NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.7% → 98.8%

Time: 11.6s

Alternatives: 11

Speedup: 1.1×

• 38.1% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -1.99999999999999997e77
  2. eps = -2e-231

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

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: 98.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.0%

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

   1. div-sub 72.0%

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

   2. +-rgt-identity 72.0%

      \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   3. div-sub 72.0%

      \[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

3. Simplified 72.0%

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

4. Taylor expanded in eps around inf 99.7%

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

   1. mul-1-neg 99.7%

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

   2. *-commutative 99.7%

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

   3. mul-1-neg 99.7%

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

   4. exp-prod 99.7%

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

   5. +-commutative 99.7%

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

   6. remove-double-neg 99.7%

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

   7. mul-1-neg 99.7%

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

   8. sub-neg 99.7%

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

   9. exp-prod 99.7%

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

   10. mul-1-neg 99.7%

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

   11. *-commutative 99.7%

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

   12. sub-neg 99.7%

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

   13. mul-1-neg 99.7%

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

   14. remove-double-neg 99.7%

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

   15. distribute-rgt-neg-in 99.7%

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

   16. +-commutative 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 88.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{x \cdot \left(-\varepsilon\right)}\\ \mathbf{if}\;x \leq 7.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + t_0}{2}\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{+84}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon - x} + t_0}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (* x (- eps)))))
   (if (<= x 7.2e+18)
     (/ (+ (exp (* x eps)) t_0) 2.0)
     (if (<= x 1.22e+84) 0.0 (/ (+ (exp (- (* x eps) x)) t_0) 2.0)))))

C

double code(double x, double eps) {
	double t_0 = exp((x * -eps));
	double tmp;
	if (x <= 7.2e+18) {
		tmp = (exp((x * eps)) + t_0) / 2.0;
	} else if (x <= 1.22e+84) {
		tmp = 0.0;
	} else {
		tmp = (exp(((x * eps) - x)) + t_0) / 2.0;
	}
	return tmp;
}

Fortran

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 = exp((x * -eps))
    if (x <= 7.2d+18) then
        tmp = (exp((x * eps)) + t_0) / 2.0d0
    else if (x <= 1.22d+84) then
        tmp = 0.0d0
    else
        tmp = (exp(((x * eps) - x)) + t_0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.exp((x * -eps));
	double tmp;
	if (x <= 7.2e+18) {
		tmp = (Math.exp((x * eps)) + t_0) / 2.0;
	} else if (x <= 1.22e+84) {
		tmp = 0.0;
	} else {
		tmp = (Math.exp(((x * eps) - x)) + t_0) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.exp((x * -eps))
	tmp = 0
	if x <= 7.2e+18:
		tmp = (math.exp((x * eps)) + t_0) / 2.0
	elif x <= 1.22e+84:
		tmp = 0.0
	else:
		tmp = (math.exp(((x * eps) - x)) + t_0) / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = exp(Float64(x * Float64(-eps)))
	tmp = 0.0
	if (x <= 7.2e+18)
		tmp = Float64(Float64(exp(Float64(x * eps)) + t_0) / 2.0);
	elseif (x <= 1.22e+84)
		tmp = 0.0;
	else
		tmp = Float64(Float64(exp(Float64(Float64(x * eps) - x)) + t_0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = exp((x * -eps));
	tmp = 0.0;
	if (x <= 7.2e+18)
		tmp = (exp((x * eps)) + t_0) / 2.0;
	elseif (x <= 1.22e+84)
		tmp = 0.0;
	else
		tmp = (exp(((x * eps) - x)) + t_0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Exp[N[(x * (-eps)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 7.2e+18], N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.22e+84], 0.0, N[(N[(N[Exp[N[(N[(x * eps), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 7.2e18

   1. Initial program 62.1%

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

      1. div-sub 62.1%

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

      2. +-rgt-identity 62.1%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      3. div-sub 62.1%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Simplified 62.1%

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

   4. Taylor expanded in eps around inf 99.6%

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

      1. mul-1-neg 99.6%

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

      2. *-commutative 99.6%

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

      3. mul-1-neg 99.6%

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

      4. exp-prod 99.6%

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

      5. +-commutative 99.6%

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

      6. remove-double-neg 99.6%

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

      7. mul-1-neg 99.6%

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

      8. sub-neg 99.6%

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

      9. exp-prod 99.6%

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

      10. mul-1-neg 99.6%

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

      11. *-commutative 99.6%

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

      12. sub-neg 99.6%

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

      13. mul-1-neg 99.6%

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

      14. remove-double-neg 99.6%

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

      15. distribute-rgt-neg-in 99.6%

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

      16. +-commutative 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in eps around inf 99.2%

