NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.7% → 98.9%

Time: 6.9s

Alternatives: 17

Speedup: 1.2×

• 38.8% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -3.230000979644222e+54
  2. eps = 2.13640831400891e-53

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.7% 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, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, eps)
	return Float64(Float64(exp(Float64(x * Float64(-1.0 + eps))) - Float64(-1.0 / exp(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[(-1.0 / N[Exp[N[(x * eps + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 73.9%

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

3. Taylor expanded in eps around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 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. Step-by-step derivation

   1. lift-exp.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. rec-exp N/A

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

   5. lift-fma.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. lift-/.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Final simplification 99.7%

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

Alternative 2: 84.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if eps < 1

   1. Initial program 64.6%

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

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

   6. Taylor expanded in eps around 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 87.2

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

   8. Applied rewrites 87.2%

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

   if 1 < eps

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

4. Final simplification 90.5%

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

Alternative 3: 69.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -2.8 \cdot 10^{-233}:\\ \;\;\;\;\left(1 + e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+26}:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - \frac{-1}{\mathsf{fma}\left(\varepsilon - -1, x, 1\right)}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 + t\_0\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -2.8e-233)
     (* (+ 1.0 (exp (- (fma x eps x)))) 0.5)
     (if (<= x 3.1e+26)
       (* (- (exp (* x eps)) (/ -1.0 (fma (- eps -1.0) x 1.0))) 0.5)
       (* (+ t_0 t_0) 0.5)))))

C

double code(double x, double eps) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -2.8e-233) {
		tmp = (1.0 + exp(-fma(x, eps, x))) * 0.5;
	} else if (x <= 3.1e+26) {
		tmp = (exp((x * eps)) - (-1.0 / fma((eps - -1.0), x, 1.0))) * 0.5;
	} else {
		tmp = (t_0 + t_0) * 0.5;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -2.8e-233)
		tmp = Float64(Float64(1.0 + exp(Float64(-fma(x, eps, x)))) * 0.5);
	elseif (x <= 3.1e+26)
		tmp = Float64(Float64(exp(Float64(x * eps)) - Float64(-1.0 / fma(Float64(eps - -1.0), x, 1.0))) * 0.5);
	else
		tmp = Float64(Float64(t_0 + t_0) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -2.8e-233], N[(N[(1.0 + N[Exp[(-N[(x * eps + x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 3.1e+26], N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] - N[(-1.0 / N[(N[(eps - -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(t$95$0 + t$95$0), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.8000000000000001e-233

   1. Initial program 74.1%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 99.4%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative 68.7

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

      2. distribute-rgt-neg-in 68.7

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

   8. Applied rewrites 68.7%

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

   if -2.8000000000000001e-233 < x < 3.1e26

   1. Initial program 53.7%

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

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. rec-exp N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-/.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 96.5

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

   10. Applied rewrites 96.5%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-+.f64 81.8

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

   13. Applied rewrites 81.8%

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

   if 3.1e26 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in eps around 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 66.2

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

   8. Applied rewrites 66.2%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-233}:\\ \;\;\;\;\left(1 + e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+26}:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - \frac{-1}{\mathsf{fma}\left(\varepsilon - -1, x, 1\right)}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-x} + e^{-x}\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 73.9%

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

3. Taylor expanded in eps around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 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. Final simplification 99.7%

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

Alternative 5: 69.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-233}:\\ \;\;\;\;\left(1 + e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+26}:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - \frac{-1}{\mathsf{fma}\left(\varepsilon - -1, x, 1\right)}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left({\varepsilon}^{-1} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.8e-233)
   (* (+ 1.0 (exp (- (fma x eps x)))) 0.5)
   (if (<= x 3.1e+26)
     (* (- (exp (* x eps)) (/ -1.0 (fma (- eps -1.0) x 1.0))) 0.5)
     (/ (- (- (pow eps -1.0) -1.0) (* (- (/ 1.0 eps) 1.0) 1.0)) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.8e-233) {
		tmp = (1.0 + exp(-fma(x, eps, x))) * 0.5;
	} else if (x <= 3.1e+26) {
		tmp = (exp((x * eps)) - (-1.0 / fma((eps - -1.0), x, 1.0))) * 0.5;
	} else {
		tmp = ((pow(eps, -1.0) - -1.0) - (((1.0 / eps) - 1.0) * 1.0)) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.8e-233)
		tmp = Float64(Float64(1.0 + exp(Float64(-fma(x, eps, x)))) * 0.5);
	elseif (x <= 3.1e+26)
		tmp = Float64(Float64(exp(Float64(x * eps)) - Float64(-1.0 / fma(Float64(eps - -1.0), x, 1.0))) * 0.5);
	else
		tmp = Float64(Float64(Float64((eps ^ -1.0) - -1.0) - Float64(Float64(Float64(1.0 / eps) - 1.0) * 1.0)) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.8e-233], N[(N[(1.0 + N[Exp[(-N[(x * eps + x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 3.1e+26], N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] - N[(-1.0 / N[(N[(eps - -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[Power[eps, -1.0], $MachinePrecision] - -1.0), $MachinePrecision] - N[(N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.8000000000000001e-233

