NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.8% → 98.7%

Time: 26.6s

Alternatives: 16

Speedup: 1.8×

• 39.1% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.6%

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

3. Simplified 74.6%

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

4. Taylor expanded in eps around inf 99.3%

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

5. Final simplification 99.3%

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

Alternative 2: 83.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.8 \cdot 10^{-93}:\\ \;\;\;\;\frac{e^{-x} + \left(x + \left(1 - x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x - x} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 1.8e-93)
   (/ (+ (exp (- x)) (+ x (- 1.0 x))) 2.0)
   (/ (+ (exp (- (* eps x) x)) (exp (* eps (- x)))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 1.8e-93) {
		tmp = (exp(-x) + (x + (1.0 - x))) / 2.0;
	} else {
		tmp = (exp(((eps * x) - x)) + exp((eps * -x))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 1.8d-93) then
        tmp = (exp(-x) + (x + (1.0d0 - x))) / 2.0d0
    else
        tmp = (exp(((eps * x) - x)) + exp((eps * -x))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= 1.8e-93) {
		tmp = (Math.exp(-x) + (x + (1.0 - x))) / 2.0;
	} else {
		tmp = (Math.exp(((eps * x) - x)) + Math.exp((eps * -x))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 1.8e-93:
		tmp = (math.exp(-x) + (x + (1.0 - x))) / 2.0
	else:
		tmp = (math.exp(((eps * x) - x)) + math.exp((eps * -x))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 1.8e-93)
		tmp = Float64(Float64(exp(Float64(-x)) + Float64(x + Float64(1.0 - x))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(Float64(eps * x) - x)) + exp(Float64(eps * Float64(-x)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 1.8e-93)
		tmp = (exp(-x) + (x + (1.0 - x))) / 2.0;
	else
		tmp = (exp(((eps * x) - x)) + exp((eps * -x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 1.8e-93], N[(N[(N[Exp[(-x)], $MachinePrecision] + N[(x + N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(N[(eps * x), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(eps * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.8000000000000001e-93

   1. Initial program 63.2%

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

   3. Simplified 63.2%

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

   4. Taylor expanded in x around 0 27.5%

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

      1. associate-*r* 27.5%

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

      2. mul-1-neg 27.5%

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

      3. distribute-lft-out 27.5%

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

      4. *-rgt-identity 27.5%

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

      5. cancel-sign-sub-inv 27.5%

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

   6. Simplified 27.5%

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

   7. Taylor expanded in eps around inf 63.2%

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

   8. Taylor expanded in eps around 0 79.1%

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

      1. mul-1-neg 79.1%

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

      2. distribute-lft-in 79.1%

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

      3. *-lft-identity 79.1%

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

      4. cancel-sign-sub-inv 79.1%

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

      5. metadata-eval 79.1%

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

      6. *-lft-identity 79.1%

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

      7. *-lft-identity 79.1%

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

   10. Simplified 79.1%

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

   if 1.8000000000000001e-93 < eps

   1. Initial program 90.8%

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

   3. Simplified 90.9%

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

   4. Taylor expanded in eps around inf 99.3%

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

      1. div-inv 99.3%

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

      2. rec-exp 99.3%

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

      3. fma-udef 99.3%

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

      4. *-commutative 99.3%

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

      5. distribute-neg-in 99.3%

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

      6. +-commutative 99.3%

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

      7. sub-neg 99.3%

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

      8. neg-mul-1 99.3%

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

      9. *-commutative 99.3%

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

      10. distribute-rgt-out-- 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. *-lft-identity 99.3%

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

   8. Simplified 99.3%

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

   9. Taylor expanded in eps around inf 99.3%

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

       1. mul-1-neg 99.3%

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

       2. *-commutative 99.3%

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

   11. Simplified 99.3%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.8 \cdot 10^{-93}:\\ \;\;\;\;\frac{e^{-x} + \left(x + \left(1 - x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x - x} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \end{array} \]

Alternative 3: 98.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.6%

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

3. Simplified 74.6%

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

4. Taylor expanded in eps around inf 99.3%

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

   1. div-inv 99.3%

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

   2. rec-exp 99.3%

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

   3. fma-udef 99.3%

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

   4. *-commutative 99.3%

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

   5. distribute-neg-in 99.3%

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

   6. +-commutative 99.3%

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

   7. sub-neg 99.3%

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

   8. neg-mul-1 99.3%

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

   9. *-commutative 99.3%

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

   10. distribute-rgt-out-- 99.3%

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

6. Applied egg-rr 99.3%

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

   1. *-lft-identity 99.3%

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

8. Simplified 99.3%

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

9. Final simplification 99.3%

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

Alternative 4: 79.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{1}{\varepsilon}\\ t_1 := e^{-x}\\ \mathbf{if}\;x \leq -250000000:\\ \;\;\;\;\frac{1 + t_1}{2}\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-162}:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(0.5, \left(\left(x \cdot x\right) \cdot t_0\right) \cdot \left(\varepsilon \cdot \varepsilon\right), \varepsilon \cdot \left(x \cdot t_0\right)\right)}{2}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+60}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+253}:\\ \;\;\;\;\frac{t_0 \cdot e^{x \cdot \left(\varepsilon + -1\right)} + \left(x + \left(\left(1 - x\right) - \varepsilon \cdot x\right)\right)}{2}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+283}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x - x} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 + \left(x + \left(1 - x\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ 1.0 eps))) (t_1 (exp (- x))))
   (if (<= x -250000000.0)
     (/ (+ 1.0 t_1) 2.0)
     (if (<= x -1.55e-162)
       (/
        (+ 2.0 (fma 0.5 (* (* (* x x) t_0) (* eps eps)) (* eps (* x t_0))))
        2.0)
       (if (<= x 9e+60)
         (/ (+ 1.0 (exp (* eps x))) 2.0)
         (if (<= x 7e+253)
           (/
            (+ (* t_0 (exp (* x (+ eps -1.0)))) (+ x (- (- 1.0 x) (* eps x))))
            2.0)
           (if (<= x 2.8e+283)
             (/ (+ (exp (- (* eps x) x)) 1.0) 2.0)
             (/ (+ t_1 (+ x (- 1.0 x))) 2.0))))))))

