NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.4% → 98.7%

Time: 6.3s

Alternatives: 15

Speedup: 3.8×

• 39.1% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -1.2972457497143217e+287
  2. eps = 1.4107308733521866e-265

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: 73.4% 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: 98.7% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := N[(N[(N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Power[N[Exp[-1.0], $MachinePrecision], N[(x * eps$95$m + 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 99.4%

   \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]
6. 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. mul-1-neg N/A

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

   5. pow-exp N/A

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

   6. lift-exp.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lift-fma.f64 99.4

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

7. Applied rewrites 99.4%

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

8. Final simplification 99.4%

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

Alternative 2: 98.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 2.15 \cdot 10^{-5}:\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x \cdot eps\_m} + e^{-\mathsf{fma}\left(x, eps\_m, x\right)}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 2.15e-5)
   (exp (- x))
   (* (+ (exp (* x eps_m)) (exp (- (fma x eps_m x)))) 0.5)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 2.15e-5) {
		tmp = exp(-x);
	} else {
		tmp = (exp((x * eps_m)) + exp(-fma(x, eps_m, x))) * 0.5;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 2.15e-5)
		tmp = exp(Float64(-x));
	else
		tmp = Float64(Float64(exp(Float64(x * eps_m)) + exp(Float64(-fma(x, eps_m, x)))) * 0.5);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 2.15e-5], N[Exp[(-x)], $MachinePrecision], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] + N[Exp[(-N[(x * eps$95$m + x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 2.1500000000000001e-5

   1. Initial program 62.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 99.1%

      \[\leadsto \color{blue}{\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)\right) \cdot 0.5} \]
   6. 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. mul-1-neg N/A

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

      5. pow-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-fma.f64 99.1

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

   7. Applied rewrites 99.1%

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

   8. Taylor expanded in eps around 0

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

      1. *-commutative N/A

         \[\leadsto e^{-1 \cdot x} \]

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

         \[\leadsto e^{-1 \cdot x} \]

      3. pow-exp N/A

         \[\leadsto e^{-1 \cdot x} \]

      4. +-commutative N/A

         \[\leadsto e^{-1 \cdot x} \]

      5. *-commutative N/A

         \[\leadsto e^{-1 \cdot x} \]

      6. mul-1-neg N/A

         \[\leadsto e^{-1 \cdot x} \]

      7. *-commutative N/A

         \[\leadsto e^{-1 \cdot x} \]

      8. +-commutative N/A

         \[\leadsto e^{-1 \cdot x} \]

      9. mul-1-neg N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 79.1

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

   10. Applied rewrites 79.1%

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

   if 2.1500000000000001e-5 < 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. distribute-lft-neg-in N/A

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

      2. *-commutative 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 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2.15 \cdot 10^{-5}:\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} + e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.9%

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

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

Alternative 4: 84.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \left(e^{\left(-x\right) \cdot \left(-eps\_m\right)} - -1\right) \cdot 0.5\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{-272}:\\ \;\;\;\;\left(1 + e^{-x \cdot eps\_m}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 970000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.56 \cdot 10^{+100}:\\ \;\;\;\;e^{-x}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+275}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{eps\_m} - -1\right) - \left(\frac{1}{eps\_m} - 1\right) \cdot 1}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* (- (exp (* (- x) (- eps_m))) -1.0) 0.5)))
   (if (<= x -1.35e-272)
     (* (+ 1.0 (exp (- (* x eps_m)))) 0.5)
     (if (<= x 970000.0)
       t_0
       (if (<= x 1.56e+100)
         (exp (- x))
         (if (<= x 3.3e+275)
           t_0
           (/
            (- (- (/ 1.0 eps_m) -1.0) (* (- (/ 1.0 eps_m) 1.0) 1.0))
            2.0)))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (exp((-x * -eps_m)) - -1.0) * 0.5;
	double tmp;
	if (x <= -1.35e-272) {
		tmp = (1.0 + exp(-(x * eps_m))) * 0.5;
	} else if (x <= 970000.0) {
		tmp = t_0;
	} else if (x <= 1.56e+100) {
		tmp = exp(-x);
	} else if (x <= 3.3e+275) {
		tmp = t_0;
	} else {
		tmp = (((1.0 / eps_m) - -1.0) - (((1.0 / eps_m) - 1.0) * 1.0)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (exp((-x * -eps_m)) - (-1.0d0)) * 0.5d0
    if (x <= (-1.35d-272)) then
        tmp = (1.0d0 + exp(-(x * eps_m))) * 0.5d0
    else if (x <= 970000.0d0) then
        tmp = t_0
    else if (x <= 1.56d+100) then
        tmp = exp(-x)
    else if (x <= 3.3d+275) then
        tmp = t_0
    else
        tmp = (((1.0d0 / eps_m) - (-1.0d0)) - (((1.0d0 / eps_m) - 1.0d0) * 1.0d0)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = (Math.exp((-x * -eps_m)) - -1.0) * 0.5;
	double tmp;
	if (x <= -1.35e-272) {
		tmp = (1.0 + Math.exp(-(x * eps_m))) * 0.5;
	} else if (x <= 970000.0) {
		tmp = t_0;
	} else if (x <= 1.56e+100) {
		tmp = Math.exp(-x);
	} else if (x <= 3.3e+275) {
		tmp = t_0;
	} else {
		tmp = (((1.0 / eps_m) - -1.0) - (((1.0 / eps_m) - 1.0) * 1.0)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (math.exp((-x * -eps_m)) - -1.0) * 0.5
	tmp = 0
	if x <= -1.35e-272:
		tmp = (1.0 + math.exp(-(x * eps_m))) * 0.5
	elif x <= 970000.0:
		tmp = t_0
	elif x <= 1.56e+100:
		tmp = math.exp(-x)
	elif x <= 3.3e+275:
		tmp = t_0
	else:
		tmp = (((1.0 / eps_m) - -1.0) - (((1.0 / eps_m) - 1.0) * 1.0)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(exp(Float64(Float64(-x) * Float64(-eps_m))) - -1.0) * 0.5)
	tmp = 0.0
	if (x <= -1.35e-272)
		tmp = Float64(Float64(1.0 + exp(Float64(-Float64(x * eps_m)))) * 0.5);
	elseif (x <= 970000.0)
		tmp = t_0;
	elseif (x <= 1.56e+100)
		tmp = exp(Float64(-x));
	elseif (x <= 3.3e+275)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / eps_m) - -1.0) - Float64(Float64(Float64(1.0 / eps_m) - 1.0) * 1.0)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (exp((-x * -eps_m)) - -1.0) * 0.5;
	tmp = 0.0;
	if (x <= -1.35e-272)
		tmp = (1.0 + exp(-(x * eps_m))) * 0.5;
	elseif (x <= 970000.0)
		tmp = t_0;
	elseif (x <= 1.56e+100)
		tmp = exp(-x);
	elseif (x <= 3.3e+275)
		tmp = t_0;
	else
		tmp = (((1.0 / eps_m) - -1.0) - (((1.0 / eps_m) - 1.0) * 1.0)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[N[((-x) * (-eps$95$m)), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[x, -1.35e-272], N[(N[(1.0 + N[Exp[(-N[(x * eps$95$m), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 970000.0], t$95$0, If[LessEqual[x, 1.56e+100], N[Exp[(-x)], $MachinePrecision], If[LessEqual[x, 3.3e+275], t$95$0, N[(N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision] - N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.34999999999999996e-272

