NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.0% → 99.6%

Time: 7.6s

Alternatives: 12

Speedup: 1.6×

• 37.4% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -1.1052865409304529e+31
  2. eps = 6.094196227351208e-117

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: 73.0% 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: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0], N[(N[(N[(x - -2.0), $MachinePrecision] * t$95$0 + N[(t$95$0 * 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[(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 41.9%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 42.0%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 98.1%

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

   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%

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

   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 100.0%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in eps around inf

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

   14. Applied rewrites 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 79.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\varepsilon \leq -1.45 \cdot 10^{+158}:\\ \;\;\;\;\frac{1 \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - -1 \cdot e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;\varepsilon \leq -1 \lor \neg \left(\varepsilon \leq 1\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(1 - \mathsf{fma}\left(x, \varepsilon, x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - -2, t\_0, t\_0 \cdot x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= eps -1.45e+158)
     (/ (- (* 1.0 (fma (- eps 1.0) x 1.0)) (* -1.0 (exp (* eps (- x))))) 2.0)
     (if (or (<= eps -1.0) (not (<= eps 1.0)))
       (/
        (-
         (* (+ 1.0 (/ 1.0 eps)) (exp (* (+ -1.0 eps) x)))
         (* (- (/ 1.0 eps) 1.0) (- 1.0 (fma x eps x))))
        2.0)
       (* (fma (- x -2.0) t_0 (* t_0 x)) 0.5)))))

C

double code(double x, double eps) {
	double t_0 = exp(-x);
	double tmp;
	if (eps <= -1.45e+158) {
		tmp = ((1.0 * fma((eps - 1.0), x, 1.0)) - (-1.0 * exp((eps * -x)))) / 2.0;
	} else if ((eps <= -1.0) || !(eps <= 1.0)) {
		tmp = (((1.0 + (1.0 / eps)) * exp(((-1.0 + eps) * x))) - (((1.0 / eps) - 1.0) * (1.0 - fma(x, eps, x)))) / 2.0;
	} else {
		tmp = fma((x - -2.0), t_0, (t_0 * x)) * 0.5;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (eps <= -1.45e+158)
		tmp = Float64(Float64(Float64(1.0 * fma(Float64(eps - 1.0), x, 1.0)) - Float64(-1.0 * exp(Float64(eps * Float64(-x))))) / 2.0);
	elseif ((eps <= -1.0) || !(eps <= 1.0))
		tmp = 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) * Float64(1.0 - fma(x, eps, x)))) / 2.0);
	else
		tmp = Float64(fma(Float64(x - -2.0), t_0, Float64(t_0 * x)) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[eps, -1.45e+158], N[(N[(N[(1.0 * N[(N[(eps - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[Exp[N[(eps * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[eps, -1.0], N[Not[LessEqual[eps, 1.0]], $MachinePrecision]], 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[(1.0 - N[(x * eps + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(x - -2.0), $MachinePrecision] * t$95$0 + N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.45000000000000012e158

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 85.5%

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

   12. Taylor expanded in eps around inf

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

   14. Applied rewrites 85.5%

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

   if -1.45000000000000012e158 < eps < -1 or 1 < 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 69.8%

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

   if -1 < eps < 1

   1. Initial program 41.9%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 42.0%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 98.1%

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

   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%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.45 \cdot 10^{+158}:\\ \;\;\;\;\frac{1 \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - -1 \cdot e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;\varepsilon \leq -1 \lor \neg \left(\varepsilon \leq 1\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(1 - \mathsf{fma}\left(x, \varepsilon, x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - -2, e^{-x}, e^{-x} \cdot x\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return ((1.0 * exp(((-1.0 + eps) * x))) - (-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 * exp((((-1.0d0) + eps) * x))) - ((-1.0d0) * exp((((-1.0d0) - eps) * x)))) / 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