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

      1. associate-*r* 99.2%

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

      2. neg-mul-1 99.2%

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

   9. Simplified 99.2%

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

   10. Taylor expanded in x around inf 99.2%

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

   11. Taylor expanded in eps around inf 99.5%

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

       1. *-commutative 99.5%

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

   13. Simplified 99.5%

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

   if 7.2e18 < x < 1.2200000000000001e84

   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. Step-by-step derivation

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 80.3%

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

      1. div-sub 80.3%

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

      2. rec-exp 80.3%

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

      3. mul-1-neg 80.3%

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

      4. +-inverses 80.3%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   6. Simplified 80.3%

      \[\leadsto \frac{\color{blue}{0}}{2} \]

   if 1.2200000000000001e84 < x

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      3. div-sub 100.0%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. mul-1-neg 100.0%

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

      4. exp-prod 100.0%

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

      5. +-commutative 100.0%

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

      6. remove-double-neg 100.0%

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

      7. mul-1-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. exp-prod 100.0%

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

      10. mul-1-neg 100.0%

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

      11. *-commutative 100.0%

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

      12. sub-neg 100.0%

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

      13. mul-1-neg 100.0%

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

      14. remove-double-neg 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in eps around inf 64.7%

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

      1. associate-*r* 64.7%

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

      2. neg-mul-1 64.7%

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

   9. Simplified 64.7%

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

   10. Taylor expanded in x around inf 64.7%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{+84}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon - x} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \end{array} \]

Alternative 3: 82.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{+17}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+84}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+213}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon - x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 7.2e+17)
   (/ (+ (exp (* x eps)) (exp (* x (- eps)))) 2.0)
   (if (<= x 3e+84)
     0.0
     (if (<= x 3.3e+213) (/ (+ 1.0 (exp (- (* x eps) x))) 2.0) 0.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 7.2e+17) {
		tmp = (exp((x * eps)) + exp((x * -eps))) / 2.0;
	} else if (x <= 3e+84) {
		tmp = 0.0;
	} else if (x <= 3.3e+213) {
		tmp = (1.0 + exp(((x * eps) - x))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 7.2d+17) then
        tmp = (exp((x * eps)) + exp((x * -eps))) / 2.0d0
    else if (x <= 3d+84) then
        tmp = 0.0d0
    else if (x <= 3.3d+213) then
        tmp = (1.0d0 + exp(((x * eps) - x))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 7.2e+17) {
		tmp = (Math.exp((x * eps)) + Math.exp((x * -eps))) / 2.0;
	} else if (x <= 3e+84) {
		tmp = 0.0;
	} else if (x <= 3.3e+213) {
		tmp = (1.0 + Math.exp(((x * eps) - x))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 7.2e+17:
		tmp = (math.exp((x * eps)) + math.exp((x * -eps))) / 2.0
	elif x <= 3e+84:
		tmp = 0.0
	elif x <= 3.3e+213:
		tmp = (1.0 + math.exp(((x * eps) - x))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 7.2e+17)
		tmp = Float64(Float64(exp(Float64(x * eps)) + exp(Float64(x * Float64(-eps)))) / 2.0);
	elseif (x <= 3e+84)
		tmp = 0.0;
	elseif (x <= 3.3e+213)
		tmp = Float64(Float64(1.0 + exp(Float64(Float64(x * eps) - x))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 7.2e+17)
		tmp = (exp((x * eps)) + exp((x * -eps))) / 2.0;
	elseif (x <= 3e+84)
		tmp = 0.0;
	elseif (x <= 3.3e+213)
		tmp = (1.0 + exp(((x * eps) - x))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 7.2e+17], N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * (-eps)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3e+84], 0.0, If[LessEqual[x, 3.3e+213], N[(N[(1.0 + N[Exp[N[(N[(x * eps), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 7.2e17

   1. Initial program 62.1%

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

      1. div-sub 62.1%

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

      2. +-rgt-identity 62.1%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      3. div-sub 62.1%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Simplified 62.1%

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

   4. Taylor expanded in eps around inf 99.6%

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

      1. mul-1-neg 99.6%

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

      2. *-commutative 99.6%

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

      3. mul-1-neg 99.6%

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

      4. exp-prod 99.6%

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

      5. +-commutative 99.6%

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

      6. remove-double-neg 99.6%

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

      7. mul-1-neg 99.6%

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

      8. sub-neg 99.6%

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

      9. exp-prod 99.6%

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

      10. mul-1-neg 99.6%

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

      11. *-commutative 99.6%

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

      12. sub-neg 99.6%

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

      13. mul-1-neg 99.6%

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

      14. remove-double-neg 99.6%

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

      15. distribute-rgt-neg-in 99.6%

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

      16. +-commutative 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in eps around inf 99.2%

