NMSE Section 6.1 mentioned, A

Percentage Accurate: 72.9% → 100.0%

Time: 7.4s

Alternatives: 16

Speedup: 2.8×

• 39.1% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -3.539356793979253e+252
  2. eps = -2.0863806904459572e-271

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: 72.9% 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: 100.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if eps < 1

   1. Initial program 61.5%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 72.8%

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

      1. distribute-rgt1-in N/A

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

      2. distribute-lft-out N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-exp.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. distribute-lft-out N/A

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

      9. distribute-rgt1-in N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-exp.f64 N/A

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

      15. lower-neg.f64 72.8

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

   7. Applied rewrites 72.8%

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

   if 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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

      1. exp-neg N/A

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

      2. lower-/.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 100.0

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

   10. Applied rewrites 100.0%

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

       1. lower-*.f64 N/A

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

   12. Applied rewrites 100.0%

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

4. Final simplification 79.9%

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

Alternative 2: 98.9% accurate, 1.2× speedup

Math

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

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 1.0)
   (* (* (exp (- x)) 2.0) 0.5)
   (* (+ (exp (* eps_m x)) (exp (- (fma x eps_m x)))) 0.5)))

C

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

Julia

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 1.0], N[(N[(N[Exp[(-x)], $MachinePrecision] * 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Exp[N[(eps$95$m * x), $MachinePrecision]], $MachinePrecision] + N[Exp[(-N[(x * eps$95$m + x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1

   1. Initial program 61.5%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.6%

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

      1. exp-neg N/A

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

      2. lower-/.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-fma.f64 97.6

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

   7. Applied rewrites 97.6%

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

   8. Taylor expanded in eps around 0

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

      1. rec-exp N/A

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

      2. mul-1-neg N/A

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

      3. rec-exp N/A

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

      4. count-2-rev N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 81.6

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

   10. Applied rewrites 81.6%

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

   if 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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

4. Final simplification 86.4%

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

Alternative 3: 98.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

3. Taylor expanded in eps around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 98.3%

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

6. Final simplification 98.3%

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

Alternative 4: 81.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{-x} \cdot 2\\ \mathbf{if}\;x \leq -360000000:\\ \;\;\;\;t\_0 \cdot 0.5\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-285}:\\ \;\;\;\;\left(1 - \left(x \cdot \frac{1 - eps\_m \cdot eps\_m}{1 - eps\_m} - 1\right)\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+103} \lor \neg \left(x \leq 2.8 \cdot 10^{+196}\right):\\ \;\;\;\;\left(e^{x \cdot eps\_m} - -1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot t\_0\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* (exp (- x)) 2.0)))
   (if (<= x -360000000.0)
     (* t_0 0.5)
     (if (<= x -6e-285)
       (* (- 1.0 (- (* x (/ (- 1.0 (* eps_m eps_m)) (- 1.0 eps_m))) 1.0)) 0.5)
       (if (or (<= x 1.25e+103) (not (<= x 2.8e+196)))
         (* (- (exp (* x eps_m)) -1.0) 0.5)
         (* (* x t_0) 0.5))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp(-x) * 2.0;
	double tmp;
	if (x <= -360000000.0) {
		tmp = t_0 * 0.5;
	} else if (x <= -6e-285) {
		tmp = (1.0 - ((x * ((1.0 - (eps_m * eps_m)) / (1.0 - eps_m))) - 1.0)) * 0.5;
	} else if ((x <= 1.25e+103) || !(x <= 2.8e+196)) {
		tmp = (exp((x * eps_m)) - -1.0) * 0.5;
	} else {
		tmp = (x * t_0) * 0.5;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, eps_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x) * 2.0d0
    if (x <= (-360000000.0d0)) then
        tmp = t_0 * 0.5d0
    else if (x <= (-6d-285)) then
        tmp = (1.0d0 - ((x * ((1.0d0 - (eps_m * eps_m)) / (1.0d0 - eps_m))) - 1.0d0)) * 0.5d0
    else if ((x <= 1.25d+103) .or. (.not. (x <= 2.8d+196))) then
        tmp = (exp((x * eps_m)) - (-1.0d0)) * 0.5d0
    else
        tmp = (x * t_0) * 0.5d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp(-x) * 2.0;
	double tmp;
	if (x <= -360000000.0) {
		tmp = t_0 * 0.5;
	} else if (x <= -6e-285) {
		tmp = (1.0 - ((x * ((1.0 - (eps_m * eps_m)) / (1.0 - eps_m))) - 1.0)) * 0.5;
	} else if ((x <= 1.25e+103) || !(x <= 2.8e+196)) {
		tmp = (Math.exp((x * eps_m)) - -1.0) * 0.5;
	} else {
		tmp = (x * t_0) * 0.5;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp(-x) * 2.0
	tmp = 0
	if x <= -360000000.0:
		tmp = t_0 * 0.5
	elif x <= -6e-285:
		tmp = (1.0 - ((x * ((1.0 - (eps_m * eps_m)) / (1.0 - eps_m))) - 1.0)) * 0.5
	elif (x <= 1.25e+103) or not (x <= 2.8e+196):
		tmp = (math.exp((x * eps_m)) - -1.0) * 0.5
	else:
		tmp = (x * t_0) * 0.5
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(exp(Float64(-x)) * 2.0)
	tmp = 0.0
	if (x <= -360000000.0)
		tmp = Float64(t_0 * 0.5);
	elseif (x <= -6e-285)
		tmp = Float64(Float64(1.0 - Float64(Float64(x * Float64(Float64(1.0 - Float64(eps_m * eps_m)) / Float64(1.0 - eps_m))) - 1.0)) * 0.5);
	elseif ((x <= 1.25e+103) || !(x <= 2.8e+196))
		tmp = Float64(Float64(exp(Float64(x * eps_m)) - -1.0) * 0.5);
	else
		tmp = Float64(Float64(x * t_0) * 0.5);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp(-x) * 2.0;
	tmp = 0.0;
	if (x <= -360000000.0)
		tmp = t_0 * 0.5;
	elseif (x <= -6e-285)
		tmp = (1.0 - ((x * ((1.0 - (eps_m * eps_m)) / (1.0 - eps_m))) - 1.0)) * 0.5;
	elseif ((x <= 1.25e+103) || ~((x <= 2.8e+196)))
		tmp = (exp((x * eps_m)) - -1.0) * 0.5;
	else
		tmp = (x * t_0) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[Exp[(-x)], $MachinePrecision] * 2.0), $MachinePrecision]}, If[LessEqual[x, -360000000.0], N[(t$95$0 * 0.5), $MachinePrecision], If[LessEqual[x, -6e-285], N[(N[(1.0 - N[(N[(x * N[(N[(1.0 - N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[Or[LessEqual[x, 1.25e+103], N[Not[LessEqual[x, 2.8e+196]], $MachinePrecision]], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(x * t$95$0), $MachinePrecision] * 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.6e8

