NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.3% → 99.8%

Time: 5.1s

Alternatives: 12

Speedup: 2.2×

• 37.9% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -2.0500434668730143e+65
  2. eps = 5.232068672313939e-51

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: 99.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 5 \cdot 10^{-33}:\\ \;\;\;\;\left(x - -1\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-eps\_m \cdot x} + e^{\left(eps\_m - 1\right) \cdot x}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 5e-33)
   (* (- x -1.0) (exp (- x)))
   (* (+ (exp (- (* eps_m x))) (exp (* (- eps_m 1.0) x))) 0.5)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 5e-33) {
		tmp = (x - -1.0) * exp(-x);
	} else {
		tmp = (exp(-(eps_m * x)) + exp(((eps_m - 1.0) * x))) * 0.5;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, eps_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 5d-33) then
        tmp = (x - (-1.0d0)) * exp(-x)
    else
        tmp = (exp(-(eps_m * x)) + exp(((eps_m - 1.0d0) * x))) * 0.5d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 5e-33) {
		tmp = (x - -1.0) * Math.exp(-x);
	} else {
		tmp = (Math.exp(-(eps_m * x)) + Math.exp(((eps_m - 1.0) * x))) * 0.5;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 5e-33:
		tmp = (x - -1.0) * math.exp(-x)
	else:
		tmp = (math.exp(-(eps_m * x)) + math.exp(((eps_m - 1.0) * x))) * 0.5
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 5e-33)
		tmp = Float64(Float64(x - -1.0) * exp(Float64(-x)));
	else
		tmp = Float64(Float64(exp(Float64(-Float64(eps_m * x))) + exp(Float64(Float64(eps_m - 1.0) * x))) * 0.5);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 5e-33)
		tmp = (x - -1.0) * exp(-x);
	else
		tmp = (exp(-(eps_m * x)) + exp(((eps_m - 1.0) * x))) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 5e-33], N[(N[(x - -1.0), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[(-N[(eps$95$m * x), $MachinePrecision])], $MachinePrecision] + N[Exp[N[(N[(eps$95$m - 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 5.00000000000000028e-33

   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 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 57.3%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

   6. Applied rewrites 57.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 57.3

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

   8. Applied rewrites 57.4%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{x - -1}{e^{x}}\right) \cdot 0.5} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. div-flip N/A

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

      8. mult-flip N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{x}}{x - -1}}} \]

      9. div-flip N/A

         \[\leadsto \frac{x - -1}{\color{blue}{e^{x}}} \]

      10. mult-flip N/A

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

      11. lift-exp.f64 N/A

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

      12. rec-exp N/A

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

      13. lower-*.f64 N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 57.4

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

   10. Applied rewrites 57.4%

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

   if 5.00000000000000028e-33 < eps

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.1

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in eps around inf

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

      1. lower-*.f64 92.1

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

   9. Applied rewrites 92.1%

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

Alternative 2: 99.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 5 \cdot 10^{-36}:\\ \;\;\;\;\left(x - -1\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-\mathsf{fma}\left(x, eps\_m, x\right)} + e^{\left(eps\_m - 1\right) \cdot x}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 5e-36)
   (* (- x -1.0) (exp (- x)))
   (* (+ (exp (- (fma x eps_m x))) (exp (* (- eps_m 1.0) x))) 0.5)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 5e-36) {
		tmp = (x - -1.0) * exp(-x);
	} else {
		tmp = (exp(-fma(x, eps_m, x)) + exp(((eps_m - 1.0) * x))) * 0.5;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 5e-36)
		tmp = Float64(Float64(x - -1.0) * exp(Float64(-x)));
	else
		tmp = Float64(Float64(exp(Float64(-fma(x, eps_m, x))) + exp(Float64(Float64(eps_m - 1.0) * x))) * 0.5);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 5e-36], N[(N[(x - -1.0), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[(-N[(x * eps$95$m + x), $MachinePrecision])], $MachinePrecision] + N[Exp[N[(N[(eps$95$m - 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 5.00000000000000004e-36

   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 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 57.3%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

   6. Applied rewrites 57.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 57.3

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

   8. Applied rewrites 57.4%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{x - -1}{e^{x}}\right) \cdot 0.5} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. div-flip N/A

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

      8. mult-flip N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{x}}{x - -1}}} \]

      9. div-flip N/A

         \[\leadsto \frac{x - -1}{\color{blue}{e^{x}}} \]

