NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.6% → 99.8%

Time: 15.7s

Alternatives: 10

Speedup: 2.0×

• 37.4% 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: 73.6% 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: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.000235:\\ \;\;\;\;\frac{2 \cdot \frac{x + 1}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= eps 0.000235)
   (/ (* 2.0 (/ (+ x 1.0) (exp x))) 2.0)
   (/ (+ (exp (* eps x)) (exp (* eps (- x)))) 2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (eps <= 0.000235) {
		tmp = (2.0 * ((x + 1.0) / exp(x))) / 2.0;
	} else {
		tmp = (exp((eps * x)) + exp((eps * -x))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 0.000235d0) then
        tmp = (2.0d0 * ((x + 1.0d0) / exp(x))) / 2.0d0
    else
        tmp = (exp((eps * x)) + exp((eps * -x))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (eps <= 0.000235) {
		tmp = (2.0 * ((x + 1.0) / Math.exp(x))) / 2.0;
	} else {
		tmp = (Math.exp((eps * x)) + Math.exp((eps * -x))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if eps <= 0.000235:
		tmp = (2.0 * ((x + 1.0) / math.exp(x))) / 2.0
	else:
		tmp = (math.exp((eps * x)) + math.exp((eps * -x))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (eps <= 0.000235)
		tmp = Float64(Float64(2.0 * Float64(Float64(x + 1.0) / exp(x))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(eps * x)) + exp(Float64(eps * Float64(-x)))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 0.000235)
		tmp = (2.0 * ((x + 1.0) / exp(x))) / 2.0;
	else
		tmp = (exp((eps * x)) + exp((eps * -x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[eps, 0.000235], N[(N[(2.0 * N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(eps * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 2.34999999999999993e-4

   1. Initial program 57.1%

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

      1. sub-neg 57.1%

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

      2. neg-sub0 57.1%

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

      3. associate-+r- 57.1%

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

   3. Simplified 57.1%

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

   4. Taylor expanded in eps around 0 77.0%

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

      1. associate--r+ 76.9%

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

      2. associate-*r* 76.9%

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

      3. neg-mul-1 76.9%

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

      4. cancel-sign-sub 76.9%

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

      5. distribute-rgt1-in 76.9%

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

      6. distribute-rgt-out-- 76.9%

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

      7. neg-mul-1 76.9%

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

      8. neg-mul-1 76.9%

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

   6. Simplified 76.9%

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

   7. Taylor expanded in x around inf 77.0%

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

      1. distribute-lft-out 77.0%

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

      2. distribute-rgt1-in 76.9%

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

      3. +-commutative 76.9%

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

      4. rec-exp 77.0%

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

      5. associate-*r/ 77.0%

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

      6. *-commutative 77.0%

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

      7. *-lft-identity 77.0%

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

   9. Simplified 77.0%

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

   if 2.34999999999999993e-4 < eps

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   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%

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

      1. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

      3. *-commutative 100.0%

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

      4. mul-1-neg 100.0%

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

      5. mul-1-neg 100.0%

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

      6. distribute-lft-neg-out 100.0%

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

      7. exp-prod 82.6%

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

      8. remove-double-neg 82.6%

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

      9. mul-1-neg 82.6%

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

      10. sub-neg 82.6%

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

      11. exp-prod 100.0%

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

      12. neg-mul-1 100.0%

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

      13. associate-*r* 100.0%

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

      14. mul-1-neg 100.0%

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

      15. distribute-rgt-neg-in 100.0%

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

      16. sub-neg 100.0%

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

      17. mul-1-neg 100.0%

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

      18. remove-double-neg 100.0%

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

      19. distribute-rgt-neg-in 100.0%

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

      20. distribute-lft-in 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in eps around inf 100.0%

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

       1. *-commutative 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.000235:\\ \;\;\;\;\frac{2 \cdot \frac{x + 1}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \end{array} \]

Alternative 2: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \frac{e^{\varepsilon \cdot x - x} + \frac{1}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}}{2} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (/ (+ (exp (- (* eps x) x)) (/ 1.0 (exp (fma eps x x)))) 2.0))

C

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

Julia

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

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := N[(N[(N[Exp[N[(N[(eps * x), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[Exp[N[(eps * x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 69.0%