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

      1. exp-neg N/A

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

      2. lower-/.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in eps around 0

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

      1. rec-exp N/A

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

      2. mul-1-neg N/A

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

      3. rec-exp N/A

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

      4. count-2-rev N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 100.0

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

   10. Applied rewrites 100.0%

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

   if -3.6e8 < x < -6.00000000000000007e-285

   1. Initial program 54.0%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.0%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 81.5

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

   8. Applied rewrites 81.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 65.3%

       \[\leadsto \left(1 - \left(x \cdot \left(1 + \varepsilon\right) - 1\right)\right) \cdot 0.5 \]
   12. Step-by-step derivation

       1. flip-+ N/A

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

       2. lower-/.f64 N/A

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

       3. metadata-eval N/A

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

       4. unpow2 N/A

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

       5. lower--.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. lower--.f64 72.0

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

   13. Applied rewrites 72.0%

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

   if -6.00000000000000007e-285 < x < 1.25e103 or 2.8000000000000002e196 < x

   1. Initial program 67.4%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.8%

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

      1. exp-neg N/A

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

      2. lower-/.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-fma.f64 98.8

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

   7. Applied rewrites 98.8%

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

   8. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 87.7

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

   10. Applied rewrites 87.7%

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

   11. Taylor expanded in x around 0

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

       1. rec-exp 68.5

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

   13. Applied rewrites 68.5%

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

   if 1.25e103 < x < 2.8000000000000002e196

   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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 78.9%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 78.9

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

   8. Applied rewrites 78.9%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-exp.f64 N/A

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

       4. lower-neg.f64 78.9

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

   11. Applied rewrites 78.9%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -360000000:\\ \;\;\;\;\left(e^{-x} \cdot 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-285}:\\ \;\;\;\;\left(1 - \left(x \cdot \frac{1 - \varepsilon \cdot \varepsilon}{1 - \varepsilon} - 1\right)\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+103} \lor \neg \left(x \leq 2.8 \cdot 10^{+196}\right):\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - -1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(e^{-x} \cdot 2\right)\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \left(e^{-x} \cdot 2\right) \cdot 0.5\\ \mathbf{if}\;x \leq -360000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-285}:\\ \;\;\;\;\left(1 - \left(x \cdot \frac{1 - eps\_m \cdot eps\_m}{1 - eps\_m} - 1\right)\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+103} \lor \neg \left(x \leq 8.5 \cdot 10^{+236}\right):\\ \;\;\;\;\left(e^{x \cdot eps\_m} - -1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* (* (exp (- x)) 2.0) 0.5)))
   (if (<= x -360000000.0)
     t_0
     (if (<= x -6e-285)
       (* (- 1.0 (- (* x (/ (- 1.0 (* eps_m eps_m)) (- 1.0 eps_m))) 1.0)) 0.5)
       (if (or (<= x 1.25e+103) (not (<= x 8.5e+236)))
         (* (- (exp (* x eps_m)) -1.0) 0.5)
         t_0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (exp(-x) * 2.0) * 0.5;
	double tmp;
	if (x <= -360000000.0) {
		tmp = t_0;
	} else if (x <= -6e-285) {
		tmp = (1.0 - ((x * ((1.0 - (eps_m * eps_m)) / (1.0 - eps_m))) - 1.0)) * 0.5;
	} else if ((x <= 1.25e+103) || !(x <= 8.5e+236)) {
		tmp = (exp((x * eps_m)) - -1.0) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, eps_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (exp(-x) * 2.0d0) * 0.5d0
    if (x <= (-360000000.0d0)) then
        tmp = t_0
    else if (x <= (-6d-285)) then
        tmp = (1.0d0 - ((x * ((1.0d0 - (eps_m * eps_m)) / (1.0d0 - eps_m))) - 1.0d0)) * 0.5d0
    else if ((x <= 1.25d+103) .or. (.not. (x <= 8.5d+236))) then
        tmp = (exp((x * eps_m)) - (-1.0d0)) * 0.5d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = (Math.exp(-x) * 2.0) * 0.5;
	double tmp;
	if (x <= -360000000.0) {
		tmp = t_0;
	} else if (x <= -6e-285) {
		tmp = (1.0 - ((x * ((1.0 - (eps_m * eps_m)) / (1.0 - eps_m))) - 1.0)) * 0.5;
	} else if ((x <= 1.25e+103) || !(x <= 8.5e+236)) {
		tmp = (Math.exp((x * eps_m)) - -1.0) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (math.exp(-x) * 2.0) * 0.5
	tmp = 0
	if x <= -360000000.0:
		tmp = t_0
	elif x <= -6e-285:
		tmp = (1.0 - ((x * ((1.0 - (eps_m * eps_m)) / (1.0 - eps_m))) - 1.0)) * 0.5
	elif (x <= 1.25e+103) or not (x <= 8.5e+236):
		tmp = (math.exp((x * eps_m)) - -1.0) * 0.5
	else:
		tmp = t_0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(exp(Float64(-x)) * 2.0) * 0.5)
	tmp = 0.0
	if (x <= -360000000.0)
		tmp = t_0;
	elseif (x <= -6e-285)
		tmp = Float64(Float64(1.0 - Float64(Float64(x * Float64(Float64(1.0 - Float64(eps_m * eps_m)) / Float64(1.0 - eps_m))) - 1.0)) * 0.5);
	elseif ((x <= 1.25e+103) || !(x <= 8.5e+236))
		tmp = Float64(Float64(exp(Float64(x * eps_m)) - -1.0) * 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (exp(-x) * 2.0) * 0.5;
	tmp = 0.0;
	if (x <= -360000000.0)
		tmp = t_0;
	elseif (x <= -6e-285)
		tmp = (1.0 - ((x * ((1.0 - (eps_m * eps_m)) / (1.0 - eps_m))) - 1.0)) * 0.5;
	elseif ((x <= 1.25e+103) || ~((x <= 8.5e+236)))
		tmp = (exp((x * eps_m)) - -1.0) * 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(N[Exp[(-x)], $MachinePrecision] * 2.0), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[x, -360000000.0], t$95$0, If[LessEqual[x, -6e-285], N[(N[(1.0 - N[(N[(x * N[(N[(1.0 - N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[Or[LessEqual[x, 1.25e+103], N[Not[LessEqual[x, 8.5e+236]], $MachinePrecision]], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.6e8 or 1.25e103 < x < 8.5000000000000008e236