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

      1. associate-*r* 99.2%

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

      2. neg-mul-1 99.2%

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

   9. Simplified 99.2%

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

   10. Taylor expanded in x around inf 99.2%

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

   11. Taylor expanded in eps around inf 99.5%

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

       1. *-commutative 99.5%

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

   13. Simplified 99.5%

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

   if 7.2e17 < x < 2.99999999999999996e84 or 3.3000000000000001e213 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 70.7%

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

      1. div-sub 70.7%

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

      2. rec-exp 70.7%

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

      3. mul-1-neg 70.7%

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

      4. +-inverses 70.7%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   6. Simplified 70.7%

      \[\leadsto \frac{\color{blue}{0}}{2} \]

   if 2.99999999999999996e84 < x < 3.3000000000000001e213

   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. Step-by-step derivation

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      3. div-sub 100.0%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 28.6%

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

   5. Taylor expanded in eps around inf 29.0%

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

      1. exp-prod 29.0%

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

      2. sub-neg 29.0%

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

      3. neg-mul-1 29.0%

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

      4. exp-prod 29.0%

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

      5. +-commutative 29.0%

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

      6. neg-mul-1 29.0%

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

      7. *-commutative 29.0%

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

      8. neg-mul-1 29.0%

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

      9. sub-neg 29.0%

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

      10. distribute-rgt-neg-in 29.0%

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

   7. Simplified 29.0%

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

   8. Taylor expanded in x around 0 29.0%

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

      1. sub-neg 29.0%

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

      2. metadata-eval 29.0%

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

      3. *-commutative 29.0%

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

      4. distribute-rgt-in 29.0%

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

      5. neg-mul-1 29.0%

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

      6. sub-neg 29.0%

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

   10. Simplified 29.0%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{+17}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+84}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+213}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon - x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4: 66.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -400:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+18} \lor \neg \left(x \leq 3.8 \cdot 10^{+84}\right) \land x \leq 1.4 \cdot 10^{+217}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon - x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -400.0)
   (/ (/ (expm1 (- x)) eps) 2.0)
   (if (or (<= x 6.8e+18) (and (not (<= x 3.8e+84)) (<= x 1.4e+217)))
     (/ (+ 1.0 (exp (- (* x eps) x))) 2.0)
     0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -400.0) {
		tmp = (expm1(-x) / eps) / 2.0;
	} else if ((x <= 6.8e+18) || (!(x <= 3.8e+84) && (x <= 1.4e+217))) {
		tmp = (1.0 + exp(((x * eps) - x))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -400.0) {
		tmp = (Math.expm1(-x) / eps) / 2.0;
	} else if ((x <= 6.8e+18) || (!(x <= 3.8e+84) && (x <= 1.4e+217))) {
		tmp = (1.0 + Math.exp(((x * eps) - x))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -400.0:
		tmp = (math.expm1(-x) / eps) / 2.0
	elif (x <= 6.8e+18) or (not (x <= 3.8e+84) and (x <= 1.4e+217)):
		tmp = (1.0 + math.exp(((x * eps) - x))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -400.0)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps) / 2.0);
	elseif ((x <= 6.8e+18) || (!(x <= 3.8e+84) && (x <= 1.4e+217)))
		tmp = Float64(Float64(1.0 + exp(Float64(Float64(x * eps) - x))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -400.0], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 6.8e+18], And[N[Not[LessEqual[x, 3.8e+84]], $MachinePrecision], LessEqual[x, 1.4e+217]]], N[(N[(1.0 + N[Exp[N[(N[(x * eps), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -400

   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. Step-by-step derivation

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      3. div-sub 100.0%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 66.1%

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

   5. Taylor expanded in eps around 0 35.0%

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

      1. expm1-def 35.0%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(-1 \cdot x\right)}}{\varepsilon}}{2} \]

      2. neg-mul-1 35.0%

         \[\leadsto \frac{\frac{\mathsf{expm1}\left(\color{blue}{-x}\right)}{\varepsilon}}{2} \]

   7. Simplified 35.0%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}}{2} \]

   if -400 < x < 6.8e18 or 3.8000000000000001e84 < x < 1.39999999999999997e217

   1. Initial program 59.9%

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

      1. div-sub 59.9%

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

      2. +-rgt-identity 59.9%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      3. div-sub 59.9%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Simplified 59.9%

