NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.7% → 99.6%

Time: 10.2s

Alternatives: 12

Speedup: 1.1×

• 37.3% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -2.0858671606645182e+160
  2. eps = 2.5827413663171862e-64

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 49.6%

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

   3. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. *-lft-identity N/A

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

      10. lower-fma.f64 46.0

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

   5. Applied rewrites 46.0%

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

   6. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 100.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. *-lft-identity N/A

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

      10. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 100.0

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

   10. Applied rewrites 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 64.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 50.3%

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

   3. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. *-lft-identity N/A

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

      10. lower-fma.f64 46.7

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

   5. Applied rewrites 46.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 83.5%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. *-lft-identity N/A

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

      10. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

      1. *-inverses N/A

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

      2. div-add N/A

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

      3. +-commutative N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower--.f64 57.0

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

   10. Applied rewrites 57.0%

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

4. Final simplification 72.5%

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

Alternative 3: 51.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 50.3%

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

   3. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. *-lft-identity N/A

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

      10. lower-fma.f64 46.7

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

   5. Applied rewrites 46.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 83.5%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. *-lft-identity N/A

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

      10. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. lft-mult-inverse N/A

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

      2. associate-*l/ N/A

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

      3. *-lft-identity N/A

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

      4. div-add-rev N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower--.f64 57.0

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

   8. Applied rewrites 57.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 23.5%

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

4. Final simplification 58.6%

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

Alternative 4: 79.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{1}{\varepsilon}\\ \mathbf{if}\;\varepsilon \leq -7200000000:\\ \;\;\;\;\frac{t\_0 \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left(\mathsf{fma}\left(x, \varepsilon, x\right) - 1\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 0.0135:\\ \;\;\;\;\mathsf{fma}\left(e^{-x}, \left(1 + x\right) - -1, \frac{x}{e^{x}}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ 1.0 eps))))
   (if (<= eps -7200000000.0)
     (/ (- (* t_0 (exp (* (+ -1.0 eps) x))) (- (fma x eps x) 1.0)) 2.0)
     (if (<= eps 0.0135)
       (* (fma (exp (- x)) (- (+ 1.0 x) -1.0) (/ x (exp x))) 0.5)
       (/
        (-
         (* t_0 (fma (- eps 1.0) x 1.0))
         (* (- (/ 1.0 eps) 1.0) (exp (* (- -1.0 eps) x))))
        2.0)))))

C

double code(double x, double eps) {
	double t_0 = 1.0 + (1.0 / eps);
	double tmp;
	if (eps <= -7200000000.0) {
		tmp = ((t_0 * exp(((-1.0 + eps) * x))) - (fma(x, eps, x) - 1.0)) / 2.0;
	} else if (eps <= 0.0135) {
		tmp = fma(exp(-x), ((1.0 + x) - -1.0), (x / exp(x))) * 0.5;
	} else {
		tmp = ((t_0 * fma((eps - 1.0), x, 1.0)) - (((1.0 / eps) - 1.0) * exp(((-1.0 - eps) * x)))) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(1.0 + Float64(1.0 / eps))
	tmp = 0.0
	if (eps <= -7200000000.0)
		tmp = Float64(Float64(Float64(t_0 * exp(Float64(Float64(-1.0 + eps) * x))) - Float64(fma(x, eps, x) - 1.0)) / 2.0);
	elseif (eps <= 0.0135)
		tmp = Float64(fma(exp(Float64(-x)), Float64(Float64(1.0 + x) - -1.0), Float64(x / exp(x))) * 0.5);
	else
		tmp = Float64(Float64(Float64(t_0 * fma(Float64(eps - 1.0), x, 1.0)) - Float64(Float64(Float64(1.0 / eps) - 1.0) * exp(Float64(Float64(-1.0 - eps) * x)))) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -7200000000.0], N[(N[(N[(t$95$0 * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(x * eps + x), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps, 0.0135], N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(N[(1.0 + x), $MachinePrecision] - -1.0), $MachinePrecision] + N[(x / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(t$95$0 * N[(N[(eps - 1.0), $MachinePrecision] * x + 1.0), $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]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -7.2e9