C

double code(double x, double eps) {
	double t_0 = 1.0 + (1.0 / eps);
	double t_1 = exp(-x);
	double tmp;
	if (x <= -250000000.0) {
		tmp = (1.0 + t_1) / 2.0;
	} else if (x <= -1.55e-162) {
		tmp = (2.0 + fma(0.5, (((x * x) * t_0) * (eps * eps)), (eps * (x * t_0)))) / 2.0;
	} else if (x <= 9e+60) {
		tmp = (1.0 + exp((eps * x))) / 2.0;
	} else if (x <= 7e+253) {
		tmp = ((t_0 * exp((x * (eps + -1.0)))) + (x + ((1.0 - x) - (eps * x)))) / 2.0;
	} else if (x <= 2.8e+283) {
		tmp = (exp(((eps * x) - x)) + 1.0) / 2.0;
	} else {
		tmp = (t_1 + (x + (1.0 - x))) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(1.0 + Float64(1.0 / eps))
	t_1 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -250000000.0)
		tmp = Float64(Float64(1.0 + t_1) / 2.0);
	elseif (x <= -1.55e-162)
		tmp = Float64(Float64(2.0 + fma(0.5, Float64(Float64(Float64(x * x) * t_0) * Float64(eps * eps)), Float64(eps * Float64(x * t_0)))) / 2.0);
	elseif (x <= 9e+60)
		tmp = Float64(Float64(1.0 + exp(Float64(eps * x))) / 2.0);
	elseif (x <= 7e+253)
		tmp = Float64(Float64(Float64(t_0 * exp(Float64(x * Float64(eps + -1.0)))) + Float64(x + Float64(Float64(1.0 - x) - Float64(eps * x)))) / 2.0);
	elseif (x <= 2.8e+283)
		tmp = Float64(Float64(exp(Float64(Float64(eps * x) - x)) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(t_1 + Float64(x + Float64(1.0 - x))) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -250000000.0], N[(N[(1.0 + t$95$1), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -1.55e-162], N[(N[(2.0 + N[(0.5 * N[(N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 9e+60], N[(N[(1.0 + N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 7e+253], N[(N[(N[(t$95$0 * N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(x + N[(N[(1.0 - x), $MachinePrecision] - N[(eps * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.8e+283], N[(N[(N[Exp[N[(N[(eps * x), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$1 + N[(x + N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < -2.5e8

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in x around 0 52.9%

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

   6. Taylor expanded in eps around 0 100.0%

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

   if -2.5e8 < x < -1.5499999999999999e-162

   1. Initial program 56.3%

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

   3. Simplified 56.3%

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

   4. Taylor expanded in x around 0 34.9%

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

   5. Taylor expanded in eps around inf 32.5%

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

      1. *-commutative 32.5%

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

   7. Simplified 32.5%

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

   8. Taylor expanded in x around 0 88.3%

      \[\leadsto \frac{\color{blue}{2 + \left(0.5 \cdot \left({\varepsilon}^{2} \cdot \left({x}^{2} \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)\right) + \varepsilon \cdot \left(x \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)\right)}}{2} \]
   9. Step-by-step derivation

      1. fma-def 88.3%

         \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(0.5, {\varepsilon}^{2} \cdot \left({x}^{2} \cdot \left(1 + \frac{1}{\varepsilon}\right)\right), \varepsilon \cdot \left(x \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)\right)}}{2} \]

      2. *-commutative 88.3%

         \[\leadsto \frac{2 + \mathsf{fma}\left(0.5, \color{blue}{\left({x}^{2} \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) \cdot {\varepsilon}^{2}}, \varepsilon \cdot \left(x \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)\right)}{2} \]

      3. unpow2 88.3%

         \[\leadsto \frac{2 + \mathsf{fma}\left(0.5, \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) \cdot {\varepsilon}^{2}, \varepsilon \cdot \left(x \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)\right)}{2} \]

      4. unpow2 88.3%

         \[\leadsto \frac{2 + \mathsf{fma}\left(0.5, \left(\left(x \cdot x\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}, \varepsilon \cdot \left(x \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)\right)}{2} \]

   10. Simplified 88.3%

       \[\leadsto \frac{\color{blue}{2 + \mathsf{fma}\left(0.5, \left(\left(x \cdot x\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right), \varepsilon \cdot \left(x \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)\right)}}{2} \]

   if -1.5499999999999999e-162 < x < 9.00000000000000026e60

   1. Initial program 58.0%

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

   3. Simplified 58.0%

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

   4. Taylor expanded in x around 0 40.9%

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

   5. Taylor expanded in eps around inf 43.4%

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

      1. *-commutative 43.4%

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

   7. Simplified 43.4%

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

   8. Taylor expanded in eps around inf 82.3%

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

      1. *-commutative 82.3%

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

   10. Simplified 82.3%

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

   if 9.00000000000000026e60 < x < 6.99999999999999955e253

   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%

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

   4. Taylor expanded in x around 0 21.5%

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

      1. associate-*r* 21.5%

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

      2. mul-1-neg 21.5%

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

      3. distribute-lft-out 21.5%

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

      4. *-rgt-identity 21.5%

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

      5. cancel-sign-sub-inv 21.5%

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

   6. Simplified 21.5%

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

   7. Taylor expanded in eps around inf 71.5%

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

   if 6.99999999999999955e253 < x < 2.80000000000000004e283

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in x around 0 46.2%

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

   if 2.80000000000000004e283 < 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%

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

   4. Taylor expanded in x around 0 1.0%

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

      1. associate-*r* 1.0%

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

      2. mul-1-neg 1.0%

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

      3. distribute-lft-out 1.0%

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

      4. *-rgt-identity 1.0%

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

      5. cancel-sign-sub-inv 1.0%

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

   6. Simplified 1.0%

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

   7. Taylor expanded in eps around inf 1.4%

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

   8. Taylor expanded in eps around 0 67.2%

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

      1. mul-1-neg 67.2%

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

      2. distribute-lft-in 67.2%

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

      3. *-lft-identity 67.2%

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

      4. cancel-sign-sub-inv 67.2%

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

      5. metadata-eval 67.2%

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

      6. *-lft-identity 67.2%

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

      7. *-lft-identity 67.2%

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

   10. Simplified 67.2%

       \[\leadsto \frac{\color{blue}{e^{-x} + \left(x + \left(1 - x\right)\right)}}{2} \]
3. Recombined 6 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -250000000:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-162}:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(0.5, \left(\left(x \cdot x\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right), \varepsilon \cdot \left(x \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)\right)}{2}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+60}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+253}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + \left(x + \left(\left(1 - x\right) - \varepsilon \cdot x\right)\right)}{2}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+283}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x - x} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} + \left(x + \left(1 - x\right)\right)}{2}\\ \end{array} \]