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

   4. Taylor expanded in x around 0 36.9%

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

   5. Taylor expanded in eps around inf 77.1%

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

      1. exp-prod 77.1%

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

      2. sub-neg 77.1%

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

      3. neg-mul-1 77.1%

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

      4. exp-prod 77.1%

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

      5. +-commutative 77.1%

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

      6. neg-mul-1 77.1%

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

      7. *-commutative 77.1%

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

      8. neg-mul-1 77.1%

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

      9. sub-neg 77.1%

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

      10. distribute-rgt-neg-in 77.1%

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

   7. Simplified 77.1%

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

   8. Taylor expanded in x around 0 77.1%

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

      1. sub-neg 77.1%

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

      2. metadata-eval 77.1%

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

      3. *-commutative 77.1%

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

      4. distribute-rgt-in 77.1%

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

      5. neg-mul-1 77.1%

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

      6. sub-neg 77.1%

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

   10. Simplified 77.1%

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

   if 6.8e18 < x < 3.8000000000000001e84 or 1.39999999999999997e217 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 70.7%

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

      1. div-sub 70.7%

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

      2. rec-exp 70.7%

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

      3. mul-1-neg 70.7%

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

      4. +-inverses 70.7%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   6. Simplified 70.7%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -400:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+18} \lor \neg \left(x \leq 3.8 \cdot 10^{+84}\right) \land x \leq 1.4 \cdot 10^{+217}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon - x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5: 67.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-307}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+15} \lor \neg \left(x \leq 3.65 \cdot 10^{+84}\right) \land x \leq 2.8 \cdot 10^{+215}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon - x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -4e-307)
   (/ (+ 1.0 (exp (* x (- eps)))) 2.0)
   (if (or (<= x 2.5e+15) (and (not (<= x 3.65e+84)) (<= x 2.8e+215)))
     (/ (+ 1.0 (exp (- (* x eps) x))) 2.0)
     0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -4e-307) {
		tmp = (1.0 + exp((x * -eps))) / 2.0;
	} else if ((x <= 2.5e+15) || (!(x <= 3.65e+84) && (x <= 2.8e+215))) {
		tmp = (1.0 + exp(((x * eps) - x))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-4d-307)) then
        tmp = (1.0d0 + exp((x * -eps))) / 2.0d0
    else if ((x <= 2.5d+15) .or. (.not. (x <= 3.65d+84)) .and. (x <= 2.8d+215)) then
        tmp = (1.0d0 + exp(((x * eps) - x))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -4e-307) {
		tmp = (1.0 + Math.exp((x * -eps))) / 2.0;
	} else if ((x <= 2.5e+15) || (!(x <= 3.65e+84) && (x <= 2.8e+215))) {
		tmp = (1.0 + Math.exp(((x * eps) - x))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -4e-307:
		tmp = (1.0 + math.exp((x * -eps))) / 2.0
	elif (x <= 2.5e+15) or (not (x <= 3.65e+84) and (x <= 2.8e+215)):
		tmp = (1.0 + math.exp(((x * eps) - x))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -4e-307)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-eps)))) / 2.0);
	elseif ((x <= 2.5e+15) || (!(x <= 3.65e+84) && (x <= 2.8e+215)))
		tmp = Float64(Float64(1.0 + exp(Float64(Float64(x * eps) - x))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -4e-307)
		tmp = (1.0 + exp((x * -eps))) / 2.0;
	elseif ((x <= 2.5e+15) || (~((x <= 3.65e+84)) && (x <= 2.8e+215)))
		tmp = (1.0 + exp(((x * eps) - x))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -4e-307], N[(N[(1.0 + N[Exp[N[(x * (-eps)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 2.5e+15], And[N[Not[LessEqual[x, 3.65e+84]], $MachinePrecision], LessEqual[x, 2.8e+215]]], N[(N[(1.0 + N[Exp[N[(N[(x * eps), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -3.99999999999999964e-307

   1. Initial program 72.9%

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

      1. div-sub 72.9%

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

      2. +-rgt-identity 72.9%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      3. div-sub 72.9%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Simplified 72.9%

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

   4. Taylor expanded in eps around inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. *-commutative 99.8%

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

      3. mul-1-neg 99.8%

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

      4. exp-prod 99.8%

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

      5. +-commutative 99.8%

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

      6. remove-double-neg 99.8%

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

      7. mul-1-neg 99.8%

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

      8. sub-neg 99.8%

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

      9. exp-prod 99.8%

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

      10. mul-1-neg 99.8%

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

      11. *-commutative 99.8%

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

      12. sub-neg 99.8%

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

      13. mul-1-neg 99.8%

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

      14. remove-double-neg 99.8%

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

      15. distribute-rgt-neg-in 99.8%

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

      16. +-commutative 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in eps around inf 99.8%

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

      1. associate-*r* 99.8%

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

      2. neg-mul-1 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in x around inf 99.8%

