NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.1% → 99.8%

Time: 17.9s

Alternatives: 12

Speedup: 2.1×

• 38.7% 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: 73.1% 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, 1.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\varepsilon \leq 53:\\ \;\;\;\;\frac{t_0 \cdot \left(\left(x + 1\right) - -1\right) + x \cdot t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= eps 53.0)
     (/ (+ (* t_0 (- (+ x 1.0) -1.0)) (* x t_0)) 2.0)
     (/ (+ (exp (* x (+ eps -1.0))) (exp (* x (- eps)))) 2.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double t_0 = exp(-x);
	double tmp;
	if (eps <= 53.0) {
		tmp = ((t_0 * ((x + 1.0) - -1.0)) + (x * t_0)) / 2.0;
	} else {
		tmp = (exp((x * (eps + -1.0))) + exp((x * -eps))) / 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) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (eps <= 53.0d0) then
        tmp = ((t_0 * ((x + 1.0d0) - (-1.0d0))) + (x * t_0)) / 2.0d0
    else
        tmp = (exp((x * (eps + (-1.0d0)))) + exp((x * -eps))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double t_0 = Math.exp(-x);
	double tmp;
	if (eps <= 53.0) {
		tmp = ((t_0 * ((x + 1.0) - -1.0)) + (x * t_0)) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps + -1.0))) + Math.exp((x * -eps))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	t_0 = math.exp(-x)
	tmp = 0
	if eps <= 53.0:
		tmp = ((t_0 * ((x + 1.0) - -1.0)) + (x * t_0)) / 2.0
	else:
		tmp = (math.exp((x * (eps + -1.0))) + math.exp((x * -eps))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (eps <= 53.0)
		tmp = Float64(Float64(Float64(t_0 * Float64(Float64(x + 1.0) - -1.0)) + Float64(x * t_0)) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps + -1.0))) + exp(Float64(x * Float64(-eps)))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	t_0 = exp(-x);
	tmp = 0.0;
	if (eps <= 53.0)
		tmp = ((t_0 * ((x + 1.0) - -1.0)) + (x * t_0)) / 2.0;
	else
		tmp = (exp((x * (eps + -1.0))) + exp((x * -eps))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[eps, 53.0], N[(N[(N[(t$95$0 * N[(N[(x + 1.0), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] + N[(x * t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * (-eps)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 53

   1. Initial program 61.4%

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

   3. Simplified 61.4%

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

   4. Taylor expanded in eps around 0 65.3%

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

      1. associate--r+ 65.3%

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

      2. associate-*r* 65.3%

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

      3. mul-1-neg 65.3%

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

      4. cancel-sign-sub 65.3%

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

      5. distribute-rgt1-in 65.3%

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

      6. distribute-rgt-out-- 65.3%

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

      7. mul-1-neg 65.3%

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

      8. mul-1-neg 65.3%

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

   6. Simplified 65.3%

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

   if 53 < eps

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in eps around inf 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. mul-1-neg 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*r* 99.9%

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

      6. mul-1-neg 99.9%

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

      7. mul-1-neg 99.9%

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

      8. sub-neg 99.9%

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

      9. mul-1-neg 99.9%

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

      10. exp-prod 99.9%

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

      11. *-lft-identity 99.9%

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

      12. metadata-eval 99.9%

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

      13. cancel-sign-sub-inv 99.9%

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

      14. exp-prod 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in eps around inf 99.9%

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

      1. associate-*r* 99.9%

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

      2. neg-mul-1 99.9%

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

   9. Simplified 99.9%

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

   10. Taylor expanded in x around inf 99.9%

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

       1. mul-1-neg 99.9%

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

       2. *-commutative 99.9%

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

       3. distribute-rgt-neg-in 99.9%

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

       4. mul-1-neg 99.9%

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

       5. distribute-rgt-neg-in 99.9%

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

       6. sub-neg 99.9%

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

       7. mul-1-neg 99.9%

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

       8. distribute-neg-in 99.9%

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

       9. metadata-eval 99.9%

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

       10. mul-1-neg 99.9%

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

       11. remove-double-neg 99.9%

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

   12. Simplified 99.9%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 53:\\ \;\;\;\;\frac{e^{-x} \cdot \left(\left(x + 1\right) - -1\right) + x \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \end{array} \]

