NMSE Section 6.1 mentioned, A

Percentage Accurate: 72.6% → 98.9%

Time: 6.9s

Alternatives: 12

Speedup: 1.2×

• 37.8% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -8.098726218933734e+183
  2. eps = 2.3513239784486256e-56

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: 72.6% 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: 98.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (((((1.0d0 + (1.0d0 / eps)) * exp((((-1.0d0) + eps) * x))) - (((1.0d0 / eps) - 1.0d0) * exp((((-1.0d0) - eps) * x)))) / 2.0d0) <= 0.0d0) then
        tmp = (t_0 + t_0) * 0.5d0
    else
        tmp = (exp((x * eps)) + exp((-eps * x))) * 0.5d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[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], 0.0], N[(N[(t$95$0 + t$95$0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] + N[Exp[N[((-eps) * x), $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 42.6%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 99.6%

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

   6. Taylor expanded in eps around 0

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

      1. mul-1-neg N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lift-neg.f64 99.6

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

   8. Applied rewrites 99.6%

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

   if 0.0 < (/.f64 (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) #s(literal 2 binary64))

   1. Initial program 98.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 98.7%

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

   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

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

      2. lower-*.f64 98.7

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

   8. Applied rewrites 98.7%

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

   9. Taylor expanded in eps around inf

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

       1. associate-*r* N/A

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

       2. mul-1-neg N/A

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

       3. lower-*.f64 N/A

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

       4. lower-neg.f64 98.7

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

   11. Applied rewrites 98.7%

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

4. Final simplification 99.1%

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

Alternative 2: 98.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.7%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
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 99.0%

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

6. Final simplification 99.0%

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

Alternative 3: 70.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{1}{\varepsilon}\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{-250}:\\ \;\;\;\;\left(1 + e^{\left(-\varepsilon\right) \cdot x}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 0.00039:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - \left(x \cdot \varepsilon - 1\right)\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+137}:\\ \;\;\;\;\frac{t\_0 \cdot e^{\left(-\varepsilon\right) \cdot \left(-x\right)} + \left(\frac{-1}{\varepsilon} - -1\right) \cdot \mathsf{fma}\left(-1, x \cdot \varepsilon, 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, x, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ 1.0 eps))))
   (if (<= x -7.5e-250)
     (* (+ 1.0 (exp (* (- eps) x))) 0.5)
     (if (<= x 0.00039)
       (* (- (exp (* x eps)) (- (* x eps) 1.0)) 0.5)
       (if (<= x 3.2e+137)
         (/
          (+
           (* t_0 (exp (* (- eps) (- x))))
           (* (- (/ -1.0 eps) -1.0) (fma -1.0 (* x eps) 1.0)))
          2.0)
         (/
          (-
           (* t_0 (fma (- eps 1.0) x 1.0))
           (* (- (/ 1.0 eps) 1.0) (fma -1.0 x 1.0)))
          2.0))))))

C

double code(double x, double eps) {
	double t_0 = 1.0 + (1.0 / eps);
	double tmp;
	if (x <= -7.5e-250) {
		tmp = (1.0 + exp((-eps * x))) * 0.5;
	} else if (x <= 0.00039) {
		tmp = (exp((x * eps)) - ((x * eps) - 1.0)) * 0.5;
	} else if (x <= 3.2e+137) {
		tmp = ((t_0 * exp((-eps * -x))) + (((-1.0 / eps) - -1.0) * fma(-1.0, (x * eps), 1.0))) / 2.0;
	} else {
		tmp = ((t_0 * fma((eps - 1.0), x, 1.0)) - (((1.0 / eps) - 1.0) * fma(-1.0, x, 1.0))) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(1.0 + Float64(1.0 / eps))
	tmp = 0.0
	if (x <= -7.5e-250)
		tmp = Float64(Float64(1.0 + exp(Float64(Float64(-eps) * x))) * 0.5);
	elseif (x <= 0.00039)
		tmp = Float64(Float64(exp(Float64(x * eps)) - Float64(Float64(x * eps) - 1.0)) * 0.5);
	elseif (x <= 3.2e+137)
		tmp = Float64(Float64(Float64(t_0 * exp(Float64(Float64(-eps) * Float64(-x)))) + Float64(Float64(Float64(-1.0 / eps) - -1.0) * fma(-1.0, Float64(x * eps), 1.0))) / 2.0);
	else
		tmp = Float64(Float64(Float64(t_0 * fma(Float64(eps - 1.0), x, 1.0)) - Float64(Float64(Float64(1.0 / eps) - 1.0) * fma(-1.0, x, 1.0))) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.5e-250], N[(N[(1.0 + N[Exp[N[((-eps) * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 0.00039], N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] - N[(N[(x * eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 3.2e+137], N[(N[(N[(t$95$0 * N[Exp[N[((-eps) * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-1.0 / eps), $MachinePrecision] - -1.0), $MachinePrecision] * N[(-1.0 * N[(x * eps), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(t$95$0 * N[(N[(eps - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision] * N[(-1.0 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.50000000000000009e-250

