NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.4% → 99.7%

Time: 8.9s

Alternatives: 16

Speedup: 1.8×

• 37.8% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -1.3445248722624216e+279
  2. eps = 3.916781233355594e-130

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, x, 2\right)}{e^{x}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0
         (/
          (-
           (* (+ 1.0 (/ 1.0 eps)) (exp (* (+ -1.0 eps) x)))
           (* (- (/ 1.0 eps) 1.0) (exp (* (- -1.0 eps) x))))
          2.0)))
   (if (<= t_0 0.0) (* (/ (fma 2.0 x 2.0) (exp x)) 0.5) t_0)))

C

double code(double x, double eps) {
	double t_0 = (((1.0 + (1.0 / eps)) * exp(((-1.0 + eps) * x))) - (((1.0 / eps) - 1.0) * exp(((-1.0 - eps) * x)))) / 2.0;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (fma(2.0, x, 2.0) / exp(x)) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

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

Wolfram

code[x_, eps_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, 0.0], N[(N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]

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)) < 0.0

   1. Initial program 35.1%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 34.8%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 10.0%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in x around 0

       \[\leadsto \frac{2 + 2 \cdot x}{e^{x}} \cdot \frac{1}{2} \]
   13. Step-by-step derivation

   14. Applied rewrites 100.0%

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

   if 0.0 < (/.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 98.6%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, x, 2\right)}{e^{x}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, x, 2\right)}{e^{x}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot e^{x \cdot \varepsilon} - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2}\\ \end{array} \end{array} \]

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)
      0.0)
   (* (/ (fma 2.0 x 2.0) (exp x)) 0.5)
   (/ (- (* 1.0 (exp (* x eps))) (/ -1.0 (exp (fma x 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) <= 0.0) {
		tmp = (fma(2.0, x, 2.0) / exp(x)) * 0.5;
	} else {
		tmp = ((1.0 * exp((x * eps))) - (-1.0 / exp(fma(x, 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) <= 0.0)
		tmp = Float64(Float64(fma(2.0, x, 2.0) / exp(x)) * 0.5);
	else
		tmp = Float64(Float64(Float64(1.0 * exp(Float64(x * eps))) - Float64(-1.0 / exp(fma(x, eps, x)))) / 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], 0.0], N[(N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(1.0 * N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(-1.0 / N[Exp[N[(x * 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)) < 0.0

   1. Initial program 35.1%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 34.8%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 10.0%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in x around 0

       \[\leadsto \frac{2 + 2 \cdot x}{e^{x}} \cdot \frac{1}{2} \]
   13. Step-by-step derivation

   14. Applied rewrites 100.0%

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

   if 0.0 < (/.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 98.6%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 95.7%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 95.7%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 98.6%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2} \leq 0:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, x, 2\right)}{e^{x}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot e^{x \cdot \varepsilon} - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2} \leq 0:\\ \;\;\;\;\frac{\left(x - -1\right) + \left(x - -1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (/
       (-
        (* (+ 1.0 (/ 1.0 eps)) (exp (* (+ -1.0 eps) x)))
        (* (- (/ 1.0 eps) 1.0) (exp (* (- -1.0 eps) x))))
       2.0)
      0.0)
   (* (/ (+ (- x -1.0) (- x -1.0)) (fma (fma 0.5 x 1.0) x 1.0)) 0.5)
   (fma (- (* 0.3333333333333333 x) 0.5) (* x x) 1.0)))

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) <= 0.0) {
		tmp = (((x - -1.0) + (x - -1.0)) / fma(fma(0.5, x, 1.0), x, 1.0)) * 0.5;
	} else {
		tmp = fma(((0.3333333333333333 * x) - 0.5), (x * x), 1.0);
	}
	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) <= 0.0)
		tmp = Float64(Float64(Float64(Float64(x - -1.0) + Float64(x - -1.0)) / fma(fma(0.5, x, 1.0), x, 1.0)) * 0.5);
	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[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], 0.0], N[(N[(N[(N[(x - -1.0), $MachinePrecision] + N[(x - -1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $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 (/.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)) < 0.0

   1. Initial program 35.1%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 34.8%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 10.0%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 82.4%

