NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.7% → 99.5%

Time: 9.9s

Alternatives: 15

Speedup: 0.7×

• 38.2% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -1.384654996992273e+232
  2. eps = 2.9281826587269914e-168

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 48.2%

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

   3. Taylor expanded in eps around 0

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

      1. distribute-lft-out N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. distribute-rgt1-in N/A

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

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

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

      5. distribute-lft-out N/A

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

      6. distribute-neg-in N/A

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

      7. exp-neg N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. distribute-frac-neg N/A

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

      11. metadata-eval N/A

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

      12. exp-neg N/A

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

      13. distribute-lft-neg-out N/A

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

      14. metadata-eval N/A

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

      15. *-lft-identity N/A

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

   5. Applied rewrites 100.0%

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

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

   1. Initial program 99.8%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 99.8

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in eps around 0

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

       1. lower--.f64 N/A

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

       2. lower-*.f64 99.8

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

   11. Applied rewrites 99.8%

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

4. Final simplification 99.9%

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

Alternative 2: 78.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\varepsilon} - 1\\ t_1 := 1 + \frac{1}{\varepsilon}\\ \mathbf{if}\;\frac{t\_1 \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - t\_0 \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2} \leq 100:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 + x, e^{-x}, \frac{1 + x}{e^{x}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 \cdot e^{x \cdot \varepsilon - x} - t\_0}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 eps) 1.0)) (t_1 (+ 1.0 (/ 1.0 eps))))
   (if (<=
        (/
         (- (* t_1 (exp (* (+ -1.0 eps) x))) (* t_0 (exp (* (- -1.0 eps) x))))
         2.0)
        100.0)
     (/ (fma (+ 1.0 x) (exp (- x)) (/ (+ 1.0 x) (exp x))) 2.0)
     (/ (- (* t_1 (exp (- (* x eps) x))) t_0) 2.0))))

C

double code(double x, double eps) {
	double t_0 = (1.0 / eps) - 1.0;
	double t_1 = 1.0 + (1.0 / eps);
	double tmp;
	if ((((t_1 * exp(((-1.0 + eps) * x))) - (t_0 * exp(((-1.0 - eps) * x)))) / 2.0) <= 100.0) {
		tmp = fma((1.0 + x), exp(-x), ((1.0 + x) / exp(x))) / 2.0;
	} else {
		tmp = ((t_1 * exp(((x * eps) - x))) - t_0) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(1.0 / eps) - 1.0)
	t_1 = Float64(1.0 + Float64(1.0 / eps))
	tmp = 0.0
	if (Float64(Float64(Float64(t_1 * exp(Float64(Float64(-1.0 + eps) * x))) - Float64(t_0 * exp(Float64(Float64(-1.0 - eps) * x)))) / 2.0) <= 100.0)
		tmp = Float64(fma(Float64(1.0 + x), exp(Float64(-x)), Float64(Float64(1.0 + x) / exp(x))) / 2.0);
	else
		tmp = Float64(Float64(Float64(t_1 * exp(Float64(Float64(x * eps) - x))) - t_0) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$1 * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 100.0], N[(N[(N[(1.0 + x), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision] + N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(t$95$1 * N[Exp[N[(N[(x * eps), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) #s(literal 2 binary64)) < 100

   1. Initial program 48.5%

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

   3. Taylor expanded in eps around 0

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

      1. distribute-lft-out N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

      3. distribute-rgt1-in N/A

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

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

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

      5. distribute-lft-out N/A

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

      6. distribute-neg-in N/A

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

      7. exp-neg N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. distribute-frac-neg N/A

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

      11. metadata-eval N/A

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

      12. exp-neg N/A

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

      13. distribute-lft-neg-out N/A

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

      14. metadata-eval N/A

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

      15. *-lft-identity N/A

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

   5. Applied rewrites 99.5%

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

   if 100 < (/.f64 (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) #s(literal 2 binary64))