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

3. Simplified 68.9%

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

4. Taylor expanded in eps around inf 98.6%

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

5. Final simplification 98.6%

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

Alternative 3: 98.8% accurate, 1.1× speedup

Math

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

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (/ (+ (exp (* x (+ eps -1.0))) (exp (- (- x) (* eps x)))) 2.0))

C

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

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (exp((x * (eps + (-1.0d0)))) + exp((-x - (eps * x)))) / 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := N[(N[(N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[((-x) - N[(eps * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 69.0%

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

   1. sub-neg 69.0%

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

   2. neg-sub0 69.0%

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

   3. associate-+r- 69.0%

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

3. Simplified 69.0%

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

4. Taylor expanded in eps around inf 98.6%

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

   1. associate-*r* 98.6%

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

   2. neg-mul-1 98.6%

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

   3. *-commutative 98.6%

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

   4. mul-1-neg 98.6%

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

   5. mul-1-neg 98.6%

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

   6. distribute-lft-neg-out 98.6%

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

   7. exp-prod 91.6%

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

   8. remove-double-neg 91.6%

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

   9. mul-1-neg 91.6%

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

   10. sub-neg 91.6%

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

   11. exp-prod 98.6%

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

   12. neg-mul-1 98.6%

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

   13. associate-*r* 98.6%

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

   14. mul-1-neg 98.6%

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

   15. distribute-rgt-neg-in 98.6%

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

   16. sub-neg 98.6%

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

   17. mul-1-neg 98.6%

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

   18. remove-double-neg 98.6%

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

   19. distribute-rgt-neg-in 98.6%

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

   20. distribute-lft-in 98.6%

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

6. Simplified 98.6%

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

7. Final simplification 98.6%

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

Alternative 4: 84.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-275}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot \left(-x\right)} + 1}{2}\\ \mathbf{elif}\;x \leq 3.55 \cdot 10^{+49}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x - x} + \left(x \cdot \left(-1 - \varepsilon\right) + 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{x + 1}{e^{x}}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -5e-275)
   (/ (+ (exp (* eps (- x))) 1.0) 2.0)
   (if (<= x 3.55e+49)
     (/ (+ (exp (- (* eps x) x)) (+ (* x (- -1.0 eps)) 1.0)) 2.0)
     (/ (* 2.0 (/ (+ x 1.0) (exp x))) 2.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -5e-275) {
		tmp = (exp((eps * -x)) + 1.0) / 2.0;
	} else if (x <= 3.55e+49) {
		tmp = (exp(((eps * x) - x)) + ((x * (-1.0 - eps)) + 1.0)) / 2.0;
	} else {
		tmp = (2.0 * ((x + 1.0) / exp(x))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-5d-275)) then
        tmp = (exp((eps * -x)) + 1.0d0) / 2.0d0
    else if (x <= 3.55d+49) then
        tmp = (exp(((eps * x) - x)) + ((x * ((-1.0d0) - eps)) + 1.0d0)) / 2.0d0
    else
        tmp = (2.0d0 * ((x + 1.0d0) / exp(x))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= -5e-275) {
		tmp = (Math.exp((eps * -x)) + 1.0) / 2.0;
	} else if (x <= 3.55e+49) {
		tmp = (Math.exp(((eps * x) - x)) + ((x * (-1.0 - eps)) + 1.0)) / 2.0;
	} else {
		tmp = (2.0 * ((x + 1.0) / Math.exp(x))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= -5e-275:
		tmp = (math.exp((eps * -x)) + 1.0) / 2.0
	elif x <= 3.55e+49:
		tmp = (math.exp(((eps * x) - x)) + ((x * (-1.0 - eps)) + 1.0)) / 2.0
	else:
		tmp = (2.0 * ((x + 1.0) / math.exp(x))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= -5e-275)
		tmp = Float64(Float64(exp(Float64(eps * Float64(-x))) + 1.0) / 2.0);
	elseif (x <= 3.55e+49)
		tmp = Float64(Float64(exp(Float64(Float64(eps * x) - x)) + Float64(Float64(x * Float64(-1.0 - eps)) + 1.0)) / 2.0);
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(x + 1.0) / exp(x))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -5e-275)
		tmp = (exp((eps * -x)) + 1.0) / 2.0;
	elseif (x <= 3.55e+49)
		tmp = (exp(((eps * x) - x)) + ((x * (-1.0 - eps)) + 1.0)) / 2.0;
	else
		tmp = (2.0 * ((x + 1.0) / exp(x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, -5e-275], N[(N[(N[Exp[N[(eps * (-x)), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3.55e+49], N[(N[(N[Exp[N[(N[(eps * x), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision] + N[(N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 * N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.99999999999999983e-275