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

   if 0.0 < (/.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 98.6%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 95.7%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 95.7%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 31.8%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 37.9%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2} \leq 0:\\ \;\;\;\;\frac{\left(x - -1\right) + \left(x - -1\right)}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\varepsilon} - 1\\ t_1 := 1 + \frac{1}{\varepsilon}\\ \mathbf{if}\;\varepsilon \leq -1:\\ \;\;\;\;\frac{t\_1 \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - t\_0 \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, x, 2\right)}{e^{x}} \cdot 0.5\\ \mathbf{elif}\;\varepsilon \leq 1.35 \cdot 10^{+90}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1, e^{-x}, \frac{1 + x}{1}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - t\_0 \cdot \left(1 - \mathsf{fma}\left(x, \varepsilon, x\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 eps) 1.0)) (t_1 (+ 1.0 (/ 1.0 eps))))
   (if (<= eps -1.0)
     (/
      (- (* t_1 (fma (- eps 1.0) x 1.0)) (* t_0 (exp (* (- -1.0 eps) x))))
      2.0)
     (if (<= eps 1.0)
       (* (/ (fma 2.0 x 2.0) (exp x)) 0.5)
       (if (<= eps 1.35e+90)
         (/ (fma 1.0 (exp (- x)) (/ (+ 1.0 x) 1.0)) 2.0)
         (/
          (- (* t_1 (exp (* (+ -1.0 eps) x))) (* t_0 (- 1.0 (fma x eps x))))
          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 (eps <= -1.0) {
		tmp = ((t_1 * fma((eps - 1.0), x, 1.0)) - (t_0 * exp(((-1.0 - eps) * x)))) / 2.0;
	} else if (eps <= 1.0) {
		tmp = (fma(2.0, x, 2.0) / exp(x)) * 0.5;
	} else if (eps <= 1.35e+90) {
		tmp = fma(1.0, exp(-x), ((1.0 + x) / 1.0)) / 2.0;
	} else {
		tmp = ((t_1 * exp(((-1.0 + eps) * x))) - (t_0 * (1.0 - fma(x, eps, x)))) / 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 (eps <= -1.0)
		tmp = Float64(Float64(Float64(t_1 * fma(Float64(eps - 1.0), x, 1.0)) - Float64(t_0 * exp(Float64(Float64(-1.0 - eps) * x)))) / 2.0);
	elseif (eps <= 1.0)
		tmp = Float64(Float64(fma(2.0, x, 2.0) / exp(x)) * 0.5);
	elseif (eps <= 1.35e+90)
		tmp = Float64(fma(1.0, exp(Float64(-x)), Float64(Float64(1.0 + x) / 1.0)) / 2.0);
	else
		tmp = Float64(Float64(Float64(t_1 * exp(Float64(Float64(-1.0 + eps) * x))) - Float64(t_0 * Float64(1.0 - fma(x, eps, x)))) / 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[eps, -1.0], N[(N[(N[(t$95$1 * N[(N[(eps - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps, 1.0], N[(N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[eps, 1.35e+90], N[(N[(1.0 * N[Exp[(-x)], $MachinePrecision] + N[(N[(1.0 + x), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(t$95$1 * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[(1.0 - N[(x * eps + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if eps < -1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 70.8%

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

   if -1 < eps < 1

   1. Initial program 36.3%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 33.2%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 9.6%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in x around 0

       \[\leadsto \frac{2 + 2 \cdot x}{e^{x}} \cdot \frac{1}{2} \]
   13. Step-by-step derivation

   14. Applied rewrites 100.0%

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

   if 1 < eps < 1.35e90

   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 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

   5. Applied rewrites 50.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 51.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 85.8%

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

   if 1.35e90 < eps

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 68.3%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, x, 2\right)}{e^{x}} \cdot 0.5\\ \mathbf{elif}\;\varepsilon \leq 1.35 \cdot 10^{+90}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1, e^{-x}, \frac{1 + x}{1}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(1 - \mathsf{fma}\left(x, \varepsilon, x\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{x \cdot \varepsilon}\\ t_1 := 1 + \frac{1}{\varepsilon}\\ t_2 := \frac{1}{\varepsilon} - 1\\ \mathbf{if}\;x \leq -700:\\ \;\;\;\;\frac{t\_1 \cdot e^{-x} - t\_2}{2}\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;\frac{1 \cdot t\_0 - \frac{-1}{1}}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+133}:\\ \;\;\;\;\frac{t\_1 \cdot t\_0 - t\_2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (* x eps)))
        (t_1 (+ 1.0 (/ 1.0 eps)))
        (t_2 (- (/ 1.0 eps) 1.0)))
   (if (<= x -700.0)
     (/ (- (* t_1 (exp (- x))) t_2) 2.0)
     (if (<= x 1.42)
       (/ (- (* 1.0 t_0) (/ -1.0 1.0)) 2.0)
       (if (<= x 2e+133) (/ (- (* t_1 t_0) t_2) 2.0) (/ x (exp x)))))))