Alternative 2: 98.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

eps = abs(eps);
double code(double x, double eps) {
	return (exp((x * (eps + -1.0))) + exp((x * (-1.0 - eps)))) / 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 * ((-1.0d0) - eps)))) / 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 * (-1.0 - eps)))) / 2.0;
}

Python

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

Julia

eps = abs(eps)
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

eps = abs(eps)
function tmp = code(x, eps)
	tmp = (exp((x * (eps + -1.0))) + exp((x * (-1.0 - eps)))) / 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[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 71.6%

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

3. Simplified 71.6%

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

4. Taylor expanded in eps around inf 99.0%

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

   1. *-commutative 99.0%

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

   2. sub-neg 99.0%

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

   3. mul-1-neg 99.0%

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

   4. *-commutative 99.0%

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

   5. associate-*r* 99.0%

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

   6. mul-1-neg 99.0%

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

   7. mul-1-neg 99.0%

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

   8. sub-neg 99.0%

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

   9. mul-1-neg 99.0%

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

   10. exp-prod 99.0%

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

   11. *-lft-identity 99.0%

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

   12. metadata-eval 99.0%

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

   13. cancel-sign-sub-inv 99.0%

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

   14. exp-prod 99.0%

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

6. Simplified 99.0%

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

7. Final simplification 99.0%

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

Alternative 3: 92.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

eps = abs(eps)
function tmp = code(x, eps)
	tmp = (exp((x * (eps + -1.0))) + exp((x * -eps))) / 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 * (-eps)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 71.6%

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

3. Simplified 71.6%

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

4. Taylor expanded in eps around inf 99.0%

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

   1. *-commutative 99.0%

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

   2. sub-neg 99.0%

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

   3. mul-1-neg 99.0%

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

   4. *-commutative 99.0%

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

   5. associate-*r* 99.0%

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

   6. mul-1-neg 99.0%

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

   7. mul-1-neg 99.0%

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

   8. sub-neg 99.0%

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

   9. mul-1-neg 99.0%

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

   10. exp-prod 99.0%

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

   11. *-lft-identity 99.0%

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

   12. metadata-eval 99.0%

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

   13. cancel-sign-sub-inv 99.0%

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

   14. exp-prod 99.0%

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

6. Simplified 99.0%

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

7. Taylor expanded in eps around inf 93.0%

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

   1. associate-*r* 93.0%

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

   2. neg-mul-1 93.0%

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

9. Simplified 93.0%

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

10. Taylor expanded in x around inf 93.0%

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

    1. mul-1-neg 93.0%

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

    2. *-commutative 93.0%

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

    3. distribute-rgt-neg-in 93.0%

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

    4. mul-1-neg 93.0%

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

    5. distribute-rgt-neg-in 93.0%

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

    6. sub-neg 93.0%

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

    7. mul-1-neg 93.0%

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

    8. distribute-neg-in 93.0%

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

    9. metadata-eval 93.0%

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

    10. mul-1-neg 93.0%

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

    11. remove-double-neg 93.0%

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

12. Simplified 93.0%

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

13. Final simplification 93.0%

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

Alternative 4: 84.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-260}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + {e}^{\left(x \cdot \left(\varepsilon + -1\right)\right)}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -1.2e-260)
   (/ (+ 1.0 (exp (* x (- eps)))) 2.0)
   (/ (+ 1.0 (pow E (* x (+ eps -1.0)))) 2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -1.2e-260) {
		tmp = (1.0 + exp((x * -eps))) / 2.0;
	} else {
		tmp = (1.0 + pow(((double) M_E), (x * (eps + -1.0)))) / 2.0;
	}
	return tmp;
}