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

      1. exp-neg N/A

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

      2. lower-/.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in eps around 0

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

      1. rec-exp N/A

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

      2. mul-1-neg N/A

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

      3. rec-exp N/A

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

      4. count-2-rev N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 87.5

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

   10. Applied rewrites 87.5%

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

   if -3.6e8 < x < -6.00000000000000007e-285

   1. Initial program 54.0%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.0%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 81.5

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

   8. Applied rewrites 81.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 65.3%

       \[\leadsto \left(1 - \left(x \cdot \left(1 + \varepsilon\right) - 1\right)\right) \cdot 0.5 \]
   12. Step-by-step derivation

       1. flip-+ N/A

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

       2. lower-/.f64 N/A

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

       3. metadata-eval N/A

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

       4. unpow2 N/A

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

       5. lower--.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. lower--.f64 72.0

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

   13. Applied rewrites 72.0%

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

   if -6.00000000000000007e-285 < x < 1.25e103 or 8.5000000000000008e236 < x

   1. Initial program 65.6%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.8%

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

      1. exp-neg N/A

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

      2. lower-/.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-fma.f64 98.8

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

   7. Applied rewrites 98.8%

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

   8. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 87.9

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

   10. Applied rewrites 87.9%

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

   11. Taylor expanded in x around 0

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

       1. rec-exp 71.3

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

   13. Applied rewrites 71.3%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -360000000:\\ \;\;\;\;\left(e^{-x} \cdot 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-285}:\\ \;\;\;\;\left(1 - \left(x \cdot \frac{1 - \varepsilon \cdot \varepsilon}{1 - \varepsilon} - 1\right)\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+103} \lor \neg \left(x \leq 8.5 \cdot 10^{+236}\right):\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - -1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-x} \cdot 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-288}:\\ \;\;\;\;\left(1 + e^{-\mathsf{fma}\left(x, eps\_m, x\right)}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 6.8:\\ \;\;\;\;\left(e^{x \cdot eps\_m} - \left(x \cdot \left(1 + eps\_m\right) - 1\right)\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+196}:\\ \;\;\;\;\left(x \cdot \left(e^{-x} \cdot 2\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x \cdot \left(-1 + eps\_m\right)} - -1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2e-288)
   (* (+ 1.0 (exp (- (fma x eps_m x)))) 0.5)
   (if (<= x 6.8)
     (* (- (exp (* x eps_m)) (- (* x (+ 1.0 eps_m)) 1.0)) 0.5)
     (if (<= x 2.8e+196)
       (* (* x (* (exp (- x)) 2.0)) 0.5)
       (* (- (exp (* x (+ -1.0 eps_m))) -1.0) 0.5)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -2e-288) {
		tmp = (1.0 + exp(-fma(x, eps_m, x))) * 0.5;
	} else if (x <= 6.8) {
		tmp = (exp((x * eps_m)) - ((x * (1.0 + eps_m)) - 1.0)) * 0.5;
	} else if (x <= 2.8e+196) {
		tmp = (x * (exp(-x) * 2.0)) * 0.5;
	} else {
		tmp = (exp((x * (-1.0 + eps_m))) - -1.0) * 0.5;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -2e-288)
		tmp = Float64(Float64(1.0 + exp(Float64(-fma(x, eps_m, x)))) * 0.5);
	elseif (x <= 6.8)
		tmp = Float64(Float64(exp(Float64(x * eps_m)) - Float64(Float64(x * Float64(1.0 + eps_m)) - 1.0)) * 0.5);
	elseif (x <= 2.8e+196)
		tmp = Float64(Float64(x * Float64(exp(Float64(-x)) * 2.0)) * 0.5);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 + eps_m))) - -1.0) * 0.5);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -2e-288], N[(N[(1.0 + N[Exp[(-N[(x * eps$95$m + x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 6.8], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] - N[(N[(x * N[(1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 2.8e+196], N[(N[(x * N[(N[Exp[(-x)], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.00000000000000012e-288