   1. Initial program 65.0%

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

      1. sub-neg 65.0%

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

      2. neg-sub0 65.0%

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

      3. associate-+r- 65.0%

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

   3. Simplified 65.0%

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

   4. Taylor expanded in x around 0 40.3%

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

      1. metadata-eval 40.3%

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

      2. distribute-neg-frac 40.3%

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

      3. sub-neg 40.3%

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

   6. Simplified 40.3%

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

   7. Taylor expanded in eps around inf 73.9%

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

      1. mul-1-neg 73.9%

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

      2. associate-*r* 73.9%

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

      3. neg-mul-1 73.9%

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

      4. *-commutative 73.9%

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

   9. Simplified 73.9%

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

   10. Taylor expanded in eps around inf 74.2%

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

       1. associate-*r* 74.2%

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

       2. mul-1-neg 74.2%

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

   12. Simplified 74.2%

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

   if -4.99999999999999983e-275 < x < 3.54999999999999986e49

   1. Initial program 57.1%

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

   3. Simplified 57.0%

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

   4. Taylor expanded in eps around inf 97.9%

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

   5. Taylor expanded in x around 0 84.6%

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

      1. associate-*r* 84.6%

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

      2. neg-mul-1 84.6%

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

      3. *-commutative 84.6%

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

   7. Simplified 84.6%

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

   if 3.54999999999999986e49 < 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. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   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 0 63.3%

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

      1. associate--r+ 63.3%

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

      2. associate-*r* 63.3%

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

      3. neg-mul-1 63.3%

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

      4. cancel-sign-sub 63.3%

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

      5. distribute-rgt1-in 63.3%

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

      6. distribute-rgt-out-- 63.3%

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

      7. neg-mul-1 63.3%

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

      8. neg-mul-1 63.3%

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

   6. Simplified 63.3%

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

   7. Taylor expanded in x around inf 63.3%

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

      1. distribute-lft-out 63.3%

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

      2. distribute-rgt1-in 63.3%

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

      3. +-commutative 63.3%

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

      4. rec-exp 63.3%

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

      5. associate-*r/ 63.3%

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

      6. *-commutative 63.3%

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

      7. *-lft-identity 63.3%

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

   9. Simplified 63.3%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-275}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot \left(-x\right)} + 1}{2}\\ \mathbf{elif}\;x \leq 3.55 \cdot 10^{+49}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x - x} + \left(x \cdot \left(-1 - \varepsilon\right) + 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{x + 1}{e^{x}}}{2}\\ \end{array} \]