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.2e-260) {
		tmp = (1.0 + Math.exp((x * -eps))) / 2.0;
	} else {
		tmp = (1.0 + Math.pow(Math.E, (x * (eps + -1.0)))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= -1.2e-260:
		tmp = (1.0 + math.exp((x * -eps))) / 2.0
	else:
		tmp = (1.0 + math.pow(math.e, (x * (eps + -1.0)))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= -1.2e-260)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-eps)))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + (exp(1) ^ Float64(x * Float64(eps + -1.0)))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.2e-260)
		tmp = (1.0 + exp((x * -eps))) / 2.0;
	else
		tmp = (1.0 + (2.71828182845904523536 ^ (x * (eps + -1.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, -1.2e-260], N[(N[(1.0 + N[Exp[N[(x * (-eps)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Power[E, N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2e-260

   1. Initial program 71.9%

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

   3. Simplified 71.9%

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

   4. Taylor expanded in eps around inf 97.6%

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

      1. *-commutative 97.6%

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

      2. sub-neg 97.6%

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

      3. mul-1-neg 97.6%

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

      4. *-commutative 97.6%

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

      5. associate-*r* 97.6%

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

      6. mul-1-neg 97.6%

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

      7. mul-1-neg 97.6%

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

      8. sub-neg 97.6%

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

      9. mul-1-neg 97.6%

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

      10. exp-prod 97.6%

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

      11. *-lft-identity 97.6%

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

      12. metadata-eval 97.6%

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

      13. cancel-sign-sub-inv 97.6%

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

      14. exp-prod 97.6%

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

   6. Simplified 97.6%

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

   7. Taylor expanded in eps around inf 97.7%

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

      1. associate-*r* 97.7%

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

      2. neg-mul-1 97.7%

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

   9. Simplified 97.7%

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

   10. Taylor expanded in x around inf 97.7%

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

       1. mul-1-neg 97.7%

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

       2. *-commutative 97.7%

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

       3. distribute-rgt-neg-in 97.7%

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

       4. mul-1-neg 97.7%

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

       5. distribute-rgt-neg-in 97.7%

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

       6. sub-neg 97.7%

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

       7. mul-1-neg 97.7%

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

       8. distribute-neg-in 97.7%

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

       9. metadata-eval 97.7%

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

       10. mul-1-neg 97.7%

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

       11. remove-double-neg 97.7%

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

   12. Simplified 97.7%

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

   13. Taylor expanded in x around 0 68.1%

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

   if -1.2e-260 < x

   1. Initial program 71.4%

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

   3. Simplified 71.4%

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

   4. Taylor expanded in x around 0 36.5%

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

   5. Taylor expanded in eps around inf 65.2%

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

      1. mul-1-neg 65.2%

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

      2. distribute-rgt-neg-in 65.2%

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

      3. sub-neg 65.2%

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

      4. neg-mul-1 65.2%

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

      5. distribute-neg-in 65.2%

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

      6. metadata-eval 65.2%

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

      7. neg-mul-1 65.2%

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

      8. remove-double-neg 65.2%

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

   7. Simplified 65.2%

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

      1. *-un-lft-identity 65.2%

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

      2. exp-prod 65.2%

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

   9. Applied egg-rr 65.2%

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

       1. exp-1-e 65.2%

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

       2. +-commutative 65.2%

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

   11. Simplified 65.2%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-260}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + {e}^{\left(x \cdot \left(\varepsilon + -1\right)\right)}}{2}\\ \end{array} \]

Alternative 5: 84.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-260}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -1.2e-260)
   (/ (+ 1.0 (exp (* x (- eps)))) 2.0)
   (/ (+ 1.0 (exp (* x (+ eps -1.0)))) 2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -1.2e-260) {
		tmp = (1.0 + exp((x * -eps))) / 2.0;
	} else {
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 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 <= (-1.2d-260)) then
        tmp = (1.0d0 + exp((x * -eps))) / 2.0d0
    else
        tmp = (1.0d0 + exp((x * (eps + (-1.0d0))))) / 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 <= -1.2e-260) {
		tmp = (1.0 + Math.exp((x * -eps))) / 2.0;
	} else {
		tmp = (1.0 + Math.exp((x * (eps + -1.0)))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= -1.2e-260:
		tmp = (1.0 + math.exp((x * -eps))) / 2.0
	else:
		tmp = (1.0 + math.exp((x * (eps + -1.0)))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= -1.2e-260)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-eps)))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps + -1.0)))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.2e-260)
		tmp = (1.0 + exp((x * -eps))) / 2.0;
	else
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, -1.2e-260], N[(N[(1.0 + N[Exp[N[(x * (-eps)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2e-260