   1. Initial program 72.4%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.4%

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

   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. distribute-lft-neg-in 72.0

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

      2. exp-neg 72.0

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

      3. *-commutative 72.0

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

      4. exp-neg 72.0

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

   8. Applied rewrites 72.0%

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

   9. Taylor expanded in eps around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 72.2

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

   11. Applied rewrites 72.2%

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

   if -1.34999999999999996e-272 < x < 9.7e5 or 1.55999999999999998e100 < x < 3.30000000000000022e275

   1. Initial program 70.4%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 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. 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. mul-1-neg N/A

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

      5. pow-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 69.2%

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

   11. Taylor expanded in eps around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 69.1

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

   13. Applied rewrites 69.1%

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

   if 9.7e5 < x < 1.55999999999999998e100

   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. 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. mul-1-neg N/A

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

      5. pow-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in eps around 0

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

      1. *-commutative N/A

         \[\leadsto e^{-1 \cdot x} \]

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

         \[\leadsto e^{-1 \cdot x} \]

      3. pow-exp N/A

         \[\leadsto e^{-1 \cdot x} \]

      4. +-commutative N/A

         \[\leadsto e^{-1 \cdot x} \]

      5. *-commutative N/A

         \[\leadsto e^{-1 \cdot x} \]

      6. mul-1-neg N/A

         \[\leadsto e^{-1 \cdot x} \]

      7. *-commutative N/A

         \[\leadsto e^{-1 \cdot x} \]

      8. +-commutative N/A

         \[\leadsto e^{-1 \cdot x} \]

      9. mul-1-neg N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 78.1

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

   10. Applied rewrites 78.1%

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

   if 3.30000000000000022e275 < 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 2.4%

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

   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 100.0

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

   8. Applied rewrites 100.0%

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

      1. lift-pow.f64 N/A

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

      2. inv-pow N/A

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

      3. lift-/.f64 100.0

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

   10. Applied rewrites 100.0%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-272}:\\ \;\;\;\;\left(1 + e^{-x \cdot \varepsilon}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 970000:\\ \;\;\;\;\left(e^{\left(-x\right) \cdot \left(-\varepsilon\right)} - -1\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.56 \cdot 10^{+100}:\\ \;\;\;\;e^{-x}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+275}:\\ \;\;\;\;\left(e^{\left(-x\right) \cdot \left(-\varepsilon\right)} - -1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \left(-1 + eps\_m\right)}\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{-272}:\\ \;\;\;\;\left(1 + e^{-x \cdot eps\_m}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 970000:\\ \;\;\;\;\left(t\_0 - \left(\mathsf{fma}\left(x, eps\_m, x\right) - 1\right)\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.56 \cdot 10^{+100}:\\ \;\;\;\;e^{-x}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+275}:\\ \;\;\;\;\left(t\_0 - -1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{eps\_m} - -1\right) - \left(\frac{1}{eps\_m} - 1\right) \cdot 1}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (+ -1.0 eps_m)))))
   (if (<= x -1.35e-272)
     (* (+ 1.0 (exp (- (* x eps_m)))) 0.5)
     (if (<= x 970000.0)
       (* (- t_0 (- (fma x eps_m x) 1.0)) 0.5)
       (if (<= x 1.56e+100)
         (exp (- x))
         (if (<= x 7.2e+275)
           (* (- t_0 -1.0) 0.5)
           (/
            (- (- (/ 1.0 eps_m) -1.0) (* (- (/ 1.0 eps_m) 1.0) 1.0))
            2.0)))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * (-1.0 + eps_m)));
	double tmp;
	if (x <= -1.35e-272) {
		tmp = (1.0 + exp(-(x * eps_m))) * 0.5;
	} else if (x <= 970000.0) {
		tmp = (t_0 - (fma(x, eps_m, x) - 1.0)) * 0.5;
	} else if (x <= 1.56e+100) {
		tmp = exp(-x);
	} else if (x <= 7.2e+275) {
		tmp = (t_0 - -1.0) * 0.5;
	} else {
		tmp = (((1.0 / eps_m) - -1.0) - (((1.0 / eps_m) - 1.0) * 1.0)) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * Float64(-1.0 + eps_m)))
	tmp = 0.0
	if (x <= -1.35e-272)
		tmp = Float64(Float64(1.0 + exp(Float64(-Float64(x * eps_m)))) * 0.5);
	elseif (x <= 970000.0)
		tmp = Float64(Float64(t_0 - Float64(fma(x, eps_m, x) - 1.0)) * 0.5);
	elseif (x <= 1.56e+100)
		tmp = exp(Float64(-x));
	elseif (x <= 7.2e+275)
		tmp = Float64(Float64(t_0 - -1.0) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / eps_m) - -1.0) - Float64(Float64(Float64(1.0 / eps_m) - 1.0) * 1.0)) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1.35e-272], N[(N[(1.0 + N[Exp[(-N[(x * eps$95$m), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 970000.0], N[(N[(t$95$0 - N[(N[(x * eps$95$m + x), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 1.56e+100], N[Exp[(-x)], $MachinePrecision], If[LessEqual[x, 7.2e+275], N[(N[(t$95$0 - -1.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision] - N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1.34999999999999996e-272

   1. Initial program 72.4%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.4%

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

   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. distribute-lft-neg-in 72.0

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

      2. exp-neg 72.0

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

      3. *-commutative 72.0

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

      4. exp-neg 72.0

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

   8. Applied rewrites 72.0%

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

   9. Taylor expanded in eps around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 72.2

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

   11. Applied rewrites 72.2%

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

   if -1.34999999999999996e-272 < x < 9.7e5

   1. Initial program 53.3%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt1-in N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. lower--.f64 N/A

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

      7. lift-fma.f64 88.4

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

   8. Applied rewrites 88.4%

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

   if 9.7e5 < x < 1.55999999999999998e100

   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. 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. mul-1-neg N/A

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

      5. pow-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in eps around 0

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

      1. *-commutative N/A

         \[\leadsto e^{-1 \cdot x} \]

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

         \[\leadsto e^{-1 \cdot x} \]

      3. pow-exp N/A

         \[\leadsto e^{-1 \cdot x} \]

      4. +-commutative N/A

         \[\leadsto e^{-1 \cdot x} \]

      5. *-commutative N/A

         \[\leadsto e^{-1 \cdot x} \]

      6. mul-1-neg N/A

         \[\leadsto e^{-1 \cdot x} \]

      7. *-commutative N/A

         \[\leadsto e^{-1 \cdot x} \]

      8. +-commutative N/A

         \[\leadsto e^{-1 \cdot x} \]

      9. mul-1-neg N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 78.1

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

   10. Applied rewrites 78.1%

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

   if 1.55999999999999998e100 < x < 7.1999999999999996e275

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 37.4%

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

   if 7.1999999999999996e275 < 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 2.4%

