NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.8% → 99.9%

Time: 17.3s

Precision: binary64

Cost: 14024

• 38.4% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} t_0 := \left(x + 1\right) \cdot e^{-x}\\ \mathbf{if}\;\varepsilon \leq -1:\\ \;\;\;\;\frac{e^{x - \varepsilon \cdot x} + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;\varepsilon \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{t_0 + t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \end{array} \]

FPCore

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

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (+ x 1.0) (exp (- x)))))
   (if (<= eps -1.0)
     (/ (+ (exp (- x (* eps x))) (exp (* eps x))) 2.0)
     (if (<= eps 2e-18)
       (/ (+ t_0 t_0) 2.0)
       (/ (+ (exp (* x (+ -1.0 eps))) (exp (* x (- eps)))) 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;
}

↓

double code(double x, double eps) {
	double t_0 = (x + 1.0) * exp(-x);
	double tmp;
	if (eps <= -1.0) {
		tmp = (exp((x - (eps * x))) + exp((eps * x))) / 2.0;
	} else if (eps <= 2e-18) {
		tmp = (t_0 + t_0) / 2.0;
	} else {
		tmp = (exp((x * (-1.0 + eps))) + exp((x * -eps))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    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

↓

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x + 1.0d0) * exp(-x)
    if (eps <= (-1.0d0)) then
        tmp = (exp((x - (eps * x))) + exp((eps * x))) / 2.0d0
    else if (eps <= 2d-18) then
        tmp = (t_0 + t_0) / 2.0d0
    else
        tmp = (exp((x * ((-1.0d0) + eps))) + exp((x * -eps))) / 2.0d0
    end if
    code = tmp
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;
}

↓

public static double code(double x, double eps) {
	double t_0 = (x + 1.0) * Math.exp(-x);
	double tmp;
	if (eps <= -1.0) {
		tmp = (Math.exp((x - (eps * x))) + Math.exp((eps * x))) / 2.0;
	} else if (eps <= 2e-18) {
		tmp = (t_0 + t_0) / 2.0;
	} else {
		tmp = (Math.exp((x * (-1.0 + eps))) + Math.exp((x * -eps))) / 2.0;
	}
	return tmp;
}

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

↓

def code(x, eps):
	t_0 = (x + 1.0) * math.exp(-x)
	tmp = 0
	if eps <= -1.0:
		tmp = (math.exp((x - (eps * x))) + math.exp((eps * x))) / 2.0
	elif eps <= 2e-18:
		tmp = (t_0 + t_0) / 2.0
	else:
		tmp = (math.exp((x * (-1.0 + eps))) + math.exp((x * -eps))) / 2.0
	return tmp

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

↓

function code(x, eps)
	t_0 = Float64(Float64(x + 1.0) * exp(Float64(-x)))
	tmp = 0.0
	if (eps <= -1.0)
		tmp = Float64(Float64(exp(Float64(x - Float64(eps * x))) + exp(Float64(eps * x))) / 2.0);
	elseif (eps <= 2e-18)
		tmp = Float64(Float64(t_0 + t_0) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 + eps))) + exp(Float64(x * Float64(-eps)))) / 2.0);
	end
	return tmp
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

↓

function tmp_2 = code(x, eps)
	t_0 = (x + 1.0) * exp(-x);
	tmp = 0.0;
	if (eps <= -1.0)
		tmp = (exp((x - (eps * x))) + exp((eps * x))) / 2.0;
	elseif (eps <= 2e-18)
		tmp = (t_0 + t_0) / 2.0;
	else
		tmp = (exp((x * (-1.0 + eps))) + exp((x * -eps))) / 2.0;
	end
	tmp_2 = tmp;
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]

↓

code[x_, eps_] := Block[{t$95$0 = N[(N[(x + 1.0), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -1.0], N[(N[(N[Exp[N[(x - N[(eps * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps, 2e-18], N[(N[(t$95$0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(-1.0 + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * (-eps)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1

   1. Initial program 100.0%

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

   2. Simplified 100.0%

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

      [Start] 100.0	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
      div-sub [=>] 100.0	\[ \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]
      +-rgt-identity [<=] 100.0	\[ \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
      div-sub [<=] 100.0	\[ \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Taylor expanded in eps around inf 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Simplified 100.0%

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

      [Start] 100.0	\[ \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}{2} \]
      *-commutative [=>] 100.0	\[ \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(x \cdot \varepsilon\right)}}}{2} \]