   1. Initial program 71.9%

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

   3. Simplified 71.9%

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

   4. Taylor expanded in eps around inf 97.6%

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

      1. *-commutative 97.6%

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

      2. sub-neg 97.6%

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

      3. mul-1-neg 97.6%

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

      4. *-commutative 97.6%

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

      5. associate-*r* 97.6%

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

      6. mul-1-neg 97.6%

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

      7. mul-1-neg 97.6%

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

      8. sub-neg 97.6%

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

      9. mul-1-neg 97.6%

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

      10. exp-prod 97.6%

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

      11. *-lft-identity 97.6%

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

      12. metadata-eval 97.6%

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

      13. cancel-sign-sub-inv 97.6%

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

      14. exp-prod 97.6%

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

   6. Simplified 97.6%

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

   7. Taylor expanded in eps around inf 97.7%

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

      1. associate-*r* 97.7%

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

      2. neg-mul-1 97.7%

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

   9. Simplified 97.7%

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

   10. Taylor expanded in x around inf 97.7%

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

       1. mul-1-neg 97.7%

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

       2. *-commutative 97.7%

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

       3. distribute-rgt-neg-in 97.7%

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

       4. mul-1-neg 97.7%

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

       5. distribute-rgt-neg-in 97.7%

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

       6. sub-neg 97.7%

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

       7. mul-1-neg 97.7%

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

       8. distribute-neg-in 97.7%

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

       9. metadata-eval 97.7%

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

       10. mul-1-neg 97.7%

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

       11. remove-double-neg 97.7%

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

   12. Simplified 97.7%

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

   13. Taylor expanded in x around 0 68.1%

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

   if -1.2e-260 < x

   1. Initial program 71.4%

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

   3. Simplified 71.4%

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

   4. Taylor expanded in x around 0 36.5%

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

   5. Taylor expanded in eps around inf 65.2%

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

      1. mul-1-neg 65.2%

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

      2. distribute-rgt-neg-in 65.2%

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

      3. sub-neg 65.2%

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

      4. neg-mul-1 65.2%

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

      5. distribute-neg-in 65.2%

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

      6. metadata-eval 65.2%

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

      7. neg-mul-1 65.2%

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

      8. remove-double-neg 65.2%

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

   7. Simplified 65.2%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-260}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]

Alternative 6: 77.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 350:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 350.0)
   (/ (+ 1.0 (exp (* x (- eps)))) 2.0)
   (/ (/ (expm1 x) eps) 2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 350.0) {
		tmp = (1.0 + exp((x * -eps))) / 2.0;
	} else {
		tmp = (expm1(x) / eps) / 2.0;
	}
	return tmp;
}

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 350.0) {
		tmp = (1.0 + Math.exp((x * -eps))) / 2.0;
	} else {
		tmp = (Math.expm1(x) / eps) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 350.0:
		tmp = (1.0 + math.exp((x * -eps))) / 2.0
	else:
		tmp = (math.expm1(x) / eps) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 350.0)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-eps)))) / 2.0);
	else
		tmp = Float64(Float64(expm1(x) / eps) / 2.0);
	end
	return tmp
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 350.0], N[(N[(1.0 + N[Exp[N[(x * (-eps)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(Exp[x] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 350