   1. Initial program 74.3%

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

   3. Taylor expanded in eps around 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 97.7%

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

   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

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

      2. lower-*.f64 97.7

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

   8. Applied rewrites 97.7%

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

   9. Taylor expanded in eps around inf

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

       1. associate-*r* N/A

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

       2. mul-1-neg N/A

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

       3. lower-*.f64 N/A

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

       4. lower-neg.f64 97.8

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

   11. Applied rewrites 97.8%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 61.0%

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

   if -7.50000000000000009e-250 < x < 3.89999999999999993e-4

   1. Initial program 52.7%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 99.5%

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

   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

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

      2. lower-*.f64 99.5

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

   8. Applied rewrites 99.5%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(e^{x \cdot \varepsilon} - \left(x \cdot \left(1 + \varepsilon\right) - 1\right)\right) \cdot \frac{1}{2} \]
   10. 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 94.8

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

   11. Applied rewrites 94.8%

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

   12. Taylor expanded in eps around inf

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

       1. *-commutative N/A

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

       2. lift-*.f64 95.2

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

   14. Applied rewrites 95.2%

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

   if 3.89999999999999993e-4 < x < 3.20000000000000019e137

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 16.7

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

   5. Applied rewrites 16.7%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 16.8

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

   8. Applied rewrites 16.8%

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

   9. Taylor expanded in eps around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 57.4

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

   11. Applied rewrites 57.4%

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

   if 3.20000000000000019e137 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 32.3

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

   5. Applied rewrites 32.3%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 14.4

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

   8. Applied rewrites 14.4%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 31.3%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-250}:\\ \;\;\;\;\left(1 + e^{\left(-\varepsilon\right) \cdot x}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 0.00039:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - \left(x \cdot \varepsilon - 1\right)\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+137}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(-\varepsilon\right) \cdot \left(-x\right)} + \left(\frac{-1}{\varepsilon} - -1\right) \cdot \mathsf{fma}\left(-1, x \cdot \varepsilon, 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, x, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{+21}:\\ \;\;\;\;\left(e^{\left(-x\right) \cdot 1} - -1\right) \cdot 0.5\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-183}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - -1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.35e+21)
   (* (- (exp (* (- x) 1.0)) -1.0) 0.5)
   (if (<= x -6.2e-183)
     (* (fma (fma -1.0 (/ (- (* eps eps) 1.0) (- eps 1.0)) eps) x 2.0) 0.5)
     (* (- (exp (* x eps)) -1.0) 0.5))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.35e+21) {
		tmp = (exp((-x * 1.0)) - -1.0) * 0.5;
	} else if (x <= -6.2e-183) {
		tmp = fma(fma(-1.0, (((eps * eps) - 1.0) / (eps - 1.0)), eps), x, 2.0) * 0.5;
	} else {
		tmp = (exp((x * eps)) - -1.0) * 0.5;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.35e+21)
		tmp = Float64(Float64(exp(Float64(Float64(-x) * 1.0)) - -1.0) * 0.5);
	elseif (x <= -6.2e-183)
		tmp = Float64(fma(fma(-1.0, Float64(Float64(Float64(eps * eps) - 1.0) / Float64(eps - 1.0)), eps), x, 2.0) * 0.5);
	else
		tmp = Float64(Float64(exp(Float64(x * eps)) - -1.0) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.35e+21], N[(N[(N[Exp[N[((-x) * 1.0), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, -6.2e-183], N[(N[(N[(-1.0 * N[(N[(N[(eps * eps), $MachinePrecision] - 1.0), $MachinePrecision] / N[(eps - 1.0), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.35e21

   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 50.3%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 100.0%