   6. Applied egg-rr 61.3%

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

      [Start] 100.0	\[ \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(x \cdot \varepsilon\right)}}{2} \]
      sub-neg [=>] 100.0	\[ \frac{\color{blue}{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} + \left(--1 \cdot e^{-1 \cdot \left(x \cdot \varepsilon\right)}\right)}}{2} \]
      add-sqr-sqrt [=>] 43.5	\[ \frac{e^{\color{blue}{\sqrt{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} \cdot \sqrt{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)}}} + \left(--1 \cdot e^{-1 \cdot \left(x \cdot \varepsilon\right)}\right)}{2} \]
      sqrt-unprod [=>] 100.0	\[ \frac{e^{\color{blue}{\sqrt{\left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)\right) \cdot \left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)\right)}}} + \left(--1 \cdot e^{-1 \cdot \left(x \cdot \varepsilon\right)}\right)}{2} \]
      mul-1-neg [=>] 100.0	\[ \frac{e^{\sqrt{\color{blue}{\left(-\left(1 - \varepsilon\right) \cdot x\right)} \cdot \left(-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)\right)}} + \left(--1 \cdot e^{-1 \cdot \left(x \cdot \varepsilon\right)}\right)}{2} \]
      mul-1-neg [=>] 100.0	\[ \frac{e^{\sqrt{\left(-\left(1 - \varepsilon\right) \cdot x\right) \cdot \color{blue}{\left(-\left(1 - \varepsilon\right) \cdot x\right)}}} + \left(--1 \cdot e^{-1 \cdot \left(x \cdot \varepsilon\right)}\right)}{2} \]
      sqr-neg [=>] 100.0	\[ \frac{e^{\sqrt{\color{blue}{\left(\left(1 - \varepsilon\right) \cdot x\right) \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)}}} + \left(--1 \cdot e^{-1 \cdot \left(x \cdot \varepsilon\right)}\right)}{2} \]
      sqrt-unprod [<=] 56.5	\[ \frac{e^{\color{blue}{\sqrt{\left(1 - \varepsilon\right) \cdot x} \cdot \sqrt{\left(1 - \varepsilon\right) \cdot x}}} + \left(--1 \cdot e^{-1 \cdot \left(x \cdot \varepsilon\right)}\right)}{2} \]
      add-sqr-sqrt [<=] 64.7	\[ \frac{e^{\color{blue}{\left(1 - \varepsilon\right) \cdot x}} + \left(--1 \cdot e^{-1 \cdot \left(x \cdot \varepsilon\right)}\right)}{2} \]
      *-commutative [=>] 64.7	\[ \frac{e^{\color{blue}{x \cdot \left(1 - \varepsilon\right)}} + \left(--1 \cdot e^{-1 \cdot \left(x \cdot \varepsilon\right)}\right)}{2} \]
      exp-prod [=>] 64.9	\[ \frac{\color{blue}{{\left(e^{x}\right)}^{\left(1 - \varepsilon\right)}} + \left(--1 \cdot e^{-1 \cdot \left(x \cdot \varepsilon\right)}\right)}{2} \]
      mul-1-neg [=>] 64.9	\[ \frac{{\left(e^{x}\right)}^{\left(1 - \varepsilon\right)} + \left(-\color{blue}{\left(-e^{-1 \cdot \left(x \cdot \varepsilon\right)}\right)}\right)}{2} \]
      remove-double-neg [=>] 64.9	\[ \frac{{\left(e^{x}\right)}^{\left(1 - \varepsilon\right)} + \color{blue}{e^{-1 \cdot \left(x \cdot \varepsilon\right)}}}{2} \]
      add-sqr-sqrt [=>] 56.5	\[ \frac{{\left(e^{x}\right)}^{\left(1 - \varepsilon\right)} + e^{\color{blue}{\sqrt{-1 \cdot \left(x \cdot \varepsilon\right)} \cdot \sqrt{-1 \cdot \left(x \cdot \varepsilon\right)}}}}{2} \]
      sqrt-unprod [=>] 99.0	\[ \frac{{\left(e^{x}\right)}^{\left(1 - \varepsilon\right)} + e^{\color{blue}{\sqrt{\left(-1 \cdot \left(x \cdot \varepsilon\right)\right) \cdot \left(-1 \cdot \left(x \cdot \varepsilon\right)\right)}}}}{2} \]
      mul-1-neg [=>] 99.0	\[ \frac{{\left(e^{x}\right)}^{\left(1 - \varepsilon\right)} + e^{\sqrt{\color{blue}{\left(-x \cdot \varepsilon\right)} \cdot \left(-1 \cdot \left(x \cdot \varepsilon\right)\right)}}}{2} \]