   1. Initial program 61.9%

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

   3. Simplified 61.9%

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

   4. Taylor expanded in eps around inf 98.7%

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

      1. *-commutative 98.7%

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

      2. sub-neg 98.7%

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

      3. mul-1-neg 98.7%

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

      4. *-commutative 98.7%

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

      5. associate-*r* 98.7%

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

      6. mul-1-neg 98.7%

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

      7. mul-1-neg 98.7%

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

      8. sub-neg 98.7%

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

      9. mul-1-neg 98.7%

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

      10. exp-prod 98.7%

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

      11. *-lft-identity 98.7%

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

      12. metadata-eval 98.7%

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

      13. cancel-sign-sub-inv 98.7%

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

      14. exp-prod 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in eps around inf 98.7%

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

      1. associate-*r* 98.7%

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

      2. neg-mul-1 98.7%

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

   9. Simplified 98.7%

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

   10. Taylor expanded in x around inf 98.7%

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

       1. mul-1-neg 98.7%

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

       2. *-commutative 98.7%

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

       3. distribute-rgt-neg-in 98.7%

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

       4. mul-1-neg 98.7%

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

       5. distribute-rgt-neg-in 98.7%

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

       6. sub-neg 98.7%

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

       7. mul-1-neg 98.7%

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

       8. distribute-neg-in 98.7%

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

       9. metadata-eval 98.7%

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

       10. mul-1-neg 98.7%

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

       11. remove-double-neg 98.7%

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

   12. Simplified 98.7%

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

   13. Taylor expanded in x around 0 78.7%

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

   if 350 < 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%

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

   4. Taylor expanded in x around 0 31.2%

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

   5. Taylor expanded in eps around 0 1.9%

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

      1. expm1-def 1.9%

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

      2. neg-mul-1 1.9%

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

   7. Simplified 1.9%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}}{2} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 1.5%

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

      2. expm1-udef 1.4%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}\right)} - 1}}{2} \]

      3. expm1-udef 1.4%

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

      4. expm1-udef 1.4%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{\mathsf{expm1}\left(-x\right)}}{\varepsilon}\right)} - 1}{2} \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}\right)}{\varepsilon}\right)} - 1}{2} \]

      6. sqrt-unprod 29.5%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}\right)}{\varepsilon}\right)} - 1}{2} \]

      7. sqr-neg 29.5%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(\sqrt{\color{blue}{x \cdot x}}\right)}{\varepsilon}\right)} - 1}{2} \]

      8. sqrt-unprod 29.5%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)}{\varepsilon}\right)} - 1}{2} \]

      9. add-sqr-sqrt 29.5%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(\color{blue}{x}\right)}{\varepsilon}\right)} - 1}{2} \]

   9. Applied egg-rr 29.5%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}\right)} - 1}}{2} \]
   10. Step-by-step derivation

       1. expm1-def 29.5%

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

       2. expm1-log1p 29.7%

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

   11. Simplified 29.7%

       \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 350:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \end{array} \]

Alternative 7: 70.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -510:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 350:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x -510.0)
   (/ (/ (expm1 (- x)) eps) 2.0)
   (if (<= x 350.0) 1.0 (/ (/ (expm1 x) eps) 2.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= -510.0) {
		tmp = (expm1(-x) / eps) / 2.0;
	} else if (x <= 350.0) {
		tmp = 1.0;
	} else {
		tmp = (expm1(x) / eps) / 2.0;
	}
	return tmp;
}

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= -510.0) {
		tmp = (Math.expm1(-x) / eps) / 2.0;
	} else if (x <= 350.0) {
		tmp = 1.0;
	} else {
		tmp = (Math.expm1(x) / eps) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= -510.0:
		tmp = (math.expm1(-x) / eps) / 2.0
	elif x <= 350.0:
		tmp = 1.0
	else:
		tmp = (math.expm1(x) / eps) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= -510.0)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps) / 2.0);
	elseif (x <= 350.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(expm1(x) / eps) / 2.0);
	end
	return tmp
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, -510.0], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 350.0], 1.0, N[(N[(N[(Exp[x] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -510