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

   11. Taylor expanded in x around 0 71.3%

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

   if -3.99999999999999964e-307 < x < 2.5e15 or 3.65e84 < x < 2.8e215

   1. Initial program 62.5%

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

      1. div-sub 62.5%

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

      2. +-rgt-identity 62.5%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      3. div-sub 62.5%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Simplified 62.5%

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

   4. Taylor expanded in x around 0 34.9%

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

   5. Taylor expanded in eps around inf 72.8%

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

      1. exp-prod 72.8%

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

      2. sub-neg 72.8%

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

      3. neg-mul-1 72.8%

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

      4. exp-prod 72.8%

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

      5. +-commutative 72.8%

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

      6. neg-mul-1 72.8%

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

      7. *-commutative 72.8%

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

      8. neg-mul-1 72.8%

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

      9. sub-neg 72.8%

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

      10. distribute-rgt-neg-in 72.8%

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

   7. Simplified 72.8%

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

   8. Taylor expanded in x around 0 72.8%

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

      1. sub-neg 72.8%

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

      2. metadata-eval 72.8%

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

      3. *-commutative 72.8%

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

      4. distribute-rgt-in 72.8%

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

      5. neg-mul-1 72.8%

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

      6. sub-neg 72.8%

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

   10. Simplified 72.8%

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

   if 2.5e15 < x < 3.65e84 or 2.8e215 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 70.7%

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

      1. div-sub 70.7%

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

      2. rec-exp 70.7%

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

      3. mul-1-neg 70.7%

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

      4. +-inverses 70.7%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   6. Simplified 70.7%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-307}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+15} \lor \neg \left(x \leq 3.65 \cdot 10^{+84}\right) \land x \leq 2.8 \cdot 10^{+215}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon - x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6: 67.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{x \cdot \varepsilon - x}\\ \mathbf{if}\;x \leq -1 \cdot 10^{-297}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{+19}:\\ \;\;\;\;\frac{t_0 + \left(1 - x \cdot \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+84}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+216}:\\ \;\;\;\;\frac{1 + t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (- (* x eps) x))))
   (if (<= x -1e-297)
     (/ (+ 1.0 (exp (* x (- eps)))) 2.0)
     (if (<= x 7.6e+19)
       (/ (+ t_0 (- 1.0 (* x eps))) 2.0)
       (if (<= x 6e+84) 0.0 (if (<= x 8.2e+216) (/ (+ 1.0 t_0) 2.0) 0.0))))))

C

double code(double x, double eps) {
	double t_0 = exp(((x * eps) - x));
	double tmp;
	if (x <= -1e-297) {
		tmp = (1.0 + exp((x * -eps))) / 2.0;
	} else if (x <= 7.6e+19) {
		tmp = (t_0 + (1.0 - (x * eps))) / 2.0;
	} else if (x <= 6e+84) {
		tmp = 0.0;
	} else if (x <= 8.2e+216) {
		tmp = (1.0 + t_0) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

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 = exp(((x * eps) - x))
    if (x <= (-1d-297)) then
        tmp = (1.0d0 + exp((x * -eps))) / 2.0d0
    else if (x <= 7.6d+19) then
        tmp = (t_0 + (1.0d0 - (x * eps))) / 2.0d0
    else if (x <= 6d+84) then
        tmp = 0.0d0
    else if (x <= 8.2d+216) then
        tmp = (1.0d0 + t_0) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.exp(((x * eps) - x));
	double tmp;
	if (x <= -1e-297) {
		tmp = (1.0 + Math.exp((x * -eps))) / 2.0;
	} else if (x <= 7.6e+19) {
		tmp = (t_0 + (1.0 - (x * eps))) / 2.0;
	} else if (x <= 6e+84) {
		tmp = 0.0;
	} else if (x <= 8.2e+216) {
		tmp = (1.0 + t_0) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.exp(((x * eps) - x))
	tmp = 0
	if x <= -1e-297:
		tmp = (1.0 + math.exp((x * -eps))) / 2.0
	elif x <= 7.6e+19:
		tmp = (t_0 + (1.0 - (x * eps))) / 2.0
	elif x <= 6e+84:
		tmp = 0.0
	elif x <= 8.2e+216:
		tmp = (1.0 + t_0) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	t_0 = exp(Float64(Float64(x * eps) - x))
	tmp = 0.0
	if (x <= -1e-297)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-eps)))) / 2.0);
	elseif (x <= 7.6e+19)
		tmp = Float64(Float64(t_0 + Float64(1.0 - Float64(x * eps))) / 2.0);
	elseif (x <= 6e+84)
		tmp = 0.0;
	elseif (x <= 8.2e+216)
		tmp = Float64(Float64(1.0 + t_0) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = exp(((x * eps) - x));
	tmp = 0.0;
	if (x <= -1e-297)
		tmp = (1.0 + exp((x * -eps))) / 2.0;
	elseif (x <= 7.6e+19)
		tmp = (t_0 + (1.0 - (x * eps))) / 2.0;
	elseif (x <= 6e+84)
		tmp = 0.0;
	elseif (x <= 8.2e+216)
		tmp = (1.0 + t_0) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Exp[N[(N[(x * eps), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -1e-297], N[(N[(1.0 + N[Exp[N[(x * (-eps)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 7.6e+19], N[(N[(t$95$0 + N[(1.0 - N[(x * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 6e+84], 0.0, If[LessEqual[x, 8.2e+216], N[(N[(1.0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.00000000000000004e-297