      10. mult-flip N/A

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

      11. lift-exp.f64 N/A

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

      12. rec-exp N/A

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

      13. lower-*.f64 N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 57.4

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

   10. Applied rewrites 57.4%

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

   if 5.00000000000000004e-36 < eps

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.1

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

   6. Applied rewrites 99.1%

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

Alternative 3: 84.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-248}:\\ \;\;\;\;\left(e^{-\mathsf{fma}\left(x, eps\_m, x\right)} + \left(1 + x \cdot -1\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-x \cdot \left(1 - eps\_m\right)} - -1\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2.2e-248)
   (* (+ (exp (- (fma x eps_m x))) (+ 1.0 (* x -1.0))) 0.5)
   (* 0.5 (- (exp (- (* x (- 1.0 eps_m)))) -1.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -2.2e-248) {
		tmp = (exp(-fma(x, eps_m, x)) + (1.0 + (x * -1.0))) * 0.5;
	} else {
		tmp = 0.5 * (exp(-(x * (1.0 - eps_m))) - -1.0);
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -2.2e-248)
		tmp = Float64(Float64(exp(Float64(-fma(x, eps_m, x))) + Float64(1.0 + Float64(x * -1.0))) * 0.5);
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-Float64(x * Float64(1.0 - eps_m)))) - -1.0));
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -2.2e-248], N[(N[(N[Exp[(-N[(x * eps$95$m + x), $MachinePrecision])], $MachinePrecision] + N[(1.0 + N[(x * -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[(N[Exp[(-N[(x * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision])], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.19999999999999999e-248

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.1

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 65.0

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

   9. Applied rewrites 65.0%

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

   10. Taylor expanded in eps around 0

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

   12. Applied rewrites 64.0%

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

   if -2.19999999999999999e-248 < x

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 63.8%

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

Alternative 4: 77.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-247}:\\ \;\;\;\;0.5 \cdot \left(\left(1 + x\right) + e^{-x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-x \cdot \left(1 - eps\_m\right)} - -1\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 2.2e-247)
   (* 0.5 (+ (+ 1.0 x) (exp (- x))))
   (* 0.5 (- (exp (- (* x (- 1.0 eps_m)))) -1.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.2e-247) {
		tmp = 0.5 * ((1.0 + x) + exp(-x));
	} else {
		tmp = 0.5 * (exp(-(x * (1.0 - eps_m))) - -1.0);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, eps_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 2.2d-247) then
        tmp = 0.5d0 * ((1.0d0 + x) + exp(-x))
    else
        tmp = 0.5d0 * (exp(-(x * (1.0d0 - eps_m))) - (-1.0d0))
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.2e-247) {
		tmp = 0.5 * ((1.0 + x) + Math.exp(-x));
	} else {
		tmp = 0.5 * (Math.exp(-(x * (1.0 - eps_m))) - -1.0);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 2.2e-247:
		tmp = 0.5 * ((1.0 + x) + math.exp(-x))
	else:
		tmp = 0.5 * (math.exp(-(x * (1.0 - eps_m))) - -1.0)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 2.2e-247)
		tmp = Float64(0.5 * Float64(Float64(1.0 + x) + exp(Float64(-x))));
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-Float64(x * Float64(1.0 - eps_m)))) - -1.0));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 2.2e-247)
		tmp = 0.5 * ((1.0 + x) + exp(-x));
	else
		tmp = 0.5 * (exp(-(x * (1.0 - eps_m))) - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 2.2e-247], N[(0.5 * N[(N[(1.0 + x), $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Exp[(-N[(x * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision])], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.19999999999999992e-247

   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 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 57.3%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