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. *-lft-identity N/A

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

      10. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 72.8%

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

   if -7.2e9 < eps < 0.0134999999999999998

   1. Initial program 29.6%

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

   3. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. *-lft-identity N/A

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

      10. lower-fma.f64 26.6

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

   5. Applied rewrites 26.6%

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

   6. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 100.0%

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

   if 0.0134999999999999998 < eps

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 71.4

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

   5. Applied rewrites 71.4%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -7200000000:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left(\mathsf{fma}\left(x, \varepsilon, x\right) - 1\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 0.0135:\\ \;\;\;\;\mathsf{fma}\left(e^{-x}, \left(1 + x\right) - -1, \frac{x}{e^{x}}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{1}{\varepsilon}\\ \mathbf{if}\;\varepsilon \leq -7200000000:\\ \;\;\;\;\frac{t\_0 \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left(\mathsf{fma}\left(x, \varepsilon, x\right) - 1\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 0.0135:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ 1.0 eps))))
   (if (<= eps -7200000000.0)
     (/ (- (* t_0 (exp (* (+ -1.0 eps) x))) (- (fma x eps x) 1.0)) 2.0)
     (if (<= eps 0.0135)
       1.0
       (/
        (-
         (* t_0 (fma (- eps 1.0) x 1.0))
         (* (- (/ 1.0 eps) 1.0) (exp (* (- -1.0 eps) x))))
        2.0)))))

C

double code(double x, double eps) {
	double t_0 = 1.0 + (1.0 / eps);
	double tmp;
	if (eps <= -7200000000.0) {
		tmp = ((t_0 * exp(((-1.0 + eps) * x))) - (fma(x, eps, x) - 1.0)) / 2.0;
	} else if (eps <= 0.0135) {
		tmp = 1.0;
	} else {
		tmp = ((t_0 * fma((eps - 1.0), x, 1.0)) - (((1.0 / eps) - 1.0) * exp(((-1.0 - eps) * x)))) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(1.0 + Float64(1.0 / eps))
	tmp = 0.0
	if (eps <= -7200000000.0)
		tmp = Float64(Float64(Float64(t_0 * exp(Float64(Float64(-1.0 + eps) * x))) - Float64(fma(x, eps, x) - 1.0)) / 2.0);
	elseif (eps <= 0.0135)
		tmp = 1.0;
	else
		tmp = Float64(Float64(Float64(t_0 * fma(Float64(eps - 1.0), x, 1.0)) - Float64(Float64(Float64(1.0 / eps) - 1.0) * exp(Float64(Float64(-1.0 - eps) * x)))) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -7200000000.0], N[(N[(N[(t$95$0 * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(x * eps + x), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps, 0.0135], 1.0, N[(N[(N[(t$95$0 * N[(N[(eps - 1.0), $MachinePrecision] * x + 1.0), $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]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -7.2e9

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. *-lft-identity N/A

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

      10. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 72.8%

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

   if -7.2e9 < eps < 0.0134999999999999998

   1. Initial program 29.6%

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

   3. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. *-lft-identity N/A

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

      10. lower-fma.f64 26.6

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

   5. Applied rewrites 26.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 77.2%

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

   if 0.0134999999999999998 < eps

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 71.4

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

   5. Applied rewrites 71.4%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -7200000000:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left(\mathsf{fma}\left(x, \varepsilon, x\right) - 1\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 0.0135:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{1}{\varepsilon}\\ \mathbf{if}\;\varepsilon \leq -7200000000:\\ \;\;\;\;\frac{t\_0 \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left(\mathsf{fma}\left(x, \varepsilon, x\right) - 1\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 1000000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{\varepsilon} - 1, 1 + \varepsilon, \varepsilon - \frac{1}{\varepsilon}\right), x, 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ 1.0 eps))))
   (if (<= eps -7200000000.0)
     (/ (- (* t_0 (exp (* (+ -1.0 eps) x))) (- (fma x eps x) 1.0)) 2.0)
     (if (<= eps 1000000000000.0)
       (/
        (fma (fma (- (/ 1.0 eps) 1.0) (+ 1.0 eps) (- eps (/ 1.0 eps))) x 2.0)
        2.0)
       (/
        (- (* t_0 (fma (- eps 1.0) x 1.0)) (- (exp (- (fma x eps x)))))
        2.0)))))