Alternative 5: 77.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-252}:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+81}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x - x} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{x + 1}{e^{x}}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 2e-252)
   (/ (/ 2.0 (exp x)) 2.0)
   (if (<= x 5.2e+81)
     (/ (+ (exp (- (* eps x) x)) 1.0) 2.0)
     (/ (* 2.0 (/ (+ x 1.0) (exp x))) 2.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 2e-252) {
		tmp = (2.0 / exp(x)) / 2.0;
	} else if (x <= 5.2e+81) {
		tmp = (exp(((eps * x) - x)) + 1.0) / 2.0;
	} else {
		tmp = (2.0 * ((x + 1.0) / exp(x))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 2d-252) then
        tmp = (2.0d0 / exp(x)) / 2.0d0
    else if (x <= 5.2d+81) then
        tmp = (exp(((eps * x) - x)) + 1.0d0) / 2.0d0
    else
        tmp = (2.0d0 * ((x + 1.0d0) / exp(x))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 2e-252) {
		tmp = (2.0 / Math.exp(x)) / 2.0;
	} else if (x <= 5.2e+81) {
		tmp = (Math.exp(((eps * x) - x)) + 1.0) / 2.0;
	} else {
		tmp = (2.0 * ((x + 1.0) / Math.exp(x))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 2e-252:
		tmp = (2.0 / math.exp(x)) / 2.0
	elif x <= 5.2e+81:
		tmp = (math.exp(((eps * x) - x)) + 1.0) / 2.0
	else:
		tmp = (2.0 * ((x + 1.0) / math.exp(x))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 2e-252)
		tmp = Float64(Float64(2.0 / exp(x)) / 2.0);
	elseif (x <= 5.2e+81)
		tmp = Float64(Float64(exp(Float64(Float64(eps * x) - x)) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(x + 1.0) / exp(x))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 2e-252)
		tmp = (2.0 / exp(x)) / 2.0;
	elseif (x <= 5.2e+81)
		tmp = (exp(((eps * x) - x)) + 1.0) / 2.0;
	else
		tmp = (2.0 * ((x + 1.0) / exp(x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 2e-252], N[(N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5.2e+81], N[(N[(N[Exp[N[(N[(eps * x), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 * N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.99999999999999989e-252

   1. Initial program 62.6%

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

   3. Simplified 62.5%

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

   4. Taylor expanded in eps around inf 98.8%

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

   5. Taylor expanded in eps around 0 80.9%

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

      1. rec-exp 80.9%

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

      2. count-2 80.9%

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

      3. rec-exp 80.9%

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

      4. associate-*r/ 80.9%

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

      5. metadata-eval 80.9%

         \[\leadsto \frac{\frac{\color{blue}{2}}{e^{x}}}{2} \]

   7. Simplified 80.9%

      \[\leadsto \frac{\color{blue}{\frac{2}{e^{x}}}}{2} \]

   if 1.99999999999999989e-252 < x < 5.19999999999999984e81

   1. Initial program 62.0%

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

   3. Simplified 61.9%

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

   4. Taylor expanded in eps around inf 97.4%

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

   5. Taylor expanded in x around 0 76.2%

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

   if 5.19999999999999984e81 < 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. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   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 0 65.0%

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

      1. associate--r+ 65.0%

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

      2. associate-*r* 65.0%

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

      3. neg-mul-1 65.0%

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

      4. cancel-sign-sub 65.0%

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

      5. distribute-rgt1-in 65.0%

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

      6. distribute-rgt-out-- 65.0%

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

      7. neg-mul-1 65.0%

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

      8. neg-mul-1 65.0%

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

   6. Simplified 65.0%

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

   7. Taylor expanded in x around inf 65.0%

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

      1. distribute-lft-out 65.0%

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

      2. distribute-rgt1-in 65.0%

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

      3. +-commutative 65.0%

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

      4. rec-exp 65.0%

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

      5. associate-*r/ 65.0%

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

      6. *-commutative 65.0%

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

      7. *-lft-identity 65.0%

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

   9. Simplified 65.0%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-252}:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+81}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x - x} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{x + 1}{e^{x}}}{2}\\ \end{array} \]