   6. Applied rewrites 57.3%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 58.1

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

   9. Applied rewrites 58.1%

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

   if 2.19999999999999992e-247 < x

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 63.8%

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

Alternative 5: 72.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;eps\_m \leq 0.082:\\ \;\;\;\;\left(x - -1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(1 + x\right) + t\_0\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= eps_m 0.082) (* (- x -1.0) t_0) (* 0.5 (+ (+ 1.0 x) t_0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp(-x);
	double tmp;
	if (eps_m <= 0.082) {
		tmp = (x - -1.0) * t_0;
	} else {
		tmp = 0.5 * ((1.0 + x) + t_0);
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, eps_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (eps_m <= 0.082d0) then
        tmp = (x - (-1.0d0)) * t_0
    else
        tmp = 0.5d0 * ((1.0d0 + x) + t_0)
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp(-x);
	double tmp;
	if (eps_m <= 0.082) {
		tmp = (x - -1.0) * t_0;
	} else {
		tmp = 0.5 * ((1.0 + x) + t_0);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp(-x)
	tmp = 0
	if eps_m <= 0.082:
		tmp = (x - -1.0) * t_0
	else:
		tmp = 0.5 * ((1.0 + x) + t_0)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (eps_m <= 0.082)
		tmp = Float64(Float64(x - -1.0) * t_0);
	else
		tmp = Float64(0.5 * Float64(Float64(1.0 + x) + t_0));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp(-x);
	tmp = 0.0;
	if (eps_m <= 0.082)
		tmp = (x - -1.0) * t_0;
	else
		tmp = 0.5 * ((1.0 + x) + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[eps$95$m, 0.082], N[(N[(x - -1.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(0.5 * N[(N[(1.0 + x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 0.0820000000000000034

   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 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 57.3%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

   6. Applied rewrites 57.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 57.3

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

   8. Applied rewrites 57.4%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{x - -1}{e^{x}}\right) \cdot 0.5} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. div-flip N/A

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

      8. mult-flip N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{x}}{x - -1}}} \]

      9. div-flip N/A

         \[\leadsto \frac{x - -1}{\color{blue}{e^{x}}} \]

      10. mult-flip N/A

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

      11. lift-exp.f64 N/A

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

      12. rec-exp N/A

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

      13. lower-*.f64 N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 57.4

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

   10. Applied rewrites 57.4%

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

   if 0.0820000000000000034 < eps

   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 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 57.3%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

   6. Applied rewrites 57.3%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 58.1

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

   9. Applied rewrites 58.1%

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

Alternative 6: 66.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, -1\right)}{-1 - x}\\ \mathbf{elif}\;x \leq -8000:\\ \;\;\;\;\frac{1 - \left(x \cdot x\right) \cdot x}{\mathsf{fma}\left(x, x - -1, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x - -1\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.35e+154)
   (/ (fma x x -1.0) (- -1.0 x))
   (if (<= x -8000.0)
     (/ (- 1.0 (* (* x x) x)) (fma x (- x -1.0) 1.0))
     (* (- x -1.0) (exp (- x))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.35e+154) {
		tmp = fma(x, x, -1.0) / (-1.0 - x);
	} else if (x <= -8000.0) {
		tmp = (1.0 - ((x * x) * x)) / fma(x, (x - -1.0), 1.0);
	} else {
		tmp = (x - -1.0) * exp(-x);
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.35e+154)
		tmp = Float64(fma(x, x, -1.0) / Float64(-1.0 - x));
	elseif (x <= -8000.0)
		tmp = Float64(Float64(1.0 - Float64(Float64(x * x) * x)) / fma(x, Float64(x - -1.0), 1.0));
	else
		tmp = Float64(Float64(x - -1.0) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.35e+154], N[(N[(x * x + -1.0), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -8000.0], N[(N[(1.0 - N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[(x * N[(x - -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x - -1.0), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.35000000000000003e154

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{-1 \cdot x} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{x} \]

      2. lower-*.f64 43.5

         \[\leadsto 1 + -1 \cdot x \]

   7. Applied rewrites 43.5%

      \[\leadsto 1 + \color{blue}{-1 \cdot x} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{x} \]

      2. flip-+ N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lift-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sqr-neg-rev N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. metadata-eval N/A

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

      16. lift-*.f64 N/A

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

      17. mul-1-neg N/A

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

      18. add-flip-rev N/A

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

      19. +-commutative N/A

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

      20. metadata-eval N/A

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

      21. sub-flip N/A

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

      22. sub-negate-rev N/A

          \[\leadsto \frac{x \cdot x - -1 \cdot -1}{-1 - x} \]

      23. lower-/.f64 N/A

          \[\leadsto \frac{x \cdot x - -1 \cdot -1}{-1 - \color{blue}{x}} \]

   9. Applied rewrites 50.0%

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

   if -1.35000000000000003e154 < x < -8e3

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{-1 \cdot x} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{x} \]

      2. lower-*.f64 43.5

         \[\leadsto 1 + -1 \cdot x \]

   7. Applied rewrites 43.5%

      \[\leadsto 1 + \color{blue}{-1 \cdot x} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{x} \]

      2. lift-*.f64 N/A

         \[\leadsto 1 + -1 \cdot x \]

      3. mul-1-neg N/A

         \[\leadsto 1 + \left(\mathsf{neg}\left(x\right)\right) \]

      4. sub-flip-reverse N/A

         \[\leadsto 1 - x \]

