NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.4% → 98.9%

Time: 5.2s

Alternatives: 12

Speedup: 2.3×

• 37.6% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -2.6600819792370447e+205
  2. eps = 6.330960394108097e-226

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.4% accurate, 1.0× speedup

Math

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

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: 98.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

2. Taylor expanded in eps around inf

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-exp.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-+.f64 98.9

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

4. Applied rewrites 98.9%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. exp-neg N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. mult-flip-rev N/A

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

   9. lower-/.f64 N/A

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

   10. lower-exp.f64 98.9

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-+.f64 N/A

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

   14. distribute-lft-in N/A

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

   15. *-rgt-identity N/A

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

   16. +-commutative N/A

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

   17. lower-fma.f64 98.9

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

6. Applied rewrites 98.9%

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

Alternative 2: 98.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|\varepsilon\right| \leq 0.65:\\ \;\;\;\;\left(2 \cdot e^{-x}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{\left|\varepsilon\right| \cdot x} - \frac{-1}{e^{\mathsf{fma}\left(x, \left|\varepsilon\right|, x\right)}}\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (fabs eps) 0.65)
   (* (* 2.0 (exp (- x))) 0.5)
   (* 0.5 (- (exp (* (fabs eps) x)) (/ -1.0 (exp (fma x (fabs eps) x)))))))

C

double code(double x, double eps) {
	double tmp;
	if (fabs(eps) <= 0.65) {
		tmp = (2.0 * exp(-x)) * 0.5;
	} else {
		tmp = 0.5 * (exp((fabs(eps) * x)) - (-1.0 / exp(fma(x, fabs(eps), x))));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (abs(eps) <= 0.65)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) * 0.5);
	else
		tmp = Float64(0.5 * Float64(exp(Float64(abs(eps) * x)) - Float64(-1.0 / exp(fma(x, abs(eps), x)))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[Abs[eps], $MachinePrecision], 0.65], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[(N[Exp[N[(N[Abs[eps], $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] - N[(-1.0 / N[Exp[N[(x * N[Abs[eps], $MachinePrecision] + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 0.650000000000000022

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in eps around 0

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

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 70.8

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

   7. Applied rewrites 70.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 70.8

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 70.8

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

   9. Applied rewrites 70.8%

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

   if 0.650000000000000022 < eps

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 98.9

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

   4. Applied rewrites 98.9%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. lower-/.f64 N/A

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

      10. lower-exp.f64 98.9

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft-in N/A

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

      15. *-rgt-identity N/A

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

      16. +-commutative N/A

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

      17. lower-fma.f64 98.9

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

   6. Applied rewrites 98.9%

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

   7. Taylor expanded in eps around inf

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

      1. lower-*.f64 88.3

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

   9. Applied rewrites 88.3%

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

Alternative 3: 98.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 73.4%

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

2. Taylor expanded in eps around inf

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-exp.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-+.f64 98.9

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

4. Applied rewrites 98.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 98.9

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

6. Applied rewrites 98.9%

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

Alternative 4: 98.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|\varepsilon\right| \leq 0.65:\\ \;\;\;\;\left(2 \cdot e^{-x}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-\mathsf{fma}\left(x, \left|\varepsilon\right|, x\right)} + e^{x \cdot \left|\varepsilon\right|}\right) \cdot 0.5\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (fabs eps) 0.65)
   (* (* 2.0 (exp (- x))) 0.5)
   (* (+ (exp (- (fma x (fabs eps) x))) (exp (* x (fabs eps)))) 0.5)))

C

double code(double x, double eps) {
	double tmp;
	if (fabs(eps) <= 0.65) {
		tmp = (2.0 * exp(-x)) * 0.5;
	} else {
		tmp = (exp(-fma(x, fabs(eps), x)) + exp((x * fabs(eps)))) * 0.5;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (abs(eps) <= 0.65)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) * 0.5);
	else
		tmp = Float64(Float64(exp(Float64(-fma(x, abs(eps), x))) + exp(Float64(x * abs(eps)))) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[Abs[eps], $MachinePrecision], 0.65], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Exp[(-N[(x * N[Abs[eps], $MachinePrecision] + x), $MachinePrecision])], $MachinePrecision] + N[Exp[N[(x * N[Abs[eps], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 0.650000000000000022