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in x around 0

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

      1. distribute-lft-neg-in 69.2

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

      2. sinh---cosh-rev 69.2

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

   8. Applied rewrites 69.2%

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

   if -2.00000000000000012e-288 < x < 6.79999999999999982

   1. Initial program 49.5%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 87.0

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

   8. Applied rewrites 87.0%

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

   9. Taylor expanded in eps around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 87.1

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

   11. Applied rewrites 87.1%

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

   if 6.79999999999999982 < x < 2.8000000000000002e196

   1. Initial program 97.9%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 68.6%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 66.9

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

   8. Applied rewrites 66.9%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-exp.f64 N/A

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

       4. lower-neg.f64 66.9

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

   11. Applied rewrites 66.9%

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

   if 2.8000000000000002e196 < 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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 36.8%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-288}:\\ \;\;\;\;\left(1 + e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 6.8:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - \left(x \cdot \left(1 + \varepsilon\right) - 1\right)\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+196}:\\ \;\;\;\;\left(x \cdot \left(e^{-x} \cdot 2\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x \cdot \left(-1 + \varepsilon\right)} - -1\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-296}:\\ \;\;\;\;\left(1 + e^{-\mathsf{fma}\left(x, eps\_m, x\right)}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+103} \lor \neg \left(x \leq 2.8 \cdot 10^{+196}\right):\\ \;\;\;\;\left(e^{x \cdot eps\_m} - -1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(e^{-x} \cdot 2\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2e-296)
   (* (+ 1.0 (exp (- (fma x eps_m x)))) 0.5)
   (if (or (<= x 1.25e+103) (not (<= x 2.8e+196)))
     (* (- (exp (* x eps_m)) -1.0) 0.5)
     (* (* x (* (exp (- x)) 2.0)) 0.5))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -2e-296) {
		tmp = (1.0 + exp(-fma(x, eps_m, x))) * 0.5;
	} else if ((x <= 1.25e+103) || !(x <= 2.8e+196)) {
		tmp = (exp((x * eps_m)) - -1.0) * 0.5;
	} else {
		tmp = (x * (exp(-x) * 2.0)) * 0.5;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -2e-296)
		tmp = Float64(Float64(1.0 + exp(Float64(-fma(x, eps_m, x)))) * 0.5);
	elseif ((x <= 1.25e+103) || !(x <= 2.8e+196))
		tmp = Float64(Float64(exp(Float64(x * eps_m)) - -1.0) * 0.5);
	else
		tmp = Float64(Float64(x * Float64(exp(Float64(-x)) * 2.0)) * 0.5);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -2e-296], N[(N[(1.0 + N[Exp[(-N[(x * eps$95$m + x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[Or[LessEqual[x, 1.25e+103], N[Not[LessEqual[x, 2.8e+196]], $MachinePrecision]], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(x * N[(N[Exp[(-x)], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2e-296

   1. Initial program 67.4%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in x around 0

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

      1. distribute-lft-neg-in 69.8

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

      2. sinh---cosh-rev 69.8

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

   8. Applied rewrites 69.8%

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

   if -2e-296 < x < 1.25e103 or 2.8000000000000002e196 < x

   1. Initial program 68.9%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.8%

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

      1. exp-neg N/A

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

      2. lower-/.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-fma.f64 98.8

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

   7. Applied rewrites 98.8%

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

   8. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 87.2

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

   10. Applied rewrites 87.2%

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

   11. Taylor expanded in x around 0

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

       1. rec-exp 67.8

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

   13. Applied rewrites 67.8%

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

   if 1.25e103 < x < 2.8000000000000002e196

   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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 78.9%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 78.9

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

   8. Applied rewrites 78.9%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-exp.f64 N/A

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

       4. lower-neg.f64 78.9

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

   11. Applied rewrites 78.9%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-296}:\\ \;\;\;\;\left(1 + e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+103} \lor \neg \left(x \leq 2.8 \cdot 10^{+196}\right):\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - -1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(e^{-x} \cdot 2\right)\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-296}:\\ \;\;\;\;\left(1 + e^{-\mathsf{fma}\left(x, eps\_m, x\right)}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+103}:\\ \;\;\;\;\left(e^{x \cdot eps\_m} - -1\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+196}:\\ \;\;\;\;\left(x \cdot \left(e^{-x} \cdot 2\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x \cdot \left(-1 + eps\_m\right)} - -1\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2e-296)
   (* (+ 1.0 (exp (- (fma x eps_m x)))) 0.5)
   (if (<= x 1.25e+103)
     (* (- (exp (* x eps_m)) -1.0) 0.5)
     (if (<= x 2.8e+196)
       (* (* x (* (exp (- x)) 2.0)) 0.5)
       (* (- (exp (* x (+ -1.0 eps_m))) -1.0) 0.5)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -2e-296) {
		tmp = (1.0 + exp(-fma(x, eps_m, x))) * 0.5;
	} else if (x <= 1.25e+103) {
		tmp = (exp((x * eps_m)) - -1.0) * 0.5;
	} else if (x <= 2.8e+196) {
		tmp = (x * (exp(-x) * 2.0)) * 0.5;
	} else {
		tmp = (exp((x * (-1.0 + eps_m))) - -1.0) * 0.5;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -2e-296)
		tmp = Float64(Float64(1.0 + exp(Float64(-fma(x, eps_m, x)))) * 0.5);
	elseif (x <= 1.25e+103)
		tmp = Float64(Float64(exp(Float64(x * eps_m)) - -1.0) * 0.5);
	elseif (x <= 2.8e+196)
		tmp = Float64(Float64(x * Float64(exp(Float64(-x)) * 2.0)) * 0.5);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 + eps_m))) - -1.0) * 0.5);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -2e-296], N[(N[(1.0 + N[Exp[(-N[(x * eps$95$m + x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 1.25e+103], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 2.8e+196], N[(N[(x * N[(N[Exp[(-x)], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2e-296