   1. Initial program 74.1%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 99.4%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative 68.7

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

      2. distribute-rgt-neg-in 68.7

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

   8. Applied rewrites 68.7%

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

   if -2.8000000000000001e-233 < x < 3.1e26

   1. Initial program 53.7%

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

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. rec-exp N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-/.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 96.5

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

   10. Applied rewrites 96.5%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-+.f64 81.8

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

   13. Applied rewrites 81.8%

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

   if 3.1e26 < 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

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

   5. Applied rewrites 24.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. inv-pow N/A

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

      4. lower-pow.f64 62.4

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

   8. Applied rewrites 62.4%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-233}:\\ \;\;\;\;\left(1 + e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+26}:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - \frac{-1}{\mathsf{fma}\left(\varepsilon - -1, x, 1\right)}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left({\varepsilon}^{-1} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-233}:\\ \;\;\;\;\left(1 + e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+17}:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - \frac{-1}{\mathsf{fma}\left(\varepsilon - -1, x, 1\right)}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\left(\varepsilon - 1\right) \cdot x\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, x, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.8e-233)
   (* (+ 1.0 (exp (- (fma x eps x)))) 0.5)
   (if (<= x 2.25e+17)
     (* (- (exp (* x eps)) (/ -1.0 (fma (- eps -1.0) x 1.0))) 0.5)
     (/
      (-
       (* (+ 1.0 (/ 1.0 eps)) (* (- eps 1.0) x))
       (* (- (/ 1.0 eps) 1.0) (fma -1.0 x 1.0)))
      2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.8e-233) {
		tmp = (1.0 + exp(-fma(x, eps, x))) * 0.5;
	} else if (x <= 2.25e+17) {
		tmp = (exp((x * eps)) - (-1.0 / fma((eps - -1.0), x, 1.0))) * 0.5;
	} else {
		tmp = (((1.0 + (1.0 / eps)) * ((eps - 1.0) * x)) - (((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 <= -2.8e-233)
		tmp = Float64(Float64(1.0 + exp(Float64(-fma(x, eps, x)))) * 0.5);
	elseif (x <= 2.25e+17)
		tmp = Float64(Float64(exp(Float64(x * eps)) - Float64(-1.0 / fma(Float64(eps - -1.0), x, 1.0))) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * Float64(Float64(eps - 1.0) * x)) - 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, -2.8e-233], N[(N[(1.0 + N[Exp[(-N[(x * eps + x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 2.25e+17], N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] - N[(-1.0 / N[(N[(eps - -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * x), $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 3 regimes

2. if x < -2.8000000000000001e-233

   1. Initial program 74.1%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 99.4%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative 68.7

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

      2. distribute-rgt-neg-in 68.7

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

   8. Applied rewrites 68.7%

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

   if -2.8000000000000001e-233 < x < 2.25e17

   1. Initial program 51.5%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. rec-exp N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-/.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 98.6