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in eps around 0

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

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 70.8

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

   7. Applied rewrites 70.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 70.8

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 70.8

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

   9. Applied rewrites 70.8%

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

   if 0.650000000000000022 < eps

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 98.9

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

   4. Applied rewrites 98.9%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. lower-/.f64 N/A

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

      10. lower-exp.f64 98.9

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft-in N/A

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

      15. *-rgt-identity N/A

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

      16. +-commutative N/A

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

      17. lower-fma.f64 98.9

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

   6. Applied rewrites 98.9%

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

   7. Taylor expanded in eps around inf

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

      1. lower-*.f64 88.3

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

   9. Applied rewrites 88.3%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 88.3

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

   11. Applied rewrites 88.3%

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

Alternative 5: 84.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\left|\varepsilon\right|}\\ \mathbf{if}\;x \leq -2 \cdot 10^{+55}:\\ \;\;\;\;0.5 \cdot \left(e^{-x} - -1\right)\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-263}:\\ \;\;\;\;0.5 \cdot \left(\left(1 + x \cdot \left(\left|\varepsilon\right| - 1\right)\right) - \frac{-1}{e^{\mathsf{fma}\left(x, \left|\varepsilon\right|, x\right)}}\right)\\ \mathbf{elif}\;x \leq 750:\\ \;\;\;\;0.5 \cdot \left(e^{\left|\varepsilon\right| \cdot x} - \left(x \cdot \left(1 + \left|\varepsilon\right|\right) - 1\right)\right)\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+59}:\\ \;\;\;\;\frac{\left(1 + t\_0\right) - \left(t\_0 - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-x \cdot \left(1 - \left|\varepsilon\right|\right)} - -1\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs eps))))
   (if (<= x -2e+55)
     (* 0.5 (- (exp (- x)) -1.0))
     (if (<= x -2e-263)
       (*
        0.5
        (-
         (+ 1.0 (* x (- (fabs eps) 1.0)))
         (/ -1.0 (exp (fma x (fabs eps) x)))))
       (if (<= x 750.0)
         (* 0.5 (- (exp (* (fabs eps) x)) (- (* x (+ 1.0 (fabs eps))) 1.0)))
         (if (<= x 3.2e+59)
           (/ (- (+ 1.0 t_0) (- t_0 1.0)) 2.0)
           (* 0.5 (- (exp (- (* x (- 1.0 (fabs eps))))) -1.0))))))))

C

double code(double x, double eps) {
	double t_0 = 1.0 / fabs(eps);
	double tmp;
	if (x <= -2e+55) {
		tmp = 0.5 * (exp(-x) - -1.0);
	} else if (x <= -2e-263) {
		tmp = 0.5 * ((1.0 + (x * (fabs(eps) - 1.0))) - (-1.0 / exp(fma(x, fabs(eps), x))));
	} else if (x <= 750.0) {
		tmp = 0.5 * (exp((fabs(eps) * x)) - ((x * (1.0 + fabs(eps))) - 1.0));
	} else if (x <= 3.2e+59) {
		tmp = ((1.0 + t_0) - (t_0 - 1.0)) / 2.0;
	} else {
		tmp = 0.5 * (exp(-(x * (1.0 - fabs(eps)))) - -1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(1.0 / abs(eps))
	tmp = 0.0
	if (x <= -2e+55)
		tmp = Float64(0.5 * Float64(exp(Float64(-x)) - -1.0));
	elseif (x <= -2e-263)
		tmp = Float64(0.5 * Float64(Float64(1.0 + Float64(x * Float64(abs(eps) - 1.0))) - Float64(-1.0 / exp(fma(x, abs(eps), x)))));
	elseif (x <= 750.0)
		tmp = Float64(0.5 * Float64(exp(Float64(abs(eps) * x)) - Float64(Float64(x * Float64(1.0 + abs(eps))) - 1.0)));
	elseif (x <= 3.2e+59)
		tmp = Float64(Float64(Float64(1.0 + t_0) - Float64(t_0 - 1.0)) / 2.0);
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-Float64(x * Float64(1.0 - abs(eps))))) - -1.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 / N[Abs[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2e+55], N[(0.5 * N[(N[Exp[(-x)], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2e-263], N[(0.5 * N[(N[(1.0 + N[(x * N[(N[Abs[eps], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1.0 / N[Exp[N[(x * N[Abs[eps], $MachinePrecision] + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 750.0], N[(0.5 * N[(N[Exp[N[(N[Abs[eps], $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] - N[(N[(x * N[(1.0 + N[Abs[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.2e+59], N[(N[(N[(1.0 + t$95$0), $MachinePrecision] - N[(t$95$0 - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(0.5 * N[(N[Exp[(-N[(x * N[(1.0 - N[Abs[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -2.00000000000000002e55