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

   if -2.35e21 < x < -6.19999999999999999e-183

   1. Initial program 66.6%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 94.6%

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

   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 39.2

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

   8. Applied rewrites 39.2%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   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 51.4

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

   10. Applied rewrites 51.4%

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

   11. Taylor expanded in eps around inf

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

   13. Applied rewrites 51.4%

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

   if -6.19999999999999999e-183 < x

   1. Initial program 72.7%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 99.8%

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

   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

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

      2. lower-*.f64 82.0

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

   8. Applied rewrites 82.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(e^{x \cdot \varepsilon} - \left(x \cdot \left(1 + \varepsilon\right) - 1\right)\right) \cdot \frac{1}{2} \]
   10. 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 62.7

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

   11. Applied rewrites 62.7%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 61.7%

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

Alternative 5: 65.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-250}:\\ \;\;\;\;\left(1 + e^{\left(-\varepsilon\right) \cdot x}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - -1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -6.5e-250)
   (* (+ 1.0 (exp (* (- eps) x))) 0.5)
   (* (- (exp (* x eps)) -1.0) 0.5)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -6.5e-250) {
		tmp = (1.0 + exp((-eps * x))) * 0.5;
	} else {
		tmp = (exp((x * eps)) - -1.0) * 0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-6.5d-250)) then
        tmp = (1.0d0 + exp((-eps * x))) * 0.5d0
    else
        tmp = (exp((x * eps)) - (-1.0d0)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -6.5e-250) {
		tmp = (1.0 + Math.exp((-eps * x))) * 0.5;
	} else {
		tmp = (Math.exp((x * eps)) - -1.0) * 0.5;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -6.5e-250:
		tmp = (1.0 + math.exp((-eps * x))) * 0.5
	else:
		tmp = (math.exp((x * eps)) - -1.0) * 0.5
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -6.5e-250)
		tmp = Float64(Float64(1.0 + exp(Float64(Float64(-eps) * x))) * 0.5);
	else
		tmp = Float64(Float64(exp(Float64(x * eps)) - -1.0) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -6.5e-250)
		tmp = (1.0 + exp((-eps * x))) * 0.5;
	else
		tmp = (exp((x * eps)) - -1.0) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -6.5e-250], N[(N[(1.0 + N[Exp[N[((-eps) * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.49999999999999942e-250

   1. Initial program 74.3%

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

   3. Taylor expanded in eps around 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 97.7%

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

   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

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

      2. lower-*.f64 97.7

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

   8. Applied rewrites 97.7%

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

   9. Taylor expanded in eps around inf

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

       1. associate-*r* N/A

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

       2. mul-1-neg N/A

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

       3. lower-*.f64 N/A

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

       4. lower-neg.f64 97.8

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

   11. Applied rewrites 97.8%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 61.0%

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

   if -6.49999999999999942e-250 < x

   1. Initial program 76.5%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 99.7%

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

   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

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

      2. lower-*.f64 80.3

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

   8. Applied rewrites 80.3%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(e^{x \cdot \varepsilon} - \left(x \cdot \left(1 + \varepsilon\right) - 1\right)\right) \cdot \frac{1}{2} \]
   10. 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 59.6

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

   11. Applied rewrites 59.6%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 58.5%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-250}:\\ \;\;\;\;\left(1 + e^{\left(-\varepsilon\right) \cdot x}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - -1\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-183}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - -1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -6.2e-183)
   (* (fma (fma -1.0 (/ (- (* eps eps) 1.0) (- eps 1.0)) eps) x 2.0) 0.5)
   (* (- (exp (* x eps)) -1.0) 0.5)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -6.2e-183) {
		tmp = fma(fma(-1.0, (((eps * eps) - 1.0) / (eps - 1.0)), eps), x, 2.0) * 0.5;
	} else {
		tmp = (exp((x * eps)) - -1.0) * 0.5;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -6.2e-183)
		tmp = Float64(fma(fma(-1.0, Float64(Float64(Float64(eps * eps) - 1.0) / Float64(eps - 1.0)), eps), x, 2.0) * 0.5);
	else
		tmp = Float64(Float64(exp(Float64(x * eps)) - -1.0) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -6.2e-183], N[(N[(N[(-1.0 * N[(N[(N[(eps * eps), $MachinePrecision] - 1.0), $MachinePrecision] / N[(eps - 1.0), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.19999999999999999e-183