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 56.9

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

   5. Applied rewrites 56.9%

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

   6. Taylor expanded in x around inf

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 56.9

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

   8. Applied rewrites 56.9%

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

4. Final simplification 83.2%

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

Alternative 3: 78.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\varepsilon} - 1\\ t_1 := 1 + \frac{1}{\varepsilon}\\ t_2 := e^{-x}\\ \mathbf{if}\;\frac{t\_1 \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - t\_0 \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2} \leq 100:\\ \;\;\;\;\mathsf{fma}\left(t\_2, \left(1 + x\right) - -1, t\_2 \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 \cdot e^{x \cdot \varepsilon - x} - t\_0}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 eps) 1.0)) (t_1 (+ 1.0 (/ 1.0 eps))) (t_2 (exp (- x))))
   (if (<=
        (/
         (- (* t_1 (exp (* (+ -1.0 eps) x))) (* t_0 (exp (* (- -1.0 eps) x))))
         2.0)
        100.0)
     (* (fma t_2 (- (+ 1.0 x) -1.0) (* t_2 x)) 0.5)
     (/ (- (* t_1 (exp (- (* x eps) x))) t_0) 2.0))))

C

double code(double x, double eps) {
	double t_0 = (1.0 / eps) - 1.0;
	double t_1 = 1.0 + (1.0 / eps);
	double t_2 = exp(-x);
	double tmp;
	if ((((t_1 * exp(((-1.0 + eps) * x))) - (t_0 * exp(((-1.0 - eps) * x)))) / 2.0) <= 100.0) {
		tmp = fma(t_2, ((1.0 + x) - -1.0), (t_2 * x)) * 0.5;
	} else {
		tmp = ((t_1 * exp(((x * eps) - x))) - t_0) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(1.0 / eps) - 1.0)
	t_1 = Float64(1.0 + Float64(1.0 / eps))
	t_2 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(Float64(Float64(t_1 * exp(Float64(Float64(-1.0 + eps) * x))) - Float64(t_0 * exp(Float64(Float64(-1.0 - eps) * x)))) / 2.0) <= 100.0)
		tmp = Float64(fma(t_2, Float64(Float64(1.0 + x) - -1.0), Float64(t_2 * x)) * 0.5);
	else
		tmp = Float64(Float64(Float64(t_1 * exp(Float64(Float64(x * eps) - x))) - t_0) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$1 * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 100.0], N[(N[(t$95$2 * N[(N[(1.0 + x), $MachinePrecision] - -1.0), $MachinePrecision] + N[(t$95$2 * x), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(t$95$1 * N[Exp[N[(N[(x * eps), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) #s(literal 2 binary64)) < 100

   1. Initial program 48.5%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 45.0%

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

   6. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 99.4%

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

   if 100 < (/.f64 (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) #s(literal 2 binary64))

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 56.9

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

   5. Applied rewrites 56.9%

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

   6. Taylor expanded in x around inf

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 56.9

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

   8. Applied rewrites 56.9%

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

4. Final simplification 83.2%

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

Alternative 4: 78.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\varepsilon} - 1\\ t_1 := 1 + \frac{1}{\varepsilon}\\ \mathbf{if}\;\frac{t\_1 \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - t\_0 \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2} \leq 100:\\ \;\;\;\;\left(\left(\left(x + 2\right) + x\right) \cdot e^{-x}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 \cdot e^{x \cdot \varepsilon - x} - t\_0}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 eps) 1.0)) (t_1 (+ 1.0 (/ 1.0 eps))))
   (if (<=
        (/
         (- (* t_1 (exp (* (+ -1.0 eps) x))) (* t_0 (exp (* (- -1.0 eps) x))))
         2.0)
        100.0)
     (* (* (+ (+ x 2.0) x) (exp (- x))) 0.5)
     (/ (- (* t_1 (exp (- (* x eps) x))) t_0) 2.0))))