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

   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 100.0

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

   8. Applied rewrites 100.0%

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

      1. lift-pow.f64 N/A

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

      2. inv-pow N/A

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

      3. lift-/.f64 100.0

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

   10. Applied rewrites 100.0%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-272}:\\ \;\;\;\;\left(1 + e^{-x \cdot \varepsilon}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 970000:\\ \;\;\;\;\left(e^{x \cdot \left(-1 + \varepsilon\right)} - \left(\mathsf{fma}\left(x, \varepsilon, x\right) - 1\right)\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.56 \cdot 10^{+100}:\\ \;\;\;\;e^{-x}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+275}:\\ \;\;\;\;\left(e^{x \cdot \left(-1 + \varepsilon\right)} - -1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-272}:\\ \;\;\;\;\left(1 + e^{-x \cdot eps\_m}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 970000:\\ \;\;\;\;\left(e^{x \cdot eps\_m} - \left(\mathsf{fma}\left(x, eps\_m, x\right) - 1\right)\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.56 \cdot 10^{+100}:\\ \;\;\;\;e^{-x}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+275}:\\ \;\;\;\;\left(e^{x \cdot \left(-1 + eps\_m\right)} - -1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{eps\_m} - -1\right) - \left(\frac{1}{eps\_m} - 1\right) \cdot 1}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.35e-272)
   (* (+ 1.0 (exp (- (* x eps_m)))) 0.5)
   (if (<= x 970000.0)
     (* (- (exp (* x eps_m)) (- (fma x eps_m x) 1.0)) 0.5)
     (if (<= x 1.56e+100)
       (exp (- x))
       (if (<= x 7.2e+275)
         (* (- (exp (* x (+ -1.0 eps_m))) -1.0) 0.5)
         (/ (- (- (/ 1.0 eps_m) -1.0) (* (- (/ 1.0 eps_m) 1.0) 1.0)) 2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.35e-272) {
		tmp = (1.0 + exp(-(x * eps_m))) * 0.5;
	} else if (x <= 970000.0) {
		tmp = (exp((x * eps_m)) - (fma(x, eps_m, x) - 1.0)) * 0.5;
	} else if (x <= 1.56e+100) {
		tmp = exp(-x);
	} else if (x <= 7.2e+275) {
		tmp = (exp((x * (-1.0 + eps_m))) - -1.0) * 0.5;
	} else {
		tmp = (((1.0 / eps_m) - -1.0) - (((1.0 / eps_m) - 1.0) * 1.0)) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.35e-272)
		tmp = Float64(Float64(1.0 + exp(Float64(-Float64(x * eps_m)))) * 0.5);
	elseif (x <= 970000.0)
		tmp = Float64(Float64(exp(Float64(x * eps_m)) - Float64(fma(x, eps_m, x) - 1.0)) * 0.5);
	elseif (x <= 1.56e+100)
		tmp = exp(Float64(-x));
	elseif (x <= 7.2e+275)
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 + eps_m))) - -1.0) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / eps_m) - -1.0) - Float64(Float64(Float64(1.0 / eps_m) - 1.0) * 1.0)) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.35e-272], N[(N[(1.0 + N[Exp[(-N[(x * eps$95$m), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 970000.0], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] - N[(N[(x * eps$95$m + x), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 1.56e+100], N[Exp[(-x)], $MachinePrecision], If[LessEqual[x, 7.2e+275], N[(N[(N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision] - N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1.34999999999999996e-272

   1. Initial program 72.4%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.4%

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

   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. distribute-lft-neg-in 72.0

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

      2. exp-neg 72.0

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

      3. *-commutative 72.0

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

      4. exp-neg 72.0

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

   8. Applied rewrites 72.0%

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

   9. Taylor expanded in eps around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 72.2

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

   11. Applied rewrites 72.2%

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

   if -1.34999999999999996e-272 < x < 9.7e5

   1. Initial program 53.3%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 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. 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. mul-1-neg N/A

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

      5. pow-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 100.0

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

   10. Applied rewrites 100.0%

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

   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 88.4

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

   13. Applied rewrites 88.4%

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

   if 9.7e5 < x < 1.55999999999999998e100

   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. 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. mul-1-neg N/A

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

      5. pow-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in eps around 0

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

      1. *-commutative N/A

         \[\leadsto e^{-1 \cdot x} \]

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

         \[\leadsto e^{-1 \cdot x} \]

      3. pow-exp N/A

         \[\leadsto e^{-1 \cdot x} \]

      4. +-commutative N/A

         \[\leadsto e^{-1 \cdot x} \]

      5. *-commutative N/A

         \[\leadsto e^{-1 \cdot x} \]

      6. mul-1-neg N/A

         \[\leadsto e^{-1 \cdot x} \]

      7. *-commutative N/A

         \[\leadsto e^{-1 \cdot x} \]

      8. +-commutative N/A

         \[\leadsto e^{-1 \cdot x} \]

      9. mul-1-neg N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 78.1

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

   10. Applied rewrites 78.1%

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

   if 1.55999999999999998e100 < x < 7.1999999999999996e275

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 37.4%

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

   if 7.1999999999999996e275 < 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 2.4%

