NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.3% → 99.2%

Time: 5.5s

Alternatives: 13

Speedup: 2.1×

• 39.0% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -1.6585799718311462e+28
  2. eps = -1.7559145769467168e-238

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: 73.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

2. Taylor expanded in eps around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
3. 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

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

4. Applied rewrites 99.2%

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

Alternative 2: 83.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.1 \cdot 10^{-31}:\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - \left(-e^{-x \cdot \varepsilon}\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 1.1e-31)
   (exp (- x))
   (* (- (exp (* x eps)) (- (exp (- (* x eps))))) 0.5)))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 1.1e-31) {
		tmp = exp(-x);
	} else {
		tmp = (exp((x * eps)) - -exp(-(x * eps))) * 0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 1.1d-31) then
        tmp = exp(-x)
    else
        tmp = (exp((x * eps)) - -exp(-(x * eps))) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= 1.1e-31) {
		tmp = Math.exp(-x);
	} else {
		tmp = (Math.exp((x * eps)) - -Math.exp(-(x * eps))) * 0.5;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 1.1e-31:
		tmp = math.exp(-x)
	else:
		tmp = (math.exp((x * eps)) - -math.exp(-(x * eps))) * 0.5
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 1.1e-31)
		tmp = exp(Float64(-x));
	else
		tmp = Float64(Float64(exp(Float64(x * eps)) - Float64(-exp(Float64(-Float64(x * eps))))) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 1.1e-31)
		tmp = exp(-x);
	else
		tmp = (exp((x * eps)) - -exp(-(x * eps))) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 1.1e-31], N[Exp[(-x)], $MachinePrecision], N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] - (-N[Exp[(-N[(x * eps), $MachinePrecision])], $MachinePrecision])), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.10000000000000005e-31

   1. Initial program 62.6%

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

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. 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

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. lower-*.f64 85.4

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

   7. Applied rewrites 85.4%

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

   8. Taylor expanded in eps around 0

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

      1. lower-exp.f64 N/A

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

      2. lift-neg.f64 77.6

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

   10. Applied rewrites 77.6%

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

   if 1.10000000000000005e-31 < eps

   1. Initial program 97.5%

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

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. 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

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. lower-*.f64 97.6

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

   7. Applied rewrites 97.6%

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

   8. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 97.7

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

   10. Applied rewrites 97.7%

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

Alternative 3: 69.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-295}:\\ \;\;\;\;\left(1 - \left(-e^{-x \cdot \varepsilon}\right)\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 3800000:\\ \;\;\;\;\left(e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(x + \frac{x - 1}{\varepsilon}\right) \cdot \varepsilon\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot 1 - \frac{\mathsf{fma}\left(-1, \varepsilon, 1\right)}{\varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1e-295)
   (* (- 1.0 (- (exp (- (* x eps))))) 0.5)
   (if (<= x 3800000.0)
     (* (- (exp (* (- x) (- 1.0 eps))) (* (+ x (/ (- x 1.0) eps)) eps)) 0.5)
     (/ (- (* (+ 1.0 (/ 1.0 eps)) 1.0) (/ (fma -1.0 eps 1.0) eps)) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1e-295) {
		tmp = (1.0 - -exp(-(x * eps))) * 0.5;
	} else if (x <= 3800000.0) {
		tmp = (exp((-x * (1.0 - eps))) - ((x + ((x - 1.0) / eps)) * eps)) * 0.5;
	} else {
		tmp = (((1.0 + (1.0 / eps)) * 1.0) - (fma(-1.0, eps, 1.0) / eps)) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1e-295)
		tmp = Float64(Float64(1.0 - Float64(-exp(Float64(-Float64(x * eps))))) * 0.5);
	elseif (x <= 3800000.0)
		tmp = Float64(Float64(exp(Float64(Float64(-x) * Float64(1.0 - eps))) - Float64(Float64(x + Float64(Float64(x - 1.0) / eps)) * eps)) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * 1.0) - Float64(fma(-1.0, eps, 1.0) / eps)) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1e-295], N[(N[(1.0 - (-N[Exp[(-N[(x * eps), $MachinePrecision])], $MachinePrecision])), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 3800000.0], N[(N[(N[Exp[N[((-x) * N[(1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(N[(x + N[(N[(x - 1.0), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] - N[(N[(-1.0 * eps + 1.0), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.00000000000000006e-295