Alternative 5: 77.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\varepsilon \cdot x}\\ t_1 := 1 + \frac{1}{\varepsilon}\\ t_2 := e^{-x}\\ \mathbf{if}\;x \leq -250000000:\\ \;\;\;\;\frac{1 + t_2}{2}\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-162}:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(0.5, \left(\left(x \cdot x\right) \cdot t_1\right) \cdot \left(\varepsilon \cdot \varepsilon\right), \varepsilon \cdot \left(x \cdot t_1\right)\right)}{2}\\ \mathbf{elif}\;x \leq 300:\\ \;\;\;\;\frac{1 + t_0}{2}\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{+228}:\\ \;\;\;\;\frac{\left(1 + t_1 \cdot t_0\right) + \frac{-1}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+253}:\\ \;\;\;\;\frac{t_2 + \left(x + \left(1 - x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x - x} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (* eps x))) (t_1 (+ 1.0 (/ 1.0 eps))) (t_2 (exp (- x))))
   (if (<= x -250000000.0)
     (/ (+ 1.0 t_2) 2.0)
     (if (<= x -1.55e-162)
       (/
        (+ 2.0 (fma 0.5 (* (* (* x x) t_1) (* eps eps)) (* eps (* x t_1))))
        2.0)
       (if (<= x 300.0)
         (/ (+ 1.0 t_0) 2.0)
         (if (<= x 7.6e+228)
           (/ (+ (+ 1.0 (* t_1 t_0)) (/ -1.0 eps)) 2.0)
           (if (<= x 7e+253)
             (/ (+ t_2 (+ x (- 1.0 x))) 2.0)
             (/ (+ (exp (- (* eps x) x)) 1.0) 2.0))))))))

C

double code(double x, double eps) {
	double t_0 = exp((eps * x));
	double t_1 = 1.0 + (1.0 / eps);
	double t_2 = exp(-x);
	double tmp;
	if (x <= -250000000.0) {
		tmp = (1.0 + t_2) / 2.0;
	} else if (x <= -1.55e-162) {
		tmp = (2.0 + fma(0.5, (((x * x) * t_1) * (eps * eps)), (eps * (x * t_1)))) / 2.0;
	} else if (x <= 300.0) {
		tmp = (1.0 + t_0) / 2.0;
	} else if (x <= 7.6e+228) {
		tmp = ((1.0 + (t_1 * t_0)) + (-1.0 / eps)) / 2.0;
	} else if (x <= 7e+253) {
		tmp = (t_2 + (x + (1.0 - x))) / 2.0;
	} else {
		tmp = (exp(((eps * x) - x)) + 1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = exp(Float64(eps * x))
	t_1 = Float64(1.0 + Float64(1.0 / eps))
	t_2 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -250000000.0)
		tmp = Float64(Float64(1.0 + t_2) / 2.0);
	elseif (x <= -1.55e-162)
		tmp = Float64(Float64(2.0 + fma(0.5, Float64(Float64(Float64(x * x) * t_1) * Float64(eps * eps)), Float64(eps * Float64(x * t_1)))) / 2.0);
	elseif (x <= 300.0)
		tmp = Float64(Float64(1.0 + t_0) / 2.0);
	elseif (x <= 7.6e+228)
		tmp = Float64(Float64(Float64(1.0 + Float64(t_1 * t_0)) + Float64(-1.0 / eps)) / 2.0);
	elseif (x <= 7e+253)
		tmp = Float64(Float64(t_2 + Float64(x + Float64(1.0 - x))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(Float64(eps * x) - x)) + 1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -250000000.0], N[(N[(1.0 + t$95$2), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -1.55e-162], N[(N[(2.0 + N[(0.5 * N[(N[(N[(x * x), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 300.0], N[(N[(1.0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 7.6e+228], N[(N[(N[(1.0 + N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 7e+253], N[(N[(t$95$2 + N[(x + N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(N[(eps * x), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < -2.5e8

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in x around 0 52.9%

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

   6. Taylor expanded in eps around 0 100.0%

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

   if -2.5e8 < x < -1.5499999999999999e-162

   1. Initial program 56.3%

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

   3. Simplified 56.3%

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

   4. Taylor expanded in x around 0 34.9%

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

   5. Taylor expanded in eps around inf 32.5%

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

      1. *-commutative 32.5%

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

   7. Simplified 32.5%

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

   8. Taylor expanded in x around 0 88.3%

      \[\leadsto \frac{\color{blue}{2 + \left(0.5 \cdot \left({\varepsilon}^{2} \cdot \left({x}^{2} \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)\right) + \varepsilon \cdot \left(x \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)\right)}}{2} \]
   9. Step-by-step derivation

      1. fma-def 88.3%

         \[\leadsto \frac{2 + \color{blue}{\mathsf{fma}\left(0.5, {\varepsilon}^{2} \cdot \left({x}^{2} \cdot \left(1 + \frac{1}{\varepsilon}\right)\right), \varepsilon \cdot \left(x \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)\right)}}{2} \]

      2. *-commutative 88.3%

         \[\leadsto \frac{2 + \mathsf{fma}\left(0.5, \color{blue}{\left({x}^{2} \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) \cdot {\varepsilon}^{2}}, \varepsilon \cdot \left(x \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)\right)}{2} \]

      3. unpow2 88.3%

         \[\leadsto \frac{2 + \mathsf{fma}\left(0.5, \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) \cdot {\varepsilon}^{2}, \varepsilon \cdot \left(x \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)\right)}{2} \]

      4. unpow2 88.3%

         \[\leadsto \frac{2 + \mathsf{fma}\left(0.5, \left(\left(x \cdot x\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}, \varepsilon \cdot \left(x \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)\right)}{2} \]