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

   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 100.0

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

   8. Applied rewrites 100.0%

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

      1. lift-pow.f64 N/A

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

      2. inv-pow N/A

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

      3. lift-/.f64 100.0

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

   10. Applied rewrites 100.0%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-272}:\\ \;\;\;\;\left(1 + e^{-x \cdot \varepsilon}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 970000:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - \left(\mathsf{fma}\left(x, \varepsilon, x\right) - 1\right)\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.56 \cdot 10^{+100}:\\ \;\;\;\;e^{-x}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+275}:\\ \;\;\;\;\left(e^{x \cdot \left(-1 + \varepsilon\right)} - -1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-272}:\\ \;\;\;\;\left(1 + e^{-x \cdot eps\_m}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 970000:\\ \;\;\;\;\left(e^{\left(-x\right) \cdot \left(-eps\_m\right)} - -1\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.56 \cdot 10^{+100}:\\ \;\;\;\;e^{-x}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+275}:\\ \;\;\;\;\left(e^{x \cdot \left(-1 + eps\_m\right)} - -1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{eps\_m} - -1\right) - \left(\frac{1}{eps\_m} - 1\right) \cdot 1}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.35e-272)
   (* (+ 1.0 (exp (- (* x eps_m)))) 0.5)
   (if (<= x 970000.0)
     (* (- (exp (* (- x) (- eps_m))) -1.0) 0.5)
     (if (<= x 1.56e+100)
       (exp (- x))
       (if (<= x 7.2e+275)
         (* (- (exp (* x (+ -1.0 eps_m))) -1.0) 0.5)
         (/ (- (- (/ 1.0 eps_m) -1.0) (* (- (/ 1.0 eps_m) 1.0) 1.0)) 2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.35e-272) {
		tmp = (1.0 + exp(-(x * eps_m))) * 0.5;
	} else if (x <= 970000.0) {
		tmp = (exp((-x * -eps_m)) - -1.0) * 0.5;
	} else if (x <= 1.56e+100) {
		tmp = exp(-x);
	} else if (x <= 7.2e+275) {
		tmp = (exp((x * (-1.0 + eps_m))) - -1.0) * 0.5;
	} else {
		tmp = (((1.0 / eps_m) - -1.0) - (((1.0 / eps_m) - 1.0) * 1.0)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.35d-272)) then
        tmp = (1.0d0 + exp(-(x * eps_m))) * 0.5d0
    else if (x <= 970000.0d0) then
        tmp = (exp((-x * -eps_m)) - (-1.0d0)) * 0.5d0
    else if (x <= 1.56d+100) then
        tmp = exp(-x)
    else if (x <= 7.2d+275) then
        tmp = (exp((x * ((-1.0d0) + eps_m))) - (-1.0d0)) * 0.5d0
    else
        tmp = (((1.0d0 / eps_m) - (-1.0d0)) - (((1.0d0 / eps_m) - 1.0d0) * 1.0d0)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.35e-272) {
		tmp = (1.0 + Math.exp(-(x * eps_m))) * 0.5;
	} else if (x <= 970000.0) {
		tmp = (Math.exp((-x * -eps_m)) - -1.0) * 0.5;
	} else if (x <= 1.56e+100) {
		tmp = Math.exp(-x);
	} else if (x <= 7.2e+275) {
		tmp = (Math.exp((x * (-1.0 + eps_m))) - -1.0) * 0.5;
	} else {
		tmp = (((1.0 / eps_m) - -1.0) - (((1.0 / eps_m) - 1.0) * 1.0)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.35e-272:
		tmp = (1.0 + math.exp(-(x * eps_m))) * 0.5
	elif x <= 970000.0:
		tmp = (math.exp((-x * -eps_m)) - -1.0) * 0.5
	elif x <= 1.56e+100:
		tmp = math.exp(-x)
	elif x <= 7.2e+275:
		tmp = (math.exp((x * (-1.0 + eps_m))) - -1.0) * 0.5
	else:
		tmp = (((1.0 / eps_m) - -1.0) - (((1.0 / eps_m) - 1.0) * 1.0)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.35e-272)
		tmp = Float64(Float64(1.0 + exp(Float64(-Float64(x * eps_m)))) * 0.5);
	elseif (x <= 970000.0)
		tmp = Float64(Float64(exp(Float64(Float64(-x) * Float64(-eps_m))) - -1.0) * 0.5);
	elseif (x <= 1.56e+100)
		tmp = exp(Float64(-x));
	elseif (x <= 7.2e+275)
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 + eps_m))) - -1.0) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / eps_m) - -1.0) - Float64(Float64(Float64(1.0 / eps_m) - 1.0) * 1.0)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.35e-272)
		tmp = (1.0 + exp(-(x * eps_m))) * 0.5;
	elseif (x <= 970000.0)
		tmp = (exp((-x * -eps_m)) - -1.0) * 0.5;
	elseif (x <= 1.56e+100)
		tmp = exp(-x);
	elseif (x <= 7.2e+275)
		tmp = (exp((x * (-1.0 + eps_m))) - -1.0) * 0.5;
	else
		tmp = (((1.0 / eps_m) - -1.0) - (((1.0 / eps_m) - 1.0) * 1.0)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.35e-272], N[(N[(1.0 + N[Exp[(-N[(x * eps$95$m), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 970000.0], N[(N[(N[Exp[N[((-x) * (-eps$95$m)), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 1.56e+100], N[Exp[(-x)], $MachinePrecision], If[LessEqual[x, 7.2e+275], N[(N[(N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision] - N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1.34999999999999996e-272

   1. Initial program 72.4%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.4%

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

   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. distribute-lft-neg-in 72.0

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

      2. exp-neg 72.0

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

      3. *-commutative 72.0

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

      4. exp-neg 72.0

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

   8. Applied rewrites 72.0%

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

   9. Taylor expanded in eps around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 72.2

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

   11. Applied rewrites 72.2%

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

   if -1.34999999999999996e-272 < x < 9.7e5

   1. Initial program 53.3%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 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. 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. mul-1-neg N/A

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

      5. pow-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 87.5%

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

   11. Taylor expanded in eps around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 87.5

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

   13. Applied rewrites 87.5%

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

   if 9.7e5 < x < 1.55999999999999998e100

   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. 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. mul-1-neg N/A

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

      5. pow-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in eps around 0

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

      1. *-commutative N/A

         \[\leadsto e^{-1 \cdot x} \]

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

         \[\leadsto e^{-1 \cdot x} \]

      3. pow-exp N/A

         \[\leadsto e^{-1 \cdot x} \]

      4. +-commutative N/A

         \[\leadsto e^{-1 \cdot x} \]

      5. *-commutative N/A

         \[\leadsto e^{-1 \cdot x} \]

      6. mul-1-neg N/A

         \[\leadsto e^{-1 \cdot x} \]

      7. *-commutative N/A

         \[\leadsto e^{-1 \cdot x} \]

      8. +-commutative N/A

         \[\leadsto e^{-1 \cdot x} \]

      9. mul-1-neg N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 78.1

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

   10. Applied rewrites 78.1%

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

   if 1.55999999999999998e100 < x < 7.1999999999999996e275

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 37.4%

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

   if 7.1999999999999996e275 < 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 2.4%

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

   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 100.0

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

   8. Applied rewrites 100.0%

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

      1. lift-pow.f64 N/A

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

      2. inv-pow N/A

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

      3. lift-/.f64 100.0

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

   10. Applied rewrites 100.0%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-272}:\\ \;\;\;\;\left(1 + e^{-x \cdot \varepsilon}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 970000:\\ \;\;\;\;\left(e^{\left(-x\right) \cdot \left(-\varepsilon\right)} - -1\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.56 \cdot 10^{+100}:\\ \;\;\;\;e^{-x}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+275}:\\ \;\;\;\;\left(e^{x \cdot \left(-1 + \varepsilon\right)} - -1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -10500000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-223}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{eps\_m \cdot eps\_m - 1}{eps\_m - 1}, -1 + eps\_m\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-253}:\\ \;\;\;\;\mathsf{fma}\left(-2, x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 900000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, eps\_m - -1, \frac{-1 + eps\_m \cdot eps\_m}{eps\_m - -1}\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -10500000000000.0)
     t_0
     (if (<= x -6e-223)
       (*
        (fma
         (fma -1.0 (/ (- (* eps_m eps_m) 1.0) (- eps_m 1.0)) (+ -1.0 eps_m))
         x
         2.0)
        0.5)
       (if (<= x 5.2e-253)
         (* (fma -2.0 x 2.0) 0.5)
         (if (<= x 900000.0)
           (*
            (fma
             (fma
              -1.0
              (- eps_m -1.0)
              (/ (+ -1.0 (* eps_m eps_m)) (- eps_m -1.0)))
             x
             2.0)
            0.5)
           t_0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -10500000000000.0) {
		tmp = t_0;
	} else if (x <= -6e-223) {
		tmp = fma(fma(-1.0, (((eps_m * eps_m) - 1.0) / (eps_m - 1.0)), (-1.0 + eps_m)), x, 2.0) * 0.5;
	} else if (x <= 5.2e-253) {
		tmp = fma(-2.0, x, 2.0) * 0.5;
	} else if (x <= 900000.0) {
		tmp = fma(fma(-1.0, (eps_m - -1.0), ((-1.0 + (eps_m * eps_m)) / (eps_m - -1.0))), x, 2.0) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -10500000000000.0)
		tmp = t_0;
	elseif (x <= -6e-223)
		tmp = Float64(fma(fma(-1.0, Float64(Float64(Float64(eps_m * eps_m) - 1.0) / Float64(eps_m - 1.0)), Float64(-1.0 + eps_m)), x, 2.0) * 0.5);
	elseif (x <= 5.2e-253)
		tmp = Float64(fma(-2.0, x, 2.0) * 0.5);
	elseif (x <= 900000.0)
		tmp = Float64(fma(fma(-1.0, Float64(eps_m - -1.0), Float64(Float64(-1.0 + Float64(eps_m * eps_m)) / Float64(eps_m - -1.0))), x, 2.0) * 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -10500000000000.0], t$95$0, If[LessEqual[x, -6e-223], N[(N[(N[(-1.0 * N[(N[(N[(eps$95$m * eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision] / N[(eps$95$m - 1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 5.2e-253], N[(N[(-2.0 * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 900000.0], N[(N[(N[(-1.0 * N[(eps$95$m - -1.0), $MachinePrecision] + N[(N[(-1.0 + N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision] / N[(eps$95$m - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.05e13 or 9e5 < 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. 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. mul-1-neg N/A