   1. Initial program 69.1%

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

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. 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

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. lower-*.f64 99.0

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 99.4

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

   10. Applied rewrites 99.4%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 74.1%

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

   if -1.00000000000000006e-295 < x < 3.8e6

   1. Initial program 55.0%

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

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. 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

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. lift-fma.f64 85.1

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

   7. Applied rewrites 85.1%

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

   8. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. div-sub N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 85.1

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

   10. Applied rewrites 85.1%

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

   if 3.8e6 < 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. Taylor expanded in x around 0

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

   4. Applied rewrites 28.6%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. inv-pow N/A

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

      3. lower-pow.f64 46.8

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

   7. Applied rewrites 46.8%

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

   8. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 46.8

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

   10. Applied rewrites 46.8%

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

Alternative 4: 69.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-295}:\\ \;\;\;\;\left(1 - \left(-e^{-x \cdot \varepsilon}\right)\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 3800000:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - \left(\mathsf{fma}\left(x, \varepsilon, x\right) - 1\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot 1 - \frac{\mathsf{fma}\left(-1, \varepsilon, 1\right)}{\varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1e-295)
   (* (- 1.0 (- (exp (- (* x eps))))) 0.5)
   (if (<= x 3800000.0)
     (* (- (exp (* x eps)) (- (fma x eps x) 1.0)) 0.5)
     (/ (- (* (+ 1.0 (/ 1.0 eps)) 1.0) (/ (fma -1.0 eps 1.0) eps)) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1e-295) {
		tmp = (1.0 - -exp(-(x * eps))) * 0.5;
	} else if (x <= 3800000.0) {
		tmp = (exp((x * eps)) - (fma(x, eps, x) - 1.0)) * 0.5;
	} else {
		tmp = (((1.0 + (1.0 / eps)) * 1.0) - (fma(-1.0, eps, 1.0) / eps)) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1e-295)
		tmp = Float64(Float64(1.0 - Float64(-exp(Float64(-Float64(x * eps))))) * 0.5);
	elseif (x <= 3800000.0)
		tmp = Float64(Float64(exp(Float64(x * eps)) - Float64(fma(x, eps, x) - 1.0)) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * 1.0) - Float64(fma(-1.0, eps, 1.0) / eps)) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1e-295], N[(N[(1.0 - (-N[Exp[(-N[(x * eps), $MachinePrecision])], $MachinePrecision])), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 3800000.0], N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] - N[(N[(x * eps + x), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] - N[(N[(-1.0 * eps + 1.0), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.00000000000000006e-295

   1. Initial program 69.1%

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

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. 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

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. lower-*.f64 99.0

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 99.4

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

   10. Applied rewrites 99.4%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 74.1%

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

   if -1.00000000000000006e-295 < x < 3.8e6

   1. Initial program 55.0%

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

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. 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

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. lower-*.f64 98.1

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

   7. Applied rewrites 98.1%

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

   8. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 98.6

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

   10. Applied rewrites 98.6%

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

   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. lower--.f64 N/A

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

       2. *-commutative N/A

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

       3. +-commutative N/A

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

       4. distribute-rgt1-in N/A

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 85.1

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

   13. Applied rewrites 85.1%

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

   if 3.8e6 < 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. Taylor expanded in x around 0