   10. Simplified 88.3%

       \[\leadsto \frac{\color{blue}{2 + \mathsf{fma}\left(0.5, \left(\left(x \cdot x\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right), \varepsilon \cdot \left(x \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)\right)}}{2} \]

   if -1.5499999999999999e-162 < x < 300

   1. Initial program 53.4%

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

   3. Simplified 53.4%

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

   4. Taylor expanded in x around 0 41.1%

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

   5. Taylor expanded in eps around inf 41.2%

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

      1. *-commutative 41.2%

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

   7. Simplified 41.2%

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

   8. Taylor expanded in eps around inf 87.0%

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

      1. *-commutative 87.0%

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

   10. Simplified 87.0%

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

   if 300 < x < 7.6000000000000004e228

   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%

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

   4. Taylor expanded in x around 0 36.3%

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

   5. Taylor expanded in eps around inf 57.5%

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

      1. *-commutative 57.5%

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

   7. Simplified 57.5%

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

   8. Taylor expanded in x around inf 57.5%

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

   if 7.6000000000000004e228 < x < 6.99999999999999955e253

   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%

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

   4. Taylor expanded in x around 0 1.6%

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

      1. associate-*r* 1.6%

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

      2. mul-1-neg 1.6%

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

      3. distribute-lft-out 1.6%

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

      4. *-rgt-identity 1.6%

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

      5. cancel-sign-sub-inv 1.6%

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

   6. Simplified 1.6%

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

   7. Taylor expanded in eps around inf 2.6%

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

   8. Taylor expanded in eps around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. *-lft-identity 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. metadata-eval 100.0%

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

      6. *-lft-identity 100.0%

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

      7. *-lft-identity 100.0%

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

   10. Simplified 100.0%

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

   if 6.99999999999999955e253 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in x around 0 41.9%

      \[\leadsto \frac{e^{\varepsilon \cdot x - x} + \color{blue}{1}}{2} \]
3. Recombined 6 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -250000000:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-162}:\\ \;\;\;\;\frac{2 + \mathsf{fma}\left(0.5, \left(\left(x \cdot x\right) \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right), \varepsilon \cdot \left(x \cdot \left(1 + \frac{1}{\varepsilon}\right)\right)\right)}{2}\\ \mathbf{elif}\;x \leq 300:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{+228}:\\ \;\;\;\;\frac{\left(1 + \left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\varepsilon \cdot x}\right) + \frac{-1}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+253}:\\ \;\;\;\;\frac{e^{-x} + \left(x + \left(1 - x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x - x} + 1}{2}\\ \end{array} \]

Alternative 6: 76.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -250000000:\\ \;\;\;\;\frac{1 + t_0}{2}\\ \mathbf{elif}\;x \leq 300:\\ \;\;\;\;\frac{\left(1 + e^{x \cdot \left(\varepsilon + -1\right)}\right) - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+227}:\\ \;\;\;\;\frac{\left(1 + \left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\varepsilon \cdot x}\right) + \frac{-1}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+253}:\\ \;\;\;\;\frac{t_0 + \left(x + \left(1 - x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x - x} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -250000000.0)
     (/ (+ 1.0 t_0) 2.0)
     (if (<= x 300.0)
       (/ (- (+ 1.0 (exp (* x (+ eps -1.0)))) (* eps x)) 2.0)
       (if (<= x 2.8e+227)
         (/
          (+ (+ 1.0 (* (+ 1.0 (/ 1.0 eps)) (exp (* eps x)))) (/ -1.0 eps))
          2.0)
         (if (<= x 7e+253)
           (/ (+ t_0 (+ x (- 1.0 x))) 2.0)
           (/ (+ (exp (- (* eps x) x)) 1.0) 2.0)))))))

C

double code(double x, double eps) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -250000000.0) {
		tmp = (1.0 + t_0) / 2.0;
	} else if (x <= 300.0) {
		tmp = ((1.0 + exp((x * (eps + -1.0)))) - (eps * x)) / 2.0;
	} else if (x <= 2.8e+227) {
		tmp = ((1.0 + ((1.0 + (1.0 / eps)) * exp((eps * x)))) + (-1.0 / eps)) / 2.0;
	} else if (x <= 7e+253) {
		tmp = (t_0 + (x + (1.0 - x))) / 2.0;
	} else {
		tmp = (exp(((eps * x) - x)) + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (x <= (-250000000.0d0)) then
        tmp = (1.0d0 + t_0) / 2.0d0
    else if (x <= 300.0d0) then
        tmp = ((1.0d0 + exp((x * (eps + (-1.0d0))))) - (eps * x)) / 2.0d0
    else if (x <= 2.8d+227) then
        tmp = ((1.0d0 + ((1.0d0 + (1.0d0 / eps)) * exp((eps * x)))) + ((-1.0d0) / eps)) / 2.0d0
    else if (x <= 7d+253) then
        tmp = (t_0 + (x + (1.0d0 - x))) / 2.0d0
    else
        tmp = (exp(((eps * x) - x)) + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.exp(-x);
	double tmp;
	if (x <= -250000000.0) {
		tmp = (1.0 + t_0) / 2.0;
	} else if (x <= 300.0) {
		tmp = ((1.0 + Math.exp((x * (eps + -1.0)))) - (eps * x)) / 2.0;
	} else if (x <= 2.8e+227) {
		tmp = ((1.0 + ((1.0 + (1.0 / eps)) * Math.exp((eps * x)))) + (-1.0 / eps)) / 2.0;
	} else if (x <= 7e+253) {
		tmp = (t_0 + (x + (1.0 - x))) / 2.0;
	} else {
		tmp = (Math.exp(((eps * x) - x)) + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.exp(-x)
	tmp = 0
	if x <= -250000000.0:
		tmp = (1.0 + t_0) / 2.0
	elif x <= 300.0:
		tmp = ((1.0 + math.exp((x * (eps + -1.0)))) - (eps * x)) / 2.0
	elif x <= 2.8e+227:
		tmp = ((1.0 + ((1.0 + (1.0 / eps)) * math.exp((eps * x)))) + (-1.0 / eps)) / 2.0
	elif x <= 7e+253:
		tmp = (t_0 + (x + (1.0 - x))) / 2.0
	else:
		tmp = (math.exp(((eps * x) - x)) + 1.0) / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -250000000.0)
		tmp = Float64(Float64(1.0 + t_0) / 2.0);
	elseif (x <= 300.0)
		tmp = Float64(Float64(Float64(1.0 + exp(Float64(x * Float64(eps + -1.0)))) - Float64(eps * x)) / 2.0);
	elseif (x <= 2.8e+227)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(eps * x)))) + Float64(-1.0 / eps)) / 2.0);
	elseif (x <= 7e+253)
		tmp = Float64(Float64(t_0 + Float64(x + Float64(1.0 - x))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(Float64(eps * x) - x)) + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = exp(-x);
	tmp = 0.0;
	if (x <= -250000000.0)
		tmp = (1.0 + t_0) / 2.0;
	elseif (x <= 300.0)
		tmp = ((1.0 + exp((x * (eps + -1.0)))) - (eps * x)) / 2.0;
	elseif (x <= 2.8e+227)
		tmp = ((1.0 + ((1.0 + (1.0 / eps)) * exp((eps * x)))) + (-1.0 / eps)) / 2.0;
	elseif (x <= 7e+253)
		tmp = (t_0 + (x + (1.0 - x))) / 2.0;
	else
		tmp = (exp(((eps * x) - x)) + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -250000000.0], N[(N[(1.0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 300.0], N[(N[(N[(1.0 + N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.8e+227], N[(N[(N[(1.0 + N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 7e+253], N[(N[(t$95$0 + N[(x + N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(N[(eps * x), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -2.5e8