   1. Initial program 67.4%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in x around 0

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

      1. distribute-lft-neg-in 69.8

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

      2. sinh---cosh-rev 69.8

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

   8. Applied rewrites 69.8%

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

   if -2e-296 < x < 1.25e103

   1. Initial program 60.6%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.5%

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

      1. exp-neg N/A

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

      2. lower-/.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-fma.f64 98.5

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

   7. Applied rewrites 98.5%

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

   8. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 89.5

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

   10. Applied rewrites 89.5%

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

   11. Taylor expanded in x around 0

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

       1. rec-exp 76.2

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

   13. Applied rewrites 76.2%

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

   if 1.25e103 < x < 2.8000000000000002e196

   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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 78.9%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 78.9

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

   8. Applied rewrites 78.9%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-exp.f64 N/A

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

       4. lower-neg.f64 78.9

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

   11. Applied rewrites 78.9%

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

   if 2.8000000000000002e196 < 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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 36.8%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-296}:\\ \;\;\;\;\left(1 + e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+103}:\\ \;\;\;\;\left(e^{x \cdot \varepsilon} - -1\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+196}:\\ \;\;\;\;\left(x \cdot \left(e^{-x} \cdot 2\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x \cdot \left(-1 + \varepsilon\right)} - -1\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 5.2 \cdot 10^{+199}:\\ \;\;\;\;\left(e^{-x} \cdot 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \left(x \cdot \frac{1 - eps\_m \cdot eps\_m}{1 - eps\_m} - 1\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 5.2e+199)
   (* (* (exp (- x)) 2.0) 0.5)
   (* (- 1.0 (- (* x (/ (- 1.0 (* eps_m eps_m)) (- 1.0 eps_m))) 1.0)) 0.5)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 5.2e+199) {
		tmp = (exp(-x) * 2.0) * 0.5;
	} else {
		tmp = (1.0 - ((x * ((1.0 - (eps_m * eps_m)) / (1.0 - eps_m))) - 1.0)) * 0.5;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, eps_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 5.2d+199) then
        tmp = (exp(-x) * 2.0d0) * 0.5d0
    else
        tmp = (1.0d0 - ((x * ((1.0d0 - (eps_m * eps_m)) / (1.0d0 - eps_m))) - 1.0d0)) * 0.5d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 5.2e+199) {
		tmp = (Math.exp(-x) * 2.0) * 0.5;
	} else {
		tmp = (1.0 - ((x * ((1.0 - (eps_m * eps_m)) / (1.0 - eps_m))) - 1.0)) * 0.5;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 5.2e+199:
		tmp = (math.exp(-x) * 2.0) * 0.5
	else:
		tmp = (1.0 - ((x * ((1.0 - (eps_m * eps_m)) / (1.0 - eps_m))) - 1.0)) * 0.5
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 5.2e+199)
		tmp = Float64(Float64(exp(Float64(-x)) * 2.0) * 0.5);
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(x * Float64(Float64(1.0 - Float64(eps_m * eps_m)) / Float64(1.0 - eps_m))) - 1.0)) * 0.5);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 5.2e+199)
		tmp = (exp(-x) * 2.0) * 0.5;
	else
		tmp = (1.0 - ((x * ((1.0 - (eps_m * eps_m)) / (1.0 - eps_m))) - 1.0)) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 5.2e+199], N[(N[(N[Exp[(-x)], $MachinePrecision] * 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(1.0 - N[(N[(x * N[(N[(1.0 - N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 5.2000000000000003e199

   1. Initial program 68.5%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 98.1%

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

      1. exp-neg N/A

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

      2. lower-/.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-fma.f64 98.1

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

   7. Applied rewrites 98.1%

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

   8. Taylor expanded in eps around 0

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

      1. rec-exp N/A

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

      2. mul-1-neg N/A

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

      3. rec-exp N/A

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

      4. count-2-rev N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-neg.f64 77.3

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

   10. Applied rewrites 77.3%

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

   if 5.2000000000000003e199 < eps

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 44.3

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

   8. Applied rewrites 44.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 31.3%

       \[\leadsto \left(1 - \left(x \cdot \left(1 + \varepsilon\right) - 1\right)\right) \cdot 0.5 \]
   12. Step-by-step derivation

       1. flip-+ N/A

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

       2. lower-/.f64 N/A

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

       3. metadata-eval N/A

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

       4. unpow2 N/A

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

       5. lower--.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. lower--.f64 56.3