   7. Simplified 100.0%

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

      [Start] 61.3	\[ \frac{{\left(e^{x}\right)}^{\left(1 - \varepsilon\right)} + {\left(e^{\varepsilon}\right)}^{x}}{2} \]
      exp-prod [<=] 81.0	\[ \frac{\color{blue}{e^{x \cdot \left(1 - \varepsilon\right)}} + {\left(e^{\varepsilon}\right)}^{x}}{2} \]
      sub-neg [=>] 81.0	\[ \frac{e^{x \cdot \color{blue}{\left(1 + \left(-\varepsilon\right)\right)}} + {\left(e^{\varepsilon}\right)}^{x}}{2} \]
      distribute-lft-in [=>] 81.0	\[ \frac{e^{\color{blue}{x \cdot 1 + x \cdot \left(-\varepsilon\right)}} + {\left(e^{\varepsilon}\right)}^{x}}{2} \]
      distribute-rgt-neg-in [<=] 81.0	\[ \frac{e^{x \cdot 1 + \color{blue}{\left(-x \cdot \varepsilon\right)}} + {\left(e^{\varepsilon}\right)}^{x}}{2} \]
      *-commutative [<=] 81.0	\[ \frac{e^{x \cdot 1 + \left(-\color{blue}{\varepsilon \cdot x}\right)} + {\left(e^{\varepsilon}\right)}^{x}}{2} \]
      unsub-neg [=>] 81.0	\[ \frac{e^{\color{blue}{x \cdot 1 - \varepsilon \cdot x}} + {\left(e^{\varepsilon}\right)}^{x}}{2} \]
      *-rgt-identity [=>] 81.0	\[ \frac{e^{\color{blue}{x} - \varepsilon \cdot x} + {\left(e^{\varepsilon}\right)}^{x}}{2} \]
      *-commutative [=>] 81.0	\[ \frac{e^{x - \color{blue}{x \cdot \varepsilon}} + {\left(e^{\varepsilon}\right)}^{x}}{2} \]
      exp-prod [<=] 100.0	\[ \frac{e^{x - x \cdot \varepsilon} + \color{blue}{e^{\varepsilon \cdot x}}}{2} \]
      *-commutative [=>] 100.0	\[ \frac{e^{x - x \cdot \varepsilon} + e^{\color{blue}{x \cdot \varepsilon}}}{2} \]

   if -1 < eps < 2.0000000000000001e-18

   1. Initial program 40.4%

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

   2. Simplified 40.4%

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

      [Start] 40.4	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
      div-sub [=>] 40.4	\[ \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]
      +-rgt-identity [<=] 40.4	\[ \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
      div-sub [<=] 40.4	\[ \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Taylor expanded in eps around 0 100.0%

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

   4. Simplified 100.0%

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

      [Start] 100.0	\[ \frac{\left(e^{-1 \cdot x} \cdot x + e^{-1 \cdot x}\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]
      *-commutative [=>] 100.0	\[ \frac{\left(\color{blue}{x \cdot e^{-1 \cdot x}} + e^{-1 \cdot x}\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]
      distribute-lft1-in [=>] 100.0	\[ \frac{\color{blue}{\left(x + 1\right) \cdot e^{-1 \cdot x}} - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]
      mul-1-neg [=>] 100.0	\[ \frac{\left(x + 1\right) \cdot e^{\color{blue}{-x}} - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]
      distribute-lft-out [=>] 100.0	\[ \frac{\left(x + 1\right) \cdot e^{-x} - \color{blue}{-1 \cdot \left(e^{-1 \cdot x} \cdot x + e^{-1 \cdot x}\right)}}{2} \]
      mul-1-neg [=>] 100.0	\[ \frac{\left(x + 1\right) \cdot e^{-x} - \color{blue}{\left(-\left(e^{-1 \cdot x} \cdot x + e^{-1 \cdot x}\right)\right)}}{2} \]
      *-commutative [=>] 100.0	\[ \frac{\left(x + 1\right) \cdot e^{-x} - \left(-\left(\color{blue}{x \cdot e^{-1 \cdot x}} + e^{-1 \cdot x}\right)\right)}{2} \]
      distribute-lft1-in [=>] 100.0	\[ \frac{\left(x + 1\right) \cdot e^{-x} - \left(-\color{blue}{\left(x + 1\right) \cdot e^{-1 \cdot x}}\right)}{2} \]
      mul-1-neg [=>] 100.0	\[ \frac{\left(x + 1\right) \cdot e^{-x} - \left(-\left(x + 1\right) \cdot e^{\color{blue}{-x}}\right)}{2} \]

   if 2.0000000000000001e-18 < eps

   1. Initial program 98.6%

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

   2. Simplified 98.6%

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

      [Start] 98.6	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
      div-sub [=>] 98.6	\[ \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]
      +-rgt-identity [<=] 98.6	\[ \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
      div-sub [<=] 98.6	\[ \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Taylor expanded in eps around inf 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Simplified 100.0%