   1. Initial program 83.1%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 21.6

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

   8. Applied rewrites 21.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   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 43.0

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

   10. Applied rewrites 43.0%

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

   11. Taylor expanded in eps around inf

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

   13. Applied rewrites 43.0%

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

   if -6.19999999999999999e-183 < x

   1. Initial program 72.7%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 99.8%

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

   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

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

      2. lower-*.f64 82.0

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

   8. Applied rewrites 82.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(e^{x \cdot \varepsilon} - \left(x \cdot \left(1 + \varepsilon\right) - 1\right)\right) \cdot \frac{1}{2} \]
   10. 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 62.7

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

   11. Applied rewrites 62.7%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 61.7%

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

Alternative 7: 60.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-183}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-248}:\\ \;\;\;\;\left(1 - \left(\mathsf{fma}\left(x, \varepsilon, x\right) - 1\right)\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 820:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, \frac{-1 + \varepsilon \cdot \varepsilon}{\varepsilon - -1}\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, x, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -6.2e-183)
   (* (fma (fma -1.0 (/ (- (* eps eps) 1.0) (- eps 1.0)) eps) x 2.0) 0.5)
   (if (<= x 1.9e-248)
     (* (- 1.0 (- (fma x eps x) 1.0)) 0.5)
     (if (<= x 820.0)
       (*
        (fma
         (fma -1.0 (- eps -1.0) (/ (+ -1.0 (* eps eps)) (- eps -1.0)))
         x
         2.0)
        0.5)
       (/
        (-
         (* (+ 1.0 (/ 1.0 eps)) (fma (- eps 1.0) x 1.0))
         (* (- (/ 1.0 eps) 1.0) (fma -1.0 x 1.0)))
        2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -6.2e-183) {
		tmp = fma(fma(-1.0, (((eps * eps) - 1.0) / (eps - 1.0)), eps), x, 2.0) * 0.5;
	} else if (x <= 1.9e-248) {
		tmp = (1.0 - (fma(x, eps, x) - 1.0)) * 0.5;
	} else if (x <= 820.0) {
		tmp = fma(fma(-1.0, (eps - -1.0), ((-1.0 + (eps * eps)) / (eps - -1.0))), x, 2.0) * 0.5;
	} else {
		tmp = (((1.0 + (1.0 / eps)) * fma((eps - 1.0), x, 1.0)) - (((1.0 / eps) - 1.0) * fma(-1.0, x, 1.0))) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -6.2e-183)
		tmp = Float64(fma(fma(-1.0, Float64(Float64(Float64(eps * eps) - 1.0) / Float64(eps - 1.0)), eps), x, 2.0) * 0.5);
	elseif (x <= 1.9e-248)
		tmp = Float64(Float64(1.0 - Float64(fma(x, eps, x) - 1.0)) * 0.5);
	elseif (x <= 820.0)
		tmp = Float64(fma(fma(-1.0, Float64(eps - -1.0), Float64(Float64(-1.0 + Float64(eps * eps)) / Float64(eps - -1.0))), x, 2.0) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * fma(Float64(eps - 1.0), x, 1.0)) - Float64(Float64(Float64(1.0 / eps) - 1.0) * fma(-1.0, x, 1.0))) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -6.2e-183], N[(N[(N[(-1.0 * N[(N[(N[(eps * eps), $MachinePrecision] - 1.0), $MachinePrecision] / N[(eps - 1.0), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 1.9e-248], N[(N[(1.0 - N[(N[(x * eps + x), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 820.0], N[(N[(N[(-1.0 * N[(eps - -1.0), $MachinePrecision] + N[(N[(-1.0 + N[(eps * eps), $MachinePrecision]), $MachinePrecision] / N[(eps - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision] * N[(-1.0 * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6.19999999999999999e-183

   1. Initial program 83.1%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 21.6

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

   8. Applied rewrites 21.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   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 43.0

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

   10. Applied rewrites 43.0%

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

   11. Taylor expanded in eps around inf

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

   13. Applied rewrites 43.0%

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

   if -6.19999999999999999e-183 < x < 1.8999999999999999e-248

   1. Initial program 52.6%

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

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

      2. 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

      \[\leadsto \left(e^{x \cdot \varepsilon} - \left(x \cdot \left(1 + \varepsilon\right) - 1\right)\right) \cdot \frac{1}{2} \]
   10. 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 98.4