Alternative 6: 84.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-278}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot \left(-x\right)} + 1}{2}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+82}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x - x} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{x + 1}{e^{x}}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -1e-278)
   (/ (+ (exp (* eps (- x))) 1.0) 2.0)
   (if (<= x 1.1e+82)
     (/ (+ (exp (- (* eps x) x)) 1.0) 2.0)
     (/ (* 2.0 (/ (+ x 1.0) (exp x))) 2.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -1e-278) {
		tmp = (exp((eps * -x)) + 1.0) / 2.0;
	} else if (x <= 1.1e+82) {
		tmp = (exp(((eps * x) - x)) + 1.0) / 2.0;
	} else {
		tmp = (2.0 * ((x + 1.0) / exp(x))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-1d-278)) then
        tmp = (exp((eps * -x)) + 1.0d0) / 2.0d0
    else if (x <= 1.1d+82) then
        tmp = (exp(((eps * x) - x)) + 1.0d0) / 2.0d0
    else
        tmp = (2.0d0 * ((x + 1.0d0) / exp(x))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= -1e-278) {
		tmp = (Math.exp((eps * -x)) + 1.0) / 2.0;
	} else if (x <= 1.1e+82) {
		tmp = (Math.exp(((eps * x) - x)) + 1.0) / 2.0;
	} else {
		tmp = (2.0 * ((x + 1.0) / Math.exp(x))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= -1e-278:
		tmp = (math.exp((eps * -x)) + 1.0) / 2.0
	elif x <= 1.1e+82:
		tmp = (math.exp(((eps * x) - x)) + 1.0) / 2.0
	else:
		tmp = (2.0 * ((x + 1.0) / math.exp(x))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= -1e-278)
		tmp = Float64(Float64(exp(Float64(eps * Float64(-x))) + 1.0) / 2.0);
	elseif (x <= 1.1e+82)
		tmp = Float64(Float64(exp(Float64(Float64(eps * x) - x)) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(2.0 * Float64(Float64(x + 1.0) / exp(x))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1e-278)
		tmp = (exp((eps * -x)) + 1.0) / 2.0;
	elseif (x <= 1.1e+82)
		tmp = (exp(((eps * x) - x)) + 1.0) / 2.0;
	else
		tmp = (2.0 * ((x + 1.0) / exp(x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, -1e-278], N[(N[(N[Exp[N[(eps * (-x)), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.1e+82], N[(N[(N[Exp[N[(N[(eps * x), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 * N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.99999999999999938e-279

   1. Initial program 65.0%

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

      1. sub-neg 65.0%

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

      2. neg-sub0 65.0%

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

      3. associate-+r- 65.0%

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

   3. Simplified 65.0%

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

   4. Taylor expanded in x around 0 40.3%

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

      1. metadata-eval 40.3%

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

      2. distribute-neg-frac 40.3%

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

      3. sub-neg 40.3%

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

   6. Simplified 40.3%

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

   7. Taylor expanded in eps around inf 73.9%

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

      1. mul-1-neg 73.9%

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

      2. associate-*r* 73.9%

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

      3. neg-mul-1 73.9%

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

      4. *-commutative 73.9%

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

   9. Simplified 73.9%

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

   10. Taylor expanded in eps around inf 74.2%

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

       1. associate-*r* 74.2%

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

       2. mul-1-neg 74.2%

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

   12. Simplified 74.2%

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

   if -9.99999999999999938e-279 < x < 1.1000000000000001e82

   1. Initial program 59.6%

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

   3. Simplified 59.5%

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

   4. Taylor expanded in eps around inf 98.0%

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

   5. Taylor expanded in x around 0 81.2%

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

   if 1.1000000000000001e82 < 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. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   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 0 65.0%

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

      1. associate--r+ 65.0%

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

      2. associate-*r* 65.0%

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

      3. neg-mul-1 65.0%

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

      4. cancel-sign-sub 65.0%

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

      5. distribute-rgt1-in 65.0%

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

      6. distribute-rgt-out-- 65.0%

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

      7. neg-mul-1 65.0%

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

      8. neg-mul-1 65.0%

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

   6. Simplified 65.0%

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

   7. Taylor expanded in x around inf 65.0%

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

      1. distribute-lft-out 65.0%

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

      2. distribute-rgt1-in 65.0%

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

      3. +-commutative 65.0%

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

      4. rec-exp 65.0%

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

      5. associate-*r/ 65.0%

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

      6. *-commutative 65.0%

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

      7. *-lft-identity 65.0%

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

   9. Simplified 65.0%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-278}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot \left(-x\right)} + 1}{2}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+82}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x - x} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{x + 1}{e^{x}}}{2}\\ \end{array} \]

Alternative 7: 71.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \frac{\frac{2}{e^{x}}}{2} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps) :precision binary64 (/ (/ 2.0 (exp x)) 2.0))

C

eps = abs(eps);
double code(double x, double eps) {
	return (2.0 / exp(x)) / 2.0;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (2.0d0 / exp(x)) / 2.0d0
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	return (2.0 / Math.exp(x)) / 2.0;
}

Python

eps = abs(eps)
def code(x, eps):
	return (2.0 / math.exp(x)) / 2.0

Julia

eps = abs(eps)
function code(x, eps)
	return Float64(Float64(2.0 / exp(x)) / 2.0)
end