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

      5. pow-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in eps around 0

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

      1. *-commutative N/A

         \[\leadsto e^{-1 \cdot x} \]

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

         \[\leadsto e^{-1 \cdot x} \]

      3. pow-exp N/A

         \[\leadsto e^{-1 \cdot x} \]

      4. +-commutative N/A

         \[\leadsto e^{-1 \cdot x} \]

      5. *-commutative N/A

         \[\leadsto e^{-1 \cdot x} \]

      6. mul-1-neg N/A

         \[\leadsto e^{-1 \cdot x} \]

      7. *-commutative N/A

         \[\leadsto e^{-1 \cdot x} \]

      8. +-commutative N/A

         \[\leadsto e^{-1 \cdot x} \]

      9. mul-1-neg N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 68.5

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

   10. Applied rewrites 68.5%

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

   if -1.05e13 < x < -5.99999999999999983e-223

   1. Initial program 63.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 97.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lift--.f64 54.9

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

   8. Applied rewrites 54.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   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. lower--.f64 72.0

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

   10. Applied rewrites 72.0%

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

   if -5.99999999999999983e-223 < x < 5.2e-253

   1. Initial program 59.5%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 99.9%

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

   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 89.5

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

   8. Applied rewrites 89.5%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 89.5%

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

   if 5.2e-253 < x < 9e5

   1. Initial program 48.4%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 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 62.1

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

   8. Applied rewrites 62.1%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      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 77.1

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

   10. Applied rewrites 77.1%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10500000000000:\\ \;\;\;\;e^{-x}\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-223}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, -1 + \varepsilon\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-253}:\\ \;\;\;\;\mathsf{fma}\left(-2, x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 900000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, \frac{-1 + \varepsilon \cdot \varepsilon}{\varepsilon - -1}\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{-253}:\\ \;\;\;\;\left(1 + e^{-x \cdot eps\_m}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 900000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, eps\_m - -1, \frac{-1 + eps\_m \cdot eps\_m}{eps\_m - -1}\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 5.2e-253)
   (* (+ 1.0 (exp (- (* x eps_m)))) 0.5)
   (if (<= x 900000.0)
     (*
      (fma
       (fma -1.0 (- eps_m -1.0) (/ (+ -1.0 (* eps_m eps_m)) (- eps_m -1.0)))
       x
       2.0)
      0.5)
     (exp (- x)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 5.2e-253) {
		tmp = (1.0 + exp(-(x * eps_m))) * 0.5;
	} else if (x <= 900000.0) {
		tmp = fma(fma(-1.0, (eps_m - -1.0), ((-1.0 + (eps_m * eps_m)) / (eps_m - -1.0))), x, 2.0) * 0.5;
	} else {
		tmp = exp(-x);
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 5.2e-253)
		tmp = Float64(Float64(1.0 + exp(Float64(-Float64(x * eps_m)))) * 0.5);
	elseif (x <= 900000.0)
		tmp = Float64(fma(fma(-1.0, Float64(eps_m - -1.0), Float64(Float64(-1.0 + Float64(eps_m * eps_m)) / Float64(eps_m - -1.0))), x, 2.0) * 0.5);
	else
		tmp = exp(Float64(-x));
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 5.2e-253], N[(N[(1.0 + N[Exp[(-N[(x * eps$95$m), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 900000.0], N[(N[(N[(-1.0 * N[(eps$95$m - -1.0), $MachinePrecision] + N[(N[(-1.0 + N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision] / N[(eps$95$m - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[Exp[(-x)], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.2e-253

   1. Initial program 71.6%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in x around 0

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

      1. distribute-lft-neg-in 75.7

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

      2. exp-neg 75.7

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

      3. *-commutative 75.7

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

      4. exp-neg 75.7

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

   8. Applied rewrites 75.7%

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

   9. Taylor expanded in eps around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 75.9

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

   11. Applied rewrites 75.9%

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

   if 5.2e-253 < x < 9e5

   1. Initial program 48.4%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 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 62.1

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

   8. Applied rewrites 62.1%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      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 77.1

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

   10. Applied rewrites 77.1%

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

   if 9e5 < 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. 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. mul-1-neg N/A

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

      5. pow-exp N/A

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

      6. lift-exp.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in eps around 0

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

      1. *-commutative N/A

         \[\leadsto e^{-1 \cdot x} \]

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

         \[\leadsto e^{-1 \cdot x} \]

      3. pow-exp N/A

         \[\leadsto e^{-1 \cdot x} \]

      4. +-commutative N/A

         \[\leadsto e^{-1 \cdot x} \]

      5. *-commutative N/A

         \[\leadsto e^{-1 \cdot x} \]

      6. mul-1-neg N/A

         \[\leadsto e^{-1 \cdot x} \]

      7. *-commutative N/A

         \[\leadsto e^{-1 \cdot x} \]

      8. +-commutative N/A

         \[\leadsto e^{-1 \cdot x} \]