C

double code(double x, double eps) {
	double t_0 = (1.0 / eps) - 1.0;
	double t_1 = 1.0 + (1.0 / eps);
	double tmp;
	if ((((t_1 * exp(((-1.0 + eps) * x))) - (t_0 * exp(((-1.0 - eps) * x)))) / 2.0) <= 100.0) {
		tmp = (((x + 2.0) + x) * exp(-x)) * 0.5;
	} else {
		tmp = ((t_1 * exp(((x * eps) - x))) - t_0) / 2.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (1.0d0 / eps) - 1.0d0
    t_1 = 1.0d0 + (1.0d0 / eps)
    if ((((t_1 * exp((((-1.0d0) + eps) * x))) - (t_0 * exp((((-1.0d0) - eps) * x)))) / 2.0d0) <= 100.0d0) then
        tmp = (((x + 2.0d0) + x) * exp(-x)) * 0.5d0
    else
        tmp = ((t_1 * exp(((x * eps) - x))) - t_0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = (1.0 / eps) - 1.0;
	double t_1 = 1.0 + (1.0 / eps);
	double tmp;
	if ((((t_1 * Math.exp(((-1.0 + eps) * x))) - (t_0 * Math.exp(((-1.0 - eps) * x)))) / 2.0) <= 100.0) {
		tmp = (((x + 2.0) + x) * Math.exp(-x)) * 0.5;
	} else {
		tmp = ((t_1 * Math.exp(((x * eps) - x))) - t_0) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (1.0 / eps) - 1.0
	t_1 = 1.0 + (1.0 / eps)
	tmp = 0
	if (((t_1 * math.exp(((-1.0 + eps) * x))) - (t_0 * math.exp(((-1.0 - eps) * x)))) / 2.0) <= 100.0:
		tmp = (((x + 2.0) + x) * math.exp(-x)) * 0.5
	else:
		tmp = ((t_1 * math.exp(((x * eps) - x))) - t_0) / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(1.0 / eps) - 1.0)
	t_1 = Float64(1.0 + Float64(1.0 / eps))
	tmp = 0.0
	if (Float64(Float64(Float64(t_1 * exp(Float64(Float64(-1.0 + eps) * x))) - Float64(t_0 * exp(Float64(Float64(-1.0 - eps) * x)))) / 2.0) <= 100.0)
		tmp = Float64(Float64(Float64(Float64(x + 2.0) + x) * exp(Float64(-x))) * 0.5);
	else
		tmp = Float64(Float64(Float64(t_1 * exp(Float64(Float64(x * eps) - x))) - t_0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (1.0 / eps) - 1.0;
	t_1 = 1.0 + (1.0 / eps);
	tmp = 0.0;
	if ((((t_1 * exp(((-1.0 + eps) * x))) - (t_0 * exp(((-1.0 - eps) * x)))) / 2.0) <= 100.0)
		tmp = (((x + 2.0) + x) * exp(-x)) * 0.5;
	else
		tmp = ((t_1 * exp(((x * eps) - x))) - t_0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(t$95$1 * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 100.0], N[(N[(N[(N[(x + 2.0), $MachinePrecision] + x), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(t$95$1 * N[Exp[N[(N[(x * eps), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) #s(literal 2 binary64)) < 100

   1. Initial program 48.5%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 45.0%

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

   6. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 99.4%

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

   10. Applied rewrites 99.4%

       \[\leadsto \left(\left(\left(x + 2\right) + x\right) \cdot e^{-x}\right) \cdot 0.5 \]

   if 100 < (/.f64 (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) #s(literal 2 binary64))

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 56.9

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

   5. Applied rewrites 56.9%

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

   6. Taylor expanded in x around inf

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

      1. lower-exp.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 56.9

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

   8. Applied rewrites 56.9%

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

4. Final simplification 83.2%

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

Alternative 5: 78.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 48.2%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 44.6%

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

   6. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 100.0%

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

   10. Applied rewrites 100.0%

       \[\leadsto \left(\left(\left(x + 2\right) + x\right) \cdot e^{-x}\right) \cdot 0.5 \]

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

   1. Initial program 99.8%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 45.5

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

   8. Applied rewrites 45.5%

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

4. Final simplification 78.9%

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

Alternative 6: 78.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 48.2%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 44.6%

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

   6. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 100.0%

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

   10. Applied rewrites 100.0%

       \[\leadsto \left(\left(\left(x + 2\right) + x\right) \cdot e^{-x}\right) \cdot 0.5 \]