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in x around 0 52.9%

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

   6. Taylor expanded in eps around 0 100.0%

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

   if -2.5e8 < x < 300

   1. Initial program 54.3%

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

   3. Simplified 54.3%

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

   4. Taylor expanded in x around 0 40.2%

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

      1. associate-*r* 40.2%

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

      2. mul-1-neg 40.2%

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

      3. distribute-lft-out 40.2%

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

      4. *-rgt-identity 40.2%

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

      5. cancel-sign-sub-inv 40.2%

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

   6. Simplified 40.2%

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

   7. Taylor expanded in eps around inf 84.0%

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

   8. Taylor expanded in x around inf 84.0%

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

      1. mul-1-neg 84.0%

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

      2. distribute-lft-neg-in 84.0%

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

      3. *-commutative 84.0%

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

      4. associate-+r+ 84.0%

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

      5. associate-*r* 84.0%

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

      6. sub-neg 84.0%

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

      7. mul-1-neg 84.0%

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

      8. associate-*r* 84.0%

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

      9. mul-1-neg 84.0%

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

      10. *-commutative 84.0%

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

      11. mul-1-neg 84.0%

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

      12. sub-neg 84.0%

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

      13. distribute-rgt-neg-in 84.0%

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

      14. *-commutative 84.0%

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

      15. distribute-lft-neg-in 84.0%

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

      16. distribute-rgt-neg-in 84.0%

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

   10. Simplified 84.0%

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

   if 300 < x < 2.79999999999999984e227

   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%

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

   4. Taylor expanded in x around 0 36.3%

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

   5. Taylor expanded in eps around inf 57.5%

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

      1. *-commutative 57.5%

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

   7. Simplified 57.5%

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

   8. Taylor expanded in x around inf 57.5%

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

   if 2.79999999999999984e227 < x < 6.99999999999999955e253

   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%

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

   4. Taylor expanded in x around 0 1.6%

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

      1. associate-*r* 1.6%

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

      2. mul-1-neg 1.6%

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

      3. distribute-lft-out 1.6%

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

      4. *-rgt-identity 1.6%

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

      5. cancel-sign-sub-inv 1.6%

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

   6. Simplified 1.6%

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

   7. Taylor expanded in eps around inf 2.6%

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

   8. Taylor expanded in eps around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. *-lft-identity 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. metadata-eval 100.0%

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

      6. *-lft-identity 100.0%

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

      7. *-lft-identity 100.0%

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

   10. Simplified 100.0%

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

   if 6.99999999999999955e253 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in x around 0 41.9%

      \[\leadsto \frac{e^{\varepsilon \cdot x - x} + \color{blue}{1}}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -250000000:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 300:\\ \;\;\;\;\frac{\left(1 + e^{x \cdot \left(\varepsilon + -1\right)}\right) - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+227}:\\ \;\;\;\;\frac{\left(1 + \left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\varepsilon \cdot x}\right) + \frac{-1}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+253}:\\ \;\;\;\;\frac{e^{-x} + \left(x + \left(1 - x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x - x} + 1}{2}\\ \end{array} \]

Alternative 7: 73.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(1 - x\right)\\ t_1 := e^{-x}\\ \mathbf{if}\;x \leq -2:\\ \;\;\;\;\frac{t_0 + \left(t_1 - \varepsilon \cdot x\right)}{2}\\ \mathbf{elif}\;x \leq 300:\\ \;\;\;\;\frac{\left(1 + e^{x \cdot \left(\varepsilon + -1\right)}\right) - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+229}:\\ \;\;\;\;\frac{\left(1 + \left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\varepsilon \cdot x}\right) + \frac{-1}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+253}:\\ \;\;\;\;\frac{t_1 + t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x - x} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ x (- 1.0 x))) (t_1 (exp (- x))))
   (if (<= x -2.0)
     (/ (+ t_0 (- t_1 (* eps x))) 2.0)
     (if (<= x 300.0)
       (/ (- (+ 1.0 (exp (* x (+ eps -1.0)))) (* eps x)) 2.0)
       (if (<= x 1.15e+229)
         (/
          (+ (+ 1.0 (* (+ 1.0 (/ 1.0 eps)) (exp (* eps x)))) (/ -1.0 eps))
          2.0)
         (if (<= x 7e+253)
           (/ (+ t_1 t_0) 2.0)
           (/ (+ (exp (- (* eps x) x)) 1.0) 2.0)))))))