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

   4. Applied rewrites 28.6%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. inv-pow N/A

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

      3. lower-pow.f64 46.8

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

   7. Applied rewrites 46.8%

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

   8. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 46.8

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

   10. Applied rewrites 46.8%

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

Alternative 5: 74.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -29:\\ \;\;\;\;e^{-x}\\ \mathbf{elif}\;x \leq -2.55 \cdot 10^{-250}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 7500000000000:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - -1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot 1 - \frac{\mathsf{fma}\left(-1, \varepsilon, 1\right)}{\varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -29.0)
   (exp (- x))
   (if (<= x -2.55e-250)
     (* (fma (fma -1.0 (/ (- (* eps eps) 1.0) (- eps 1.0)) eps) x 2.0) 0.5)
     (if (<= x 7500000000000.0)
       (* (- (exp (* x eps)) -1.0) 0.5)
       (/ (- (* (+ 1.0 (/ 1.0 eps)) 1.0) (/ (fma -1.0 eps 1.0) eps)) 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -29.0) {
		tmp = exp(-x);
	} else if (x <= -2.55e-250) {
		tmp = fma(fma(-1.0, (((eps * eps) - 1.0) / (eps - 1.0)), eps), x, 2.0) * 0.5;
	} else if (x <= 7500000000000.0) {
		tmp = (exp((x * eps)) - -1.0) * 0.5;
	} else {
		tmp = (((1.0 + (1.0 / eps)) * 1.0) - (fma(-1.0, eps, 1.0) / eps)) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -29.0)
		tmp = exp(Float64(-x));
	elseif (x <= -2.55e-250)
		tmp = Float64(fma(fma(-1.0, Float64(Float64(Float64(eps * eps) - 1.0) / Float64(eps - 1.0)), eps), x, 2.0) * 0.5);
	elseif (x <= 7500000000000.0)
		tmp = Float64(Float64(exp(Float64(x * eps)) - -1.0) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * 1.0) - Float64(fma(-1.0, eps, 1.0) / eps)) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -29.0], N[Exp[(-x)], $MachinePrecision], If[LessEqual[x, -2.55e-250], N[(N[(N[(-1.0 * N[(N[(N[(eps * eps), $MachinePrecision] - 1.0), $MachinePrecision] / N[(eps - 1.0), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 7500000000000.0], N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] - N[(N[(-1.0 * eps + 1.0), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -29

   1. Initial program 99.4%

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

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. 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

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. lower-*.f64 99.3

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in eps around 0

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

      1. lower-exp.f64 N/A

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

      2. lift-neg.f64 99.0

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

   10. Applied rewrites 99.0%

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

   if -29 < x < -2.5500000000000001e-250

   1. Initial program 53.6%

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

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. 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

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto \left(2 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + -1 \cdot \left(1 - \varepsilon\right)\right)\right) \cdot \frac{1}{2} \]
   6. 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 69.1

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

   7. Applied rewrites 69.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   8. 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 76.2

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

   9. Applied rewrites 76.2%

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

   10. Taylor expanded in eps around inf

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

   12. Applied rewrites 76.2%

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

   if -2.5500000000000001e-250 < x < 7.5e12

   1. Initial program 55.3%

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

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. 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

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. lower-*.f64 97.6

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

   7. Applied rewrites 97.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 85.3%

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

   if 7.5e12 < 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. Taylor expanded in x around 0

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

   4. Applied rewrites 28.6%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. inv-pow N/A

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

      3. lower-pow.f64 46.6

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

   7. Applied rewrites 46.6%

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

   8. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 46.6

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

   10. Applied rewrites 46.6%

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

Alternative 6: 69.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-295}:\\ \;\;\;\;\left(1 - \left(-e^{-x \cdot \varepsilon}\right)\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 7500000000000:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - -1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot 1 - \frac{\mathsf{fma}\left(-1, \varepsilon, 1\right)}{\varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1e-295)
   (* (- 1.0 (- (exp (- (* x eps))))) 0.5)
   (if (<= x 7500000000000.0)
     (* (- (exp (* x eps)) -1.0) 0.5)
     (/ (- (* (+ 1.0 (/ 1.0 eps)) 1.0) (/ (fma -1.0 eps 1.0) eps)) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1e-295) {
		tmp = (1.0 - -exp(-(x * eps))) * 0.5;
	} else if (x <= 7500000000000.0) {
		tmp = (exp((x * eps)) - -1.0) * 0.5;
	} else {
		tmp = (((1.0 + (1.0 / eps)) * 1.0) - (fma(-1.0, eps, 1.0) / eps)) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1e-295)
		tmp = Float64(Float64(1.0 - Float64(-exp(Float64(-Float64(x * eps))))) * 0.5);
	elseif (x <= 7500000000000.0)
		tmp = Float64(Float64(exp(Float64(x * eps)) - -1.0) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * 1.0) - Float64(fma(-1.0, eps, 1.0) / eps)) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1e-295], N[(N[(1.0 - (-N[Exp[(-N[(x * eps), $MachinePrecision])], $MachinePrecision])), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 7500000000000.0], N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] - N[(N[(-1.0 * eps + 1.0), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.00000000000000006e-295