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 64.3%

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

   8. Taylor expanded in eps around inf

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

      1. lower-*.f64 64.5

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

   10. Applied rewrites 64.5%

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

   11. Taylor expanded in eps around 0

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

       1. lower-exp.f64 N/A

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

       2. lower-neg.f64 57.2

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

   13. Applied rewrites 57.2%

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

   if -2.00000000000000002e55 < x < -2e-263

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 98.9

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

   4. Applied rewrites 98.9%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. lower-/.f64 N/A

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

      10. lower-exp.f64 98.9

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft-in N/A

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

      15. *-rgt-identity N/A

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

      16. +-commutative N/A

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

      17. lower-fma.f64 98.9

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

   6. Applied rewrites 98.9%

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

   7. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 64.0

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

   9. Applied rewrites 64.0%

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

   if -2e-263 < x < 750

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 98.9

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

   4. Applied rewrites 98.9%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. lower-/.f64 N/A

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

      10. lower-exp.f64 98.9

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft-in N/A

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

      15. *-rgt-identity N/A

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

      16. +-commutative N/A

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

      17. lower-fma.f64 98.9

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

   6. Applied rewrites 98.9%

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

   7. Taylor expanded in eps around inf

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

      1. lower-*.f64 88.3

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

   9. Applied rewrites 88.3%

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

   10. Taylor expanded in x around 0

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

       1. lower--.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-+.f64 64.5

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

   12. Applied rewrites 64.5%

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

   if 750 < x < 3.19999999999999982e59

   1. Initial program 73.4%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 38.5

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

   4. Applied rewrites 38.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-/.f64 31.3

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

   7. Applied rewrites 31.3%

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

   if 3.19999999999999982e59 < x

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 64.3%

      \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1\right) \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 6: 77.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|\varepsilon\right| \leq 3.95 \cdot 10^{+65}:\\ \;\;\;\;\left(2 \cdot e^{-x}\right) \cdot 0.5\\ \mathbf{elif}\;\left|\varepsilon\right| \leq 1.8 \cdot 10^{+255}:\\ \;\;\;\;0.5 \cdot \left(e^{\left|\varepsilon\right| \cdot x} - \left(x \cdot \left(1 + \left|\varepsilon\right|\right) - 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-x \cdot \left(1 - \left|\varepsilon\right|\right)} - -1\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (fabs eps) 3.95e+65)
   (* (* 2.0 (exp (- x))) 0.5)
   (if (<= (fabs eps) 1.8e+255)
     (* 0.5 (- (exp (* (fabs eps) x)) (- (* x (+ 1.0 (fabs eps))) 1.0)))
     (* 0.5 (- (exp (- (* x (- 1.0 (fabs eps))))) -1.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (fabs(eps) <= 3.95e+65) {
		tmp = (2.0 * exp(-x)) * 0.5;
	} else if (fabs(eps) <= 1.8e+255) {
		tmp = 0.5 * (exp((fabs(eps) * x)) - ((x * (1.0 + fabs(eps))) - 1.0));
	} else {
		tmp = 0.5 * (exp(-(x * (1.0 - fabs(eps)))) - -1.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (abs(eps) <= 3.95d+65) then
        tmp = (2.0d0 * exp(-x)) * 0.5d0
    else if (abs(eps) <= 1.8d+255) then
        tmp = 0.5d0 * (exp((abs(eps) * x)) - ((x * (1.0d0 + abs(eps))) - 1.0d0))
    else
        tmp = 0.5d0 * (exp(-(x * (1.0d0 - abs(eps)))) - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (Math.abs(eps) <= 3.95e+65) {
		tmp = (2.0 * Math.exp(-x)) * 0.5;
	} else if (Math.abs(eps) <= 1.8e+255) {
		tmp = 0.5 * (Math.exp((Math.abs(eps) * x)) - ((x * (1.0 + Math.abs(eps))) - 1.0));
	} else {
		tmp = 0.5 * (Math.exp(-(x * (1.0 - Math.abs(eps)))) - -1.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if math.fabs(eps) <= 3.95e+65:
		tmp = (2.0 * math.exp(-x)) * 0.5
	elif math.fabs(eps) <= 1.8e+255:
		tmp = 0.5 * (math.exp((math.fabs(eps) * x)) - ((x * (1.0 + math.fabs(eps))) - 1.0))
	else:
		tmp = 0.5 * (math.exp(-(x * (1.0 - math.fabs(eps)))) - -1.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (abs(eps) <= 3.95e+65)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) * 0.5);
	elseif (abs(eps) <= 1.8e+255)
		tmp = Float64(0.5 * Float64(exp(Float64(abs(eps) * x)) - Float64(Float64(x * Float64(1.0 + abs(eps))) - 1.0)));
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-Float64(x * Float64(1.0 - abs(eps))))) - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (abs(eps) <= 3.95e+65)
		tmp = (2.0 * exp(-x)) * 0.5;
	elseif (abs(eps) <= 1.8e+255)
		tmp = 0.5 * (exp((abs(eps) * x)) - ((x * (1.0 + abs(eps))) - 1.0));
	else
		tmp = 0.5 * (exp(-(x * (1.0 - abs(eps)))) - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[Abs[eps], $MachinePrecision], 3.95e+65], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[N[Abs[eps], $MachinePrecision], 1.8e+255], N[(0.5 * N[(N[Exp[N[(N[Abs[eps], $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] - N[(N[(x * N[(1.0 + N[Abs[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Exp[(-N[(x * N[(1.0 - N[Abs[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 3.9499999999999999e65