C

double code(double x, double eps) {
	double t_0 = x + (1.0 - x);
	double t_1 = exp(-x);
	double tmp;
	if (x <= -2.0) {
		tmp = (t_0 + (t_1 - (eps * x))) / 2.0;
	} else if (x <= 300.0) {
		tmp = ((1.0 + exp((x * (eps + -1.0)))) - (eps * x)) / 2.0;
	} else if (x <= 1.15e+229) {
		tmp = ((1.0 + ((1.0 + (1.0 / eps)) * exp((eps * x)))) + (-1.0 / eps)) / 2.0;
	} else if (x <= 7e+253) {
		tmp = (t_1 + t_0) / 2.0;
	} else {
		tmp = (exp(((eps * x) - x)) + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x + (1.0d0 - x)
    t_1 = exp(-x)
    if (x <= (-2.0d0)) then
        tmp = (t_0 + (t_1 - (eps * x))) / 2.0d0
    else if (x <= 300.0d0) then
        tmp = ((1.0d0 + exp((x * (eps + (-1.0d0))))) - (eps * x)) / 2.0d0
    else if (x <= 1.15d+229) then
        tmp = ((1.0d0 + ((1.0d0 + (1.0d0 / eps)) * exp((eps * x)))) + ((-1.0d0) / eps)) / 2.0d0
    else if (x <= 7d+253) then
        tmp = (t_1 + t_0) / 2.0d0
    else
        tmp = (exp(((eps * x) - x)) + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x + (1.0 - x);
	double t_1 = Math.exp(-x);
	double tmp;
	if (x <= -2.0) {
		tmp = (t_0 + (t_1 - (eps * x))) / 2.0;
	} else if (x <= 300.0) {
		tmp = ((1.0 + Math.exp((x * (eps + -1.0)))) - (eps * x)) / 2.0;
	} else if (x <= 1.15e+229) {
		tmp = ((1.0 + ((1.0 + (1.0 / eps)) * Math.exp((eps * x)))) + (-1.0 / eps)) / 2.0;
	} else if (x <= 7e+253) {
		tmp = (t_1 + t_0) / 2.0;
	} else {
		tmp = (Math.exp(((eps * x) - x)) + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x + (1.0 - x)
	t_1 = math.exp(-x)
	tmp = 0
	if x <= -2.0:
		tmp = (t_0 + (t_1 - (eps * x))) / 2.0
	elif x <= 300.0:
		tmp = ((1.0 + math.exp((x * (eps + -1.0)))) - (eps * x)) / 2.0
	elif x <= 1.15e+229:
		tmp = ((1.0 + ((1.0 + (1.0 / eps)) * math.exp((eps * x)))) + (-1.0 / eps)) / 2.0
	elif x <= 7e+253:
		tmp = (t_1 + t_0) / 2.0
	else:
		tmp = (math.exp(((eps * x) - x)) + 1.0) / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x + Float64(1.0 - x))
	t_1 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -2.0)
		tmp = Float64(Float64(t_0 + Float64(t_1 - Float64(eps * x))) / 2.0);
	elseif (x <= 300.0)
		tmp = Float64(Float64(Float64(1.0 + exp(Float64(x * Float64(eps + -1.0)))) - Float64(eps * x)) / 2.0);
	elseif (x <= 1.15e+229)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(eps * x)))) + Float64(-1.0 / eps)) / 2.0);
	elseif (x <= 7e+253)
		tmp = Float64(Float64(t_1 + t_0) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(Float64(eps * x) - x)) + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x + (1.0 - x);
	t_1 = exp(-x);
	tmp = 0.0;
	if (x <= -2.0)
		tmp = (t_0 + (t_1 - (eps * x))) / 2.0;
	elseif (x <= 300.0)
		tmp = ((1.0 + exp((x * (eps + -1.0)))) - (eps * x)) / 2.0;
	elseif (x <= 1.15e+229)
		tmp = ((1.0 + ((1.0 + (1.0 / eps)) * exp((eps * x)))) + (-1.0 / eps)) / 2.0;
	elseif (x <= 7e+253)
		tmp = (t_1 + t_0) / 2.0;
	else
		tmp = (exp(((eps * x) - x)) + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x + N[(1.0 - x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -2.0], N[(N[(t$95$0 + N[(t$95$1 - N[(eps * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 300.0], N[(N[(N[(1.0 + N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.15e+229], N[(N[(N[(1.0 + N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 7e+253], N[(N[(t$95$1 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(N[(eps * x), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -2

   1. Initial program 97.4%

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

   3. Simplified 97.4%

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

   4. Taylor expanded in x around 0 50.6%

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

      1. associate-*r* 50.6%

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

      2. mul-1-neg 50.6%

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

      3. distribute-lft-out 50.6%

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

      4. *-rgt-identity 50.6%

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

      5. cancel-sign-sub-inv 50.6%

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

   6. Simplified 50.6%

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

   7. Taylor expanded in eps around inf 48.0%

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

   8. Taylor expanded in eps around 0 69.6%

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

   if -2 < x < 300

   1. Initial program 54.3%

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

   3. Simplified 54.3%

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

   4. Taylor expanded in x around 0 40.0%

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

      1. associate-*r* 40.0%

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

      2. mul-1-neg 40.0%

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

      3. distribute-lft-out 40.0%

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

      4. *-rgt-identity 40.0%

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

      5. cancel-sign-sub-inv 40.0%

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

   6. Simplified 40.0%

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

   7. Taylor expanded in eps around inf 85.1%

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

   8. Taylor expanded in x around inf 85.1%

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

      1. mul-1-neg 85.1%

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

      2. distribute-lft-neg-in 85.1%

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

      3. *-commutative 85.1%

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

      4. associate-+r+ 85.1%

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

      5. associate-*r* 85.1%

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

      6. sub-neg 85.1%

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

      7. mul-1-neg 85.1%

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

      8. associate-*r* 85.1%

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

      9. mul-1-neg 85.1%

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

      10. *-commutative 85.1%

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

      11. mul-1-neg 85.1%

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

      12. sub-neg 85.1%

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

      13. distribute-rgt-neg-in 85.1%

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

      14. *-commutative 85.1%

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

      15. distribute-lft-neg-in 85.1%

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

      16. distribute-rgt-neg-in 85.1%

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

   10. Simplified 85.1%

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

   if 300 < x < 1.15e229

   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%

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

   4. Taylor expanded in x around 0 36.3%

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

   5. Taylor expanded in eps around inf 57.5%

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

      1. *-commutative 57.5%

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

   7. Simplified 57.5%

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

   8. Taylor expanded in x around inf 57.5%

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

   if 1.15e229 < x < 6.99999999999999955e253

   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%

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

   4. Taylor expanded in x around 0 1.6%

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

      1. associate-*r* 1.6%

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

      2. mul-1-neg 1.6%

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

      3. distribute-lft-out 1.6%

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

      4. *-rgt-identity 1.6%

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

      5. cancel-sign-sub-inv 1.6%

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

   6. Simplified 1.6%

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

   7. Taylor expanded in eps around inf 2.6%

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

   8. Taylor expanded in eps around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. *-lft-identity 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. metadata-eval 100.0%