C

double code(double x, double eps) {
	double t_0 = 1.0 + (1.0 / eps);
	double tmp;
	if (eps <= -7200000000.0) {
		tmp = ((t_0 * exp(((-1.0 + eps) * x))) - (fma(x, eps, x) - 1.0)) / 2.0;
	} else if (eps <= 1000000000000.0) {
		tmp = fma(fma(((1.0 / eps) - 1.0), (1.0 + eps), (eps - (1.0 / eps))), x, 2.0) / 2.0;
	} else {
		tmp = ((t_0 * fma((eps - 1.0), x, 1.0)) - -exp(-fma(x, eps, x))) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(1.0 + Float64(1.0 / eps))
	tmp = 0.0
	if (eps <= -7200000000.0)
		tmp = Float64(Float64(Float64(t_0 * exp(Float64(Float64(-1.0 + eps) * x))) - Float64(fma(x, eps, x) - 1.0)) / 2.0);
	elseif (eps <= 1000000000000.0)
		tmp = Float64(fma(fma(Float64(Float64(1.0 / eps) - 1.0), Float64(1.0 + eps), Float64(eps - Float64(1.0 / eps))), x, 2.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(t_0 * fma(Float64(eps - 1.0), x, 1.0)) - Float64(-exp(Float64(-fma(x, eps, x))))) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -7200000000.0], N[(N[(N[(t$95$0 * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(x * eps + x), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps, 1000000000000.0], N[(N[(N[(N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision] * N[(1.0 + eps), $MachinePrecision] + N[(eps - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(t$95$0 * N[(N[(eps - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] - (-N[Exp[(-N[(x * eps + x), $MachinePrecision])], $MachinePrecision])), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -7.2e9

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. *-lft-identity N/A

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

      10. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 72.8%

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

   if -7.2e9 < eps < 1e12

   1. Initial program 32.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 77.1%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 77.1%

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

   if 1e12 < eps

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. *-lft-identity N/A

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

      10. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 71.1

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

   10. Applied rewrites 71.1%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -7200000000:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left(\mathsf{fma}\left(x, \varepsilon, x\right) - 1\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 1000000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{\varepsilon} - 1, 1 + \varepsilon, \varepsilon - \frac{1}{\varepsilon}\right), x, 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \mathsf{fma}\left(\varepsilon - 1, x, 1\right) - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\varepsilon} - 1\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, 1 + \varepsilon, \frac{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(2 - \varepsilon, \varepsilon \cdot \varepsilon, -2\right), \varepsilon, 1\right)}{\varepsilon}}{\varepsilon}}{2 - \frac{1}{\varepsilon}}\right), x, 2\right)}{2}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+17}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \frac{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}{\varepsilon - 1}, \varepsilon - \frac{1}{\varepsilon}\right), x, 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\varepsilon - -1}{\varepsilon} - t\_0}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 eps) 1.0)))
   (if (<= x -7.2e-9)
     (/
      (fma
       (fma
        t_0
        (+ 1.0 eps)
        (/
         (/ (/ (fma (fma (- 2.0 eps) (* eps eps) -2.0) eps 1.0) eps) eps)
         (- 2.0 (/ 1.0 eps))))
       x
       2.0)
      2.0)
     (if (<= x 8e+17)
       (/
        (fma
         (fma t_0 (/ (fma eps eps -1.0) (- eps 1.0)) (- eps (/ 1.0 eps)))
         x
         2.0)
        2.0)
       (/ (- (/ (- eps -1.0) eps) t_0) 2.0)))))

C

double code(double x, double eps) {
	double t_0 = (1.0 / eps) - 1.0;
	double tmp;
	if (x <= -7.2e-9) {
		tmp = fma(fma(t_0, (1.0 + eps), (((fma(fma((2.0 - eps), (eps * eps), -2.0), eps, 1.0) / eps) / eps) / (2.0 - (1.0 / eps)))), x, 2.0) / 2.0;
	} else if (x <= 8e+17) {
		tmp = fma(fma(t_0, (fma(eps, eps, -1.0) / (eps - 1.0)), (eps - (1.0 / eps))), x, 2.0) / 2.0;
	} else {
		tmp = (((eps - -1.0) / eps) - t_0) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(1.0 / eps) - 1.0)
	tmp = 0.0
	if (x <= -7.2e-9)
		tmp = Float64(fma(fma(t_0, Float64(1.0 + eps), Float64(Float64(Float64(fma(fma(Float64(2.0 - eps), Float64(eps * eps), -2.0), eps, 1.0) / eps) / eps) / Float64(2.0 - Float64(1.0 / eps)))), x, 2.0) / 2.0);
	elseif (x <= 8e+17)
		tmp = Float64(fma(fma(t_0, Float64(fma(eps, eps, -1.0) / Float64(eps - 1.0)), Float64(eps - Float64(1.0 / eps))), x, 2.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(eps - -1.0) / eps) - t_0) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -7.2e-9], N[(N[(N[(t$95$0 * N[(1.0 + eps), $MachinePrecision] + N[(N[(N[(N[(N[(N[(2.0 - eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + -2.0), $MachinePrecision] * eps + 1.0), $MachinePrecision] / eps), $MachinePrecision] / eps), $MachinePrecision] / N[(2.0 - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 8e+17], N[(N[(N[(t$95$0 * N[(N[(eps * eps + -1.0), $MachinePrecision] / N[(eps - 1.0), $MachinePrecision]), $MachinePrecision] + N[(eps - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(eps - -1.0), $MachinePrecision] / eps), $MachinePrecision] - t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.2e-9