   1. Initial program 69.1%

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

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. 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

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. lower-*.f64 99.0

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lift-*.f64 99.4

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

   10. Applied rewrites 99.4%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 74.1%

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

   if -1.00000000000000006e-295 < x < 7.5e12

   1. Initial program 55.5%

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

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. 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

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. lower-*.f64 97.3

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

   7. Applied rewrites 97.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 84.0%

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

   if 7.5e12 < 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. Taylor expanded in x around 0

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

   4. Applied rewrites 28.6%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. inv-pow N/A

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

      3. lower-pow.f64 46.6

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

   7. Applied rewrites 46.6%

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

   8. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 46.6

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

   10. Applied rewrites 46.6%

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

Alternative 7: 71.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 4.4 \cdot 10^{+220}:\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 4.4e+220)
   (exp (- x))
   (*
    (fma (fma -1.0 (/ (- (* eps eps) 1.0) (- eps 1.0)) (- (- 1.0 eps))) x 2.0)
    0.5)))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 4.4e+220) {
		tmp = exp(-x);
	} else {
		tmp = fma(fma(-1.0, (((eps * eps) - 1.0) / (eps - 1.0)), -(1.0 - eps)), x, 2.0) * 0.5;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 4.4e+220)
		tmp = exp(Float64(-x));
	else
		tmp = Float64(fma(fma(-1.0, Float64(Float64(Float64(eps * eps) - 1.0) / Float64(eps - 1.0)), Float64(-Float64(1.0 - eps))), x, 2.0) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 4.4e+220], N[Exp[(-x)], $MachinePrecision], N[(N[(N[(-1.0 * N[(N[(N[(eps * eps), $MachinePrecision] - 1.0), $MachinePrecision] / N[(eps - 1.0), $MachinePrecision]), $MachinePrecision] + (-N[(1.0 - eps), $MachinePrecision])), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 4.39999999999999978e220

   1. Initial program 70.8%

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

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. 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

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. lower-*.f64 88.2

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

   7. Applied rewrites 88.2%

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

   8. Taylor expanded in eps around 0

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

      1. lower-exp.f64 N/A

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

      2. lift-neg.f64 73.3

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

   10. Applied rewrites 73.3%

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

   if 4.39999999999999978e220 < 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. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. 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

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \left(2 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + -1 \cdot \left(1 - \varepsilon\right)\right)\right) \cdot \frac{1}{2} \]
   6. 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 8.6

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

   7. Applied rewrites 8.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   8. 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 46.6