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

   13. Applied rewrites 56.3%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 5.2 \cdot 10^{+199}:\\ \;\;\;\;\left(e^{-x} \cdot 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \left(x \cdot \frac{1 - \varepsilon \cdot \varepsilon}{1 - \varepsilon} - 1\right)\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \left(eps\_m - 1\right) \cdot x\\ t_1 := \left(\frac{1}{eps\_m} - 1\right) \cdot 1\\ t_2 := 1 + \frac{1}{eps\_m}\\ \mathbf{if}\;x \leq -520000000:\\ \;\;\;\;\frac{t\_2 \cdot \frac{1 - t\_0 \cdot t\_0}{1 - \left(-x\right)} - t\_1}{2}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-218}:\\ \;\;\;\;\left(1 - \left(x \cdot \frac{1 - eps\_m \cdot eps\_m}{1 - eps\_m} - 1\right)\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 360:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+206}:\\ \;\;\;\;\frac{t\_2 \cdot 1 - t\_1}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* (- eps_m 1.0) x))
        (t_1 (* (- (/ 1.0 eps_m) 1.0) 1.0))
        (t_2 (+ 1.0 (/ 1.0 eps_m))))
   (if (<= x -520000000.0)
     (/ (- (* t_2 (/ (- 1.0 (* t_0 t_0)) (- 1.0 (- x)))) t_1) 2.0)
     (if (<= x -2e-218)
       (* (- 1.0 (- (* x (/ (- 1.0 (* eps_m eps_m)) (- 1.0 eps_m))) 1.0)) 0.5)
       (if (<= x 360.0)
         1.0
         (if (<= x 2.4e+206)
           (/ (- (* t_2 1.0) t_1) 2.0)
           (fma (- (* 0.3333333333333333 x) 0.5) (* x x) 1.0)))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (eps_m - 1.0) * x;
	double t_1 = ((1.0 / eps_m) - 1.0) * 1.0;
	double t_2 = 1.0 + (1.0 / eps_m);
	double tmp;
	if (x <= -520000000.0) {
		tmp = ((t_2 * ((1.0 - (t_0 * t_0)) / (1.0 - -x))) - t_1) / 2.0;
	} else if (x <= -2e-218) {
		tmp = (1.0 - ((x * ((1.0 - (eps_m * eps_m)) / (1.0 - eps_m))) - 1.0)) * 0.5;
	} else if (x <= 360.0) {
		tmp = 1.0;
	} else if (x <= 2.4e+206) {
		tmp = ((t_2 * 1.0) - t_1) / 2.0;
	} else {
		tmp = fma(((0.3333333333333333 * x) - 0.5), (x * x), 1.0);
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(eps_m - 1.0) * x)
	t_1 = Float64(Float64(Float64(1.0 / eps_m) - 1.0) * 1.0)
	t_2 = Float64(1.0 + Float64(1.0 / eps_m))
	tmp = 0.0
	if (x <= -520000000.0)
		tmp = Float64(Float64(Float64(t_2 * Float64(Float64(1.0 - Float64(t_0 * t_0)) / Float64(1.0 - Float64(-x)))) - t_1) / 2.0);
	elseif (x <= -2e-218)
		tmp = Float64(Float64(1.0 - Float64(Float64(x * Float64(Float64(1.0 - Float64(eps_m * eps_m)) / Float64(1.0 - eps_m))) - 1.0)) * 0.5);
	elseif (x <= 360.0)
		tmp = 1.0;
	elseif (x <= 2.4e+206)
		tmp = Float64(Float64(Float64(t_2 * 1.0) - t_1) / 2.0);
	else
		tmp = fma(Float64(Float64(0.3333333333333333 * x) - 0.5), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(eps$95$m - 1.0), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision] * 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -520000000.0], N[(N[(N[(t$95$2 * N[(N[(1.0 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(1.0 - (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -2e-218], N[(N[(1.0 - N[(N[(x * N[(N[(1.0 - N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 360.0], 1.0, If[LessEqual[x, 2.4e+206], N[(N[(N[(t$95$2 * 1.0), $MachinePrecision] - t$95$1), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(0.3333333333333333 * x), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -5.2e8

   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

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

   5. Applied rewrites 55.5%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 31.6

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

   8. Applied rewrites 31.6%

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

      1. flip-+ N/A

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

      2. lower-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower--.f64 19.2

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

   10. Applied rewrites 19.2%

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

   11. Taylor expanded in eps around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 84.6

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

   13. Applied rewrites 84.6%

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

   if -5.2e8 < x < -2.0000000000000001e-218

   1. Initial program 52.9%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 95.4%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 80.2

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

   8. Applied rewrites 80.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 63.1%

       \[\leadsto \left(1 - \left(x \cdot \left(1 + \varepsilon\right) - 1\right)\right) \cdot 0.5 \]
   12. Step-by-step derivation

       1. flip-+ N/A

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

       2. lower-/.f64 N/A

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

       3. metadata-eval N/A

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

       4. unpow2 N/A

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

       5. lower--.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. lower--.f64 72.3

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

   13. Applied rewrites 72.3%

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

   if -2.0000000000000001e-218 < x < 360

   1. Initial program 50.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 79.8%

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

   if 360 < x < 2.4e206

   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

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

   5. Applied rewrites 25.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 66.3%

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

   if 2.4e206 < 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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 40.1%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 0.6

         \[\leadsto 1 + -0.5 \cdot \left(x \cdot x\right) \]

   8. Applied rewrites 0.6%

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

   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

       1. +-commutative N/A

          \[\leadsto {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + 1 \]

       2. *-commutative N/A

          \[\leadsto \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot {x}^{2} + 1 \]

       3. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x - \frac{1}{2}, {x}^{\color{blue}{2}}, 1\right) \]

       4. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x - \frac{1}{2}, {x}^{2}, 1\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x - \frac{1}{2}, {x}^{2}, 1\right) \]

       6. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x - \frac{1}{2}, x \cdot x, 1\right) \]