C

double code(double x, double eps) {
	double t_0 = exp((x * eps));
	double t_1 = 1.0 + (1.0 / eps);
	double t_2 = (1.0 / eps) - 1.0;
	double tmp;
	if (x <= -700.0) {
		tmp = ((t_1 * exp(-x)) - t_2) / 2.0;
	} else if (x <= 1.42) {
		tmp = ((1.0 * t_0) - (-1.0 / 1.0)) / 2.0;
	} else if (x <= 2e+133) {
		tmp = ((t_1 * t_0) - t_2) / 2.0;
	} else {
		tmp = x / exp(x);
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = exp((x * eps))
    t_1 = 1.0d0 + (1.0d0 / eps)
    t_2 = (1.0d0 / eps) - 1.0d0
    if (x <= (-700.0d0)) then
        tmp = ((t_1 * exp(-x)) - t_2) / 2.0d0
    else if (x <= 1.42d0) then
        tmp = ((1.0d0 * t_0) - ((-1.0d0) / 1.0d0)) / 2.0d0
    else if (x <= 2d+133) then
        tmp = ((t_1 * t_0) - t_2) / 2.0d0
    else
        tmp = x / exp(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.exp((x * eps));
	double t_1 = 1.0 + (1.0 / eps);
	double t_2 = (1.0 / eps) - 1.0;
	double tmp;
	if (x <= -700.0) {
		tmp = ((t_1 * Math.exp(-x)) - t_2) / 2.0;
	} else if (x <= 1.42) {
		tmp = ((1.0 * t_0) - (-1.0 / 1.0)) / 2.0;
	} else if (x <= 2e+133) {
		tmp = ((t_1 * t_0) - t_2) / 2.0;
	} else {
		tmp = x / Math.exp(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.exp((x * eps))
	t_1 = 1.0 + (1.0 / eps)
	t_2 = (1.0 / eps) - 1.0
	tmp = 0
	if x <= -700.0:
		tmp = ((t_1 * math.exp(-x)) - t_2) / 2.0
	elif x <= 1.42:
		tmp = ((1.0 * t_0) - (-1.0 / 1.0)) / 2.0
	elif x <= 2e+133:
		tmp = ((t_1 * t_0) - t_2) / 2.0
	else:
		tmp = x / math.exp(x)
	return tmp

Julia

function code(x, eps)
	t_0 = exp(Float64(x * eps))
	t_1 = Float64(1.0 + Float64(1.0 / eps))
	t_2 = Float64(Float64(1.0 / eps) - 1.0)
	tmp = 0.0
	if (x <= -700.0)
		tmp = Float64(Float64(Float64(t_1 * exp(Float64(-x))) - t_2) / 2.0);
	elseif (x <= 1.42)
		tmp = Float64(Float64(Float64(1.0 * t_0) - Float64(-1.0 / 1.0)) / 2.0);
	elseif (x <= 2e+133)
		tmp = Float64(Float64(Float64(t_1 * t_0) - t_2) / 2.0);
	else
		tmp = Float64(x / exp(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = exp((x * eps));
	t_1 = 1.0 + (1.0 / eps);
	t_2 = (1.0 / eps) - 1.0;
	tmp = 0.0;
	if (x <= -700.0)
		tmp = ((t_1 * exp(-x)) - t_2) / 2.0;
	elseif (x <= 1.42)
		tmp = ((1.0 * t_0) - (-1.0 / 1.0)) / 2.0;
	elseif (x <= 2e+133)
		tmp = ((t_1 * t_0) - t_2) / 2.0;
	else
		tmp = x / exp(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -700.0], N[(N[(N[(t$95$1 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.42], N[(N[(N[(1.0 * t$95$0), $MachinePrecision] - N[(-1.0 / 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2e+133], N[(N[(N[(t$95$1 * t$95$0), $MachinePrecision] - t$95$2), $MachinePrecision] / 2.0), $MachinePrecision], N[(x / N[Exp[x], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -700

   1. Initial program 95.2%

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

   3. Taylor expanded in 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

   5. Applied rewrites 51.5%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 97.6%

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

   if -700 < x < 1.4199999999999999

   1. Initial program 56.5%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 53.4%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 53.4%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 96.8%