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

   10. Applied rewrites 98.6%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-+.f64 84.4

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

   13. Applied rewrites 84.4%

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

   if 2.25e17 < 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 18.3

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

   5. Applied rewrites 18.3%

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

   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 29.8

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

   8. Applied rewrites 29.8%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lift--.f64 29.8

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

   11. Applied rewrites 29.8%

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

   12. Taylor expanded in eps around 0

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

   14. Applied rewrites 43.0%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-233}:\\ \;\;\;\;\left(1 + e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+17}:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - \frac{-1}{\mathsf{fma}\left(\varepsilon - -1, x, 1\right)}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\left(\varepsilon - 1\right) \cdot x\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, x, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-233}:\\ \;\;\;\;\left(1 + e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+17}:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - \left(\mathsf{fma}\left(x, \varepsilon, x\right) - 1\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\left(\varepsilon - 1\right) \cdot x\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, x, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.8e-233)
   (* (+ 1.0 (exp (- (fma x eps x)))) 0.5)
   (if (<= x 1.95e+17)
     (* (- (exp (* x eps)) (- (fma x eps x) 1.0)) 0.5)
     (/
      (-
       (* (+ 1.0 (/ 1.0 eps)) (* (- eps 1.0) x))
       (* (- (/ 1.0 eps) 1.0) (fma -1.0 x 1.0)))
      2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.8e-233) {
		tmp = (1.0 + exp(-fma(x, eps, x))) * 0.5;
	} else if (x <= 1.95e+17) {
		tmp = (exp((x * eps)) - (fma(x, eps, x) - 1.0)) * 0.5;
	} else {
		tmp = (((1.0 + (1.0 / eps)) * ((eps - 1.0) * x)) - (((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 <= -2.8e-233)
		tmp = Float64(Float64(1.0 + exp(Float64(-fma(x, eps, x)))) * 0.5);
	elseif (x <= 1.95e+17)
		tmp = Float64(Float64(exp(Float64(x * eps)) - Float64(fma(x, eps, x) - 1.0)) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * Float64(Float64(eps - 1.0) * x)) - 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, -2.8e-233], N[(N[(1.0 + N[Exp[(-N[(x * eps + x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 1.95e+17], N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] - N[(N[(x * eps + x), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * x), $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 3 regimes

2. if x < -2.8000000000000001e-233

   1. Initial program 74.1%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 99.4%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative 68.7

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

      2. distribute-rgt-neg-in 68.7

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

   8. Applied rewrites 68.7%

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

   if -2.8000000000000001e-233 < x < 1.95e17

   1. Initial program 51.5%

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

      1. lift-exp.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. rec-exp N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. lift-/.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 98.6

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

   10. Applied rewrites 98.6%

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

   11. Taylor expanded in x around 0

       \[\leadsto \left(e^{x \cdot \varepsilon} - \left(x \cdot \left(1 + \varepsilon\right) - 1\right)\right) \cdot \frac{1}{2} \]
   12. 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

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

       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 84.8

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

   13. Applied rewrites 84.8%

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

   if 1.95e17 < 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 18.3

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

   5. Applied rewrites 18.3%

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

   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 29.8

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

   8. Applied rewrites 29.8%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lift--.f64 29.8

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

   11. Applied rewrites 29.8%

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

   12. Taylor expanded in eps around 0

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

   14. Applied rewrites 43.0%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-233}:\\ \;\;\;\;\left(1 + e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+17}:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - \left(\mathsf{fma}\left(x, \varepsilon, x\right) - 1\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\left(\varepsilon - 1\right) \cdot x\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, x, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00036:\\ \;\;\;\;\left(e^{-x} - -1\right) \cdot 0.5\\ \mathbf{elif}\;x \leq -3.7 \cdot 10^{-233}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, -1\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+17}:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - -1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\left(\varepsilon - 1\right) \cdot x\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, x, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -0.00036)
   (* (- (exp (- x)) -1.0) 0.5)
   (if (<= x -3.7e-233)
     (* (fma (fma -1.0 (/ (- (* eps eps) 1.0) (- eps 1.0)) -1.0) x 2.0) 0.5)
     (if (<= x 2.25e+17)
       (* (- (exp (* x eps)) -1.0) 0.5)
       (/
        (-
         (* (+ 1.0 (/ 1.0 eps)) (* (- eps 1.0) x))
         (* (- (/ 1.0 eps) 1.0) (fma -1.0 x 1.0)))
        2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -0.00036) {
		tmp = (exp(-x) - -1.0) * 0.5;
	} else if (x <= -3.7e-233) {
		tmp = fma(fma(-1.0, (((eps * eps) - 1.0) / (eps - 1.0)), -1.0), x, 2.0) * 0.5;
	} else if (x <= 2.25e+17) {
		tmp = (exp((x * eps)) - -1.0) * 0.5;
	} else {
		tmp = (((1.0 + (1.0 / eps)) * ((eps - 1.0) * x)) - (((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 <= -0.00036)
		tmp = Float64(Float64(exp(Float64(-x)) - -1.0) * 0.5);
	elseif (x <= -3.7e-233)
		tmp = Float64(fma(fma(-1.0, Float64(Float64(Float64(eps * eps) - 1.0) / Float64(eps - 1.0)), -1.0), x, 2.0) * 0.5);
	elseif (x <= 2.25e+17)
		tmp = Float64(Float64(exp(Float64(x * eps)) - -1.0) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * Float64(Float64(eps - 1.0) * x)) - 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, -0.00036], N[(N[(N[Exp[(-x)], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, -3.7e-233], N[(N[(N[(-1.0 * N[(N[(N[(eps * eps), $MachinePrecision] - 1.0), $MachinePrecision] / N[(eps - 1.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 2.25e+17], N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * x), $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 < -3.60000000000000023e-4