3. Taylor expanded in eps around inf

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

5. Applied rewrites 74.2%

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

6. Taylor expanded in eps around inf

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

8. Applied rewrites 99.2%

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

9. Final simplification 99.2%

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

Alternative 4: 75.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.45 \cdot 10^{+158}:\\ \;\;\;\;\frac{1 \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - -1 \cdot e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;\varepsilon \leq -1 \lor \neg \left(\varepsilon \leq 1\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(1 - \mathsf{fma}\left(x, \varepsilon, x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(e^{-x}, \left(1 + x\right) - -1, \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -1.45e+158)
   (/ (- (* 1.0 (fma (- eps 1.0) x 1.0)) (* -1.0 (exp (* eps (- x))))) 2.0)
   (if (or (<= eps -1.0) (not (<= eps 1.0)))
     (/
      (-
       (* (+ 1.0 (/ 1.0 eps)) (exp (* (+ -1.0 eps) x)))
       (* (- (/ 1.0 eps) 1.0) (- 1.0 (fma x eps x))))
      2.0)
     (*
      (fma
       (exp (- x))
       (- (+ 1.0 x) -1.0)
       (/ x (fma (fma (fma 0.16666666666666666 x 0.5) x 1.0) x 1.0)))
      0.5))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -1.45e+158) {
		tmp = ((1.0 * fma((eps - 1.0), x, 1.0)) - (-1.0 * exp((eps * -x)))) / 2.0;
	} else if ((eps <= -1.0) || !(eps <= 1.0)) {
		tmp = (((1.0 + (1.0 / eps)) * exp(((-1.0 + eps) * x))) - (((1.0 / eps) - 1.0) * (1.0 - fma(x, eps, x)))) / 2.0;
	} else {
		tmp = fma(exp(-x), ((1.0 + x) - -1.0), (x / fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0))) * 0.5;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -1.45e+158)
		tmp = Float64(Float64(Float64(1.0 * fma(Float64(eps - 1.0), x, 1.0)) - Float64(-1.0 * exp(Float64(eps * Float64(-x))))) / 2.0);
	elseif ((eps <= -1.0) || !(eps <= 1.0))
		tmp = 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) * Float64(1.0 - fma(x, eps, x)))) / 2.0);
	else
		tmp = Float64(fma(exp(Float64(-x)), Float64(Float64(1.0 + x) - -1.0), Float64(x / fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0))) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -1.45e+158], N[(N[(N[(1.0 * N[(N[(eps - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[Exp[N[(eps * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[eps, -1.0], N[Not[LessEqual[eps, 1.0]], $MachinePrecision]], 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[(1.0 - N[(x * eps + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(N[(1.0 + x), $MachinePrecision] - -1.0), $MachinePrecision] + N[(x / N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.45000000000000012e158

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 85.5%

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

   12. Taylor expanded in eps around inf

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

   14. Applied rewrites 85.5%

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

   if -1.45000000000000012e158 < eps < -1 or 1 < 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 69.8%