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 3.1%

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

   7. Applied rewrites 22.1%

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

   8. Taylor expanded in eps around 0

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

   10. Applied rewrites 25.1%

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

   11. Taylor expanded in eps around 0

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

   13. Applied rewrites 80.6%

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

   if -7.2e-9 < x < 8e17

   1. Initial program 54.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 76.2%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 76.2%

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

   10. Applied rewrites 81.7%

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

   if 8e17 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 32.4

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

   5. Applied rewrites 32.4%

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

   6. Taylor expanded in x around 0

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

      1. lft-mult-inverse N/A

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

      2. associate-*l/ N/A

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

      3. *-lft-identity N/A

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

      4. div-add-rev N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower--.f64 40.6

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

   8. Applied rewrites 40.6%

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

Alternative 8: 65.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\varepsilon} - 1\\ t_1 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, 1 + \varepsilon, \frac{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}{\varepsilon}\right), x, 2\right)}{2}\\ t_2 := \frac{\varepsilon - -1}{\varepsilon}\\ \mathbf{if}\;x \leq -6.6 \cdot 10^{+44}:\\ \;\;\;\;\frac{t\_2 - \left(\mathsf{fma}\left(x, \varepsilon, x\right) - 1\right)}{2}\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-197}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{-221}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 720:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2 - t\_0}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 eps) 1.0))
        (t_1
         (/ (fma (fma t_0 (+ 1.0 eps) (/ (fma eps eps -1.0) eps)) x 2.0) 2.0))
        (t_2 (/ (- eps -1.0) eps)))
   (if (<= x -6.6e+44)
     (/ (- t_2 (- (fma x eps x) 1.0)) 2.0)
     (if (<= x -1.8e-197)
       t_1
       (if (<= x 4.4e-221) 1.0 (if (<= x 720.0) t_1 (/ (- t_2 t_0) 2.0)))))))

C

double code(double x, double eps) {
	double t_0 = (1.0 / eps) - 1.0;
	double t_1 = fma(fma(t_0, (1.0 + eps), (fma(eps, eps, -1.0) / eps)), x, 2.0) / 2.0;
	double t_2 = (eps - -1.0) / eps;
	double tmp;
	if (x <= -6.6e+44) {
		tmp = (t_2 - (fma(x, eps, x) - 1.0)) / 2.0;
	} else if (x <= -1.8e-197) {
		tmp = t_1;
	} else if (x <= 4.4e-221) {
		tmp = 1.0;
	} else if (x <= 720.0) {
		tmp = t_1;
	} else {
		tmp = (t_2 - t_0) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(1.0 / eps) - 1.0)
	t_1 = Float64(fma(fma(t_0, Float64(1.0 + eps), Float64(fma(eps, eps, -1.0) / eps)), x, 2.0) / 2.0)
	t_2 = Float64(Float64(eps - -1.0) / eps)
	tmp = 0.0
	if (x <= -6.6e+44)
		tmp = Float64(Float64(t_2 - Float64(fma(x, eps, x) - 1.0)) / 2.0);
	elseif (x <= -1.8e-197)
		tmp = t_1;
	elseif (x <= 4.4e-221)
		tmp = 1.0;
	elseif (x <= 720.0)
		tmp = t_1;
	else
		tmp = Float64(Float64(t_2 - t_0) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(t$95$0 * N[(1.0 + eps), $MachinePrecision] + N[(N[(eps * eps + -1.0), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(eps - -1.0), $MachinePrecision] / eps), $MachinePrecision]}, If[LessEqual[x, -6.6e+44], N[(N[(t$95$2 - N[(N[(x * eps + x), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -1.8e-197], t$95$1, If[LessEqual[x, 4.4e-221], 1.0, If[LessEqual[x, 720.0], t$95$1, N[(N[(t$95$2 - t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6.60000000000000027e44