   1. Initial program 97.8%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 98.6%

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

   6. Taylor expanded in 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 63.4%

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

   9. Taylor expanded in eps around 0

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

       1. mul-1-neg N/A

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

       2. lift-neg.f64 96.4

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

   11. Applied rewrites 96.4%

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

   if -3.60000000000000023e-4 < x < -3.6999999999999998e-233

   1. Initial program 55.8%

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

      \[\leadsto \left(2 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + -1 \cdot \left(1 - \varepsilon\right)\right)\right) \cdot \frac{1}{2} \]
   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 77.9

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

   8. Applied rewrites 77.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   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. lift--.f64 87.9

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

   10. Applied rewrites 87.9%

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

   11. Taylor expanded in eps around 0

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

   13. Applied rewrites 87.9%

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

   if -3.6999999999999998e-233 < x < 2.25e17

   1. Initial program 51.5%

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

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

   8. Applied rewrites 83.8%

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

   9. Taylor expanded in eps around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 84.0

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

   11. Applied rewrites 84.0%

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

   if 2.25e17 < 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 18.3

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

   5. Applied rewrites 18.3%

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

   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 29.8

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

   8. Applied rewrites 29.8%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lift--.f64 29.8

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

   11. Applied rewrites 29.8%

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

   12. Taylor expanded in eps around 0

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

   14. Applied rewrites 43.0%

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

Alternative 9: 66.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-260}:\\ \;\;\;\;\left(1 + e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+17}:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - -1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\left(\varepsilon - 1\right) \cdot x\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, x, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -5e-260)
   (* (+ 1.0 (exp (- (fma x eps x)))) 0.5)
   (if (<= x 2.25e+17)
     (* (- (exp (* x eps)) -1.0) 0.5)
     (/
      (-
       (* (+ 1.0 (/ 1.0 eps)) (* (- eps 1.0) x))
       (* (- (/ 1.0 eps) 1.0) (fma -1.0 x 1.0)))
      2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -5e-260) {
		tmp = (1.0 + exp(-fma(x, eps, x))) * 0.5;
	} else if (x <= 2.25e+17) {
		tmp = (exp((x * eps)) - -1.0) * 0.5;
	} else {
		tmp = (((1.0 + (1.0 / eps)) * ((eps - 1.0) * x)) - (((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 <= -5e-260)
		tmp = Float64(Float64(1.0 + exp(Float64(-fma(x, eps, x)))) * 0.5);
	elseif (x <= 2.25e+17)
		tmp = Float64(Float64(exp(Float64(x * eps)) - -1.0) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * Float64(Float64(eps - 1.0) * x)) - 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, -5e-260], N[(N[(1.0 + N[Exp[(-N[(x * eps + x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 2.25e+17], N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * x), $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 3 regimes

2. if x < -5.0000000000000003e-260

   1. Initial program 71.2%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative 71.3

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

      2. distribute-rgt-neg-in 71.3

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

   8. Applied rewrites 71.3%

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

   if -5.0000000000000003e-260 < x < 2.25e17

   1. Initial program 52.9%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 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