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

   if -1 < eps < 1

   1. Initial program 41.9%

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

   3. Taylor expanded in eps around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 88.2%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.45 \cdot 10^{+158}:\\ \;\;\;\;\frac{1 \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - -1 \cdot e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;\varepsilon \leq -1 \lor \neg \left(\varepsilon \leq 1\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(1 - \mathsf{fma}\left(x, \varepsilon, x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(e^{-x}, \left(1 + x\right) - -1, \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.45 \cdot 10^{+158}:\\ \;\;\;\;\frac{1 \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - -1 \cdot e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;\varepsilon \leq -1 \lor \neg \left(\varepsilon \leq 1\right):\\ \;\;\;\;\frac{1 \cdot e^{x \cdot \varepsilon} - -1 \cdot \mathsf{fma}\left(-1 - \varepsilon, x, 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(e^{-x}, \left(1 + x\right) - -1, \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -1.45e+158)
   (/ (- (* 1.0 (fma (- eps 1.0) x 1.0)) (* -1.0 (exp (* eps (- x))))) 2.0)
   (if (or (<= eps -1.0) (not (<= eps 1.0)))
     (/ (- (* 1.0 (exp (* x eps))) (* -1.0 (fma (- -1.0 eps) x 1.0))) 2.0)
     (*
      (fma
       (exp (- x))
       (- (+ 1.0 x) -1.0)
       (/ x (fma (fma (fma 0.16666666666666666 x 0.5) x 1.0) x 1.0)))
      0.5))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -1.45e+158) {
		tmp = ((1.0 * fma((eps - 1.0), x, 1.0)) - (-1.0 * exp((eps * -x)))) / 2.0;
	} else if ((eps <= -1.0) || !(eps <= 1.0)) {
		tmp = ((1.0 * exp((x * eps))) - (-1.0 * fma((-1.0 - eps), x, 1.0))) / 2.0;
	} else {
		tmp = fma(exp(-x), ((1.0 + x) - -1.0), (x / fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0))) * 0.5;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -1.45e+158)
		tmp = Float64(Float64(Float64(1.0 * fma(Float64(eps - 1.0), x, 1.0)) - Float64(-1.0 * exp(Float64(eps * Float64(-x))))) / 2.0);
	elseif ((eps <= -1.0) || !(eps <= 1.0))
		tmp = Float64(Float64(Float64(1.0 * exp(Float64(x * eps))) - Float64(-1.0 * fma(Float64(-1.0 - eps), x, 1.0))) / 2.0);
	else
		tmp = Float64(fma(exp(Float64(-x)), Float64(Float64(1.0 + x) - -1.0), Float64(x / fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0))) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -1.45e+158], N[(N[(N[(1.0 * N[(N[(eps - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[Exp[N[(eps * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[eps, -1.0], N[Not[LessEqual[eps, 1.0]], $MachinePrecision]], N[(N[(N[(1.0 * N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(-1.0 * N[(N[(-1.0 - eps), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(N[(1.0 + x), $MachinePrecision] - -1.0), $MachinePrecision] + N[(x / N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.45000000000000012e158

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 85.5%

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

   12. Taylor expanded in eps around inf

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

   14. Applied rewrites 85.5%

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

   if -1.45000000000000012e158 < eps < -1 or 1 < 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

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 69.8%

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

   if -1 < eps < 1

   1. Initial program 41.9%

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

   3. Taylor expanded in eps around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 88.2%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.45 \cdot 10^{+158}:\\ \;\;\;\;\frac{1 \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - -1 \cdot e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;\varepsilon \leq -1 \lor \neg \left(\varepsilon \leq 1\right):\\ \;\;\;\;\frac{1 \cdot e^{x \cdot \varepsilon} - -1 \cdot \mathsf{fma}\left(-1 - \varepsilon, x, 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(e^{-x}, \left(1 + x\right) - -1, \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{+21}:\\ \;\;\;\;\frac{1 \cdot e^{x \cdot \varepsilon} - -1 \cdot 1}{2}\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+213}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right) \cdot x, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 1.15e+21)
   (/ (- (* 1.0 (exp (* x eps))) (* -1.0 1.0)) 2.0)
   (if (<= x 2.05e+213)
     (/ (- (- (/ 1.0 eps) -1.0) (* (- (/ 1.0 eps) 1.0) 1.0)) 2.0)
     (fma (* (fma 0.3333333333333333 x -0.5) x) x 1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 1.15e+21) {
		tmp = ((1.0 * exp((x * eps))) - (-1.0 * 1.0)) / 2.0;
	} else if (x <= 2.05e+213) {
		tmp = (((1.0 / eps) - -1.0) - (((1.0 / eps) - 1.0) * 1.0)) / 2.0;
	} else {
		tmp = fma((fma(0.3333333333333333, x, -0.5) * x), x, 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 1.15e+21)
		tmp = Float64(Float64(Float64(1.0 * exp(Float64(x * eps))) - Float64(-1.0 * 1.0)) / 2.0);
	elseif (x <= 2.05e+213)
		tmp = Float64(Float64(Float64(Float64(1.0 / eps) - -1.0) - Float64(Float64(Float64(1.0 / eps) - 1.0) * 1.0)) / 2.0);
	else
		tmp = fma(Float64(fma(0.3333333333333333, x, -0.5) * x), x, 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 1.15e+21], 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[x, 2.05e+213], N[(N[(N[(N[(1.0 / eps), $MachinePrecision] - -1.0), $MachinePrecision] - N[(N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(0.3333333333333333 * x + -0.5), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.15e21

   1. Initial program 64.9%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 63.7%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 98.8%