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. *-lft-identity N/A

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

      10. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. lft-mult-inverse N/A

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

      2. associate-*l/ N/A

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

      3. *-lft-identity N/A

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

      4. div-add-rev N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower--.f64 67.7

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

   8. Applied rewrites 67.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 39.5%

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

   if -6.60000000000000027e44 < x < -1.7999999999999999e-197 or 4.40000000000000003e-221 < x < 720

   1. Initial program 54.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 67.8%

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

   7. Applied rewrites 52.8%

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

   9. Applied rewrites 61.2%

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

   11. Applied rewrites 79.0%

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

   if -1.7999999999999999e-197 < x < 4.40000000000000003e-221

   1. Initial program 57.2%

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

   3. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. *-lft-identity N/A

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

      10. lower-fma.f64 55.7

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

   5. Applied rewrites 55.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 90.7%

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

   if 720 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 33.0

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

   5. Applied rewrites 33.0%

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

   6. Taylor expanded in x around 0

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

      1. lft-mult-inverse N/A

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

      2. associate-*l/ N/A

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

      3. *-lft-identity N/A

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

      4. div-add-rev N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower--.f64 39.5

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

   8. Applied rewrites 39.5%

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

Alternative 9: 61.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\varepsilon} - 1\\ \mathbf{if}\;x \leq 8 \cdot 10^{+17}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \frac{\mathsf{fma}\left(\varepsilon, \varepsilon, -1\right)}{\varepsilon - 1}, \varepsilon - \frac{1}{\varepsilon}\right), x, 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\varepsilon - -1}{\varepsilon} - t\_0}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 eps) 1.0)))
   (if (<= x 8e+17)
     (/
      (fma
       (fma t_0 (/ (fma eps eps -1.0) (- eps 1.0)) (- eps (/ 1.0 eps)))
       x
       2.0)
      2.0)
     (/ (- (/ (- eps -1.0) eps) t_0) 2.0))))

C

double code(double x, double eps) {
	double t_0 = (1.0 / eps) - 1.0;
	double tmp;
	if (x <= 8e+17) {
		tmp = fma(fma(t_0, (fma(eps, eps, -1.0) / (eps - 1.0)), (eps - (1.0 / eps))), x, 2.0) / 2.0;
	} else {
		tmp = (((eps - -1.0) / eps) - t_0) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(1.0 / eps) - 1.0)
	tmp = 0.0
	if (x <= 8e+17)
		tmp = Float64(fma(fma(t_0, Float64(fma(eps, eps, -1.0) / Float64(eps - 1.0)), Float64(eps - Float64(1.0 / eps))), x, 2.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(eps - -1.0) / eps) - t_0) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, 8e+17], N[(N[(N[(t$95$0 * N[(N[(eps * eps + -1.0), $MachinePrecision] / N[(eps - 1.0), $MachinePrecision]), $MachinePrecision] + N[(eps - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(eps - -1.0), $MachinePrecision] / eps), $MachinePrecision] - t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 8e17

   1. Initial program 61.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 65.2%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 65.2%

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

   10. Applied rewrites 74.9%

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

   if 8e17 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 32.4

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

   5. Applied rewrites 32.4%

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

   6. Taylor expanded in x around 0

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

      1. lft-mult-inverse N/A

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

      2. associate-*l/ N/A

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

      3. *-lft-identity N/A

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

      4. div-add-rev N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower--.f64 40.6

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

   8. Applied rewrites 40.6%

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

Alternative 10: 60.0% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\varepsilon} - 1\\ \mathbf{if}\;x \leq 4.2 \cdot 10^{+17}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, 1 + \varepsilon, \frac{-1}{\varepsilon}\right), x, 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\varepsilon - -1}{\varepsilon} - t\_0}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 eps) 1.0)))
   (if (<= x 4.2e+17)
     (/ (fma (fma t_0 (+ 1.0 eps) (/ -1.0 eps)) x 2.0) 2.0)
     (/ (- (/ (- eps -1.0) eps) t_0) 2.0))))