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in eps around 0

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

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 70.8

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

   7. Applied rewrites 70.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 70.8

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 70.8

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

   9. Applied rewrites 70.8%

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

   if 3.9499999999999999e65 < eps < 1.7999999999999999e255

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 98.9

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

   4. Applied rewrites 98.9%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. exp-neg N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. lower-/.f64 N/A

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

      10. lower-exp.f64 98.9

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft-in N/A

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

      15. *-rgt-identity N/A

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

      16. +-commutative N/A

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

      17. lower-fma.f64 98.9

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

   6. Applied rewrites 98.9%

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

   7. Taylor expanded in eps around inf

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

      1. lower-*.f64 88.3

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

   9. Applied rewrites 88.3%

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

   10. Taylor expanded in x around 0

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

       1. lower--.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-+.f64 64.5

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

   12. Applied rewrites 64.5%

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

   if 1.7999999999999999e255 < eps

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 64.3%

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

Alternative 7: 77.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|\varepsilon\right| \leq 8.2 \cdot 10^{+62}:\\ \;\;\;\;\left(2 \cdot e^{-x}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x \cdot \left|\varepsilon\right|} - -1\right) \cdot 0.5\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (fabs eps) 8.2e+62)
   (* (* 2.0 (exp (- x))) 0.5)
   (* (- (exp (* x (fabs eps))) -1.0) 0.5)))

C

double code(double x, double eps) {
	double tmp;
	if (fabs(eps) <= 8.2e+62) {
		tmp = (2.0 * exp(-x)) * 0.5;
	} else {
		tmp = (exp((x * fabs(eps))) - -1.0) * 0.5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (abs(eps) <= 8.2d+62) then
        tmp = (2.0d0 * exp(-x)) * 0.5d0
    else
        tmp = (exp((x * abs(eps))) - (-1.0d0)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (Math.abs(eps) <= 8.2e+62) {
		tmp = (2.0 * Math.exp(-x)) * 0.5;
	} else {
		tmp = (Math.exp((x * Math.abs(eps))) - -1.0) * 0.5;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if math.fabs(eps) <= 8.2e+62:
		tmp = (2.0 * math.exp(-x)) * 0.5
	else:
		tmp = (math.exp((x * math.fabs(eps))) - -1.0) * 0.5
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (abs(eps) <= 8.2e+62)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) * 0.5);
	else
		tmp = Float64(Float64(exp(Float64(x * abs(eps))) - -1.0) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (abs(eps) <= 8.2e+62)
		tmp = (2.0 * exp(-x)) * 0.5;
	else
		tmp = (exp((x * abs(eps))) - -1.0) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[Abs[eps], $MachinePrecision], 8.2e+62], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Exp[N[(x * N[Abs[eps], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 8.19999999999999967e62