   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%

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

   4. Taylor expanded in x around 0 64.8%

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

   5. Taylor expanded in eps around 0 36.4%

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

      1. expm1-def 36.4%

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

      2. neg-mul-1 36.4%

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

   7. Simplified 36.4%

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

   if -510 < x < 350

   1. Initial program 54.0%

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

   3. Simplified 54.0%

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

   4. Taylor expanded in x around 0 73.9%

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

   if 350 < 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%

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

   4. Taylor expanded in x around 0 31.2%

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

   5. Taylor expanded in eps around 0 1.9%

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

      1. expm1-def 1.9%

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

      2. neg-mul-1 1.9%

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

   7. Simplified 1.9%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}}{2} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 1.5%

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

      2. expm1-udef 1.4%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}\right)} - 1}}{2} \]

      3. expm1-udef 1.4%

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

      4. expm1-udef 1.4%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{\mathsf{expm1}\left(-x\right)}}{\varepsilon}\right)} - 1}{2} \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}\right)}{\varepsilon}\right)} - 1}{2} \]

      6. sqrt-unprod 29.5%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}\right)}{\varepsilon}\right)} - 1}{2} \]

      7. sqr-neg 29.5%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(\sqrt{\color{blue}{x \cdot x}}\right)}{\varepsilon}\right)} - 1}{2} \]

      8. sqrt-unprod 29.5%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)}{\varepsilon}\right)} - 1}{2} \]

      9. add-sqr-sqrt 29.5%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(\color{blue}{x}\right)}{\varepsilon}\right)} - 1}{2} \]

   9. Applied egg-rr 29.5%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}\right)} - 1}}{2} \]
   10. Step-by-step derivation

       1. expm1-def 29.5%

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

       2. expm1-log1p 29.7%

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

   11. Simplified 29.7%

       \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -510:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 350:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \end{array} \]

Alternative 8: 63.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 2.0) (/ (+ 2.0 (* x (- -1.0 eps))) 2.0) (/ (/ (expm1 x) eps) 2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 2.0) {
		tmp = (2.0 + (x * (-1.0 - eps))) / 2.0;
	} else {
		tmp = (expm1(x) / eps) / 2.0;
	}
	return tmp;
}

Java

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

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 2.0:
		tmp = (2.0 + (x * (-1.0 - eps))) / 2.0
	else:
		tmp = (math.expm1(x) / eps) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 2.0)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 - eps))) / 2.0);
	else
		tmp = Float64(Float64(expm1(x) / eps) / 2.0);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 61.9%

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

   3. Simplified 61.9%

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

   4. Taylor expanded in eps around inf 98.7%

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

      1. *-commutative 98.7%

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

      2. sub-neg 98.7%

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

      3. mul-1-neg 98.7%

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

      4. *-commutative 98.7%

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

      5. associate-*r* 98.7%

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

      6. mul-1-neg 98.7%

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

      7. mul-1-neg 98.7%

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

      8. sub-neg 98.7%

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

      9. mul-1-neg 98.7%

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

      10. exp-prod 98.7%

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

      11. *-lft-identity 98.7%

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

      12. metadata-eval 98.7%

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

      13. cancel-sign-sub-inv 98.7%

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

      14. exp-prod 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in x around 0 78.5%

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

   8. Taylor expanded in x around 0 64.9%

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

      1. associate-*r* 64.9%

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

      2. neg-mul-1 64.9%

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

   10. Simplified 64.9%

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

   if 2 < 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%

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

   4. Taylor expanded in x around 0 31.2%

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

   5. Taylor expanded in eps around 0 1.9%

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

      1. expm1-def 1.9%

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

      2. neg-mul-1 1.9%

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

   7. Simplified 1.9%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}}{2} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 1.5%

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

      2. expm1-udef 1.4%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}\right)} - 1}}{2} \]

      3. expm1-udef 1.4%

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

      4. expm1-udef 1.4%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\color{blue}{\mathsf{expm1}\left(-x\right)}}{\varepsilon}\right)} - 1}{2} \]

      5. add-sqr-sqrt 0.0%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}\right)}{\varepsilon}\right)} - 1}{2} \]

      6. sqrt-unprod 29.5%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}\right)}{\varepsilon}\right)} - 1}{2} \]

      7. sqr-neg 29.5%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(\sqrt{\color{blue}{x \cdot x}}\right)}{\varepsilon}\right)} - 1}{2} \]