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

   8. Applied rewrites 81.9%

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

   9. Taylor expanded in eps around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 82.2

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

   11. Applied rewrites 82.2%

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

   if 2.25e17 < 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 18.3

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

   5. Applied rewrites 18.3%

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

   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 29.8

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

   8. Applied rewrites 29.8%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lift--.f64 29.8

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

   11. Applied rewrites 29.8%

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

   12. Taylor expanded in eps around 0

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

   14. Applied rewrites 43.0%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-260}:\\ \;\;\;\;\left(1 + e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+17}:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - -1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\left(\varepsilon - 1\right) \cdot x\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, x, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00036:\\ \;\;\;\;\left(e^{-x} - -1\right) \cdot 0.5\\ \mathbf{elif}\;x \leq -3.7 \cdot 10^{-233}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, -1\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-222}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.55 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, \frac{-1 + \varepsilon \cdot \varepsilon}{\varepsilon - -1}\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\left(\varepsilon - 1\right) \cdot x\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, x, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -0.00036)
   (* (- (exp (- x)) -1.0) 0.5)
   (if (<= x -3.7e-233)
     (* (fma (fma -1.0 (/ (- (* eps eps) 1.0) (- eps 1.0)) -1.0) x 2.0) 0.5)
     (if (<= x 5e-222)
       1.0
       (if (<= x 3.55e+16)
         (*
          (fma
           (fma -1.0 (- eps -1.0) (/ (+ -1.0 (* eps eps)) (- eps -1.0)))
           x
           2.0)
          0.5)
         (/
          (-
           (* (+ 1.0 (/ 1.0 eps)) (* (- eps 1.0) x))
           (* (- (/ 1.0 eps) 1.0) (fma -1.0 x 1.0)))
          2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -0.00036) {
		tmp = (exp(-x) - -1.0) * 0.5;
	} else if (x <= -3.7e-233) {
		tmp = fma(fma(-1.0, (((eps * eps) - 1.0) / (eps - 1.0)), -1.0), x, 2.0) * 0.5;
	} else if (x <= 5e-222) {
		tmp = 1.0;
	} else if (x <= 3.55e+16) {
		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)) * ((eps - 1.0) * x)) - (((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 <= -0.00036)
		tmp = Float64(Float64(exp(Float64(-x)) - -1.0) * 0.5);
	elseif (x <= -3.7e-233)
		tmp = Float64(fma(fma(-1.0, Float64(Float64(Float64(eps * eps) - 1.0) / Float64(eps - 1.0)), -1.0), x, 2.0) * 0.5);
	elseif (x <= 5e-222)
		tmp = 1.0;
	elseif (x <= 3.55e+16)
		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)) * Float64(Float64(eps - 1.0) * x)) - 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, -0.00036], N[(N[(N[Exp[(-x)], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, -3.7e-233], N[(N[(N[(-1.0 * N[(N[(N[(eps * eps), $MachinePrecision] - 1.0), $MachinePrecision] / N[(eps - 1.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 5e-222], 1.0, If[LessEqual[x, 3.55e+16], 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), $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 5 regimes

2. if x < -3.60000000000000023e-4

   1. Initial program 97.8%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 98.6%

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

   6. Taylor expanded in 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 63.4%

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

   9. Taylor expanded in eps around 0

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

       1. mul-1-neg N/A

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

       2. lift-neg.f64 96.4

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

   11. Applied rewrites 96.4%

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

   if -3.60000000000000023e-4 < x < -3.6999999999999998e-233

   1. Initial program 55.8%

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

      \[\leadsto \left(2 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + -1 \cdot \left(1 - \varepsilon\right)\right)\right) \cdot \frac{1}{2} \]
   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 77.9

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

   8. Applied rewrites 77.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   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. lift--.f64 87.9

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

   10. Applied rewrites 87.9%

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

   11. Taylor expanded in eps around 0

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

   13. Applied rewrites 87.9%

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

   if -3.6999999999999998e-233 < x < 5.00000000000000008e-222

   1. Initial program 46.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 95.6%

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

   if 5.00000000000000008e-222 < x < 3.55e16

   1. Initial program 55.9%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in 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 63.5

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

   8. Applied rewrites 63.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   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 73.4