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

      6. *-lft-identity 100.0%

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

      7. *-lft-identity 100.0%

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

   10. Simplified 100.0%

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

   if 6.99999999999999955e253 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in x around 0 41.9%

      \[\leadsto \frac{e^{\varepsilon \cdot x - x} + \color{blue}{1}}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\frac{\left(x + \left(1 - x\right)\right) + \left(e^{-x} - \varepsilon \cdot x\right)}{2}\\ \mathbf{elif}\;x \leq 300:\\ \;\;\;\;\frac{\left(1 + e^{x \cdot \left(\varepsilon + -1\right)}\right) - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+229}:\\ \;\;\;\;\frac{\left(1 + \left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\varepsilon \cdot x}\right) + \frac{-1}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+253}:\\ \;\;\;\;\frac{e^{-x} + \left(x + \left(1 - x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x - x} + 1}{2}\\ \end{array} \]

Alternative 8: 67.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -7 \cdot 10^{-227}:\\ \;\;\;\;\frac{1 + t_0}{2}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+229} \lor \neg \left(x \leq 7 \cdot 10^{+253}\right):\\ \;\;\;\;\frac{e^{\varepsilon \cdot x - x} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 + \left(x + \left(1 - x\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -7e-227)
     (/ (+ 1.0 t_0) 2.0)
     (if (or (<= x 1.65e+229) (not (<= x 7e+253)))
       (/ (+ (exp (- (* eps x) x)) 1.0) 2.0)
       (/ (+ t_0 (+ x (- 1.0 x))) 2.0)))))

C

double code(double x, double eps) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -7e-227) {
		tmp = (1.0 + t_0) / 2.0;
	} else if ((x <= 1.65e+229) || !(x <= 7e+253)) {
		tmp = (exp(((eps * x) - x)) + 1.0) / 2.0;
	} else {
		tmp = (t_0 + (x + (1.0 - x))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (x <= (-7d-227)) then
        tmp = (1.0d0 + t_0) / 2.0d0
    else if ((x <= 1.65d+229) .or. (.not. (x <= 7d+253))) then
        tmp = (exp(((eps * x) - x)) + 1.0d0) / 2.0d0
    else
        tmp = (t_0 + (x + (1.0d0 - x))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.exp(-x);
	double tmp;
	if (x <= -7e-227) {
		tmp = (1.0 + t_0) / 2.0;
	} else if ((x <= 1.65e+229) || !(x <= 7e+253)) {
		tmp = (Math.exp(((eps * x) - x)) + 1.0) / 2.0;
	} else {
		tmp = (t_0 + (x + (1.0 - x))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.exp(-x)
	tmp = 0
	if x <= -7e-227:
		tmp = (1.0 + t_0) / 2.0
	elif (x <= 1.65e+229) or not (x <= 7e+253):
		tmp = (math.exp(((eps * x) - x)) + 1.0) / 2.0
	else:
		tmp = (t_0 + (x + (1.0 - x))) / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -7e-227)
		tmp = Float64(Float64(1.0 + t_0) / 2.0);
	elseif ((x <= 1.65e+229) || !(x <= 7e+253))
		tmp = Float64(Float64(exp(Float64(Float64(eps * x) - x)) + 1.0) / 2.0);
	else
		tmp = Float64(Float64(t_0 + Float64(x + Float64(1.0 - x))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = exp(-x);
	tmp = 0.0;
	if (x <= -7e-227)
		tmp = (1.0 + t_0) / 2.0;
	elseif ((x <= 1.65e+229) || ~((x <= 7e+253)))
		tmp = (exp(((eps * x) - x)) + 1.0) / 2.0;
	else
		tmp = (t_0 + (x + (1.0 - x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -7e-227], N[(N[(1.0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 1.65e+229], N[Not[LessEqual[x, 7e+253]], $MachinePrecision]], N[(N[(N[Exp[N[(N[(eps * x), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + N[(x + N[(1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.0000000000000002e-227

   1. Initial program 77.3%

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

   3. Simplified 77.4%

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

   4. Taylor expanded in eps around inf 98.9%

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

   5. Taylor expanded in x around 0 65.8%

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

   6. Taylor expanded in eps around 0 82.8%

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

   if -7.0000000000000002e-227 < x < 1.65e229 or 6.99999999999999955e253 < x

   1. Initial program 72.0%

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

   3. Simplified 72.0%

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

   4. Taylor expanded in eps around inf 99.5%

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

   5. Taylor expanded in x around 0 65.9%

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

   if 1.65e229 < x < 6.99999999999999955e253

   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%

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

   4. Taylor expanded in x around 0 1.6%

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

      1. associate-*r* 1.6%

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

      2. mul-1-neg 1.6%

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

      3. distribute-lft-out 1.6%

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

      4. *-rgt-identity 1.6%

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

      5. cancel-sign-sub-inv 1.6%

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

   6. Simplified 1.6%

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

   7. Taylor expanded in eps around inf 2.6%

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

   8. Taylor expanded in eps around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-lft-in 100.0%

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

      3. *-lft-identity 100.0%

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

      4. cancel-sign-sub-inv 100.0%

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

      5. metadata-eval 100.0%

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

      6. *-lft-identity 100.0%

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

      7. *-lft-identity 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-227}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+229} \lor \neg \left(x \leq 7 \cdot 10^{+253}\right):\\ \;\;\;\;\frac{e^{\varepsilon \cdot x - x} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-x} + \left(x + \left(1 - x\right)\right)}{2}\\ \end{array} \]

Alternative 9: 67.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-227}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x - x} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -7e-227)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (/ (+ (exp (- (* eps x) x)) 1.0) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -7e-227) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else {
		tmp = (exp(((eps * x) - x)) + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-7d-227)) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else
        tmp = (exp(((eps * x) - x)) + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -7e-227) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else {
		tmp = (Math.exp(((eps * x) - x)) + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -7e-227:
		tmp = (1.0 + math.exp(-x)) / 2.0
	else:
		tmp = (math.exp(((eps * x) - x)) + 1.0) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -7e-227)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(Float64(eps * x) - x)) + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -7e-227)
		tmp = (1.0 + exp(-x)) / 2.0;
	else
		tmp = (exp(((eps * x) - x)) + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -7e-227], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(N[(eps * x), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.0000000000000002e-227