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

   12. Taylor expanded in x around 0

       \[\leadsto \frac{1 \cdot e^{x \cdot \varepsilon} - \frac{-1}{1}}{2} \]
   13. Step-by-step derivation

   14. Applied rewrites 85.2%

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

   if 1.4199999999999999 < x < 2e133

   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

   5. Applied rewrites 30.0%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 60.6%

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

   if 2e133 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 50.9%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 75.4%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 75.4%

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

Alternative 6: 79.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1:\\ \;\;\;\;\frac{1 \cdot e^{x \cdot \varepsilon} - \frac{-1}{1}}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.15 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, x, 2\right)}{e^{x}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -1.0)
   (/ (- (* 1.0 (exp (* x eps))) (/ -1.0 1.0)) 2.0)
   (if (<= eps 2.15e+15)
     (* (/ (fma 2.0 x 2.0) (exp x)) 0.5)
     (/ (- (- (/ 1.0 eps) -1.0) (/ -1.0 (exp (fma x eps x)))) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -1.0) {
		tmp = ((1.0 * exp((x * eps))) - (-1.0 / 1.0)) / 2.0;
	} else if (eps <= 2.15e+15) {
		tmp = (fma(2.0, x, 2.0) / exp(x)) * 0.5;
	} else {
		tmp = (((1.0 / eps) - -1.0) - (-1.0 / exp(fma(x, eps, x)))) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -1.0)
		tmp = Float64(Float64(Float64(1.0 * exp(Float64(x * eps))) - Float64(-1.0 / 1.0)) / 2.0);
	elseif (eps <= 2.15e+15)
		tmp = Float64(Float64(fma(2.0, x, 2.0) / exp(x)) * 0.5);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / eps) - -1.0) - Float64(-1.0 / exp(fma(x, eps, x)))) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -1.0], N[(N[(N[(1.0 * N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(-1.0 / 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps, 2.15e+15], N[(N[(N[(2.0 * x + 2.0), $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[(x * eps + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1

   1. Initial program 99.9%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 99.9%

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

   12. Taylor expanded in x around 0

       \[\leadsto \frac{1 \cdot e^{x \cdot \varepsilon} - \frac{-1}{1}}{2} \]
   13. Step-by-step derivation

   14. Applied rewrites 69.9%

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

   if -1 < eps < 2.15e15

   1. Initial program 38.5%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 34.4%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 11.6%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in x around 0

       \[\leadsto \frac{2 + 2 \cdot x}{e^{x}} \cdot \frac{1}{2} \]
   13. Step-by-step derivation

   14. Applied rewrites 100.0%

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

   if 2.15e15 < eps

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 61.7%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1:\\ \;\;\;\;\frac{1 \cdot e^{x \cdot \varepsilon} - \frac{-1}{1}}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.15 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, x, 2\right)}{e^{x}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -700:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-x} - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \mathbf{elif}\;x \leq 750:\\ \;\;\;\;\frac{1 \cdot e^{x \cdot \varepsilon} - \frac{-1}{1}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -700.0)
   (/ (- (* (+ 1.0 (/ 1.0 eps)) (exp (- x))) (- (/ 1.0 eps) 1.0)) 2.0)
   (if (<= x 750.0)
     (/ (- (* 1.0 (exp (* x eps))) (/ -1.0 1.0)) 2.0)
     (/ x (exp x)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -700.0) {
		tmp = (((1.0 + (1.0 / eps)) * exp(-x)) - ((1.0 / eps) - 1.0)) / 2.0;
	} else if (x <= 750.0) {
		tmp = ((1.0 * exp((x * eps))) - (-1.0 / 1.0)) / 2.0;
	} else {
		tmp = x / exp(x);
	}
	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 (x <= (-700.0d0)) then
        tmp = (((1.0d0 + (1.0d0 / eps)) * exp(-x)) - ((1.0d0 / eps) - 1.0d0)) / 2.0d0
    else if (x <= 750.0d0) then
        tmp = ((1.0d0 * exp((x * eps))) - ((-1.0d0) / 1.0d0)) / 2.0d0
    else
        tmp = x / exp(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -700.0) {
		tmp = (((1.0 + (1.0 / eps)) * Math.exp(-x)) - ((1.0 / eps) - 1.0)) / 2.0;
	} else if (x <= 750.0) {
		tmp = ((1.0 * Math.exp((x * eps))) - (-1.0 / 1.0)) / 2.0;
	} else {
		tmp = x / Math.exp(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -700.0:
		tmp = (((1.0 + (1.0 / eps)) * math.exp(-x)) - ((1.0 / eps) - 1.0)) / 2.0
	elif x <= 750.0:
		tmp = ((1.0 * math.exp((x * eps))) - (-1.0 / 1.0)) / 2.0
	else:
		tmp = x / math.exp(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -700.0)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(-x))) - Float64(Float64(1.0 / eps) - 1.0)) / 2.0);
	elseif (x <= 750.0)
		tmp = Float64(Float64(Float64(1.0 * exp(Float64(x * eps))) - Float64(-1.0 / 1.0)) / 2.0);
	else
		tmp = Float64(x / exp(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -700.0)
		tmp = (((1.0 + (1.0 / eps)) * exp(-x)) - ((1.0 / eps) - 1.0)) / 2.0;
	elseif (x <= 750.0)
		tmp = ((1.0 * exp((x * eps))) - (-1.0 / 1.0)) / 2.0;
	else
		tmp = x / exp(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -700.0], N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 750.0], N[(N[(N[(1.0 * N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(-1.0 / 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(x / N[Exp[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -700