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

   1. Initial program 99.8%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 45.5

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

   8. Applied rewrites 45.5%

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

   9. Taylor expanded in eps around inf

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

       1. lower-*.f64 45.5

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

   11. Applied rewrites 45.5%

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

4. Final simplification 78.9%

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

Alternative 7: 66.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2} \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\left(\left(\left(x + 2\right) + x\right) \cdot e^{-x}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right) \cdot x - 1, x, 1\right)}{\varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (/
       (-
        (* (+ 1.0 (/ 1.0 eps)) (exp (* (+ -1.0 eps) x)))
        (* (- (/ 1.0 eps) 1.0) (exp (* (- -1.0 eps) x))))
       2.0)
      2e+275)
   (* (* (+ (+ x 2.0) x) (exp (- x))) 0.5)
   (/
    (-
     (- (/ 1.0 eps) -1.0)
     (/ (fma (- (* (fma -0.16666666666666666 x 0.5) x) 1.0) x 1.0) eps))
    2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (((((1.0 + (1.0 / eps)) * exp(((-1.0 + eps) * x))) - (((1.0 / eps) - 1.0) * exp(((-1.0 - eps) * x)))) / 2.0) <= 2e+275) {
		tmp = (((x + 2.0) + x) * exp(-x)) * 0.5;
	} else {
		tmp = (((1.0 / eps) - -1.0) - (fma(((fma(-0.16666666666666666, x, 0.5) * x) - 1.0), x, 1.0) / eps)) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(Float64(-1.0 + eps) * x))) - Float64(Float64(Float64(1.0 / eps) - 1.0) * exp(Float64(Float64(-1.0 - eps) * x)))) / 2.0) <= 2e+275)
		tmp = Float64(Float64(Float64(Float64(x + 2.0) + x) * exp(Float64(-x))) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / eps) - -1.0) - Float64(fma(Float64(Float64(fma(-0.16666666666666666, x, 0.5) * x) - 1.0), x, 1.0) / eps)) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 2e+275], N[(N[(N[(N[(x + 2.0), $MachinePrecision] + x), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(1.0 / eps), $MachinePrecision] - -1.0), $MachinePrecision] - N[(N[(N[(N[(N[(-0.16666666666666666 * x + 0.5), $MachinePrecision] * x), $MachinePrecision] - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) #s(literal 2 binary64)) < 1.99999999999999992e275

   1. Initial program 49.3%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 45.9%

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

   6. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 97.7%

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

   10. Applied rewrites 97.7%

       \[\leadsto \left(\left(\left(x + 2\right) + x\right) \cdot e^{-x}\right) \cdot 0.5 \]

   if 1.99999999999999992e275 < (/.f64 (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) #s(literal 2 binary64))

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 46.0

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

   8. Applied rewrites 46.0%

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

   9. Taylor expanded in eps around 0

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

       1. lower-/.f64 N/A

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

       2. lower-exp.f64 N/A

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

       3. lower-neg.f64 23.0

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

   11. Applied rewrites 23.0%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 30.9%

       \[\leadsto \frac{\left(\frac{1}{\varepsilon} + 1\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right) \cdot x - 1, x, 1\right)}{\varepsilon}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2} \leq 2 \cdot 10^{+275}:\\ \;\;\;\;\left(\left(\left(x + 2\right) + x\right) \cdot e^{-x}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right) \cdot x - 1, x, 1\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\varepsilon} - -1\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{+117}:\\ \;\;\;\;\frac{t\_0 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right) \cdot x - 1, x, 1\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 2.7:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, t\_0, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 eps) -1.0)))
   (if (<= x -3.2e+117)
     (/
      (-
       t_0
       (/ (fma (- (* (fma -0.16666666666666666 x 0.5) x) 1.0) x 1.0) eps))
      2.0)
     (if (<= x 2.7)
       (fma (* 0.5 x) (fma (- eps 1.0) t_0 (/ (- 1.0 (* eps eps)) eps)) 1.0)
       (/ x (exp x))))))