      9. mul-1-neg N/A

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

      10. lift-exp.f64 N/A

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

      11. lift-neg.f64 54.2

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

   10. Applied rewrites 54.2%

       \[\leadsto e^{-x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{-253}:\\ \;\;\;\;\left(1 + e^{-x \cdot \varepsilon}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 900000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, \frac{-1 + \varepsilon \cdot \varepsilon}{\varepsilon - -1}\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{+124}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot -0.3333333333333333, x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-223}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{eps\_m \cdot eps\_m - 1}{eps\_m - 1}, -1 + eps\_m\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-253}:\\ \;\;\;\;\mathsf{fma}\left(-2, x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 57000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, eps\_m - -1, \frac{-1 + eps\_m \cdot eps\_m}{eps\_m - -1}\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{eps\_m} - -1\right) - \left(\frac{1}{eps\_m} - 1\right) \cdot 1}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -7.4e+124)
   (* (fma (* (* x x) -0.3333333333333333) x 2.0) 0.5)
   (if (<= x -6e-223)
     (*
      (fma
       (fma -1.0 (/ (- (* eps_m eps_m) 1.0) (- eps_m 1.0)) (+ -1.0 eps_m))
       x
       2.0)
      0.5)
     (if (<= x 5.2e-253)
       (* (fma -2.0 x 2.0) 0.5)
       (if (<= x 57000.0)
         (*
          (fma
           (fma
            -1.0
            (- eps_m -1.0)
            (/ (+ -1.0 (* eps_m eps_m)) (- eps_m -1.0)))
           x
           2.0)
          0.5)
         (/ (- (- (/ 1.0 eps_m) -1.0) (* (- (/ 1.0 eps_m) 1.0) 1.0)) 2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -7.4e+124) {
		tmp = fma(((x * x) * -0.3333333333333333), x, 2.0) * 0.5;
	} else if (x <= -6e-223) {
		tmp = fma(fma(-1.0, (((eps_m * eps_m) - 1.0) / (eps_m - 1.0)), (-1.0 + eps_m)), x, 2.0) * 0.5;
	} else if (x <= 5.2e-253) {
		tmp = fma(-2.0, x, 2.0) * 0.5;
	} else if (x <= 57000.0) {
		tmp = fma(fma(-1.0, (eps_m - -1.0), ((-1.0 + (eps_m * eps_m)) / (eps_m - -1.0))), x, 2.0) * 0.5;
	} else {
		tmp = (((1.0 / eps_m) - -1.0) - (((1.0 / eps_m) - 1.0) * 1.0)) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -7.4e+124)
		tmp = Float64(fma(Float64(Float64(x * x) * -0.3333333333333333), x, 2.0) * 0.5);
	elseif (x <= -6e-223)
		tmp = Float64(fma(fma(-1.0, Float64(Float64(Float64(eps_m * eps_m) - 1.0) / Float64(eps_m - 1.0)), Float64(-1.0 + eps_m)), x, 2.0) * 0.5);
	elseif (x <= 5.2e-253)
		tmp = Float64(fma(-2.0, x, 2.0) * 0.5);
	elseif (x <= 57000.0)
		tmp = Float64(fma(fma(-1.0, Float64(eps_m - -1.0), Float64(Float64(-1.0 + Float64(eps_m * eps_m)) / Float64(eps_m - -1.0))), x, 2.0) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / eps_m) - -1.0) - Float64(Float64(Float64(1.0 / eps_m) - 1.0) * 1.0)) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -7.4e+124], N[(N[(N[(N[(x * x), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, -6e-223], N[(N[(N[(-1.0 * N[(N[(N[(eps$95$m * eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision] / N[(eps$95$m - 1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 5.2e-253], N[(N[(-2.0 * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 57000.0], N[(N[(N[(-1.0 * N[(eps$95$m - -1.0), $MachinePrecision] + N[(N[(-1.0 + N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision] / N[(eps$95$m - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision] - N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -7.40000000000000016e124

   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. mul-1-neg N/A

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

      4. lower-exp.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-neg.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-exp.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lift-neg.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2\right) + 2\right) \cdot \frac{1}{2} \]

       2. *-commutative N/A

          \[\leadsto \left(\left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2\right) \cdot x + 2\right) \cdot \frac{1}{2} \]

       3. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2, x, 2\right) \cdot \frac{1}{2} \]

       4. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2, x, 2\right) \cdot \frac{1}{2} \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(1 + \frac{-1}{3} \cdot x\right) \cdot x - 2, x, 2\right) \cdot \frac{1}{2} \]

       6. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(1 + \frac{-1}{3} \cdot x\right) \cdot x - 2, x, 2\right) \cdot \frac{1}{2} \]

       7. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{3} \cdot x + 1\right) \cdot x - 2, x, 2\right) \cdot \frac{1}{2} \]

       8. lower-fma.f64 100.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 1\right) \cdot x - 2, x, 2\right) \cdot 0.5 \]

   11. Applied rewrites 100.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 1\right) \cdot x - 2, x, 2\right) \cdot 0.5 \]

   12. Taylor expanded in x around inf

       \[\leadsto \mathsf{fma}\left(\frac{-1}{3} \cdot {x}^{2}, x, 2\right) \cdot \frac{1}{2} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \frac{-1}{3}, x, 2\right) \cdot \frac{1}{2} \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \frac{-1}{3}, x, 2\right) \cdot \frac{1}{2} \]

       3. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{-1}{3}, x, 2\right) \cdot \frac{1}{2} \]

       4. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot -0.3333333333333333, x, 2\right) \cdot 0.5 \]

   14. Applied rewrites 100.0%

       \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot -0.3333333333333333, x, 2\right) \cdot 0.5 \]

   if -7.40000000000000016e124 < x < -5.99999999999999983e-223

   1. Initial program 66.6%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in 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 49.8

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

   8. Applied rewrites 49.8%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 66.7

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

   10. Applied rewrites 66.7%

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

   if -5.99999999999999983e-223 < x < 5.2e-253

   1. Initial program 59.5%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 99.9%

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

   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 89.5

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

   8. Applied rewrites 89.5%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 89.5%

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

   if 5.2e-253 < x < 57000

   1. Initial program 48.4%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 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 62.1

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

   8. Applied rewrites 62.1%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      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 77.1