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

   10. Applied rewrites 73.4%

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

   if 3.55e16 < 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 18.3

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

   5. Applied rewrites 18.3%

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

   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 29.8

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

   8. Applied rewrites 29.8%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lift--.f64 29.8

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

   11. Applied rewrites 29.8%

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

   12. Taylor expanded in eps around 0

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

   14. Applied rewrites 43.0%

       \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\left(\varepsilon - 1\right) \cdot x\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, x, 1\right)}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00036:\\ \;\;\;\;\left(e^{-x} - -1\right) \cdot 0.5\\ \mathbf{elif}\;x \leq -3.7 \cdot 10^{-233}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, -1\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-222}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.55 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, \frac{-1 + \varepsilon \cdot \varepsilon}{\varepsilon - -1}\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\left(\varepsilon - 1\right) \cdot x\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, x, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-233}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, -1\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-222}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.55 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, \frac{-1 + \varepsilon \cdot \varepsilon}{\varepsilon - -1}\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\left(\varepsilon - 1\right) \cdot x\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, x, 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.7e-233)
   (* (fma (fma -1.0 (/ (- (* eps eps) 1.0) (- eps 1.0)) -1.0) x 2.0) 0.5)
   (if (<= x 5e-222)
     1.0
     (if (<= x 3.55e+16)
       (*
        (fma
         (fma -1.0 (- eps -1.0) (/ (+ -1.0 (* eps eps)) (- eps -1.0)))
         x
         2.0)
        0.5)
       (/
        (-
         (* (+ 1.0 (/ 1.0 eps)) (* (- eps 1.0) x))
         (* (- (/ 1.0 eps) 1.0) (fma -1.0 x 1.0)))
        2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.7e-233) {
		tmp = fma(fma(-1.0, (((eps * eps) - 1.0) / (eps - 1.0)), -1.0), x, 2.0) * 0.5;
	} else if (x <= 5e-222) {
		tmp = 1.0;
	} else if (x <= 3.55e+16) {
		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)) * ((eps - 1.0) * x)) - (((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 <= -3.7e-233)
		tmp = Float64(fma(fma(-1.0, Float64(Float64(Float64(eps * eps) - 1.0) / Float64(eps - 1.0)), -1.0), x, 2.0) * 0.5);
	elseif (x <= 5e-222)
		tmp = 1.0;
	elseif (x <= 3.55e+16)
		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)) * Float64(Float64(eps - 1.0) * x)) - 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, -3.7e-233], N[(N[(N[(-1.0 * N[(N[(N[(eps * eps), $MachinePrecision] - 1.0), $MachinePrecision] / N[(eps - 1.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 5e-222], 1.0, If[LessEqual[x, 3.55e+16], 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), $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 < -3.6999999999999998e-233

   1. Initial program 74.1%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 99.4%

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

   6. Taylor expanded in 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 45.7

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

   8. Applied rewrites 45.7%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      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. lift--.f64 59.5

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

   10. Applied rewrites 59.5%

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

   11. Taylor expanded in eps around 0

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

   13. Applied rewrites 61.1%

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

   if -3.6999999999999998e-233 < x < 5.00000000000000008e-222

   1. Initial program 46.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 95.6%

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

   if 5.00000000000000008e-222 < x < 3.55e16

   1. Initial program 55.9%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in 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 63.5

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

   8. Applied rewrites 63.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   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 73.4