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

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 98.4%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 78.9%

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

   if 1.15e21 < x < 2.0499999999999999e213

   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

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

   5. Applied rewrites 16.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 69.0%

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

   if 2.0499999999999999e213 < 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 0

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

   5. Applied rewrites 38.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{\frac{-1}{2} \cdot {x}^{2}} \]
   7. Step-by-step derivation

   8. Applied rewrites 0.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 63.1%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), \color{blue}{x \cdot x}, 1\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 63.1%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{+21}:\\ \;\;\;\;\frac{1 \cdot e^{x \cdot \varepsilon} - -1 \cdot 1}{2}\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+213}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right) \cdot x, x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1220000000:\\ \;\;\;\;\frac{1 \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - -1 \cdot 1}{2}\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+213}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right) \cdot x, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 1220000000.0)
   (/ (- (* 1.0 (fma (- eps 1.0) x 1.0)) (* -1.0 1.0)) 2.0)
   (if (<= x 2.05e+213)
     (/ (- (- (/ 1.0 eps) -1.0) (* (- (/ 1.0 eps) 1.0) 1.0)) 2.0)
     (fma (* (fma 0.3333333333333333 x -0.5) x) x 1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 1220000000.0) {
		tmp = ((1.0 * fma((eps - 1.0), x, 1.0)) - (-1.0 * 1.0)) / 2.0;
	} else if (x <= 2.05e+213) {
		tmp = (((1.0 / eps) - -1.0) - (((1.0 / eps) - 1.0) * 1.0)) / 2.0;
	} else {
		tmp = fma((fma(0.3333333333333333, x, -0.5) * x), x, 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 1220000000.0)
		tmp = Float64(Float64(Float64(1.0 * fma(Float64(eps - 1.0), x, 1.0)) - Float64(-1.0 * 1.0)) / 2.0);
	elseif (x <= 2.05e+213)
		tmp = Float64(Float64(Float64(Float64(1.0 / eps) - -1.0) - Float64(Float64(Float64(1.0 / eps) - 1.0) * 1.0)) / 2.0);
	else
		tmp = fma(Float64(fma(0.3333333333333333, x, -0.5) * x), x, 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 1220000000.0], N[(N[(N[(1.0 * N[(N[(eps - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.05e+213], N[(N[(N[(N[(1.0 / eps), $MachinePrecision] - -1.0), $MachinePrecision] - N[(N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(0.3333333333333333 * x + -0.5), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.22e9

   1. Initial program 64.5%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 63.3%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 98.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 77.4%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 63.9%

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

   if 1.22e9 < x < 2.0499999999999999e213

   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

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

   5. Applied rewrites 15.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 68.3%

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

   if 2.0499999999999999e213 < 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 0

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

   5. Applied rewrites 38.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{\frac{-1}{2} \cdot {x}^{2}} \]
   7. Step-by-step derivation

   8. Applied rewrites 0.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 63.1%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), \color{blue}{x \cdot x}, 1\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 63.1%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1220000000:\\ \;\;\;\;\frac{1 \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - -1 \cdot 1}{2}\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+213}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} - -1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right) \cdot x, x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.5% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{+110} \lor \neg \left(x \leq 10^{+213}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right) \cdot x, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x 3.2e+110) (not (<= x 1e+213)))
   (fma (* (fma 0.3333333333333333 x -0.5) x) x 1.0)
   (/ x (fma (fma (fma 0.16666666666666666 x 0.5) x 1.0) x 1.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= 3.2e+110) || !(x <= 1e+213)) {
		tmp = fma((fma(0.3333333333333333, x, -0.5) * x), x, 1.0);
	} else {
		tmp = x / fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= 3.2e+110) || !(x <= 1e+213))
		tmp = fma(Float64(fma(0.3333333333333333, x, -0.5) * x), x, 1.0);
	else
		tmp = Float64(x / fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, 3.2e+110], N[Not[LessEqual[x, 1e+213]], $MachinePrecision]], N[(N[(N[(0.3333333333333333 * x + -0.5), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision], N[(x / N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.19999999999999994e110 or 9.99999999999999984e212 < x