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in eps around 0

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

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 70.8

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

   7. Applied rewrites 70.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 70.8

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 70.8

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

   9. Applied rewrites 70.8%

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

   if 8.19999999999999967e62 < eps

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 64.3%

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

   8. Taylor expanded in eps around inf

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

      1. lower-*.f64 64.5

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

   10. Applied rewrites 64.5%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 64.5

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 64.5

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

   12. Applied rewrites 64.5%

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

Alternative 8: 77.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|\varepsilon\right| \leq 8.2 \cdot 10^{+62}:\\ \;\;\;\;\left(2 \cdot e^{-x}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-x \cdot \left(1 - \left|\varepsilon\right|\right)} - -1\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (fabs eps) 8.2e+62)
   (* (* 2.0 (exp (- x))) 0.5)
   (* 0.5 (- (exp (- (* x (- 1.0 (fabs eps))))) -1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (fabs(eps) <= 8.2e+62) {
		tmp = (2.0 * exp(-x)) * 0.5;
	} else {
		tmp = 0.5 * (exp(-(x * (1.0 - fabs(eps)))) - -1.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (abs(eps) <= 8.2d+62) then
        tmp = (2.0d0 * exp(-x)) * 0.5d0
    else
        tmp = 0.5d0 * (exp(-(x * (1.0d0 - abs(eps)))) - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (Math.abs(eps) <= 8.2e+62) {
		tmp = (2.0 * Math.exp(-x)) * 0.5;
	} else {
		tmp = 0.5 * (Math.exp(-(x * (1.0 - Math.abs(eps)))) - -1.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if math.fabs(eps) <= 8.2e+62:
		tmp = (2.0 * math.exp(-x)) * 0.5
	else:
		tmp = 0.5 * (math.exp(-(x * (1.0 - math.fabs(eps)))) - -1.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (abs(eps) <= 8.2e+62)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) * 0.5);
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-Float64(x * Float64(1.0 - abs(eps))))) - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (abs(eps) <= 8.2e+62)
		tmp = (2.0 * exp(-x)) * 0.5;
	else
		tmp = 0.5 * (exp(-(x * (1.0 - abs(eps)))) - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[Abs[eps], $MachinePrecision], 8.2e+62], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[(N[Exp[(-N[(x * N[(1.0 - N[Abs[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 8.19999999999999967e62

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in eps around 0

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

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 70.8

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

   7. Applied rewrites 70.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 70.8

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 70.8

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

   9. Applied rewrites 70.8%

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

   if 8.19999999999999967e62 < eps

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 64.3%

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

Alternative 9: 71.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|\varepsilon\right| \leq 195000000000:\\ \;\;\;\;\left(2 \cdot e^{-x}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 + x \cdot \left(x - 2\right)\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (fabs eps) 195000000000.0)
   (* (* 2.0 (exp (- x))) 0.5)
   (* 0.5 (+ 2.0 (* x (- x 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (fabs(eps) <= 195000000000.0) {
		tmp = (2.0 * exp(-x)) * 0.5;
	} else {
		tmp = 0.5 * (2.0 + (x * (x - 2.0)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (abs(eps) <= 195000000000.0d0) then
        tmp = (2.0d0 * exp(-x)) * 0.5d0
    else
        tmp = 0.5d0 * (2.0d0 + (x * (x - 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (Math.abs(eps) <= 195000000000.0) {
		tmp = (2.0 * Math.exp(-x)) * 0.5;
	} else {
		tmp = 0.5 * (2.0 + (x * (x - 2.0)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if math.fabs(eps) <= 195000000000.0:
		tmp = (2.0 * math.exp(-x)) * 0.5
	else:
		tmp = 0.5 * (2.0 + (x * (x - 2.0)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (abs(eps) <= 195000000000.0)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) * 0.5);
	else
		tmp = Float64(0.5 * Float64(2.0 + Float64(x * Float64(x - 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (abs(eps) <= 195000000000.0)
		tmp = (2.0 * exp(-x)) * 0.5;
	else
		tmp = 0.5 * (2.0 + (x * (x - 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[Abs[eps], $MachinePrecision], 195000000000.0], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[(2.0 + N[(x * N[(x - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.95e11

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in eps around 0

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

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 70.8

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

   7. Applied rewrites 70.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 70.8

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. distribute-rgt1-in N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 70.8

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

   9. Applied rewrites 70.8%

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

   if 1.95e11 < eps

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in eps around 0

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

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 70.8

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

   7. Applied rewrites 70.8%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 57.3

         \[\leadsto 0.5 \cdot \left(2 + x \cdot \left(x - 2\right)\right) \]