      5. lower--.f64 43.5

         \[\leadsto 1 - x \]

   9. Applied rewrites 43.5%

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

       1. lift--.f64 N/A

          \[\leadsto 1 - x \]

       2. flip3-- N/A

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

       3. lower-/.f64 N/A

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

       4. metadata-eval N/A

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

       5. lower--.f64 N/A

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

       6. unpow3 N/A

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

       7. lift-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. metadata-eval N/A

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

       10. +-commutative N/A

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

       11. distribute-rgt-out N/A

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

       12. metadata-eval N/A

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

       13. sub-flip N/A

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

       14. lift--.f64 N/A

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

       15. lower-fma.f64 45.2

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

   11. Applied rewrites 45.2%

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

   if -8e3 < x

   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 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 57.3%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

   6. Applied rewrites 57.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 57.3

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

   8. Applied rewrites 57.4%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{x - -1}{e^{x}}\right) \cdot 0.5} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. div-flip N/A

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

      8. mult-flip N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{x}}{x - -1}}} \]

      9. div-flip N/A

         \[\leadsto \frac{x - -1}{\color{blue}{e^{x}}} \]

      10. mult-flip N/A

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

      11. lift-exp.f64 N/A

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

      12. rec-exp N/A

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

      13. lower-*.f64 N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 57.4

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

   10. Applied rewrites 57.4%

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

Alternative 7: 64.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -8000:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, -1\right)}{-1 - x}\\ \mathbf{else}:\\ \;\;\;\;\left(x - -1\right) \cdot e^{-x}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -8000.0) (/ (fma x x -1.0) (- -1.0 x)) (* (- x -1.0) (exp (- x)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -8000.0) {
		tmp = fma(x, x, -1.0) / (-1.0 - x);
	} else {
		tmp = (x - -1.0) * exp(-x);
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -8000.0)
		tmp = Float64(fma(x, x, -1.0) / Float64(-1.0 - x));
	else
		tmp = Float64(Float64(x - -1.0) * exp(Float64(-x)));
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -8000.0], N[(N[(x * x + -1.0), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision], N[(N[(x - -1.0), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8e3

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{-1 \cdot x} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{x} \]

      2. lower-*.f64 43.5

         \[\leadsto 1 + -1 \cdot x \]

   7. Applied rewrites 43.5%

      \[\leadsto 1 + \color{blue}{-1 \cdot x} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{x} \]

      2. flip-+ N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lift-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sqr-neg-rev N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. metadata-eval N/A

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

      16. lift-*.f64 N/A

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

      17. mul-1-neg N/A

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

      18. add-flip-rev N/A

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

      19. +-commutative N/A

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

      20. metadata-eval N/A

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

      21. sub-flip N/A

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

      22. sub-negate-rev N/A

          \[\leadsto \frac{x \cdot x - -1 \cdot -1}{-1 - x} \]

      23. lower-/.f64 N/A

          \[\leadsto \frac{x \cdot x - -1 \cdot -1}{-1 - \color{blue}{x}} \]

   9. Applied rewrites 50.0%

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

   if -8e3 < x

   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 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 57.3%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

   6. Applied rewrites 57.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 57.3

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

   8. Applied rewrites 57.4%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{x - -1}{e^{x}}\right) \cdot 0.5} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. div-flip N/A

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

      8. mult-flip N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{x}}{x - -1}}} \]

      9. div-flip N/A

         \[\leadsto \frac{x - -1}{\color{blue}{e^{x}}} \]

      10. mult-flip N/A

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

      11. lift-exp.f64 N/A

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

      12. rec-exp N/A

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

      13. lower-*.f64 N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 57.4

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

   10. Applied rewrites 57.4%

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

Alternative 8: 64.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -8000:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, -1\right)}{-1 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - -1}{e^{x}}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -8000.0) (/ (fma x x -1.0) (- -1.0 x)) (/ (- x -1.0) (exp x))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -8000.0) {
		tmp = fma(x, x, -1.0) / (-1.0 - x);
	} else {
		tmp = (x - -1.0) / exp(x);
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -8000.0)
		tmp = Float64(fma(x, x, -1.0) / Float64(-1.0 - x));
	else
		tmp = Float64(Float64(x - -1.0) / exp(x));
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -8000.0], N[(N[(x * x + -1.0), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision], N[(N[(x - -1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8e3

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{-1 \cdot x} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{x} \]

      2. lower-*.f64 43.5

         \[\leadsto 1 + -1 \cdot x \]