   1. Initial program 95.2%

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

   3. Taylor expanded in 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

   5. Applied rewrites 51.5%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 97.6%

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

   if -700 < x < 750

   1. Initial program 57.1%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 54.0%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 54.0%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 96.9%

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

   12. Taylor expanded in x around 0

       \[\leadsto \frac{1 \cdot e^{x \cdot \varepsilon} - \frac{-1}{1}}{2} \]
   13. Step-by-step derivation

   14. Applied rewrites 84.7%

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

   if 750 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 56.1%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 59.3%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 59.3%

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

Alternative 8: 78.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -700:\\ \;\;\;\;\frac{2}{e^{x}} \cdot 0.5\\ \mathbf{elif}\;x \leq 750:\\ \;\;\;\;\frac{1 \cdot e^{x \cdot \varepsilon} - \frac{-1}{1}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -700.0)
   (* (/ 2.0 (exp x)) 0.5)
   (if (<= x 750.0)
     (/ (- (* 1.0 (exp (* x eps))) (/ -1.0 1.0)) 2.0)
     (/ x (exp x)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -700.0) {
		tmp = (2.0 / exp(x)) * 0.5;
	} else if (x <= 750.0) {
		tmp = ((1.0 * exp((x * eps))) - (-1.0 / 1.0)) / 2.0;
	} else {
		tmp = x / exp(x);
	}
	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 (x <= (-700.0d0)) then
        tmp = (2.0d0 / exp(x)) * 0.5d0
    else if (x <= 750.0d0) then
        tmp = ((1.0d0 * exp((x * eps))) - ((-1.0d0) / 1.0d0)) / 2.0d0
    else
        tmp = x / exp(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -700.0) {
		tmp = (2.0 / Math.exp(x)) * 0.5;
	} else if (x <= 750.0) {
		tmp = ((1.0 * Math.exp((x * eps))) - (-1.0 / 1.0)) / 2.0;
	} else {
		tmp = x / Math.exp(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -700.0:
		tmp = (2.0 / math.exp(x)) * 0.5
	elif x <= 750.0:
		tmp = ((1.0 * math.exp((x * eps))) - (-1.0 / 1.0)) / 2.0
	else:
		tmp = x / math.exp(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -700.0)
		tmp = Float64(Float64(2.0 / exp(x)) * 0.5);
	elseif (x <= 750.0)
		tmp = Float64(Float64(Float64(1.0 * exp(Float64(x * eps))) - Float64(-1.0 / 1.0)) / 2.0);
	else
		tmp = Float64(x / exp(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -700.0)
		tmp = (2.0 / exp(x)) * 0.5;
	elseif (x <= 750.0)
		tmp = ((1.0 * exp((x * eps))) - (-1.0 / 1.0)) / 2.0;
	else
		tmp = x / exp(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -700.0], N[(N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 750.0], N[(N[(N[(1.0 * N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(-1.0 / 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(x / N[Exp[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -700

   1. Initial program 95.2%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 95.2%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 95.2%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 4.8%

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

   12. Taylor expanded in x around 0

       \[\leadsto \frac{2}{e^{x}} \cdot \frac{1}{2} \]
   13. Step-by-step derivation

   14. Applied rewrites 95.2%

       \[\leadsto \frac{2}{e^{x}} \cdot 0.5 \]

   if -700 < x < 750

   1. Initial program 57.1%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 54.0%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 54.0%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 96.9%