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

   10. Applied rewrites 77.1%

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

   if 57000 < 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 28.7%

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

   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 52.2

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

   8. Applied rewrites 52.2%

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

      1. lift-pow.f64 N/A

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

      2. inv-pow N/A

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

      3. lift-/.f64 52.2

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

   10. Applied rewrites 52.2%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{+124}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot -0.3333333333333333, x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-223}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, -1 + \varepsilon\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-253}:\\ \;\;\;\;\mathsf{fma}\left(-2, x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 57000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, \frac{-1 + \varepsilon \cdot \varepsilon}{\varepsilon - -1}\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{+124}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot -0.3333333333333333, x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-223}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{eps\_m \cdot eps\_m - 1}{eps\_m - 1}, -1 + eps\_m\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-253}:\\ \;\;\;\;\mathsf{fma}\left(-2, x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, eps\_m - -1, \frac{-1 + eps\_m \cdot eps\_m}{eps\_m - -1}\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 2, x, 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -7.4e+124)
   (* (fma (* (* x x) -0.3333333333333333) x 2.0) 0.5)
   (if (<= x -6e-223)
     (*
      (fma
       (fma -1.0 (/ (- (* eps_m eps_m) 1.0) (- eps_m 1.0)) (+ -1.0 eps_m))
       x
       2.0)
      0.5)
     (if (<= x 5.2e-253)
       (* (fma -2.0 x 2.0) 0.5)
       (if (<= x 3e+150)
         (*
          (fma
           (fma
            -1.0
            (- eps_m -1.0)
            (/ (+ -1.0 (* eps_m eps_m)) (- eps_m -1.0)))
           x
           2.0)
          0.5)
         (* (fma (- x 2.0) x 2.0) 0.5))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -7.4e+124) {
		tmp = fma(((x * x) * -0.3333333333333333), x, 2.0) * 0.5;
	} else if (x <= -6e-223) {
		tmp = fma(fma(-1.0, (((eps_m * eps_m) - 1.0) / (eps_m - 1.0)), (-1.0 + eps_m)), x, 2.0) * 0.5;
	} else if (x <= 5.2e-253) {
		tmp = fma(-2.0, x, 2.0) * 0.5;
	} else if (x <= 3e+150) {
		tmp = fma(fma(-1.0, (eps_m - -1.0), ((-1.0 + (eps_m * eps_m)) / (eps_m - -1.0))), x, 2.0) * 0.5;
	} else {
		tmp = fma((x - 2.0), x, 2.0) * 0.5;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -7.4e+124)
		tmp = Float64(fma(Float64(Float64(x * x) * -0.3333333333333333), x, 2.0) * 0.5);
	elseif (x <= -6e-223)
		tmp = Float64(fma(fma(-1.0, Float64(Float64(Float64(eps_m * eps_m) - 1.0) / Float64(eps_m - 1.0)), Float64(-1.0 + eps_m)), x, 2.0) * 0.5);
	elseif (x <= 5.2e-253)
		tmp = Float64(fma(-2.0, x, 2.0) * 0.5);
	elseif (x <= 3e+150)
		tmp = Float64(fma(fma(-1.0, Float64(eps_m - -1.0), Float64(Float64(-1.0 + Float64(eps_m * eps_m)) / Float64(eps_m - -1.0))), x, 2.0) * 0.5);
	else
		tmp = Float64(fma(Float64(x - 2.0), x, 2.0) * 0.5);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -7.4e+124], N[(N[(N[(N[(x * x), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, -6e-223], N[(N[(N[(-1.0 * N[(N[(N[(eps$95$m * eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision] / N[(eps$95$m - 1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 5.2e-253], N[(N[(-2.0 * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 3e+150], N[(N[(N[(-1.0 * N[(eps$95$m - -1.0), $MachinePrecision] + N[(N[(-1.0 + N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision] / N[(eps$95$m - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(x - 2.0), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -7.40000000000000016e124

   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. mul-1-neg N/A

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

      4. lower-exp.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-neg.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-exp.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lift-neg.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2\right) + 2\right) \cdot \frac{1}{2} \]

       2. *-commutative N/A

          \[\leadsto \left(\left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2\right) \cdot x + 2\right) \cdot \frac{1}{2} \]

       3. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2, x, 2\right) \cdot \frac{1}{2} \]

       4. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2, x, 2\right) \cdot \frac{1}{2} \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(1 + \frac{-1}{3} \cdot x\right) \cdot x - 2, x, 2\right) \cdot \frac{1}{2} \]

       6. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(1 + \frac{-1}{3} \cdot x\right) \cdot x - 2, x, 2\right) \cdot \frac{1}{2} \]

       7. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{3} \cdot x + 1\right) \cdot x - 2, x, 2\right) \cdot \frac{1}{2} \]

       8. lower-fma.f64 100.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 1\right) \cdot x - 2, x, 2\right) \cdot 0.5 \]

   11. Applied rewrites 100.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 1\right) \cdot x - 2, x, 2\right) \cdot 0.5 \]

   12. Taylor expanded in x around inf

       \[\leadsto \mathsf{fma}\left(\frac{-1}{3} \cdot {x}^{2}, x, 2\right) \cdot \frac{1}{2} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \frac{-1}{3}, x, 2\right) \cdot \frac{1}{2} \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \frac{-1}{3}, x, 2\right) \cdot \frac{1}{2} \]

       3. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{-1}{3}, x, 2\right) \cdot \frac{1}{2} \]

       4. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot -0.3333333333333333, x, 2\right) \cdot 0.5 \]

   14. Applied rewrites 100.0%

       \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot -0.3333333333333333, x, 2\right) \cdot 0.5 \]

   if -7.40000000000000016e124 < x < -5.99999999999999983e-223

   1. Initial program 66.6%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in 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 49.8

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

   8. Applied rewrites 49.8%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 66.7

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

   10. Applied rewrites 66.7%

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

   if -5.99999999999999983e-223 < x < 5.2e-253

   1. Initial program 59.5%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 99.9%

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

   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 89.5

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

   8. Applied rewrites 89.5%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 89.5%

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

   if 5.2e-253 < x < 3.00000000000000012e150

   1. Initial program 65.6%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 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 42.3

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

   8. Applied rewrites 42.3%

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

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

      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 56.4

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

   10. Applied rewrites 56.4%

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

   if 3.00000000000000012e150 < 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. mul-1-neg N/A

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

      4. lower-exp.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-neg.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-exp.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lift-neg.f64 50.8

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

   8. Applied rewrites 50.8%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower--.f64 48.6

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

   11. Applied rewrites 48.6%

       \[\leadsto \mathsf{fma}\left(x - 2, x, 2\right) \cdot 0.5 \]
3. Recombined 5 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.4 \cdot 10^{+124}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot -0.3333333333333333, x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-223}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, -1 + \varepsilon\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-253}:\\ \;\;\;\;\mathsf{fma}\left(-2, x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, \frac{-1 + \varepsilon \cdot \varepsilon}{\varepsilon - -1}\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 2, x, 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{-253}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot -0.3333333333333333, x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, eps\_m - -1, \frac{-1 + eps\_m \cdot eps\_m}{eps\_m - -1}\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 2, x, 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 5.2e-253)
   (* (fma (* (* x x) -0.3333333333333333) x 2.0) 0.5)
   (if (<= x 3e+150)
     (*
      (fma
       (fma -1.0 (- eps_m -1.0) (/ (+ -1.0 (* eps_m eps_m)) (- eps_m -1.0)))
       x
       2.0)
      0.5)
     (* (fma (- x 2.0) x 2.0) 0.5))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 5.2e-253) {
		tmp = fma(((x * x) * -0.3333333333333333), x, 2.0) * 0.5;
	} else if (x <= 3e+150) {
		tmp = fma(fma(-1.0, (eps_m - -1.0), ((-1.0 + (eps_m * eps_m)) / (eps_m - -1.0))), x, 2.0) * 0.5;
	} else {
		tmp = fma((x - 2.0), x, 2.0) * 0.5;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 5.2e-253)
		tmp = Float64(fma(Float64(Float64(x * x) * -0.3333333333333333), x, 2.0) * 0.5);
	elseif (x <= 3e+150)
		tmp = Float64(fma(fma(-1.0, Float64(eps_m - -1.0), Float64(Float64(-1.0 + Float64(eps_m * eps_m)) / Float64(eps_m - -1.0))), x, 2.0) * 0.5);
	else
		tmp = Float64(fma(Float64(x - 2.0), x, 2.0) * 0.5);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 5.2e-253], N[(N[(N[(N[(x * x), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 3e+150], N[(N[(N[(-1.0 * N[(eps$95$m - -1.0), $MachinePrecision] + N[(N[(-1.0 + N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision] / N[(eps$95$m - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(x - 2.0), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.2e-253