   1. Initial program 72.9%

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

      1. div-sub 72.9%

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

      2. +-rgt-identity 72.9%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      3. div-sub 72.9%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Simplified 72.9%

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

   4. Taylor expanded in eps around inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. *-commutative 99.8%

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

      3. mul-1-neg 99.8%

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

      4. exp-prod 99.8%

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

      5. +-commutative 99.8%

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

      6. remove-double-neg 99.8%

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

      7. mul-1-neg 99.8%

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

      8. sub-neg 99.8%

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

      9. exp-prod 99.8%

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

      10. mul-1-neg 99.8%

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

      11. *-commutative 99.8%

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

      12. sub-neg 99.8%

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

      13. mul-1-neg 99.8%

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

      14. remove-double-neg 99.8%

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

      15. distribute-rgt-neg-in 99.8%

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

      16. +-commutative 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in eps around inf 99.8%

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

      1. associate-*r* 99.8%

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

      2. neg-mul-1 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in x around inf 99.8%

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

   11. Taylor expanded in x around 0 71.3%

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

   if -1.00000000000000004e-297 < x < 7.6e19

   1. Initial program 49.9%

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

      1. div-sub 49.9%

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

      2. +-rgt-identity 49.9%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      3. div-sub 49.9%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Simplified 49.9%

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

   4. Taylor expanded in eps around inf 99.5%

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

      1. mul-1-neg 99.5%

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

      2. *-commutative 99.5%

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

      3. mul-1-neg 99.5%

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

      4. exp-prod 99.5%

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

      5. +-commutative 99.5%

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

      6. remove-double-neg 99.5%

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

      7. mul-1-neg 99.5%

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

      8. sub-neg 99.5%

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

      9. exp-prod 99.5%

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

      10. mul-1-neg 99.5%

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

      11. *-commutative 99.5%

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

      12. sub-neg 99.5%

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

      13. mul-1-neg 99.5%

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

      14. remove-double-neg 99.5%

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

      15. distribute-rgt-neg-in 99.5%

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

      16. +-commutative 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in eps around inf 98.5%

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

      1. associate-*r* 98.5%

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

      2. neg-mul-1 98.5%

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

   9. Simplified 98.5%

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

   10. Taylor expanded in x around inf 98.5%

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

   11. Taylor expanded in eps around 0 88.5%

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

       1. +-commutative 88.5%

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

       2. mul-1-neg 88.5%

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

       3. unsub-neg 88.5%

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

   13. Simplified 88.5%

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

   if 7.6e19 < x < 5.99999999999999992e84 or 8.1999999999999995e216 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 70.7%

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

      1. div-sub 70.7%

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

      2. rec-exp 70.7%

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

      3. mul-1-neg 70.7%

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

      4. +-inverses 70.7%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   6. Simplified 70.7%

      \[\leadsto \frac{\color{blue}{0}}{2} \]

   if 5.99999999999999992e84 < x < 8.1999999999999995e216

   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. Step-by-step derivation

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      3. div-sub 100.0%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 28.6%