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

   11. Applied rewrites 98.4%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 96.3%

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

   if 1.8999999999999999e-248 < x < 820

   1. Initial program 47.8%

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

   3. Taylor expanded in eps around 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 99.2%

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

   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 78.6

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

   8. Applied rewrites 78.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   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 87.8

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

   10. Applied rewrites 87.8%

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

   if 820 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 23.7

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

   5. Applied rewrites 23.7%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 21.2

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

   8. Applied rewrites 21.2%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 33.5%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-183}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-248}:\\ \;\;\;\;\left(1 - \left(\mathsf{fma}\left(x, \varepsilon, x\right) - 1\right)\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 820:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, \frac{-1 + \varepsilon \cdot \varepsilon}{\varepsilon - -1}\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, x, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.1% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \varepsilon - 1\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{-183}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{t\_0}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-244}:\\ \;\;\;\;\left(1 - \left(\mathsf{fma}\left(x, \varepsilon, x\right) - 1\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{t\_0}{-1}, -1 + \varepsilon\right), x, 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (* eps eps) 1.0)))
   (if (<= x -6.2e-183)
     (* (fma (fma -1.0 (/ t_0 (- eps 1.0)) eps) x 2.0) 0.5)
     (if (<= x 8.5e-244)
       (* (- 1.0 (- (fma x eps x) 1.0)) 0.5)
       (* (fma (fma -1.0 (/ t_0 -1.0) (+ -1.0 eps)) x 2.0) 0.5)))))

C

double code(double x, double eps) {
	double t_0 = (eps * eps) - 1.0;
	double tmp;
	if (x <= -6.2e-183) {
		tmp = fma(fma(-1.0, (t_0 / (eps - 1.0)), eps), x, 2.0) * 0.5;
	} else if (x <= 8.5e-244) {
		tmp = (1.0 - (fma(x, eps, x) - 1.0)) * 0.5;
	} else {
		tmp = fma(fma(-1.0, (t_0 / -1.0), (-1.0 + eps)), x, 2.0) * 0.5;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(eps * eps) - 1.0)
	tmp = 0.0
	if (x <= -6.2e-183)
		tmp = Float64(fma(fma(-1.0, Float64(t_0 / Float64(eps - 1.0)), eps), x, 2.0) * 0.5);
	elseif (x <= 8.5e-244)
		tmp = Float64(Float64(1.0 - Float64(fma(x, eps, x) - 1.0)) * 0.5);
	else
		tmp = Float64(fma(fma(-1.0, Float64(t_0 / -1.0), Float64(-1.0 + eps)), x, 2.0) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(eps * eps), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -6.2e-183], N[(N[(N[(-1.0 * N[(t$95$0 / N[(eps - 1.0), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 8.5e-244], N[(N[(1.0 - N[(N[(x * eps + x), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(-1.0 * N[(t$95$0 / -1.0), $MachinePrecision] + N[(-1.0 + eps), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.19999999999999999e-183

   1. Initial program 83.1%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 21.6

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

   8. Applied rewrites 21.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   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 43.0

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

   10. Applied rewrites 43.0%

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

   11. Taylor expanded in eps around inf

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

   13. Applied rewrites 43.0%

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

   if -6.19999999999999999e-183 < x < 8.4999999999999999e-244

   1. Initial program 52.6%

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

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

      2. 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

      \[\leadsto \left(e^{x \cdot \varepsilon} - \left(x \cdot \left(1 + \varepsilon\right) - 1\right)\right) \cdot \frac{1}{2} \]
   10. 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 98.4

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

   11. Applied rewrites 98.4%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 96.3%

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

   if 8.4999999999999999e-244 < x

   1. Initial program 79.8%

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

   3. Taylor expanded in eps around 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 99.7%

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

   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 32.1

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

   8. Applied rewrites 32.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   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 40.4