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

   10. Applied rewrites 73.4%

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

   if 3.55e16 < 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 18.3

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

   5. Applied rewrites 18.3%

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

   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 29.8

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

   8. Applied rewrites 29.8%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lift--.f64 29.8

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

   11. Applied rewrites 29.8%

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

   12. Taylor expanded in eps around 0

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

   14. Applied rewrites 43.0%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-233}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, -1\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-222}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.55 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, \frac{-1 + \varepsilon \cdot \varepsilon}{\varepsilon - -1}\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\left(\varepsilon - 1\right) \cdot x\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot \mathsf{fma}\left(-1, x, 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.9% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \varepsilon - 1\\ \mathbf{if}\;x \leq -3.7 \cdot 10^{-233}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{t\_0}{\varepsilon - 1}, -1\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-222}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-66}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, \frac{-1 + \varepsilon \cdot \varepsilon}{\varepsilon - -1}\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{t\_0}{-1}, -1 + \varepsilon\right), x, 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (* eps eps) 1.0)))
   (if (<= x -3.7e-233)
     (* (fma (fma -1.0 (/ t_0 (- eps 1.0)) -1.0) x 2.0) 0.5)
     (if (<= x 5e-222)
       1.0
       (if (<= x 4.6e-66)
         (*
          (fma
           (fma -1.0 (- eps -1.0) (/ (+ -1.0 (* eps eps)) (- eps -1.0)))
           x
           2.0)
          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 <= -3.7e-233) {
		tmp = fma(fma(-1.0, (t_0 / (eps - 1.0)), -1.0), x, 2.0) * 0.5;
	} else if (x <= 5e-222) {
		tmp = 1.0;
	} else if (x <= 4.6e-66) {
		tmp = fma(fma(-1.0, (eps - -1.0), ((-1.0 + (eps * eps)) / (eps - -1.0))), x, 2.0) * 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 <= -3.7e-233)
		tmp = Float64(fma(fma(-1.0, Float64(t_0 / Float64(eps - 1.0)), -1.0), x, 2.0) * 0.5);
	elseif (x <= 5e-222)
		tmp = 1.0;
	elseif (x <= 4.6e-66)
		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(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, -3.7e-233], N[(N[(N[(-1.0 * N[(t$95$0 / N[(eps - 1.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 5e-222], 1.0, If[LessEqual[x, 4.6e-66], 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[(-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 4 regimes

2. if x < -3.6999999999999998e-233

   1. Initial program 74.1%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 99.4%

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

   6. Taylor expanded in 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 45.7

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

   8. Applied rewrites 45.7%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      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. lift--.f64 59.5

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

   10. Applied rewrites 59.5%

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

   11. Taylor expanded in eps around 0

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

   13. Applied rewrites 61.1%

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

   if -3.6999999999999998e-233 < x < 5.00000000000000008e-222

   1. Initial program 46.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 95.6%

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

   if 5.00000000000000008e-222 < x < 4.59999999999999984e-66

   1. Initial program 58.4%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in 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 71.2

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

   8. Applied rewrites 71.2%

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

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

      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 80.1

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

   10. Applied rewrites 80.1%

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

   if 4.59999999999999984e-66 < x

   1. Initial program 91.1%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 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 11.3

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

   8. Applied rewrites 11.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   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. lift--.f64 19.7

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

   10. Applied rewrites 19.7%

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

   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 40.4%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-233}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, -1\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-222}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-66}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, \frac{-1 + \varepsilon \cdot \varepsilon}{\varepsilon - -1}\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{-1}, -1 + \varepsilon\right), x, 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.4% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \varepsilon - 1\\ \mathbf{if}\;x \leq -3.7 \cdot 10^{-233}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{t\_0}{\varepsilon - 1}, -1\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-183}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{t\_0}{-1}, -1 + \varepsilon\right), x, 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (* eps eps) 1.0)))
   (if (<= x -3.7e-233)
     (* (fma (fma -1.0 (/ t_0 (- eps 1.0)) -1.0) x 2.0) 0.5)
     (if (<= x 2.4e-183)
       1.0
       (* (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 <= -3.7e-233) {
		tmp = fma(fma(-1.0, (t_0 / (eps - 1.0)), -1.0), x, 2.0) * 0.5;
	} else if (x <= 2.4e-183) {
		tmp = 1.0;
	} 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 <= -3.7e-233)
		tmp = Float64(fma(fma(-1.0, Float64(t_0 / Float64(eps - 1.0)), -1.0), x, 2.0) * 0.5);
	elseif (x <= 2.4e-183)
		tmp = 1.0;
	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, -3.7e-233], N[(N[(N[(-1.0 * N[(t$95$0 / N[(eps - 1.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 2.4e-183], 1.0, 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 < -3.6999999999999998e-233

   1. Initial program 74.1%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 99.4%

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

   6. Taylor expanded in 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 45.7

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

   8. Applied rewrites 45.7%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      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. lift--.f64 59.5

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

   10. Applied rewrites 59.5%

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

   11. Taylor expanded in eps around 0

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

   13. Applied rewrites 61.1%

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

   if -3.6999999999999998e-233 < x < 2.39999999999999993e-183

   1. Initial program 50.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 90.7%

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

   if 2.39999999999999993e-183 < x

   1. Initial program 83.9%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 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 23.5

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

   8. Applied rewrites 23.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   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. lift--.f64 31.1