       7. lower-*.f64 61.5

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

   11. Applied rewrites 61.5%

       \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, \color{blue}{x \cdot x}, 1\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -520000000:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \frac{1 - \left(\left(\varepsilon - 1\right) \cdot x\right) \cdot \left(\left(\varepsilon - 1\right) \cdot x\right)}{1 - \left(-x\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot 1}{2}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-218}:\\ \;\;\;\;\left(1 - \left(x \cdot \frac{1 - \varepsilon \cdot \varepsilon}{1 - \varepsilon} - 1\right)\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 360:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+206}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot 1 - \left(\frac{1}{\varepsilon} - 1\right) \cdot 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-218}:\\ \;\;\;\;\left(1 - \left(x \cdot \frac{1 - eps\_m \cdot eps\_m}{1 - eps\_m} - 1\right)\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 360:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+206}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) \cdot 1 - \left(\frac{1}{eps\_m} - 1\right) \cdot 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2e-218)
   (* (- 1.0 (- (* x (/ (- 1.0 (* eps_m eps_m)) (- 1.0 eps_m))) 1.0)) 0.5)
   (if (<= x 360.0)
     1.0
     (if (<= x 2.4e+206)
       (/ (- (* (+ 1.0 (/ 1.0 eps_m)) 1.0) (* (- (/ 1.0 eps_m) 1.0) 1.0)) 2.0)
       (fma (- (* 0.3333333333333333 x) 0.5) (* x x) 1.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -2e-218) {
		tmp = (1.0 - ((x * ((1.0 - (eps_m * eps_m)) / (1.0 - eps_m))) - 1.0)) * 0.5;
	} else if (x <= 360.0) {
		tmp = 1.0;
	} else if (x <= 2.4e+206) {
		tmp = (((1.0 + (1.0 / eps_m)) * 1.0) - (((1.0 / eps_m) - 1.0) * 1.0)) / 2.0;
	} else {
		tmp = fma(((0.3333333333333333 * x) - 0.5), (x * x), 1.0);
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -2e-218)
		tmp = Float64(Float64(1.0 - Float64(Float64(x * Float64(Float64(1.0 - Float64(eps_m * eps_m)) / Float64(1.0 - eps_m))) - 1.0)) * 0.5);
	elseif (x <= 360.0)
		tmp = 1.0;
	elseif (x <= 2.4e+206)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * 1.0) - Float64(Float64(Float64(1.0 / eps_m) - 1.0) * 1.0)) / 2.0);
	else
		tmp = fma(Float64(Float64(0.3333333333333333 * x) - 0.5), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -2e-218], N[(N[(1.0 - N[(N[(x * N[(N[(1.0 - N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 360.0], 1.0, If[LessEqual[x, 2.4e+206], N[(N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] - N[(N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(0.3333333333333333 * x), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.0000000000000001e-218

   1. Initial program 69.5%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.0%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 66.8

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

   8. Applied rewrites 66.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 47.2%

       \[\leadsto \left(1 - \left(x \cdot \left(1 + \varepsilon\right) - 1\right)\right) \cdot 0.5 \]
   12. Step-by-step derivation

       1. flip-+ N/A

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

       2. lower-/.f64 N/A

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

       3. metadata-eval N/A

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

       4. unpow2 N/A

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

       5. lower--.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. lower--.f64 54.0

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

   13. Applied rewrites 54.0%

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

   if -2.0000000000000001e-218 < x < 360

   1. Initial program 50.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 79.8%

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

   if 360 < x < 2.4e206

   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

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

   5. Applied rewrites 25.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 66.3%

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

   if 2.4e206 < 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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 40.1%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 0.6

         \[\leadsto 1 + -0.5 \cdot \left(x \cdot x\right) \]

   8. Applied rewrites 0.6%

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

   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

       1. +-commutative N/A

          \[\leadsto {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + 1 \]

       2. *-commutative N/A

          \[\leadsto \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot {x}^{2} + 1 \]

       3. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x - \frac{1}{2}, {x}^{\color{blue}{2}}, 1\right) \]

       4. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x - \frac{1}{2}, {x}^{2}, 1\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x - \frac{1}{2}, {x}^{2}, 1\right) \]

       6. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x - \frac{1}{2}, x \cdot x, 1\right) \]

       7. lower-*.f64 61.5

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

   11. Applied rewrites 61.5%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-218}:\\ \;\;\;\;\left(1 - \left(x \cdot \frac{1 - \varepsilon \cdot \varepsilon}{1 - \varepsilon} - 1\right)\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 360:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+206}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot 1 - \left(\frac{1}{\varepsilon} - 1\right) \cdot 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.0% accurate, 6.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-218}:\\ \;\;\;\;\left(1 - \left(x \cdot \frac{1 - eps\_m \cdot eps\_m}{1 - eps\_m} - 1\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2e-218)
   (* (- 1.0 (- (* x (/ (- 1.0 (* eps_m eps_m)) (- 1.0 eps_m))) 1.0)) 0.5)
   (fma (- (* 0.3333333333333333 x) 0.5) (* x x) 1.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -2e-218) {
		tmp = (1.0 - ((x * ((1.0 - (eps_m * eps_m)) / (1.0 - eps_m))) - 1.0)) * 0.5;
	} else {
		tmp = fma(((0.3333333333333333 * x) - 0.5), (x * x), 1.0);
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -2e-218)
		tmp = Float64(Float64(1.0 - Float64(Float64(x * Float64(Float64(1.0 - Float64(eps_m * eps_m)) / Float64(1.0 - eps_m))) - 1.0)) * 0.5);
	else
		tmp = fma(Float64(Float64(0.3333333333333333 * x) - 0.5), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -2e-218], N[(N[(1.0 - N[(N[(x * N[(N[(1.0 - N[(eps$95$m * eps$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 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 x < -2.0000000000000001e-218

   1. Initial program 69.5%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.0%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 66.8

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

   8. Applied rewrites 66.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 47.2%

       \[\leadsto \left(1 - \left(x \cdot \left(1 + \varepsilon\right) - 1\right)\right) \cdot 0.5 \]
   12. Step-by-step derivation

       1. flip-+ N/A

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

       2. lower-/.f64 N/A

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

       3. metadata-eval N/A

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

       4. unpow2 N/A

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

       5. lower--.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 N/A

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

       8. lower--.f64 54.0

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

   13. Applied rewrites 54.0%

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

   if -2.0000000000000001e-218 < x

   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 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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 70.7%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 43.8

         \[\leadsto 1 + -0.5 \cdot \left(x \cdot x\right) \]

   8. Applied rewrites 43.8%

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

   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

       1. +-commutative N/A

          \[\leadsto {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + 1 \]

       2. *-commutative N/A

          \[\leadsto \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot {x}^{2} + 1 \]

       3. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x - \frac{1}{2}, {x}^{\color{blue}{2}}, 1\right) \]

       4. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x - \frac{1}{2}, {x}^{2}, 1\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x - \frac{1}{2}, {x}^{2}, 1\right) \]