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

   12. Taylor expanded in x around 0

       \[\leadsto \frac{1 \cdot e^{x \cdot \varepsilon} - \frac{-1}{1}}{2} \]
   13. Step-by-step derivation

   14. Applied rewrites 84.7%

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

   if 750 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 56.1%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 59.3%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 59.3%

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

Alternative 9: 72.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2000000:\\ \;\;\;\;\frac{2}{e^{x}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, x, 2\right)}{e^{x}} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2000000.0)
   (* (/ 2.0 (exp x)) 0.5)
   (* (/ (fma 2.0 x 2.0) (exp x)) 0.5)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2000000.0) {
		tmp = (2.0 / exp(x)) * 0.5;
	} else {
		tmp = (fma(2.0, x, 2.0) / exp(x)) * 0.5;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2000000.0)
		tmp = Float64(Float64(2.0 / exp(x)) * 0.5);
	else
		tmp = Float64(Float64(fma(2.0, x, 2.0) / exp(x)) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2000000.0], N[(N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2e6

   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

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 0.0%

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

   12. Taylor expanded in x around 0

       \[\leadsto \frac{2}{e^{x}} \cdot \frac{1}{2} \]
   13. Step-by-step derivation

   14. Applied rewrites 100.0%

       \[\leadsto \frac{2}{e^{x}} \cdot 0.5 \]

   if -2e6 < x

   1. Initial program 68.2%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 66.0%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 54.3%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 69.8%

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

   12. Taylor expanded in x around 0

       \[\leadsto \frac{2 + 2 \cdot x}{e^{x}} \cdot \frac{1}{2} \]
   13. Step-by-step derivation

   14. Applied rewrites 69.8%

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

Alternative 10: 61.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\varepsilon} - 1\\ \mathbf{if}\;x \leq -200:\\ \;\;\;\;\frac{\frac{1 + \varepsilon}{\varepsilon} - \mathsf{fma}\left(-\mathsf{fma}\left(x, \varepsilon, x\right), t\_0, t\_0\right)}{2}\\ \mathbf{elif}\;x \leq 1.55:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.3333333333333333\right) \cdot x - 0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 eps) 1.0)))
   (if (<= x -200.0)
     (/ (- (/ (+ 1.0 eps) eps) (fma (- (fma x eps x)) t_0 t_0)) 2.0)
     (if (<= x 1.55)
       (fma (- (* (fma -0.125 x 0.3333333333333333) x) 0.5) (* x x) 1.0)
       (/ x (exp x))))))

C

double code(double x, double eps) {
	double t_0 = (1.0 / eps) - 1.0;
	double tmp;
	if (x <= -200.0) {
		tmp = (((1.0 + eps) / eps) - fma(-fma(x, eps, x), t_0, t_0)) / 2.0;
	} else if (x <= 1.55) {
		tmp = fma(((fma(-0.125, x, 0.3333333333333333) * x) - 0.5), (x * x), 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 <= -200.0)
		tmp = Float64(Float64(Float64(Float64(1.0 + eps) / eps) - fma(Float64(-fma(x, eps, x)), t_0, t_0)) / 2.0);
	elseif (x <= 1.55)
		tmp = fma(Float64(Float64(fma(-0.125, x, 0.3333333333333333) * x) - 0.5), Float64(x * x), 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, -200.0], N[(N[(N[(N[(1.0 + eps), $MachinePrecision] / eps), $MachinePrecision] - N[((-N[(x * eps + x), $MachinePrecision]) * t$95$0 + t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.55], N[(N[(N[(N[(-0.125 * x + 0.3333333333333333), $MachinePrecision] * x), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(x / N[Exp[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -200

   1. Initial program 95.2%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 95.2%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 95.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 49.1%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 20.7%

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

   if -200 < x < 1.55000000000000004

   1. Initial program 56.8%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 53.7%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 53.7%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 74.3%

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

   12. Taylor expanded in x around 0

       \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(x \cdot \left(\frac{1}{3} + \frac{-1}{8} \cdot x\right) - \frac{1}{2}\right)} \]
   13. Step-by-step derivation

   14. Applied rewrites 73.5%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.3333333333333333\right) \cdot x - 0.5, \color{blue}{x \cdot x}, 1\right) \]

   if 1.55000000000000004 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 56.8%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 58.3%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 58.3%

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

Alternative 11: 71.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{2}{e^{x}} \cdot 0.5 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* (/ 2.0 (exp x)) 0.5))