      8. sqrt-unprod 29.5%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)}{\varepsilon}\right)} - 1}{2} \]

      9. add-sqr-sqrt 29.5%

         \[\leadsto \frac{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(\color{blue}{x}\right)}{\varepsilon}\right)} - 1}{2} \]

   9. Applied egg-rr 29.5%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}\right)} - 1}}{2} \]
   10. Step-by-step derivation

       1. expm1-def 29.5%

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

       2. expm1-log1p 29.7%

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

   11. Simplified 29.7%

       \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.9%

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

Alternative 9: 61.5% accurate, 20.5× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+128}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 2.0)
   (/ (+ 2.0 (* x (- -1.0 eps))) 2.0)
   (if (<= x 1.6e+128) 0.0 (/ (* x eps) 2.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 2.0) {
		tmp = (2.0 + (x * (-1.0 - eps))) / 2.0;
	} else if (x <= 1.6e+128) {
		tmp = 0.0;
	} else {
		tmp = (x * eps) / 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 <= 2.0d0) then
        tmp = (2.0d0 + (x * ((-1.0d0) - eps))) / 2.0d0
    else if (x <= 1.6d+128) then
        tmp = 0.0d0
    else
        tmp = (x * eps) / 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 <= 2.0) {
		tmp = (2.0 + (x * (-1.0 - eps))) / 2.0;
	} else if (x <= 1.6e+128) {
		tmp = 0.0;
	} else {
		tmp = (x * eps) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 2.0:
		tmp = (2.0 + (x * (-1.0 - eps))) / 2.0
	elif x <= 1.6e+128:
		tmp = 0.0
	else:
		tmp = (x * eps) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 2.0)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 - eps))) / 2.0);
	elseif (x <= 1.6e+128)
		tmp = 0.0;
	else
		tmp = Float64(Float64(x * eps) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = (2.0 + (x * (-1.0 - eps))) / 2.0;
	elseif (x <= 1.6e+128)
		tmp = 0.0;
	else
		tmp = (x * eps) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 2.0], N[(N[(2.0 + N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.6e+128], 0.0, N[(N[(x * eps), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2

   1. Initial program 61.9%

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

   3. Simplified 61.9%

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

   4. Taylor expanded in eps around inf 98.7%

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

      1. *-commutative 98.7%

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

      2. sub-neg 98.7%

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

      3. mul-1-neg 98.7%

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

      4. *-commutative 98.7%

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

      5. associate-*r* 98.7%

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

      6. mul-1-neg 98.7%

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

      7. mul-1-neg 98.7%

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

      8. sub-neg 98.7%

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

      9. mul-1-neg 98.7%

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

      10. exp-prod 98.7%

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

      11. *-lft-identity 98.7%

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

      12. metadata-eval 98.7%

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

      13. cancel-sign-sub-inv 98.7%

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

      14. exp-prod 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in x around 0 78.5%

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

   8. Taylor expanded in x around 0 64.9%

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

      1. associate-*r* 64.9%

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

      2. neg-mul-1 64.9%

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

   10. Simplified 64.9%

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

   if 2 < x < 1.59999999999999993e128

   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%

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

   4. Taylor expanded in eps around 0 39.7%

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

      1. div-sub 39.7%

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

      2. rec-exp 39.7%

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

      3. mul-1-neg 39.7%

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

      4. +-inverses 39.7%

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

   6. Simplified 39.7%

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

   if 1.59999999999999993e128 < 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%

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

   4. Taylor expanded in x around 0 28.2%

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

      1. +-commutative 28.2%

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

      2. +-commutative 28.2%

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

      3. associate-+l+ 28.2%

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

      4. mul-1-neg 28.2%

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

      5. distribute-rgt-neg-in 28.2%

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

      6. *-commutative 28.2%

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

      7. distribute-rgt-neg-in 28.2%

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

      8. mul-1-neg 28.2%

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

      9. distribute-lft-in 28.2%

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

      10. metadata-eval 28.2%

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

      11. neg-mul-1 28.2%

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

      12. distribute-neg-frac 28.2%

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

      13. metadata-eval 28.2%

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

   6. Simplified 28.2%

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

   7. Taylor expanded in eps around inf 19.7%

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

      1. *-commutative 19.7%

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

   9. Simplified 19.7%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+128}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]