       6. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x - \frac{1}{2}, x \cdot x, 1\right) \]

       7. lower-*.f64 57.4

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

   11. Applied rewrites 57.4%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-218}:\\ \;\;\;\;\left(1 - \left(x \cdot \frac{1 - \varepsilon \cdot \varepsilon}{1 - \varepsilon} - 1\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.0% accurate, 10.5× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-179}:\\ \;\;\;\;\left(1 - \left(x \cdot eps\_m - 1\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.2e-179)
   (* (- 1.0 (- (* x eps_m) 1.0)) 0.5)
   (fma (- (* 0.3333333333333333 x) 0.5) (* x x) 1.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.2e-179) {
		tmp = (1.0 - ((x * eps_m) - 1.0)) * 0.5;
	} else {
		tmp = fma(((0.3333333333333333 * x) - 0.5), (x * x), 1.0);
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.2e-179)
		tmp = Float64(Float64(1.0 - Float64(Float64(x * eps_m) - 1.0)) * 0.5);
	else
		tmp = fma(Float64(Float64(0.3333333333333333 * x) - 0.5), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.2e-179], N[(N[(1.0 - N[(N[(x * eps$95$m), $MachinePrecision] - 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 x < -1.2e-179

   1. Initial program 73.6%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.6%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 63.4

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

   8. Applied rewrites 63.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 41.3%

       \[\leadsto \left(1 - \left(x \cdot \left(1 + \varepsilon\right) - 1\right)\right) \cdot 0.5 \]

   12. Taylor expanded in eps around inf

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

   14. Applied rewrites 42.8%

       \[\leadsto \left(1 - \left(x \cdot \varepsilon - 1\right)\right) \cdot 0.5 \]

   if -1.2e-179 < x

   1. Initial program 70.5%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 72.2%

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

   6. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 47.5

         \[\leadsto 1 + -0.5 \cdot \left(x \cdot x\right) \]

   8. Applied rewrites 47.5%

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

   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

       1. +-commutative N/A

          \[\leadsto {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + 1 \]

       2. *-commutative N/A

          \[\leadsto \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot {x}^{2} + 1 \]

       3. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x - \frac{1}{2}, {x}^{\color{blue}{2}}, 1\right) \]

       4. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x - \frac{1}{2}, {x}^{2}, 1\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x - \frac{1}{2}, {x}^{2}, 1\right) \]

       6. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x - \frac{1}{2}, x \cdot x, 1\right) \]

       7. lower-*.f64 60.0

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

   11. Applied rewrites 60.0%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-179}:\\ \;\;\;\;\left(1 - \left(x \cdot \varepsilon - 1\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.5% accurate, 13.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -0.48:\\ \;\;\;\;\left(1 - x \cdot eps\_m\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -0.48) (* (- 1.0 (* x eps_m)) 0.5) 1.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -0.48) {
		tmp = (1.0 - (x * eps_m)) * 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

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

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

real(8) function code(x, eps_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-0.48d0)) then
        tmp = (1.0d0 - (x * eps_m)) * 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -0.48) {
		tmp = (1.0 - (x * eps_m)) * 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -0.48:
		tmp = (1.0 - (x * eps_m)) * 0.5
	else:
		tmp = 1.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -0.48)
		tmp = Float64(Float64(1.0 - Float64(x * eps_m)) * 0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -0.48)
		tmp = (1.0 - (x * eps_m)) * 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -0.48], N[(N[(1.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -0.47999999999999998

   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 eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.5%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 41.6

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

   8. Applied rewrites 41.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 16.6%

       \[\leadsto \left(1 - \left(x \cdot \left(1 + \varepsilon\right) - 1\right)\right) \cdot 0.5 \]

   12. Taylor expanded in eps around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 16.6

          \[\leadsto \left(1 - x \cdot \varepsilon\right) \cdot 0.5 \]

   14. Applied rewrites 16.6%

       \[\leadsto \left(1 - x \cdot \varepsilon\right) \cdot 0.5 \]

   if -0.47999999999999998 < x

   1. Initial program 66.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 52.0%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.48:\\ \;\;\;\;\left(1 - x \cdot \varepsilon\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 49.6% accurate, 16.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \left(1 - \left(x \cdot eps\_m - 1\right)\right) \cdot 0.5 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (* (- 1.0 (- (* x eps_m) 1.0)) 0.5))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return (1.0 - ((x * eps_m) - 1.0)) * 0.5;
}

Fortran

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

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

real(8) function code(x, eps_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    code = (1.0d0 - ((x * eps_m) - 1.0d0)) * 0.5d0
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return (1.0 - ((x * eps_m) - 1.0)) * 0.5;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return (1.0 - ((x * eps_m) - 1.0)) * 0.5

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return Float64(Float64(1.0 - Float64(Float64(x * eps_m) - 1.0)) * 0.5)
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = (1.0 - ((x * eps_m) - 1.0)) * 0.5;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := N[(N[(1.0 - N[(N[(x * eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 71.6%

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

3. Taylor expanded in eps around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 61.8

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

8. Applied rewrites 61.8%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 47.3%

    \[\leadsto \left(1 - \left(x \cdot \left(1 + \varepsilon\right) - 1\right)\right) \cdot 0.5 \]

12. Taylor expanded in eps around inf

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

14. Applied rewrites 48.1%

    \[\leadsto \left(1 - \left(x \cdot \varepsilon - 1\right)\right) \cdot 0.5 \]

15. Final simplification 48.1%

    \[\leadsto \left(1 - \left(x \cdot \varepsilon - 1\right)\right) \cdot 0.5 \]
16. Add Preprocessing

Alternative 16: 43.2% accurate, 273.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 44.4%

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

Reproduce

herbie shell --seed 2025044 
(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))