   1. Initial program 72.1%

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

   3. Taylor expanded in eps around 0

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

   5. Applied rewrites 54.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{\frac{-1}{2} \cdot {x}^{2}} \]
   7. Step-by-step derivation

   8. Applied rewrites 44.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 52.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), \color{blue}{x \cdot x}, 1\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 52.7%

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

   if 3.19999999999999994e110 < x < 9.99999999999999984e212

   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

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

   5. Applied rewrites 81.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 81.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 81.8%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{+110} \lor \neg \left(x \leq 10^{+213}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right) \cdot x, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.3% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.1 \cdot 10^{+109}:\\ \;\;\;\;\frac{1 \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - -1 \cdot 1}{2}\\ \mathbf{elif}\;x \leq 10^{+213}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right) \cdot x, x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 4.1e+109)
   (/ (- (* 1.0 (fma (- eps 1.0) x 1.0)) (* -1.0 1.0)) 2.0)
   (if (<= x 1e+213)
     (/ x (fma (fma (fma 0.16666666666666666 x 0.5) x 1.0) x 1.0))
     (fma (* (fma 0.3333333333333333 x -0.5) x) x 1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 4.1e+109) {
		tmp = ((1.0 * fma((eps - 1.0), x, 1.0)) - (-1.0 * 1.0)) / 2.0;
	} else if (x <= 1e+213) {
		tmp = x / fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0);
	} else {
		tmp = fma((fma(0.3333333333333333, x, -0.5) * x), x, 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 4.1e+109)
		tmp = Float64(Float64(Float64(1.0 * fma(Float64(eps - 1.0), x, 1.0)) - Float64(-1.0 * 1.0)) / 2.0);
	elseif (x <= 1e+213)
		tmp = Float64(x / fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0));
	else
		tmp = fma(Float64(fma(0.3333333333333333, x, -0.5) * x), x, 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 4.1e+109], N[(N[(N[(1.0 * N[(N[(eps - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(-1.0 * 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1e+213], N[(x / N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.3333333333333333 * x + -0.5), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 4.0999999999999997e109

   1. Initial program 68.8%

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

   3. Taylor expanded in eps around inf

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

   5. Applied rewrites 67.8%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 99.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 72.1%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 58.4%

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

   if 4.0999999999999997e109 < x < 9.99999999999999984e212

   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

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

   5. Applied rewrites 81.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 81.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 81.8%

       \[\leadsto \frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)} \]

   if 9.99999999999999984e212 < 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 0

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

   5. Applied rewrites 38.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{\frac{-1}{2} \cdot {x}^{2}} \]
   7. Step-by-step derivation

   8. Applied rewrites 0.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 63.1%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), \color{blue}{x \cdot x}, 1\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 63.1%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.1 \cdot 10^{+109}:\\ \;\;\;\;\frac{1 \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - -1 \cdot 1}{2}\\ \mathbf{elif}\;x \leq 10^{+213}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right) \cdot x, x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.0% accurate, 15.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 57.6%

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

6. Taylor expanded in x around 0

   \[\leadsto 1 + \color{blue}{\frac{-1}{2} \cdot {x}^{2}} \]
7. Step-by-step derivation

8. Applied rewrites 40.3%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 49.2%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), \color{blue}{x \cdot x}, 1\right) \]
12. Step-by-step derivation

13. Applied rewrites 49.2%

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

14. Final simplification 49.2%

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

Alternative 11: 52.9% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

3. Taylor expanded in eps around 0

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

5. Applied rewrites 57.6%

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

6. Taylor expanded in x around 0

   \[\leadsto 1 + \color{blue}{\frac{-1}{2} \cdot {x}^{2}} \]
7. Step-by-step derivation

8. Applied rewrites 40.3%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 49.2%

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

12. Taylor expanded in x around inf

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

14. Applied rewrites 49.0%

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

15. Final simplification 49.0%

    \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right) \]
16. Add Preprocessing

Alternative 12: 44.8% accurate, 273.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 1.0)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, eps):
	return 1.0

Julia

function code(x, eps)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 1.0

Derivation

1. Initial program 75.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 41.2%

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

6. Final simplification 41.2%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025018 
(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))