   1. Initial program 71.6%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in eps around 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. mul-1-neg N/A

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

      4. lower-exp.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-neg.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-exp.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lift-neg.f64 77.5

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

   8. Applied rewrites 77.5%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2\right) + 2\right) \cdot \frac{1}{2} \]

       2. *-commutative N/A

          \[\leadsto \left(\left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2\right) \cdot x + 2\right) \cdot \frac{1}{2} \]

       3. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2, x, 2\right) \cdot \frac{1}{2} \]

       4. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2, x, 2\right) \cdot \frac{1}{2} \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(1 + \frac{-1}{3} \cdot x\right) \cdot x - 2, x, 2\right) \cdot \frac{1}{2} \]

       6. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(1 + \frac{-1}{3} \cdot x\right) \cdot x - 2, x, 2\right) \cdot \frac{1}{2} \]

       7. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{3} \cdot x + 1\right) \cdot x - 2, x, 2\right) \cdot \frac{1}{2} \]

       8. lower-fma.f64 72.9

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 1\right) \cdot x - 2, x, 2\right) \cdot 0.5 \]

   11. Applied rewrites 72.9%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 1\right) \cdot x - 2, x, 2\right) \cdot 0.5 \]

   12. Taylor expanded in x around inf

       \[\leadsto \mathsf{fma}\left(\frac{-1}{3} \cdot {x}^{2}, x, 2\right) \cdot \frac{1}{2} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \frac{-1}{3}, x, 2\right) \cdot \frac{1}{2} \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \frac{-1}{3}, x, 2\right) \cdot \frac{1}{2} \]

       3. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{-1}{3}, x, 2\right) \cdot \frac{1}{2} \]

       4. lower-*.f64 73.1

          \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot -0.3333333333333333, x, 2\right) \cdot 0.5 \]

   14. Applied rewrites 73.1%

       \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot -0.3333333333333333, x, 2\right) \cdot 0.5 \]

   if 5.2e-253 < x < 3.00000000000000012e150

   1. Initial program 65.6%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 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 42.3

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

   8. Applied rewrites 42.3%

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

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

      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 56.4

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

   10. Applied rewrites 56.4%

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

   if 3.00000000000000012e150 < 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. mul-1-neg N/A

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

      4. lower-exp.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-neg.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-exp.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lift-neg.f64 50.8

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

   8. Applied rewrites 50.8%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower--.f64 48.6

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

   11. Applied rewrites 48.6%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{-253}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot -0.3333333333333333, x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon - -1, \frac{-1 + \varepsilon \cdot \varepsilon}{\varepsilon - -1}\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 2, x, 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.7% accurate, 9.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{-102}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot -0.3333333333333333, x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, 1, -1 + eps\_m\right), x, 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1.45e-102)
   (* (fma (* (* x x) -0.3333333333333333) x 2.0) 0.5)
   (* (fma (fma -1.0 1.0 (+ -1.0 eps_m)) x 2.0) 0.5)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.45e-102) {
		tmp = fma(((x * x) * -0.3333333333333333), x, 2.0) * 0.5;
	} else {
		tmp = fma(fma(-1.0, 1.0, (-1.0 + eps_m)), x, 2.0) * 0.5;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 1.45e-102)
		tmp = Float64(fma(Float64(Float64(x * x) * -0.3333333333333333), x, 2.0) * 0.5);
	else
		tmp = Float64(fma(fma(-1.0, 1.0, Float64(-1.0 + eps_m)), x, 2.0) * 0.5);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 1.45e-102], N[(N[(N[(N[(x * x), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(-1.0 * 1.0 + N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.44999999999999993e-102

   1. Initial program 66.2%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in eps around 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. mul-1-neg N/A

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

      4. lower-exp.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-neg.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-exp.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lift-neg.f64 75.0

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

   8. Applied rewrites 75.0%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

          \[\leadsto \left(x \cdot \left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2\right) + 2\right) \cdot \frac{1}{2} \]

       2. *-commutative N/A

          \[\leadsto \left(\left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2\right) \cdot x + 2\right) \cdot \frac{1}{2} \]

       3. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2, x, 2\right) \cdot \frac{1}{2} \]

       4. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2, x, 2\right) \cdot \frac{1}{2} \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(1 + \frac{-1}{3} \cdot x\right) \cdot x - 2, x, 2\right) \cdot \frac{1}{2} \]

       6. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(1 + \frac{-1}{3} \cdot x\right) \cdot x - 2, x, 2\right) \cdot \frac{1}{2} \]

       7. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{3} \cdot x + 1\right) \cdot x - 2, x, 2\right) \cdot \frac{1}{2} \]

       8. lower-fma.f64 71.4

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 1\right) \cdot x - 2, x, 2\right) \cdot 0.5 \]

   11. Applied rewrites 71.4%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 1\right) \cdot x - 2, x, 2\right) \cdot 0.5 \]

   12. Taylor expanded in x around inf

       \[\leadsto \mathsf{fma}\left(\frac{-1}{3} \cdot {x}^{2}, x, 2\right) \cdot \frac{1}{2} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \frac{-1}{3}, x, 2\right) \cdot \frac{1}{2} \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{2} \cdot \frac{-1}{3}, x, 2\right) \cdot \frac{1}{2} \]

       3. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{-1}{3}, x, 2\right) \cdot \frac{1}{2} \]

       4. lower-*.f64 71.6

          \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot -0.3333333333333333, x, 2\right) \cdot 0.5 \]

   14. Applied rewrites 71.6%

       \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot -0.3333333333333333, x, 2\right) \cdot 0.5 \]

   if 1.44999999999999993e-102 < x

   1. Initial program 86.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 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 17.6

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

   8. Applied rewrites 17.6%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 29.2%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45 \cdot 10^{-102}:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot -0.3333333333333333, x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, 1, -1 + \varepsilon\right), x, 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.1% accurate, 18.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \mathsf{fma}\left(x - 2, x, 2\right) \cdot 0.5 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (* (fma (- x 2.0) x 2.0) 0.5))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return fma((x - 2.0), x, 2.0) * 0.5;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return Float64(fma(Float64(x - 2.0), x, 2.0) * 0.5)
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := N[(N[(N[(x - 2.0), $MachinePrecision] * x + 2.0), $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 99.4%

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

6. Taylor expanded in eps around 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. mul-1-neg N/A

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

   4. lower-exp.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lift-neg.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-exp.f64 N/A

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

   9. mul-1-neg N/A

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

   10. lift-neg.f64 67.3

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

8. Applied rewrites 67.3%

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

9. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. lower--.f64 57.4

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

11. Applied rewrites 57.4%

    \[\leadsto \mathsf{fma}\left(x - 2, x, 2\right) \cdot 0.5 \]
12. Add Preprocessing

Alternative 15: 42.7% accurate, 273.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ 1 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 1.0)

C

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

Fortran

eps_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return 1.0;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return 1.0

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return 1.0
end

MATLAB

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := 1.0

Derivation

1. Initial program 73.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 40.8%

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

Reproduce

herbie shell --seed 2025061 
(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))