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

   9. Applied rewrites 46.6%

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

Alternative 8: 63.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -1\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-209}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-243}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2400000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon + 1}\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot 1 - \frac{\mathsf{fma}\left(-1, \varepsilon, 1\right)}{\varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.8e+77)
   (* (fma (fma -1.0 (+ eps 1.0) -1.0) x 2.0) 0.5)
   (if (<= x -1e-209)
     (* (fma (fma -1.0 (/ (- (* eps eps) 1.0) (- eps 1.0)) eps) x 2.0) 0.5)
     (if (<= x 2e-243)
       1.0
       (if (<= x 2400000.0)
         (*
          (fma
           (fma -1.0 (+ eps 1.0) (- (/ (- 1.0 (* eps eps)) (+ eps 1.0))))
           x
           2.0)
          0.5)
         (/
          (- (* (+ 1.0 (/ 1.0 eps)) 1.0) (/ (fma -1.0 eps 1.0) eps))
          2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.8e+77) {
		tmp = fma(fma(-1.0, (eps + 1.0), -1.0), x, 2.0) * 0.5;
	} else if (x <= -1e-209) {
		tmp = fma(fma(-1.0, (((eps * eps) - 1.0) / (eps - 1.0)), eps), x, 2.0) * 0.5;
	} else if (x <= 2e-243) {
		tmp = 1.0;
	} else if (x <= 2400000.0) {
		tmp = fma(fma(-1.0, (eps + 1.0), -((1.0 - (eps * eps)) / (eps + 1.0))), x, 2.0) * 0.5;
	} else {
		tmp = (((1.0 + (1.0 / eps)) * 1.0) - (fma(-1.0, eps, 1.0) / eps)) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.8e+77)
		tmp = Float64(fma(fma(-1.0, Float64(eps + 1.0), -1.0), x, 2.0) * 0.5);
	elseif (x <= -1e-209)
		tmp = Float64(fma(fma(-1.0, Float64(Float64(Float64(eps * eps) - 1.0) / Float64(eps - 1.0)), eps), x, 2.0) * 0.5);
	elseif (x <= 2e-243)
		tmp = 1.0;
	elseif (x <= 2400000.0)
		tmp = Float64(fma(fma(-1.0, Float64(eps + 1.0), Float64(-Float64(Float64(1.0 - Float64(eps * eps)) / Float64(eps + 1.0)))), x, 2.0) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * 1.0) - Float64(fma(-1.0, eps, 1.0) / eps)) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.8e+77], N[(N[(N[(-1.0 * N[(eps + 1.0), $MachinePrecision] + -1.0), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, -1e-209], N[(N[(N[(-1.0 * N[(N[(N[(eps * eps), $MachinePrecision] - 1.0), $MachinePrecision] / N[(eps - 1.0), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 2e-243], 1.0, If[LessEqual[x, 2400000.0], N[(N[(N[(-1.0 * N[(eps + 1.0), $MachinePrecision] + (-N[(N[(1.0 - N[(eps * eps), $MachinePrecision]), $MachinePrecision] / N[(eps + 1.0), $MachinePrecision]), $MachinePrecision])), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] - N[(N[(-1.0 * eps + 1.0), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1.7999999999999999e77

   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. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. 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

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \left(2 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + -1 \cdot \left(1 - \varepsilon\right)\right)\right) \cdot \frac{1}{2} \]
   6. 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 3.3

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

   7. Applied rewrites 3.3%

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

   8. Taylor expanded in eps around 0

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

   10. Applied rewrites 33.7%

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

   if -1.7999999999999999e77 < x < -1e-209

   1. Initial program 60.2%

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

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. 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

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lift--.f64 56.4

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

   7. Applied rewrites 56.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   8. 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 68.0

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

   9. Applied rewrites 68.0%

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

   10. Taylor expanded in eps around inf

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

   12. Applied rewrites 68.0%

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

   if -1e-209 < x < 1.99999999999999999e-243

   1. Initial program 53.1%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 92.2%

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

   if 1.99999999999999999e-243 < x < 2.4e6

   1. Initial program 56.0%

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

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. 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

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lift--.f64 65.4

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

   7. Applied rewrites 65.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   8. 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 74.8

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

   9. Applied rewrites 74.8%

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

   if 2.4e6 < 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. Taylor expanded in x around 0

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

   4. Applied rewrites 28.6%

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

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. inv-pow N/A

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

      3. lower-pow.f64 46.8

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

   7. Applied rewrites 46.8%

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

   8. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 46.9

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

   10. Applied rewrites 46.9%

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

Alternative 9: 63.4% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \varepsilon - 1\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -1\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-209}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{t\_0}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.65 \cdot 10^{-167}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{t\_0}{-1}, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (* eps eps) 1.0)))
   (if (<= x -1.8e+77)
     (* (fma (fma -1.0 (+ eps 1.0) -1.0) x 2.0) 0.5)
     (if (<= x -1e-209)
       (* (fma (fma -1.0 (/ t_0 (- eps 1.0)) eps) x 2.0) 0.5)
       (if (<= x 2.65e-167)
         1.0
         (* (fma (fma -1.0 (/ t_0 -1.0) (- (- 1.0 eps))) x 2.0) 0.5))))))