C

double code(double x, double eps) {
	double t_0 = (1.0 / eps) - -1.0;
	double tmp;
	if (x <= -3.2e+117) {
		tmp = (t_0 - (fma(((fma(-0.16666666666666666, x, 0.5) * x) - 1.0), x, 1.0) / eps)) / 2.0;
	} else if (x <= 2.7) {
		tmp = fma((0.5 * x), fma((eps - 1.0), t_0, ((1.0 - (eps * eps)) / eps)), 1.0);
	} else {
		tmp = x / exp(x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(1.0 / eps) - -1.0)
	tmp = 0.0
	if (x <= -3.2e+117)
		tmp = Float64(Float64(t_0 - Float64(fma(Float64(Float64(fma(-0.16666666666666666, x, 0.5) * x) - 1.0), x, 1.0) / eps)) / 2.0);
	elseif (x <= 2.7)
		tmp = fma(Float64(0.5 * x), fma(Float64(eps - 1.0), t_0, Float64(Float64(1.0 - Float64(eps * eps)) / eps)), 1.0);
	else
		tmp = Float64(x / exp(x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(1.0 / eps), $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[x, -3.2e+117], N[(N[(t$95$0 - N[(N[(N[(N[(N[(-0.16666666666666666 * x + 0.5), $MachinePrecision] * x), $MachinePrecision] - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.7], N[(N[(0.5 * x), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * t$95$0 + N[(N[(1.0 - N[(eps * eps), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(x / N[Exp[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.20000000000000005e117

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 38.4

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

   8. Applied rewrites 38.4%

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

   9. Taylor expanded in eps around 0

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

       1. lower-/.f64 N/A

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

       2. lower-exp.f64 N/A

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

       3. lower-neg.f64 63.6

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

   11. Applied rewrites 63.6%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 63.6%

       \[\leadsto \frac{\left(\frac{1}{\varepsilon} + 1\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right) \cdot x - 1, x, 1\right)}{\varepsilon}}{2} \]

   if -3.20000000000000005e117 < x < 2.7000000000000002

   1. Initial program 49.3%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 45.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   8. Applied rewrites 74.4%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 76.5%

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

   if 2.7000000000000002 < x

   1. Initial program 98.7%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 54.1%

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

   10. Applied rewrites 54.1%

       \[\leadsto \left(\left(\left(x + 2\right) + x\right) \cdot e^{-x}\right) \cdot 0.5 \]

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 53.0%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+117}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right) \cdot x - 1, x, 1\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 2.7:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, \frac{1}{\varepsilon} - -1, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\varepsilon} - -1\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{+117}:\\ \;\;\;\;\frac{t\_0 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right) \cdot x - 1, x, 1\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 2650000000000:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, t\_0, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 eps) -1.0)))
   (if (<= x -3.2e+117)
     (/
      (-
       t_0
       (/ (fma (- (* (fma -0.16666666666666666 x 0.5) x) 1.0) x 1.0) eps))
      2.0)
     (if (<= x 2650000000000.0)
       (fma (* 0.5 x) (fma (- eps 1.0) t_0 (/ (- 1.0 (* eps eps)) eps)) 1.0)
       (/ (- t_0 (- (/ 1.0 eps) 1.0)) 2.0)))))

C

double code(double x, double eps) {
	double t_0 = (1.0 / eps) - -1.0;
	double tmp;
	if (x <= -3.2e+117) {
		tmp = (t_0 - (fma(((fma(-0.16666666666666666, x, 0.5) * x) - 1.0), x, 1.0) / eps)) / 2.0;
	} else if (x <= 2650000000000.0) {
		tmp = fma((0.5 * x), fma((eps - 1.0), t_0, ((1.0 - (eps * eps)) / eps)), 1.0);
	} else {
		tmp = (t_0 - ((1.0 / eps) - 1.0)) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(1.0 / eps) - -1.0)
	tmp = 0.0
	if (x <= -3.2e+117)
		tmp = Float64(Float64(t_0 - Float64(fma(Float64(Float64(fma(-0.16666666666666666, x, 0.5) * x) - 1.0), x, 1.0) / eps)) / 2.0);
	elseif (x <= 2650000000000.0)
		tmp = fma(Float64(0.5 * x), fma(Float64(eps - 1.0), t_0, Float64(Float64(1.0 - Float64(eps * eps)) / eps)), 1.0);
	else
		tmp = Float64(Float64(t_0 - Float64(Float64(1.0 / eps) - 1.0)) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(1.0 / eps), $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[x, -3.2e+117], N[(N[(t$95$0 - N[(N[(N[(N[(N[(-0.16666666666666666 * x + 0.5), $MachinePrecision] * x), $MachinePrecision] - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2650000000000.0], N[(N[(0.5 * x), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * t$95$0 + N[(N[(1.0 - N[(eps * eps), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(t$95$0 - N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.20000000000000005e117

