 15: 43.8% accurate, 273.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ 1 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 1.0)

C

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

Fortran

eps_m =     private
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_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    code = 1.0d0
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return 1.0;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return 1.0

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return 1.0
end

MATLAB

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := 1.0

Derivation

1. Initial program 73.5%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 45.5%

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

Reproduce

herbie shell --seed 2025037 
(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))