   7. Applied rewrites 43.5%

      \[\leadsto 1 + \color{blue}{-1 \cdot x} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{x} \]

      2. flip-+ N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lift-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sqr-neg-rev N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. metadata-eval N/A

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

      16. lift-*.f64 N/A

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

      17. mul-1-neg N/A

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

      18. add-flip-rev N/A

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

      19. +-commutative N/A

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

      20. metadata-eval N/A

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

      21. sub-flip N/A

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

      22. sub-negate-rev N/A

          \[\leadsto \frac{x \cdot x - -1 \cdot -1}{-1 - x} \]

      23. lower-/.f64 N/A

          \[\leadsto \frac{x \cdot x - -1 \cdot -1}{-1 - \color{blue}{x}} \]

   9. Applied rewrites 50.0%

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

   if -8e3 < x

   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 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 57.3%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

   6. Applied rewrites 57.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 57.3

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

   8. Applied rewrites 57.4%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{x - -1}{e^{x}}\right) \cdot 0.5} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. lift-/.f64 N/A

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

      7. div-flip N/A

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

      8. mult-flip N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{x}}{x - -1}}} \]

      9. div-flip N/A

         \[\leadsto \frac{x - -1}{\color{blue}{e^{x}}} \]

      10. lift-/.f64 57.4

          \[\leadsto \frac{x - -1}{\color{blue}{e^{x}}} \]

   10. Applied rewrites 57.4%

       \[\leadsto \frac{x - -1}{\color{blue}{e^{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 63.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 0.66:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, -1\right)}{-1 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{x}}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 0.66) (/ (fma x x -1.0) (- -1.0 x)) (/ x (exp x))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 0.66) {
		tmp = fma(x, x, -1.0) / (-1.0 - x);
	} else {
		tmp = x / exp(x);
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 0.66)
		tmp = Float64(fma(x, x, -1.0) / Float64(-1.0 - x));
	else
		tmp = Float64(x / exp(x));
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 0.66], N[(N[(x * x + -1.0), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision], N[(x / N[Exp[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.660000000000000031

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{-1 \cdot x} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{x} \]

      2. lower-*.f64 43.5

         \[\leadsto 1 + -1 \cdot x \]

   7. Applied rewrites 43.5%

      \[\leadsto 1 + \color{blue}{-1 \cdot x} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{x} \]

      2. flip-+ N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lift-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sqr-neg-rev N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. metadata-eval N/A

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

      16. lift-*.f64 N/A

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

      17. mul-1-neg N/A

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

      18. add-flip-rev N/A

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

      19. +-commutative N/A

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

      20. metadata-eval N/A

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

      21. sub-flip N/A

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

      22. sub-negate-rev N/A

          \[\leadsto \frac{x \cdot x - -1 \cdot -1}{-1 - x} \]

      23. lower-/.f64 N/A

          \[\leadsto \frac{x \cdot x - -1 \cdot -1}{-1 - \color{blue}{x}} \]

   9. Applied rewrites 50.0%

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

   if 0.660000000000000031 < x

   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 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 57.3%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

   6. Applied rewrites 57.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 57.3

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

   8. Applied rewrites 57.4%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{x - -1}{e^{x}}\right) \cdot 0.5} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{x}{\color{blue}{e^{x}}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

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

       2. lower-exp.f64 16.0

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

   11. Applied rewrites 16.0%

       \[\leadsto \frac{x}{\color{blue}{e^{x}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 57.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-13}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, -1\right)}{-1 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + x \cdot \left(1 + x\right)}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -4e-13)
   (/ (fma x x -1.0) (- -1.0 x))
   (/ 1.0 (+ 1.0 (* x (+ 1.0 x))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -4e-13) {
		tmp = fma(x, x, -1.0) / (-1.0 - x);
	} else {
		tmp = 1.0 / (1.0 + (x * (1.0 + x)));
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -4e-13)
		tmp = Float64(fma(x, x, -1.0) / Float64(-1.0 - x));
	else
		tmp = Float64(1.0 / Float64(1.0 + Float64(x * Float64(1.0 + x))));
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -4e-13], N[(N[(x * x + -1.0), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.0000000000000001e-13

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{-1 \cdot x} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{x} \]

      2. lower-*.f64 43.5

         \[\leadsto 1 + -1 \cdot x \]

   7. Applied rewrites 43.5%

      \[\leadsto 1 + \color{blue}{-1 \cdot x} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{x} \]