C

double code(double x, double eps) {
	double t_0 = (eps * eps) - 1.0;
	double tmp;
	if (x <= -1.8e+77) {
		tmp = fma(fma(-1.0, (eps + 1.0), -1.0), x, 2.0) * 0.5;
	} else if (x <= -1e-209) {
		tmp = fma(fma(-1.0, (t_0 / (eps - 1.0)), eps), x, 2.0) * 0.5;
	} else if (x <= 2.65e-167) {
		tmp = 1.0;
	} else {
		tmp = fma(fma(-1.0, (t_0 / -1.0), -(1.0 - eps)), x, 2.0) * 0.5;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(eps * eps) - 1.0)
	tmp = 0.0
	if (x <= -1.8e+77)
		tmp = Float64(fma(fma(-1.0, Float64(eps + 1.0), -1.0), x, 2.0) * 0.5);
	elseif (x <= -1e-209)
		tmp = Float64(fma(fma(-1.0, Float64(t_0 / Float64(eps - 1.0)), eps), x, 2.0) * 0.5);
	elseif (x <= 2.65e-167)
		tmp = 1.0;
	else
		tmp = Float64(fma(fma(-1.0, Float64(t_0 / -1.0), Float64(-Float64(1.0 - eps))), x, 2.0) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(eps * eps), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -1.8e+77], N[(N[(N[(-1.0 * N[(eps + 1.0), $MachinePrecision] + -1.0), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, -1e-209], N[(N[(N[(-1.0 * N[(t$95$0 / N[(eps - 1.0), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 2.65e-167], 1.0, N[(N[(N[(-1.0 * N[(t$95$0 / -1.0), $MachinePrecision] + (-N[(1.0 - eps), $MachinePrecision])), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.7999999999999999e77

   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. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. 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

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \left(2 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + -1 \cdot \left(1 - \varepsilon\right)\right)\right) \cdot \frac{1}{2} \]
   6. 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 3.3

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

   7. Applied rewrites 3.3%

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

   8. Taylor expanded in eps around 0

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

   10. Applied rewrites 33.7%

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

   if -1.7999999999999999e77 < x < -1e-209

   1. Initial program 60.2%

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

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. 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

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lift--.f64 56.4

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

   7. Applied rewrites 56.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   8. 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 68.0

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

   9. Applied rewrites 68.0%

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

   10. Taylor expanded in eps around inf

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

   12. Applied rewrites 68.0%

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

   if -1e-209 < x < 2.65e-167

   1. Initial program 53.6%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 88.2%

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

   if 2.65e-167 < x

   1. Initial program 84.1%

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

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. 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

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lift--.f64 23.0

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

   7. Applied rewrites 23.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   8. 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 35.8

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

   9. Applied rewrites 35.8%

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

   10. Taylor expanded in eps around 0

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

   12. Applied rewrites 55.4%

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

Alternative 10: 52.4% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -1\right), x, 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq -2.55 \cdot 10^{-250}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \frac{\varepsilon \cdot \varepsilon - 1}{\varepsilon - 1}, \varepsilon\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\varepsilon - 1, x, 1\right) - -1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.8e+77)
   (* (fma (fma -1.0 (+ eps 1.0) -1.0) x 2.0) 0.5)
   (if (<= x -2.55e-250)
     (* (fma (fma -1.0 (/ (- (* eps eps) 1.0) (- eps 1.0)) eps) x 2.0) 0.5)
     (* (- (fma (- eps 1.0) x 1.0) -1.0) 0.5))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.8e+77) {
		tmp = fma(fma(-1.0, (eps + 1.0), -1.0), x, 2.0) * 0.5;
	} else if (x <= -2.55e-250) {
		tmp = fma(fma(-1.0, (((eps * eps) - 1.0) / (eps - 1.0)), eps), x, 2.0) * 0.5;
	} else {
		tmp = (fma((eps - 1.0), x, 1.0) - -1.0) * 0.5;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.8e+77)
		tmp = Float64(fma(fma(-1.0, Float64(eps + 1.0), -1.0), x, 2.0) * 0.5);
	elseif (x <= -2.55e-250)
		tmp = Float64(fma(fma(-1.0, Float64(Float64(Float64(eps * eps) - 1.0) / Float64(eps - 1.0)), eps), x, 2.0) * 0.5);
	else
		tmp = Float64(Float64(fma(Float64(eps - 1.0), x, 1.0) - -1.0) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.8e+77], N[(N[(N[(-1.0 * N[(eps + 1.0), $MachinePrecision] + -1.0), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, -2.55e-250], N[(N[(N[(-1.0 * N[(N[(N[(eps * eps), $MachinePrecision] - 1.0), $MachinePrecision] / N[(eps - 1.0), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(eps - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.7999999999999999e77