C

double code(double x, double eps) {
	double t_0 = (1.0 / eps) - 1.0;
	double tmp;
	if (x <= 4.2e+17) {
		tmp = fma(fma(t_0, (1.0 + eps), (-1.0 / eps)), x, 2.0) / 2.0;
	} else {
		tmp = (((eps - -1.0) / eps) - t_0) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(1.0 / eps) - 1.0)
	tmp = 0.0
	if (x <= 4.2e+17)
		tmp = Float64(fma(fma(t_0, Float64(1.0 + eps), Float64(-1.0 / eps)), x, 2.0) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(eps - -1.0) / eps) - t_0) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, 4.2e+17], N[(N[(N[(t$95$0 * N[(1.0 + eps), $MachinePrecision] + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision] * x + 2.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(eps - -1.0), $MachinePrecision] / eps), $MachinePrecision] - t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.2e17

   1. Initial program 61.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 65.2%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 69.1%

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

   if 4.2e17 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 32.4

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

   5. Applied rewrites 32.4%

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

   6. Taylor expanded in x around 0

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

      1. lft-mult-inverse N/A

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

      2. associate-*l/ N/A

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

      3. *-lft-identity N/A

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

      4. div-add-rev N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower--.f64 40.6

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

   8. Applied rewrites 40.6%

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

Alternative 11: 61.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\varepsilon - -1}{\varepsilon}\\ \mathbf{if}\;x \leq -2 \cdot 10^{+39}:\\ \;\;\;\;\frac{t\_0 - \left(\mathsf{fma}\left(x, \varepsilon, x\right) - 1\right)}{2}\\ \mathbf{elif}\;x \leq 370:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (- eps -1.0) eps)))
   (if (<= x -2e+39)
     (/ (- t_0 (- (fma x eps x) 1.0)) 2.0)
     (if (<= x 370.0) 1.0 (/ (- t_0 (- (/ 1.0 eps) 1.0)) 2.0)))))

C

double code(double x, double eps) {
	double t_0 = (eps - -1.0) / eps;
	double tmp;
	if (x <= -2e+39) {
		tmp = (t_0 - (fma(x, eps, x) - 1.0)) / 2.0;
	} else if (x <= 370.0) {
		tmp = 1.0;
	} else {
		tmp = (t_0 - ((1.0 / eps) - 1.0)) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(eps - -1.0) / eps)
	tmp = 0.0
	if (x <= -2e+39)
		tmp = Float64(Float64(t_0 - Float64(fma(x, eps, x) - 1.0)) / 2.0);
	elseif (x <= 370.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(t_0 - Float64(Float64(1.0 / eps) - 1.0)) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(eps - -1.0), $MachinePrecision] / eps), $MachinePrecision]}, If[LessEqual[x, -2e+39], N[(N[(t$95$0 - N[(N[(x * eps + x), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 370.0], 1.0, N[(N[(t$95$0 - N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.99999999999999988e39

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. *-lft-identity N/A

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

      10. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. lft-mult-inverse N/A

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

      2. associate-*l/ N/A

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

      3. *-lft-identity N/A

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

      4. div-add-rev N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower--.f64 67.7

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

   8. Applied rewrites 67.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 39.5%

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

   if -1.99999999999999988e39 < x < 370

   1. Initial program 55.6%

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

   3. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. *-lft-identity N/A

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

      10. lower-fma.f64 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 74.9%

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

   if 370 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 33.0

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

   5. Applied rewrites 33.0%

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

   6. Taylor expanded in x around 0

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

      1. lft-mult-inverse N/A

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

      2. associate-*l/ N/A

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

      3. *-lft-identity N/A

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

      4. div-add-rev N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower--.f64 39.5

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

   8. Applied rewrites 39.5%

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

Alternative 12: 44.8% accurate, 273.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 1.0)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, eps):
	return 1.0

Julia

function code(x, eps)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 1.0

Derivation

1. Initial program 70.9%

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

3. Taylor expanded in eps around inf

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

   1. exp-neg N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f64 N/A

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

   5. lower-exp.f64 N/A

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

   6. +-commutative N/A

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

   7. distribute-rgt-in N/A

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

   8. *-commutative N/A

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

   9. *-lft-identity N/A

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

   10. lower-fma.f64 68.8

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

5. Applied rewrites 68.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 50.2%

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

Reproduce

herbie shell --seed 2025010 
(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))