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

      [Start] 100.0	\[ \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\varepsilon \cdot x\right)}}{2} \]
      *-commutative [=>] 100.0	\[ \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \color{blue}{\left(x \cdot \varepsilon\right)}}}{2} \]

3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1:\\ \;\;\;\;\frac{e^{x - \varepsilon \cdot x} + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;\varepsilon \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(x + 1\right) \cdot e^{-x} + \left(x + 1\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	100.0%
Cost	14025

\[\begin{array}{l} t_0 := \left(x + 1\right) \cdot e^{-x}\\ \mathbf{if}\;\varepsilon \leq -1 \lor \neg \left(\varepsilon \leq 1\right):\\ \;\;\;\;\frac{e^{x - \varepsilon \cdot x} + e^{\varepsilon \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + t_0}{2}\\ \end{array} \]

Alternative 2
Accuracy	100.0%
Cost	13833

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

Alternative 3
Accuracy	98.7%
Cost	13769

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -40000 \lor \neg \left(\varepsilon \leq 1\right):\\ \;\;\;\;\frac{e^{x - \varepsilon \cdot x} + e^{\varepsilon \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \end{array} \]

Alternative 4
Accuracy	98.8%
Cost	13632

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

Alternative 5
Accuracy	86.4%
Cost	8076

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(-2 - \varepsilon\right) + x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 0.5\right)\right)\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.05:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{elif}\;\varepsilon \leq 6.4 \cdot 10^{+168}:\\ \;\;\;\;\frac{2 + x \cdot \left(x - 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(-2 - \varepsilon\right) + x \cdot \left(0.5 + 0.5 \cdot {\left(\varepsilon + 1\right)}^{2}\right)\right)}{2}\\ \end{array} \]

Alternative 6
Accuracy	88.7%
Cost	8068

\[\begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.45:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(-1 + \left(\varepsilon - \varepsilon\right)\right) + x \cdot \left(0.5 \cdot \left(\varepsilon \cdot \varepsilon + {\left(1 - \varepsilon\right)}^{2}\right)\right)\right)}{2}\\ \mathbf{elif}\;\varepsilon \leq 6.2:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(-2 - \left(\varepsilon \cdot \left(1 - x\right) - \mathsf{fma}\left(0.5, \varepsilon \cdot \left(\varepsilon \cdot x\right), x\right)\right)\right)}{2}\\ \end{array} \]

Alternative 7
Accuracy	88.5%
Cost	8008

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

Alternative 8
Accuracy	86.5%
Cost	6984

\[\begin{array}{l} t_0 := \frac{2 + x \cdot \left(\left(-2 - \varepsilon\right) + x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 0.5\right)\right)\right)}{2}\\ \mathbf{if}\;\varepsilon \leq -1.05:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1.9:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{elif}\;\varepsilon \leq 2 \cdot 10^{+164}:\\ \;\;\;\;\frac{2 + x \cdot \left(x - 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Accuracy	77.2%
Cost	1484

\[\begin{array}{l} t_0 := \frac{2 + x \cdot \left(\left(-2 - \varepsilon\right) + x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot 0.5\right)\right)\right)}{2}\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{-211}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-221}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+25}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10
Accuracy	76.2%
Cost	1228

\[\begin{array}{l} t_0 := \frac{2 + \left(\varepsilon \cdot \varepsilon\right) \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)}{2}\\ \mathbf{if}\;x \leq -1.9 \cdot 10^{-153}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-162}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+30}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11
Accuracy	62.5%
Cost	836

\[\begin{array}{l} \mathbf{if}\;x \leq 550000000000:\\ \;\;\;\;\frac{2 + x \cdot \left(x + \left(-2 - \varepsilon\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 12
Accuracy	63.3%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+115}:\\ \;\;\;\;\frac{2 + x \cdot \left(x - 2\right)}{2}\\ \mathbf{elif}\;x \leq 550000000000:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 13
Accuracy	59.1%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{+15}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 14
Accuracy	59.2%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq 550000000000:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 15
Accuracy	56.2%
Cost	196

\[\begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{+15}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 16
Accuracy	15.8%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023161 
(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))