   10. Applied rewrites 57.3%

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

Alternative 10: 65.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 3.4 \cdot 10^{-97}:\\ \;\;\;\;0.5 \cdot \left(e^{-x} - -1\right)\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{+194}:\\ \;\;\;\;\left(\mathsf{fma}\left(\left|\varepsilon\right| - 1, x, 1\right) - -1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(2 + x \cdot \left(x - 2\right)\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 3.4e-97)
   (* 0.5 (- (exp (- x)) -1.0))
   (if (<= x 5.7e+194)
     (* (- (fma (- (fabs eps) 1.0) x 1.0) -1.0) 0.5)
     (* 0.5 (+ 2.0 (* x (- x 2.0)))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 3.4e-97) {
		tmp = 0.5 * (exp(-x) - -1.0);
	} else if (x <= 5.7e+194) {
		tmp = (fma((fabs(eps) - 1.0), x, 1.0) - -1.0) * 0.5;
	} else {
		tmp = 0.5 * (2.0 + (x * (x - 2.0)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 3.4e-97)
		tmp = Float64(0.5 * Float64(exp(Float64(-x)) - -1.0));
	elseif (x <= 5.7e+194)
		tmp = Float64(Float64(fma(Float64(abs(eps) - 1.0), x, 1.0) - -1.0) * 0.5);
	else
		tmp = Float64(0.5 * Float64(2.0 + Float64(x * Float64(x - 2.0))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 3.4e-97], N[(0.5 * N[(N[Exp[(-x)], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.7e+194], N[(N[(N[(N[(N[Abs[eps], $MachinePrecision] - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] - -1.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[(2.0 + N[(x * N[(x - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.3999999999999999e-97

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 64.3%

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

   8. Taylor expanded in eps around inf

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

      1. lower-*.f64 64.5

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

   10. Applied rewrites 64.5%

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

   11. Taylor expanded in eps around 0

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

       1. lower-exp.f64 N/A

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

       2. lower-neg.f64 57.2

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

   13. Applied rewrites 57.2%

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

   if 3.3999999999999999e-97 < x < 5.69999999999999983e194

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 64.3%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 49.8

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

   10. Applied rewrites 49.8%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 49.8

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

       4. lift-+.f64 N/A

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

       5. +-commutative N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f64 49.8

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

   12. Applied rewrites 49.8%

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

   if 5.69999999999999983e194 < x

   1. Initial program 73.4%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in eps around 0

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

      1. lower--.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 70.8

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

   7. Applied rewrites 70.8%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 57.3

         \[\leadsto 0.5 \cdot \left(2 + x \cdot \left(x - 2\right)\right) \]

   10. Applied rewrites 57.3%

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

Alternative 11: 57.3% accurate, 4.8× speedup

Math

\[0.5 \cdot \left(2 + x \cdot \left(x - 2\right)\right) \]

FPCore

(FPCore (x eps) :precision binary64 (* 0.5 (+ 2.0 (* x (- x 2.0)))))

C

double code(double x, double eps) {
	return 0.5 * (2.0 + (x * (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 = 0.5d0 * (2.0d0 + (x * (x - 2.0d0)))
end function

Java

public static double code(double x, double eps) {
	return 0.5 * (2.0 + (x * (x - 2.0)));
}

Python

def code(x, eps):
	return 0.5 * (2.0 + (x * (x - 2.0)))

Julia

function code(x, eps)
	return Float64(0.5 * Float64(2.0 + Float64(x * Float64(x - 2.0))))
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.5 * (2.0 + (x * (x - 2.0)));
end

Wolfram

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

Derivation

1. Initial program 73.4%

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

2. Taylor expanded in eps around inf

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-exp.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-+.f64 98.9

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

4. Applied rewrites 98.9%

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

5. Taylor expanded in eps around 0

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

   1. lower--.f64 N/A

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

   2. lower-exp.f64 N/A

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

   3. lower-neg.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-exp.f64 N/A

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

   6. lower-neg.f64 70.8

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

7. Applied rewrites 70.8%

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

8. Taylor expanded in x around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 57.3

      \[\leadsto 0.5 \cdot \left(2 + x \cdot \left(x - 2\right)\right) \]

10. Applied rewrites 57.3%

    \[\leadsto 0.5 \cdot \left(2 + x \cdot \color{blue}{\left(x - 2\right)}\right) \]
11. Add Preprocessing

Alternative 12: 44.1% accurate, 58.4× speedup

Math

\[1 \]

FPCore

(FPCore (x eps) :precision binary64 1.0)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x, eps):
	return 1.0

Julia

function code(x, eps)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 1.0

Derivation

1. Initial program 73.4%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 44.1%

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

Reproduce

herbie shell --seed 2025170 
(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))