C

double code(double x, double eps) {
	return (2.0 / exp(x)) * 0.5;
}

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 = (2.0d0 / exp(x)) * 0.5d0
end function

Java

public static double code(double x, double eps) {
	return (2.0 / Math.exp(x)) * 0.5;
}

Python

def code(x, eps):
	return (2.0 / math.exp(x)) * 0.5

Julia

function code(x, eps)
	return Float64(Float64(2.0 / exp(x)) * 0.5)
end

MATLAB

function tmp = code(x, eps)
	tmp = (2.0 / exp(x)) * 0.5;
end

Wolfram

code[x_, eps_] := N[(N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 73.1%

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

3. Taylor expanded in eps around inf

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

5. Applied rewrites 71.2%

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

6. Taylor expanded in eps around inf

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

8. Applied rewrites 61.2%

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

9. Taylor expanded in eps around 0

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

11. Applied rewrites 59.2%

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

12. Taylor expanded in x around 0

    \[\leadsto \frac{2}{e^{x}} \cdot \frac{1}{2} \]
13. Step-by-step derivation

14. Applied rewrites 72.1%

    \[\leadsto \frac{2}{e^{x}} \cdot 0.5 \]
15. Add Preprocessing

Alternative 12: 61.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\varepsilon} - 1\\ \mathbf{if}\;x \leq -200:\\ \;\;\;\;\frac{\frac{1 + \varepsilon}{\varepsilon} - \mathsf{fma}\left(-\mathsf{fma}\left(x, \varepsilon, x\right), t\_0, t\_0\right)}{2}\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.3333333333333333\right) \cdot x - 0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - t\_0}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 eps) 1.0)))
   (if (<= x -200.0)
     (/ (- (/ (+ 1.0 eps) eps) (fma (- (fma x eps x)) t_0 t_0)) 2.0)
     (if (<= x 1.42)
       (fma (- (* (fma -0.125 x 0.3333333333333333) x) 0.5) (* x x) 1.0)
       (/ (- (- (/ 1.0 eps) -1.0) t_0) 2.0)))))

C

double code(double x, double eps) {
	double t_0 = (1.0 / eps) - 1.0;
	double tmp;
	if (x <= -200.0) {
		tmp = (((1.0 + eps) / eps) - fma(-fma(x, eps, x), t_0, t_0)) / 2.0;
	} else if (x <= 1.42) {
		tmp = fma(((fma(-0.125, x, 0.3333333333333333) * x) - 0.5), (x * x), 1.0);
	} else {
		tmp = (((1.0 / eps) - -1.0) - t_0) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(1.0 / eps) - 1.0)
	tmp = 0.0
	if (x <= -200.0)
		tmp = Float64(Float64(Float64(Float64(1.0 + eps) / eps) - fma(Float64(-fma(x, eps, x)), t_0, t_0)) / 2.0);
	elseif (x <= 1.42)
		tmp = fma(Float64(Float64(fma(-0.125, x, 0.3333333333333333) * x) - 0.5), Float64(x * x), 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / eps) - -1.0) - 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]}, If[LessEqual[x, -200.0], N[(N[(N[(N[(1.0 + eps), $MachinePrecision] / eps), $MachinePrecision] - N[((-N[(x * eps + x), $MachinePrecision]) * t$95$0 + t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.42], N[(N[(N[(N[(-0.125 * x + 0.3333333333333333), $MachinePrecision] * x), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(N[(1.0 / eps), $MachinePrecision] - -1.0), $MachinePrecision] - t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -200

   1. Initial program 95.2%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 95.2%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 95.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 49.1%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 20.7%

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

   if -200 < x < 1.4199999999999999

   1. Initial program 56.5%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 53.4%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 53.4%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 74.8%

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

   12. Taylor expanded in x around 0

       \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(x \cdot \left(\frac{1}{3} + \frac{-1}{8} \cdot x\right) - \frac{1}{2}\right)} \]
   13. Step-by-step derivation