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 38.4

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

   8. Applied rewrites 38.4%

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

   9. Taylor expanded in eps around 0

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

       1. lower-/.f64 N/A

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

       2. lower-exp.f64 N/A

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

       3. lower-neg.f64 63.6

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

   11. Applied rewrites 63.6%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 63.6%

       \[\leadsto \frac{\left(\frac{1}{\varepsilon} + 1\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right) \cdot x - 1, x, 1\right)}{\varepsilon}}{2} \]

   if -3.20000000000000005e117 < x < 2.65e12

   1. Initial program 50.2%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 46.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   8. Applied rewrites 72.3%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 74.9%

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

   if 2.65e12 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 26.1

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

   5. Applied rewrites 26.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 52.9

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

   8. Applied rewrites 52.9%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+117}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right) \cdot x - 1, x, 1\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 2650000000000:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, \frac{1}{\varepsilon} - -1, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.5% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\varepsilon} - -1\\ \mathbf{if}\;x \leq -7.8 \cdot 10^{+162}:\\ \;\;\;\;\frac{t\_0 - \frac{\mathsf{fma}\left(0.5 \cdot x - 1, x, 1\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 2650000000000:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, t\_0, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 eps) -1.0)))
   (if (<= x -7.8e+162)
     (/ (- t_0 (/ (fma (- (* 0.5 x) 1.0) x 1.0) eps)) 2.0)
     (if (<= x 2650000000000.0)
       (fma (* 0.5 x) (fma (- eps 1.0) t_0 (/ (- 1.0 (* eps eps)) eps)) 1.0)
       (/ (- t_0 (- (/ 1.0 eps) 1.0)) 2.0)))))

C

double code(double x, double eps) {
	double t_0 = (1.0 / eps) - -1.0;
	double tmp;
	if (x <= -7.8e+162) {
		tmp = (t_0 - (fma(((0.5 * x) - 1.0), x, 1.0) / eps)) / 2.0;
	} else if (x <= 2650000000000.0) {
		tmp = fma((0.5 * x), fma((eps - 1.0), t_0, ((1.0 - (eps * eps)) / eps)), 1.0);
	} else {
		tmp = (t_0 - ((1.0 / eps) - 1.0)) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(1.0 / eps) - -1.0)
	tmp = 0.0
	if (x <= -7.8e+162)
		tmp = Float64(Float64(t_0 - Float64(fma(Float64(Float64(0.5 * x) - 1.0), x, 1.0) / eps)) / 2.0);
	elseif (x <= 2650000000000.0)
		tmp = fma(Float64(0.5 * x), fma(Float64(eps - 1.0), t_0, Float64(Float64(1.0 - Float64(eps * eps)) / eps)), 1.0);
	else
		tmp = Float64(Float64(t_0 - Float64(Float64(1.0 / eps) - 1.0)) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(1.0 / eps), $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[x, -7.8e+162], N[(N[(t$95$0 - N[(N[(N[(N[(0.5 * x), $MachinePrecision] - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2650000000000.0], N[(N[(0.5 * x), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * t$95$0 + N[(N[(1.0 - N[(eps * eps), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(t$95$0 - N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.80000000000000079e162

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 38.8

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

   8. Applied rewrites 38.8%

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

   9. Taylor expanded in eps around 0

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

       1. lower-/.f64 N/A

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

       2. lower-exp.f64 N/A

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

       3. lower-neg.f64 63.2

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

   11. Applied rewrites 63.2%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 63.2%

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

   if -7.80000000000000079e162 < x < 2.65e12

   1. Initial program 51.1%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 47.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   8. Applied rewrites 71.0%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 74.2%