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

   5. Taylor expanded in eps around inf 29.0%

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

      1. exp-prod 29.0%

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

      2. sub-neg 29.0%

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

      3. neg-mul-1 29.0%

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

      4. exp-prod 29.0%

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

      5. +-commutative 29.0%

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

      6. neg-mul-1 29.0%

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

      7. *-commutative 29.0%

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

      8. neg-mul-1 29.0%

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

      9. sub-neg 29.0%

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

      10. distribute-rgt-neg-in 29.0%

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

   7. Simplified 29.0%

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

   8. Taylor expanded in x around 0 29.0%

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

      1. sub-neg 29.0%

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

      2. metadata-eval 29.0%

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

      3. *-commutative 29.0%

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

      4. distribute-rgt-in 29.0%

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

      5. neg-mul-1 29.0%

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

      6. sub-neg 29.0%

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

   10. Simplified 29.0%

       \[\leadsto \frac{1 + e^{\color{blue}{\varepsilon \cdot x - x}}}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-297}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{+19}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon - x} + \left(1 - x \cdot \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+84}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+216}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon - x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7: 63.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -480:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 30000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -480.0)
   (/ (/ (expm1 (- x)) eps) 2.0)
   (if (<= x 30000000.0) 1.0 0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -480.0) {
		tmp = (expm1(-x) / eps) / 2.0;
	} else if (x <= 30000000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -480.0) {
		tmp = (Math.expm1(-x) / eps) / 2.0;
	} else if (x <= 30000000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -480.0:
		tmp = (math.expm1(-x) / eps) / 2.0
	elif x <= 30000000.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -480.0)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps) / 2.0);
	elseif (x <= 30000000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -480.0], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 30000000.0], 1.0, 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -480

   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. Step-by-step derivation

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      3. div-sub 100.0%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 66.1%

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

   5. Taylor expanded in eps around 0 35.0%

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

      1. expm1-def 35.0%

         \[\leadsto \frac{\frac{\color{blue}{\mathsf{expm1}\left(-1 \cdot x\right)}}{\varepsilon}}{2} \]

      2. neg-mul-1 35.0%

         \[\leadsto \frac{\frac{\mathsf{expm1}\left(\color{blue}{-x}\right)}{\varepsilon}}{2} \]

   7. Simplified 35.0%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}}{2} \]

   if -480 < x < 3e7

   1. Initial program 51.2%

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

      1. div-sub 51.2%

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

      2. +-rgt-identity 51.2%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      3. div-sub 51.2%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Simplified 51.2%

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

   4. Taylor expanded in x around 0 81.6%

      \[\leadsto \frac{\color{blue}{2}}{2} \]

   if 3e7 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 58.6%

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

      1. div-sub 58.6%

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

      2. rec-exp 58.6%

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

      3. mul-1-neg 58.6%

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

      4. +-inverses 58.6%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   6. Simplified 58.6%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -480:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 30000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8: 60.1% accurate, 25.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00145:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 0.00145) (/ (- 2.0 (* x eps)) 2.0) 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 0.00145) {
		tmp = (2.0 - (x * eps)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 0.00145d0) then
        tmp = (2.0d0 - (x * eps)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 0.00145) {
		tmp = (2.0 - (x * eps)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 0.00145:
		tmp = (2.0 - (x * eps)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 0.00145)
		tmp = Float64(Float64(2.0 - Float64(x * eps)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 0.00145)
		tmp = (2.0 - (x * eps)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 0.00145], N[(N[(2.0 - N[(x * eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.00145

   1. Initial program 60.8%

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

      1. div-sub 60.8%

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

      2. +-rgt-identity 60.8%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      3. div-sub 60.8%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Simplified 60.8%

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

   4. Taylor expanded in x around 0 39.8%

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

   5. Taylor expanded in x around 0 47.4%

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

      1. sub-neg 47.4%

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

      2. metadata-eval 47.4%

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

      3. *-commutative 47.4%

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

      4. +-commutative 47.4%

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

      5. distribute-lft1-in 47.4%

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

      6. +-commutative 47.4%

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

   7. Simplified 47.4%

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

   8. Taylor expanded in eps around inf 69.7%

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

      1. +-commutative 69.7%

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

      2. neg-mul-1 69.7%

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

      3. *-rgt-identity 69.7%

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

      4. fma-udef 69.7%

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

      5. mul-1-neg 69.7%

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

      6. unsub-neg 69.7%

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

      7. fma-udef 69.7%

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

      8. *-rgt-identity 69.7%

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

      9. neg-mul-1 69.7%

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

      10. distribute-lft1-in 69.7%

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

      11. metadata-eval 69.7%

          \[\leadsto \frac{2 + \left(\color{blue}{0} \cdot x - \varepsilon \cdot x\right)}{2} \]

      12. mul0-lft 69.7%

          \[\leadsto \frac{2 + \left(\color{blue}{0} - \varepsilon \cdot x\right)}{2} \]

      13. neg-sub0 69.7%

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

      14. unsub-neg 69.7%

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

      15. *-commutative 69.7%

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

   10. Simplified 69.7%

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

   if 0.00145 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 55.5%

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

      1. div-sub 55.5%

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

      2. rec-exp 55.5%

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

      3. mul-1-neg 55.5%

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

      4. +-inverses 55.5%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   6. Simplified 55.5%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00145:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9: 60.3% accurate, 32.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;x \cdot \frac{\varepsilon}{-2}\\ \mathbf{elif}\;x \leq 30000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.0) (* x (/ eps -2.0)) (if (<= x 30000000.0) 1.0 0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.0) {
		tmp = x * (eps / -2.0);
	} else if (x <= 30000000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = x * (eps / (-2.0d0))
    else if (x <= 30000000.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.0) {
		tmp = x * (eps / -2.0);
	} else if (x <= 30000000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.0:
		tmp = x * (eps / -2.0)
	elif x <= 30000000.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(x * Float64(eps / -2.0));
	elseif (x <= 30000000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = x * (eps / -2.0);
	elseif (x <= 30000000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.0], N[(x * N[(eps / -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 30000000.0], 1.0, 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -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. Step-by-step derivation