      2. flip-+ N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lift-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sqr-neg-rev N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. metadata-eval N/A

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

      16. lift-*.f64 N/A

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

      17. mul-1-neg N/A

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

      18. add-flip-rev N/A

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

      19. +-commutative N/A

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

      20. metadata-eval N/A

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

      21. sub-flip N/A

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

      22. sub-negate-rev N/A

          \[\leadsto \frac{x \cdot x - -1 \cdot -1}{-1 - x} \]

      23. lower-/.f64 N/A

          \[\leadsto \frac{x \cdot x - -1 \cdot -1}{-1 - \color{blue}{x}} \]

   9. Applied rewrites 50.0%

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

   if -4.0000000000000001e-13 < x

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{-1 \cdot x} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{x} \]

      2. lower-*.f64 43.5

         \[\leadsto 1 + -1 \cdot x \]

   7. Applied rewrites 43.5%

      \[\leadsto 1 + \color{blue}{-1 \cdot x} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{x} \]

      2. flip-+ N/A

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

      3. div-flip N/A

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

      4. lower-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. add-flip-rev N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. sub-flip N/A

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

      11. lift--.f64 N/A

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

      12. metadata-eval N/A

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

      13. lift-*.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lift-*.f64 N/A

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

      16. mul-1-neg N/A

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

      17. sqr-neg-rev N/A

          \[\leadsto \frac{1}{\frac{x - -1}{1 - x \cdot x}} \]

      18. unpow2 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{1 - {x}^{2}}} \]

      19. lift-pow.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{1 - {x}^{2}}} \]

      20. lower-/.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{1 - \color{blue}{{x}^{2}}}} \]

      21. lower--.f64 50.0

          \[\leadsto \frac{1}{\frac{x - -1}{1 - {x}^{\color{blue}{2}}}} \]

      22. lift-pow.f64 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{1 - {x}^{2}}} \]

      23. unpow2 N/A

          \[\leadsto \frac{1}{\frac{x - -1}{1 - x \cdot x}} \]

      24. lower-*.f64 50.0

          \[\leadsto \frac{1}{\frac{x - -1}{1 - x \cdot x}} \]

   9. Applied rewrites 50.0%

      \[\leadsto \frac{1}{\frac{x - -1}{\color{blue}{1 - x \cdot x}}} \]

   10. Taylor expanded in x around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-+.f64 50.2

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

   12. Applied rewrites 50.2%

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

Alternative 11: 50.0% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return Float64(fma(x, x, -1.0) / Float64(-1.0 - x))
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := N[(N[(x * x + -1.0), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $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. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-exp.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-+.f64 99.1

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

4. Applied rewrites 99.1%

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

5. Taylor expanded in x around 0

   \[\leadsto 1 + \color{blue}{-1 \cdot x} \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto 1 + -1 \cdot \color{blue}{x} \]

   2. lower-*.f64 43.5

      \[\leadsto 1 + -1 \cdot x \]

7. Applied rewrites 43.5%

   \[\leadsto 1 + \color{blue}{-1 \cdot x} \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto 1 + -1 \cdot \color{blue}{x} \]

   2. flip-+ N/A

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

   3. frac-2neg N/A

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

   4. metadata-eval N/A

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

   5. lift-*.f64 N/A

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

   6. mul-1-neg N/A

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

   7. lift-*.f64 N/A

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

   8. mul-1-neg N/A

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

   9. sqr-neg-rev N/A

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

   10. unpow2 N/A

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

   11. lift-pow.f64 N/A

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

   12. sub-negate-rev N/A

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

   13. lift-pow.f64 N/A

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

   14. unpow2 N/A

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

   15. metadata-eval N/A

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

   16. lift-*.f64 N/A

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

   17. mul-1-neg N/A

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

   18. add-flip-rev N/A

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

   19. +-commutative N/A

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

   20. metadata-eval N/A

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

   21. sub-flip N/A

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

   22. sub-negate-rev N/A

       \[\leadsto \frac{x \cdot x - -1 \cdot -1}{-1 - x} \]

   23. lower-/.f64 N/A

       \[\leadsto \frac{x \cdot x - -1 \cdot -1}{-1 - \color{blue}{x}} \]

9. Applied rewrites 50.0%

   \[\leadsto \frac{\mathsf{fma}\left(x, x, -1\right)}{-1 - \color{blue}{x}} \]
10. Add Preprocessing

Alternative 12: 44.1% accurate, 58.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 44.1%

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

Reproduce

herbie shell --seed 2025149 
(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))