   14. Applied rewrites 74.0%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.3333333333333333\right) \cdot x - 0.5, \color{blue}{x \cdot x}, 1\right) \]

   if 1.4199999999999999 < 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

   5. Applied rewrites 23.6%

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

   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

   8. Applied rewrites 56.4%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -200:\\ \;\;\;\;\frac{\frac{1 + \varepsilon}{\varepsilon} - \mathsf{fma}\left(-\mathsf{fma}\left(x, \varepsilon, x\right), \frac{1}{\varepsilon} - 1, \frac{1}{\varepsilon} - 1\right)}{2}\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.3333333333333333\right) \cdot x - 0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.0% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\varepsilon} - 1\\ \mathbf{if}\;x \leq -3700:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - t\_0}{2}\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.3333333333333333\right) \cdot x - 0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - t\_0}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 eps) 1.0)))
   (if (<= x -3700.0)
     (/ (- (* (+ 1.0 (/ 1.0 eps)) (fma (- eps 1.0) x 1.0)) t_0) 2.0)
     (if (<= x 1.42)
       (fma (- (* (fma -0.125 x 0.3333333333333333) x) 0.5) (* x x) 1.0)
       (/ (- (- (/ 1.0 eps) -1.0) t_0) 2.0)))))

C

double code(double x, double eps) {
	double t_0 = (1.0 / eps) - 1.0;
	double tmp;
	if (x <= -3700.0) {
		tmp = (((1.0 + (1.0 / eps)) * fma((eps - 1.0), x, 1.0)) - t_0) / 2.0;
	} else if (x <= 1.42) {
		tmp = fma(((fma(-0.125, x, 0.3333333333333333) * x) - 0.5), (x * x), 1.0);
	} else {
		tmp = (((1.0 / eps) - -1.0) - t_0) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(1.0 / eps) - 1.0)
	tmp = 0.0
	if (x <= -3700.0)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * fma(Float64(eps - 1.0), x, 1.0)) - t_0) / 2.0);
	elseif (x <= 1.42)
		tmp = fma(Float64(Float64(fma(-0.125, x, 0.3333333333333333) * x) - 0.5), Float64(x * x), 1.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / eps) - -1.0) - 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]}, If[LessEqual[x, -3700.0], N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.42], N[(N[(N[(N[(-0.125 * x + 0.3333333333333333), $MachinePrecision] * x), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(N[(1.0 / eps), $MachinePrecision] - -1.0), $MachinePrecision] - t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3700

   1. Initial program 97.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 47.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 19.8%

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

   if -3700 < x < 1.4199999999999999

   1. Initial program 56.5%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 53.4%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 53.4%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 74.4%

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

   12. Taylor expanded in x around 0

       \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(x \cdot \left(\frac{1}{3} + \frac{-1}{8} \cdot x\right) - \frac{1}{2}\right)} \]
   13. Step-by-step derivation

   14. Applied rewrites 73.1%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.3333333333333333\right) \cdot x - 0.5, \color{blue}{x \cdot x}, 1\right) \]

   if 1.4199999999999999 < 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

   5. Applied rewrites 23.6%

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

   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

   8. Applied rewrites 56.4%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3700:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.3333333333333333\right) \cdot x - 0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.1% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.42:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.3333333333333333\right) \cdot x - 0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 1.42)
   (fma (- (* (fma -0.125 x 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 <= 1.42) {
		tmp = fma(((fma(-0.125, x, 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 <= 1.42)
		tmp = fma(Float64(Float64(fma(-0.125, x, 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, 1.42], N[(N[(N[(N[(-0.125 * x + 0.3333333333333333), $MachinePrecision] * 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 2 regimes

2. if x < 1.4199999999999999

   1. Initial program 64.8%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 62.4%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 62.4%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 59.8%

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

   12. Taylor expanded in x around 0

       \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(x \cdot \left(\frac{1}{3} + \frac{-1}{8} \cdot x\right) - \frac{1}{2}\right)} \]
   13. Step-by-step derivation

   14. Applied rewrites 58.2%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.3333333333333333\right) \cdot x - 0.5, \color{blue}{x \cdot x}, 1\right) \]

   if 1.4199999999999999 < 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

   5. Applied rewrites 23.6%

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

   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

   8. Applied rewrites 56.4%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.42:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.3333333333333333\right) \cdot x - 0.5, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 52.4% 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 73.1%

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

3. Taylor expanded in eps around inf

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

5. Applied rewrites 71.2%

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

6. Taylor expanded in eps around inf

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

8. Applied rewrites 61.2%

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

9. Taylor expanded in eps around 0

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

11. Applied rewrites 59.2%

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

12. Taylor expanded in x around 0

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

14. Applied rewrites 49.3%

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

Alternative 16: 44.2% accurate, 273.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 1.0)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, eps):
	return 1.0

Julia

function code(x, eps)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 1.0

Derivation

1. Initial program 73.1%

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

3. Taylor expanded in eps around inf

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

5. Applied rewrites 71.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 45.1%

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

Reproduce

herbie shell --seed 2025019 
(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))