   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. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. 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

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \left(2 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + -1 \cdot \left(1 - \varepsilon\right)\right)\right) \cdot \frac{1}{2} \]
   6. 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 3.3

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

   7. Applied rewrites 3.3%

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

   8. Taylor expanded in eps around 0

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

   10. Applied rewrites 33.7%

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

   if -1.7999999999999999e77 < x < -2.5500000000000001e-250

   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. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. 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

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto \left(2 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + -1 \cdot \left(1 - \varepsilon\right)\right)\right) \cdot \frac{1}{2} \]
   6. 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 60.3

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

   7. Applied rewrites 60.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -\left(1 - \varepsilon\right)\right), x, 2\right) \cdot 0.5 \]
   8. 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 69.7

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

   9. Applied rewrites 69.7%

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

   10. Taylor expanded in eps around inf

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

   12. Applied rewrites 69.7%

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

   if -2.5500000000000001e-250 < x

   1. Initial program 74.9%

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

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. 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

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. lower-*.f64 83.2

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

   7. Applied rewrites 83.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 60.3%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lift--.f64 48.0

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

   13. Applied rewrites 48.0%

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

Alternative 11: 49.9% accurate, 10.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-295}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1, \varepsilon + 1, -1\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\varepsilon - 1, x, 1\right) - -1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1e-295)
   (* (fma (fma -1.0 (+ eps 1.0) -1.0) x 2.0) 0.5)
   (* (- (fma (- eps 1.0) x 1.0) -1.0) 0.5)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1e-295) {
		tmp = fma(fma(-1.0, (eps + 1.0), -1.0), x, 2.0) * 0.5;
	} else {
		tmp = (fma((eps - 1.0), x, 1.0) - -1.0) * 0.5;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1e-295)
		tmp = Float64(fma(fma(-1.0, Float64(eps + 1.0), -1.0), x, 2.0) * 0.5);
	else
		tmp = Float64(Float64(fma(Float64(eps - 1.0), x, 1.0) - -1.0) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1e-295], N[(N[(N[(-1.0 * N[(eps + 1.0), $MachinePrecision] + -1.0), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(eps - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.00000000000000006e-295

   1. Initial program 69.1%

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

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. 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

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \left(2 + x \cdot \left(-1 \cdot \left(1 + \varepsilon\right) + -1 \cdot \left(1 - \varepsilon\right)\right)\right) \cdot \frac{1}{2} \]
   6. 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 49.1

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

   7. Applied rewrites 49.1%

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

   8. Taylor expanded in eps around 0

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

   10. Applied rewrites 56.8%

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

   if -1.00000000000000006e-295 < x

   1. Initial program 76.4%

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

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. 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

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. lower-*.f64 82.1

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

   7. Applied rewrites 82.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 57.9%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lift--.f64 44.9

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

   13. Applied rewrites 44.9%

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

Alternative 12: 49.7% accurate, 15.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

2. Taylor expanded in eps around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
3. 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

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in eps around inf

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

   1. *-commutative N/A

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

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

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

   3. *-commutative N/A

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

   4. lower-*.f64 89.2

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

7. Applied rewrites 89.2%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 64.6%

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

11. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. lift--.f64 49.7

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

13. Applied rewrites 49.7%

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

Alternative 13: 43.8% accurate, 273.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 1.0)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, eps):
	return 1.0

Julia

function code(x, eps)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 1.0

Derivation

1. Initial program 73.3%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 43.8%

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

Reproduce

herbie shell --seed 2025093 
(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))