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      3. div-sub 100.0%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 64.6%

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

   5. Taylor expanded in x around 0 40.6%

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

   6. Taylor expanded in eps around inf 40.6%

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

      1. expm1-log1p-u 40.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\varepsilon \cdot x}{2}\right)\right)} \]

      2. expm1-udef 40.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\varepsilon \cdot x}{2}\right)} - 1} \]

      3. frac-2neg 40.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-\varepsilon \cdot x}{-2}}\right)} - 1 \]

      4. distribute-lft-neg-out 40.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\left(-\varepsilon\right) \cdot x}}{-2}\right)} - 1 \]

      5. associate-/l* 40.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{-\varepsilon}{\frac{-2}{x}}}\right)} - 1 \]

      6. add-sqr-sqrt 40.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-\varepsilon} \cdot \sqrt{-\varepsilon}}}{\frac{-2}{x}}\right)} - 1 \]

      7. sqrt-unprod 66.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-\varepsilon\right) \cdot \left(-\varepsilon\right)}}}{\frac{-2}{x}}\right)} - 1 \]

      8. sqr-neg 66.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{\varepsilon \cdot \varepsilon}}}{\frac{-2}{x}}\right)} - 1 \]

      9. sqrt-unprod 20.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\varepsilon} \cdot \sqrt{\varepsilon}}}{\frac{-2}{x}}\right)} - 1 \]

      10. add-sqr-sqrt 20.8%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\varepsilon}}{\frac{-2}{x}}\right)} - 1 \]

      11. metadata-eval 20.8%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\varepsilon}{\frac{\color{blue}{-2}}{x}}\right)} - 1 \]

   8. Applied egg-rr 20.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\varepsilon}{\frac{-2}{x}}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 20.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\varepsilon}{\frac{-2}{x}}\right)\right)} \]

      2. expm1-log1p 20.9%

         \[\leadsto \color{blue}{\frac{\varepsilon}{\frac{-2}{x}}} \]

      3. associate-/r/ 20.9%

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

   10. Simplified 20.9%

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

   if -1 < x < 3e7

   1. Initial program 50.9%

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

      1. div-sub 50.9%

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

      2. +-rgt-identity 50.9%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      3. div-sub 50.9%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Simplified 50.9%

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

   4. Taylor expanded in x around 0 82.1%

      \[\leadsto \frac{\color{blue}{2}}{2} \]

   if 3e7 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 58.6%

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

      1. div-sub 58.6%

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

      2. rec-exp 58.6%

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

      3. mul-1-neg 58.6%

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

      4. +-inverses 58.6%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   6. Simplified 58.6%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;x \cdot \frac{\varepsilon}{-2}\\ \mathbf{elif}\;x \leq 30000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10: 57.2% accurate, 74.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 30000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (if (<= x 30000000.0) 1.0 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 30000000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 30000000.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 30000000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 30000000.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 30000000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 30000000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 30000000.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 3e7

   1. Initial program 61.6%

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

      1. div-sub 61.6%

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

      2. +-rgt-identity 61.6%

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

      3. div-sub 61.6%

         \[\leadsto \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Simplified 61.6%

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

   4. Taylor expanded in x around 0 64.8%

      \[\leadsto \frac{\color{blue}{2}}{2} \]

   if 3e7 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 58.6%

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

      1. div-sub 58.6%

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

      2. rec-exp 58.6%

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

      3. mul-1-neg 58.6%

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

      4. +-inverses 58.6%

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   6. Simplified 58.6%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 30000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11: 16.1% accurate, 227.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

double code(double x, double eps) {
	return 0.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 72.0%

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

3. Simplified 56.8%

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

4. Taylor expanded in eps around 0 17.4%

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

   1. div-sub 17.4%

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

   2. rec-exp 17.4%

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

   3. mul-1-neg 17.4%

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

   4. +-inverses 17.7%

      \[\leadsto \frac{\color{blue}{0}}{2} \]

6. Simplified 17.7%

   \[\leadsto \frac{\color{blue}{0}}{2} \]

7. Final simplification 17.7%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023229 
(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))