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

   10. Applied rewrites 40.4%

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

   11. Taylor expanded in eps around 0

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

   13. Applied rewrites 59.3%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-183}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-244}:\\ \;\;\;\;\left(1 - \left(\mathsf{fma}\left(x, \varepsilon, x\right) - 1\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{-1}, -1 + \varepsilon\right), x, 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 52.5% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-183}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, 1, -1 + \varepsilon\right), x, 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -6.2e-183)
   (* (fma (fma -1.0 (/ (- (* eps eps) 1.0) (- eps 1.0)) eps) x 2.0) 0.5)
   (* (fma (fma -1.0 1.0 (+ -1.0 eps)) x 2.0) 0.5)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -6.2e-183) {
		tmp = fma(fma(-1.0, (((eps * eps) - 1.0) / (eps - 1.0)), eps), x, 2.0) * 0.5;
	} else {
		tmp = fma(fma(-1.0, 1.0, (-1.0 + eps)), x, 2.0) * 0.5;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -6.2e-183)
		tmp = Float64(fma(fma(-1.0, Float64(Float64(Float64(eps * eps) - 1.0) / Float64(eps - 1.0)), eps), x, 2.0) * 0.5);
	else
		tmp = Float64(fma(fma(-1.0, 1.0, Float64(-1.0 + eps)), x, 2.0) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -6.2e-183], N[(N[(N[(-1.0 * N[(N[(N[(eps * eps), $MachinePrecision] - 1.0), $MachinePrecision] / N[(eps - 1.0), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(-1.0 * 1.0 + N[(-1.0 + eps), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.19999999999999999e-183

   1. Initial program 83.1%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 21.6

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

   8. Applied rewrites 21.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   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 43.0

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

   10. Applied rewrites 43.0%

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

   11. Taylor expanded in eps around inf

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

   13. Applied rewrites 43.0%

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

   if -6.19999999999999999e-183 < x

   1. Initial program 72.7%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 99.8%

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

   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 48.8

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

   8. Applied rewrites 48.8%

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

   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 54.2%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-183}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, 1, -1 + \varepsilon\right), x, 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.7% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-250}:\\ \;\;\;\;\left(1 - \left(\mathsf{fma}\left(x, \varepsilon, x\right) - 1\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, 1, -1 + \varepsilon\right), x, 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -6.5e-250)
   (* (- 1.0 (- (fma x eps x) 1.0)) 0.5)
   (* (fma (fma -1.0 1.0 (+ -1.0 eps)) x 2.0) 0.5)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -6.5e-250) {
		tmp = (1.0 - (fma(x, eps, x) - 1.0)) * 0.5;
	} else {
		tmp = fma(fma(-1.0, 1.0, (-1.0 + eps)), x, 2.0) * 0.5;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -6.5e-250)
		tmp = Float64(Float64(1.0 - Float64(fma(x, eps, x) - 1.0)) * 0.5);
	else
		tmp = Float64(fma(fma(-1.0, 1.0, Float64(-1.0 + eps)), x, 2.0) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -6.5e-250], N[(N[(1.0 - N[(N[(x * eps + x), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(-1.0 * 1.0 + N[(-1.0 + eps), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.49999999999999942e-250

   1. Initial program 74.3%

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

   3. Taylor expanded in eps around 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 97.7%

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

   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

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

      2. lower-*.f64 97.7

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

   8. Applied rewrites 97.7%

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

   9. Taylor expanded in x around 0

      \[\leadsto \left(e^{x \cdot \varepsilon} - \left(x \cdot \left(1 + \varepsilon\right) - 1\right)\right) \cdot \frac{1}{2} \]
   10. 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 70.9

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

   11. Applied rewrites 70.9%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 43.0%

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

   if -6.49999999999999942e-250 < x

   1. Initial program 76.5%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 99.7%

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

   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 44.9

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

   8. Applied rewrites 44.9%

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

   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 50.8%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-250}:\\ \;\;\;\;\left(1 - \left(\mathsf{fma}\left(x, \varepsilon, x\right) - 1\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, 1, -1 + \varepsilon\right), x, 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.7% accurate, 15.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.7%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
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 99.0%

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

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

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

   2. lower-*.f64 86.5

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

8. Applied rewrites 86.5%

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

9. Taylor expanded in x around 0

   \[\leadsto \left(e^{x \cdot \varepsilon} - \left(x \cdot \left(1 + \varepsilon\right) - 1\right)\right) \cdot \frac{1}{2} \]
10. 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 63.6

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

11. Applied rewrites 63.6%

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

12. Taylor expanded in x around 0

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

14. Applied rewrites 48.9%

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

15. Final simplification 48.9%

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

Alternative 12: 44.8% accurate, 273.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 1.0)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 1.0d0
end function

Java

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

Python

def code(x, eps):
	return 1.0

Julia

function code(x, eps)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 1.0

Derivation

1. Initial program 75.7%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 41.1%

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

Reproduce

herbie shell --seed 2025051 
(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))