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

   if 2.65e12 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 26.1

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

   5. Applied rewrites 26.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 52.9

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

   8. Applied rewrites 52.9%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{+162}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \frac{\mathsf{fma}\left(0.5 \cdot x - 1, x, 1\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 2650000000000:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, \frac{1}{\varepsilon} - -1, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{1}{\varepsilon} - -1 \cdot \mathsf{fma}\left(-1 - \varepsilon, x, 1\right)}{2}\\ \mathbf{elif}\;x \leq 20000:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -8e-18)
   (/ (- (/ 1.0 eps) (* -1.0 (fma (- -1.0 eps) x 1.0))) 2.0)
   (if (<= x 20000.0)
     (fma (- (* 0.3333333333333333 x) 0.5) (* x x) 1.0)
     (/ (- (- (/ 1.0 eps) -1.0) (- (/ 1.0 eps) 1.0)) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -8e-18) {
		tmp = ((1.0 / eps) - (-1.0 * fma((-1.0 - eps), x, 1.0))) / 2.0;
	} else if (x <= 20000.0) {
		tmp = fma(((0.3333333333333333 * x) - 0.5), (x * x), 1.0);
	} else {
		tmp = (((1.0 / eps) - -1.0) - ((1.0 / eps) - 1.0)) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -8e-18)
		tmp = Float64(Float64(Float64(1.0 / eps) - Float64(-1.0 * fma(Float64(-1.0 - eps), x, 1.0))) / 2.0);
	elseif (x <= 20000.0)
		tmp = fma(Float64(Float64(0.3333333333333333 * x) - 0.5), Float64(x * x), 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / eps) - -1.0) - Float64(Float64(1.0 / eps) - 1.0)) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -8e-18], N[(N[(N[(1.0 / eps), $MachinePrecision] - N[(-1.0 * N[(N[(-1.0 - eps), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 20000.0], N[(N[(N[(0.3333333333333333 * x), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(N[(1.0 / eps), $MachinePrecision] - -1.0), $MachinePrecision] - N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.0000000000000006e-18

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 47.6

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

   8. Applied rewrites 47.6%

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

   9. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. +-commutative N/A

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

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

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

       4. mul-1-neg N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. distribute-lft-in N/A

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

       8. metadata-eval N/A

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

       9. fp-cancel-sign-sub-inv N/A

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

       10. metadata-eval N/A

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

       11. *-lft-identity N/A

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

       12. lower--.f64 23.2

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

   11. Applied rewrites 23.2%

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

   12. Taylor expanded in eps around 0

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

   14. Applied rewrites 23.2%

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

   if -8.0000000000000006e-18 < x < 2e4

   1. Initial program 44.5%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 40.7%

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

   6. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 81.0%

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

   10. Applied rewrites 81.0%

       \[\leadsto \left(\left(\left(x + 2\right) + x\right) \cdot e^{-x}\right) \cdot 0.5 \]

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 80.5%

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

   if 2e4 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 25.5

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

   5. Applied rewrites 25.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 52.9

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

   8. Applied rewrites 52.9%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{1}{\varepsilon} - -1 \cdot \mathsf{fma}\left(-1 - \varepsilon, x, 1\right)}{2}\\ \mathbf{elif}\;x \leq 20000:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.5% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\varepsilon} - -1\\ \mathbf{if}\;x \leq 290:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, t\_0, \frac{1}{\varepsilon}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 eps) -1.0)))
   (if (<= x 290.0)
     (fma (* 0.5 x) (fma (- eps 1.0) t_0 (/ 1.0 eps)) 1.0)
     (/ (- t_0 (- (/ 1.0 eps) 1.0)) 2.0))))