Alternative 10: 54.7% accurate, 24.9× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 5.1 \cdot 10^{+14}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+130}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 5.1e+14) 1.0 (if (<= x 3.2e+130) 0.0 (/ (* x eps) 2.0))))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 5.1e+14) {
		tmp = 1.0;
	} else if (x <= 3.2e+130) {
		tmp = 0.0;
	} else {
		tmp = (x * eps) / 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 <= 5.1d+14) then
        tmp = 1.0d0
    else if (x <= 3.2d+130) then
        tmp = 0.0d0
    else
        tmp = (x * eps) / 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 <= 5.1e+14) {
		tmp = 1.0;
	} else if (x <= 3.2e+130) {
		tmp = 0.0;
	} else {
		tmp = (x * eps) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 5.1e+14:
		tmp = 1.0
	elif x <= 3.2e+130:
		tmp = 0.0
	else:
		tmp = (x * eps) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 5.1e+14)
		tmp = 1.0;
	elseif (x <= 3.2e+130)
		tmp = 0.0;
	else
		tmp = Float64(Float64(x * eps) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 5.1e+14)
		tmp = 1.0;
	elseif (x <= 3.2e+130)
		tmp = 0.0;
	else
		tmp = (x * eps) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 5.1e+14], 1.0, If[LessEqual[x, 3.2e+130], 0.0, N[(N[(x * eps), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.1e14

   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

   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 60.8%

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

   if 5.1e14 < x < 3.2e130

   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%

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

   4. Taylor expanded in eps around 0 43.8%

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

      1. div-sub 43.8%

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

      2. rec-exp 43.8%

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

      3. mul-1-neg 43.8%

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

      4. +-inverses 43.8%

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

   6. Simplified 43.8%

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

   if 3.2e130 < 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%

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

   4. Taylor expanded in x around 0 28.2%

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

      1. +-commutative 28.2%

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

      2. +-commutative 28.2%

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

      3. associate-+l+ 28.2%

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

      4. mul-1-neg 28.2%

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

      5. distribute-rgt-neg-in 28.2%

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

      6. *-commutative 28.2%

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

      7. distribute-rgt-neg-in 28.2%

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

      8. mul-1-neg 28.2%

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

      9. distribute-lft-in 28.2%

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

      10. metadata-eval 28.2%

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

      11. neg-mul-1 28.2%

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

      12. distribute-neg-frac 28.2%

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

      13. metadata-eval 28.2%

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

   6. Simplified 28.2%

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

   7. Taylor expanded in eps around inf 19.7%

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

      1. *-commutative 19.7%

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

   9. Simplified 19.7%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.1 \cdot 10^{+14}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+130}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]

Alternative 11: 56.7% accurate, 74.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 5.1 \cdot 10^{+14}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps) :precision binary64 (if (<= x 5.1e+14) 1.0 0.0))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 5.1e+14) {
		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 <= 5.1d+14) 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 <= 5.1e+14) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 5.1e+14:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 5.1e+14)
		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 <= 5.1e+14)
		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, 5.1e+14], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 5.1e14

   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

   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 60.8%

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

   if 5.1e14 < 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%

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

   4. Taylor expanded in eps around 0 33.3%

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

      1. div-sub 33.3%

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

      2. rec-exp 33.3%

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

      3. mul-1-neg 33.3%

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

      4. +-inverses 33.3%

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

   6. Simplified 33.3%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.1 \cdot 10^{+14}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 12: 15.9% 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 71.6%

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

3. Simplified 64.8%

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

4. Taylor expanded in eps around 0 9.8%

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

   1. div-sub 9.8%

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

   2. rec-exp 9.8%

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

   3. mul-1-neg 9.8%

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

   4. +-inverses 10.0%

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

6. Simplified 10.0%

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

7. Final simplification 10.0%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023320 
(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))