MATLAB

eps = abs(eps)
function tmp = code(x, eps)
	tmp = (2.0 / exp(x)) / 2.0;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := N[(N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 69.0%

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

3. Simplified 68.9%

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

4. Taylor expanded in eps around inf 98.6%

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

5. Taylor expanded in eps around 0 73.6%

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

   1. rec-exp 73.6%

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

   2. count-2 73.6%

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

   3. rec-exp 73.6%

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

   4. associate-*r/ 73.6%

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

   5. metadata-eval 73.6%

      \[\leadsto \frac{\frac{\color{blue}{2}}{e^{x}}}{2} \]

7. Simplified 73.6%

   \[\leadsto \frac{\color{blue}{\frac{2}{e^{x}}}}{2} \]

8. Final simplification 73.6%

   \[\leadsto \frac{\frac{2}{e^{x}}}{2} \]

Alternative 8: 64.8% accurate, 25.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.004:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 0.004) (/ (- 2.0 (* eps x)) 2.0) 0.0))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 0.004) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 0.004d0) then
        tmp = (2.0d0 - (eps * x)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 0.004) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 0.004:
		tmp = (2.0 - (eps * x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 0.004)
		tmp = Float64(Float64(2.0 - Float64(eps * x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 0.004)
		tmp = (2.0 - (eps * x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 0.004], N[(N[(2.0 - N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.0040000000000000001

   1. Initial program 58.7%

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

      1. sub-neg 58.7%

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

      2. neg-sub0 58.7%

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

      3. associate-+r- 58.7%

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

   3. Simplified 58.7%

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

   4. Taylor expanded in x around 0 40.8%

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

      1. metadata-eval 40.8%

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

      2. distribute-neg-frac 40.8%

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

      3. sub-neg 40.8%

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

   6. Simplified 40.8%

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

   7. Taylor expanded in x around 0 47.6%

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

      1. associate-*r* 47.6%

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

      2. sub-neg 47.6%

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

      3. metadata-eval 47.6%

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

      4. +-commutative 47.6%

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

   9. Simplified 47.6%

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

   10. Taylor expanded in eps around inf 68.0%

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

       1. mul-1-neg 68.0%

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

       2. *-commutative 68.0%

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

       3. unsub-neg 68.0%

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

   12. Simplified 68.0%

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

   if 0.0040000000000000001 < x

   1. Initial program 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in eps around 0 55.3%

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

      1. neg-mul-1 55.3%

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

      2. rec-exp 55.3%

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

      3. neg-mul-1 55.3%

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

      4. div-sub 55.3%

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

      5. +-inverses 55.3%

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

   6. Simplified 55.3%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.004:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9: 57.8% accurate, 74.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 7800000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps) :precision binary64 (if (<= x 7800000.0) 1.0 0.0))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 7800000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 7800000.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 7800000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 7800000.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 7800000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 7800000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 7800000.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 7.8e6

   1. Initial program 59.3%

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

      1. sub-neg 59.3%

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

      2. neg-sub0 59.3%

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

      3. associate-+r- 59.3%

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

   3. Simplified 59.3%

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

   4. Taylor expanded in x around 0 63.1%

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

   if 7.8e6 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 59.7%

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

      1. neg-mul-1 59.7%

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

      2. rec-exp 59.7%

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

      3. neg-mul-1 59.7%

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

      4. div-sub 59.7%

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

      5. +-inverses 59.7%

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

   6. Simplified 59.7%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7800000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10: 16.5% accurate, 227.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ 0 \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	return 0.0;
}

Python

eps = abs(eps)
def code(x, eps):
	return 0.0

Julia

eps = abs(eps)
function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := 0.0

Derivation

1. Initial program 69.0%

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

3. Simplified 68.9%

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

4. Taylor expanded in eps around 0 16.0%

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

   1. neg-mul-1 16.0%

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

   2. rec-exp 16.0%

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

   3. neg-mul-1 16.0%

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

   4. div-sub 16.0%

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

   5. +-inverses 16.2%

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

6. Simplified 16.2%

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

7. Final simplification 16.2%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023279 
(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))