C

double code(double x, double eps) {
	double t_0 = (1.0 / eps) - -1.0;
	double tmp;
	if (x <= 290.0) {
		tmp = fma((0.5 * x), fma((eps - 1.0), t_0, (1.0 / eps)), 1.0);
	} else {
		tmp = (t_0 - ((1.0 / eps) - 1.0)) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(1.0 / eps) - -1.0)
	tmp = 0.0
	if (x <= 290.0)
		tmp = fma(Float64(0.5 * x), fma(Float64(eps - 1.0), t_0, Float64(1.0 / eps)), 1.0);
	else
		tmp = Float64(Float64(t_0 - Float64(Float64(1.0 / eps) - 1.0)) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(1.0 / eps), $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[x, 290.0], N[(N[(0.5 * x), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * t$95$0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(t$95$0 - N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 290

   1. Initial program 55.2%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 52.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   8. Applied rewrites 65.5%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 71.0%

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

   if 290 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 26.2

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

   5. Applied rewrites 26.2%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 51.6

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

   8. Applied rewrites 51.6%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 290:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, \frac{1}{\varepsilon} - -1, \frac{1}{\varepsilon}\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.2% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{1}{\varepsilon} - -1 \cdot \mathsf{fma}\left(-1 - \varepsilon, x, 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -8e-18)
   (/ (- (/ 1.0 eps) (* -1.0 (fma (- -1.0 eps) x 1.0))) 2.0)
   (fma (- (* 0.3333333333333333 x) 0.5) (* x x) 1.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -8e-18) {
		tmp = ((1.0 / eps) - (-1.0 * fma((-1.0 - eps), x, 1.0))) / 2.0;
	} else {
		tmp = fma(((0.3333333333333333 * x) - 0.5), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -8e-18)
		tmp = Float64(Float64(Float64(1.0 / eps) - Float64(-1.0 * fma(Float64(-1.0 - eps), x, 1.0))) / 2.0);
	else
		tmp = fma(Float64(Float64(0.3333333333333333 * x) - 0.5), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -8e-18], N[(N[(N[(1.0 / eps), $MachinePrecision] - N[(-1.0 * N[(N[(-1.0 - eps), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(0.3333333333333333 * x), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.0000000000000006e-18

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 47.6

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

   8. Applied rewrites 47.6%

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

   9. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. +-commutative N/A

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

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

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

       4. mul-1-neg N/A

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

       5. *-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. distribute-lft-in N/A

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

       8. metadata-eval N/A

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

       9. fp-cancel-sign-sub-inv N/A

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

       10. metadata-eval N/A

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

       11. *-lft-identity N/A

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

       12. lower--.f64 23.2

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

   11. Applied rewrites 23.2%

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

   12. Taylor expanded in eps around 0

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

   14. Applied rewrites 23.2%

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

   if -8.0000000000000006e-18 < x

   1. Initial program 62.7%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 60.2%

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

   6. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 72.4%

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

   10. Applied rewrites 72.4%

       \[\leadsto \left(\left(\left(x + 2\right) + x\right) \cdot e^{-x}\right) \cdot 0.5 \]

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 65.9%

       \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, \color{blue}{x \cdot x}, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 53.1% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, x \cdot x, 1\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma (- (* 0.3333333333333333 x) 0.5) (* x x) 1.0))

C

double code(double x, double eps) {
	return fma(((0.3333333333333333 * x) - 0.5), (x * x), 1.0);
}

Julia

function code(x, eps)
	return fma(Float64(Float64(0.3333333333333333 * x) - 0.5), Float64(x * x), 1.0)
end

Wolfram

code[x_, eps_] := N[(N[(N[(0.3333333333333333 * x), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 68.1%

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

3. Taylor expanded in eps around inf

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

5. Applied rewrites 66.0%

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

6. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

8. Applied rewrites 62.0%

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

10. Applied rewrites 62.0%

    \[\leadsto \left(\left(\left(x + 2\right) + x\right) \cdot e^{-x}\right) \cdot 0.5 \]

11. Taylor expanded in x around 0

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

13. Applied rewrites 56.4%

    \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, \color{blue}{x \cdot x}, 1\right) \]
14. Add Preprocessing

Alternative 15: 44.1% accurate, 273.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 1.0)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, eps):
	return 1.0

Julia

function code(x, eps)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 1.0

Derivation

1. Initial program 68.1%

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

3. Taylor expanded in eps around inf

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

5. Applied rewrites 66.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 47.4%

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

Reproduce

herbie shell --seed 2025007 
(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))