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

   10. Applied rewrites 31.1%

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

   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 47.7%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-233}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, -1\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-183}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{-1}, -1 + \varepsilon\right), x, 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 52.3% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-233}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, -1\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-x, 1 - \varepsilon, 1\right) - -1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.7e-233)
   (* (fma (fma -1.0 (/ (- (* eps eps) 1.0) (- eps 1.0)) -1.0) x 2.0) 0.5)
   (* (- (fma (- x) (- 1.0 eps) 1.0) -1.0) 0.5)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.7e-233) {
		tmp = fma(fma(-1.0, (((eps * eps) - 1.0) / (eps - 1.0)), -1.0), x, 2.0) * 0.5;
	} else {
		tmp = (fma(-x, (1.0 - eps), 1.0) - -1.0) * 0.5;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.7e-233)
		tmp = Float64(fma(fma(-1.0, Float64(Float64(Float64(eps * eps) - 1.0) / Float64(eps - 1.0)), -1.0), x, 2.0) * 0.5);
	else
		tmp = Float64(Float64(fma(Float64(-x), Float64(1.0 - eps), 1.0) - -1.0) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.7e-233], N[(N[(N[(-1.0 * N[(N[(N[(eps * eps), $MachinePrecision] - 1.0), $MachinePrecision] / N[(eps - 1.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[((-x) * N[(1.0 - eps), $MachinePrecision] + 1.0), $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.6999999999999998e-233

   1. Initial program 74.1%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 99.4%

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

   6. Taylor expanded in 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 45.7

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

   8. Applied rewrites 45.7%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      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. lift--.f64 59.5

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

   10. Applied rewrites 59.5%

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

   11. Taylor expanded in eps around 0

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

   13. Applied rewrites 61.1%

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

   if -3.6999999999999998e-233 < x

   1. Initial program 73.7%

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

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

   8. Applied rewrites 56.8%

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

   9. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

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

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. lift-neg.f64 N/A

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

       6. lift--.f64 49.6

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

   11. Applied rewrites 49.6%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-233}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, -1\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-x, 1 - \varepsilon, 1\right) - -1\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 49.7% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-260}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, -1\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-x, 1 - \varepsilon, 1\right) - -1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -5e-260)
   (* (fma (fma -1.0 (- eps -1.0) -1.0) x 2.0) 0.5)
   (* (- (fma (- x) (- 1.0 eps) 1.0) -1.0) 0.5)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -5e-260) {
		tmp = fma(fma(-1.0, (eps - -1.0), -1.0), x, 2.0) * 0.5;
	} else {
		tmp = (fma(-x, (1.0 - eps), 1.0) - -1.0) * 0.5;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -5e-260)
		tmp = Float64(fma(fma(-1.0, Float64(eps - -1.0), -1.0), x, 2.0) * 0.5);
	else
		tmp = Float64(Float64(fma(Float64(-x), Float64(1.0 - eps), 1.0) - -1.0) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -5e-260], N[(N[(N[(-1.0 * N[(eps - -1.0), $MachinePrecision] + -1.0), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[((-x) * N[(1.0 - eps), $MachinePrecision] + 1.0), $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.0000000000000003e-260

   1. Initial program 71.2%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in 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 50.6

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

   8. Applied rewrites 50.6%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 58.4%

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

   if -5.0000000000000003e-260 < x

   1. Initial program 75.9%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 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

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

   8. Applied rewrites 54.0%

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

   9. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

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

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. lift-neg.f64 N/A

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

       6. lift--.f64 46.3

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

   11. Applied rewrites 46.3%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-260}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, -1\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(-x, 1 - \varepsilon, 1\right) - -1\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 50.0% accurate, 13.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.9%

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

3. Taylor expanded in eps around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 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

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

8. Applied rewrites 64.5%

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

9. Taylor expanded in x around 0

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

    1. mul-1-neg N/A

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

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

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

    3. +-commutative N/A

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

    4. lower-fma.f64 N/A

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

    5. lift-neg.f64 N/A

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

    6. lift--.f64 54.4

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

11. Applied rewrites 54.4%

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

12. Final simplification 54.4%

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

Alternative 17: 43.8% 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 73.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 44.9%

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

Reproduce

herbie shell --seed 2025057 
(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))