   1. Initial program 77.3%

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

   3. Simplified 77.4%

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

   4. Taylor expanded in eps around inf 98.9%

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

   5. Taylor expanded in x around 0 65.8%

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

   6. Taylor expanded in eps around 0 82.8%

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

   if -7.0000000000000002e-227 < x

   1. Initial program 73.2%

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

   3. Simplified 73.2%

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

   4. Taylor expanded in eps around inf 99.5%

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

   5. Taylor expanded in x around 0 63.3%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-227}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x - x} + 1}{2}\\ \end{array} \]

Alternative 10: 67.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-226}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -5e-226)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (/ (+ 1.0 (exp (* eps x))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -5e-226) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else {
		tmp = (1.0 + exp((eps * x))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-5d-226)) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else
        tmp = (1.0d0 + exp((eps * x))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -5e-226) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else {
		tmp = (1.0 + Math.exp((eps * x))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -5e-226:
		tmp = (1.0 + math.exp(-x)) / 2.0
	else:
		tmp = (1.0 + math.exp((eps * x))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -5e-226)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(Float64(eps * x))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -5e-226)
		tmp = (1.0 + exp(-x)) / 2.0;
	else
		tmp = (1.0 + exp((eps * x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -5e-226], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.9999999999999998e-226

   1. Initial program 77.3%

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

   3. Simplified 77.4%

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

   4. Taylor expanded in eps around inf 98.9%

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

   5. Taylor expanded in x around 0 65.8%

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

   6. Taylor expanded in eps around 0 82.8%

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

   if -4.9999999999999998e-226 < x

   1. Initial program 73.2%

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

   3. Simplified 73.2%

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

   4. Taylor expanded in x around 0 36.8%

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

   5. Taylor expanded in eps around inf 43.8%

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

      1. *-commutative 43.8%

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

   7. Simplified 43.8%

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

   8. Taylor expanded in eps around inf 63.2%

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

      1. *-commutative 63.2%

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

   10. Simplified 63.2%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-226}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \end{array} \]

Alternative 11: 61.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \frac{\left(x \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) \cdot \left(-1 + \varepsilon \cdot \varepsilon\right)}{\varepsilon + 1}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 2.6e-15)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (/
    (+ 2.0 (/ (* (* x (+ 1.0 (/ 1.0 eps))) (+ -1.0 (* eps eps))) (+ eps 1.0)))
    2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 2.6e-15) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else {
		tmp = (2.0 + (((x * (1.0 + (1.0 / eps))) * (-1.0 + (eps * eps))) / (eps + 1.0))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 2.6d-15) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else
        tmp = (2.0d0 + (((x * (1.0d0 + (1.0d0 / eps))) * ((-1.0d0) + (eps * eps))) / (eps + 1.0d0))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 2.6e-15) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else {
		tmp = (2.0 + (((x * (1.0 + (1.0 / eps))) * (-1.0 + (eps * eps))) / (eps + 1.0))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 2.6e-15:
		tmp = (1.0 + math.exp(-x)) / 2.0
	else:
		tmp = (2.0 + (((x * (1.0 + (1.0 / eps))) * (-1.0 + (eps * eps))) / (eps + 1.0))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 2.6e-15)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(Float64(x * Float64(1.0 + Float64(1.0 / eps))) * Float64(-1.0 + Float64(eps * eps))) / Float64(eps + 1.0))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 2.6e-15)
		tmp = (1.0 + exp(-x)) / 2.0;
	else
		tmp = (2.0 + (((x * (1.0 + (1.0 / eps))) * (-1.0 + (eps * eps))) / (eps + 1.0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 2.6e-15], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(N[(N[(x * N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(eps + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.60000000000000004e-15

   1. Initial program 63.2%

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

   3. Simplified 63.2%

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

   4. Taylor expanded in eps around inf 99.4%

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

   5. Taylor expanded in x around 0 77.8%

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

   6. Taylor expanded in eps around 0 82.1%

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

   if 2.60000000000000004e-15 < x

   1. Initial program 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in x around 0 34.7%

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

   5. Taylor expanded in x around 0 16.7%

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

      1. associate-*r* 16.7%

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

      2. mul-1-neg 16.7%

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

      3. *-commutative 16.7%

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

   7. Simplified 16.7%

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

      1. *-commutative 16.7%

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

      2. flip-- 20.0%

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

      3. metadata-eval 20.0%

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

      4. metadata-eval 20.0%

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

      5. associate-*r/ 20.0%

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

      6. metadata-eval 20.0%

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

      7. +-commutative 20.0%

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

   9. Applied egg-rr 20.0%

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

       1. associate-/l* 20.0%

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

       2. +-commutative 20.0%

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

   11. Simplified 20.0%

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

   12. Taylor expanded in x around 0 26.8%

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

       1. mul-1-neg 26.8%

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

       2. associate-*r* 26.8%

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

       3. unpow2 26.8%

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

   14. Simplified 26.8%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \frac{\left(x \cdot \left(1 + \frac{1}{\varepsilon}\right)\right) \cdot \left(-1 + \varepsilon \cdot \varepsilon\right)}{\varepsilon + 1}}{2}\\ \end{array} \]

Alternative 12: 48.5% accurate, 15.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := If[LessEqual[x, -2e-18], N[(N[(N[(x * N[(eps + 1.0), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(eps * x), $MachinePrecision] + 2.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.0000000000000001e-18

   1. Initial program 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in x around 0 49.4%

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

      1. associate-*r* 49.4%

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

      2. mul-1-neg 49.4%

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

      3. distribute-lft-out 49.4%

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

      4. *-rgt-identity 49.4%

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

      5. cancel-sign-sub-inv 49.4%

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

   6. Simplified 49.4%

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

   7. Taylor expanded in x around inf 24.5%

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

      1. sub-neg 24.5%

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

      2. metadata-eval 24.5%

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

      3. associate-*r* 24.5%

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

      4. +-commutative 24.5%

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

   9. Simplified 24.5%

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

   if -2.0000000000000001e-18 < x

   1. Initial program 70.4%

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

   3. Simplified 70.4%

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

   4. Taylor expanded in x around 0 37.6%

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

   5. Taylor expanded in x around 0 40.0%

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

      1. associate-*r* 40.0%

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

      2. mul-1-neg 40.0%

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

      3. *-commutative 40.0%

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

   7. Simplified 40.0%

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

   8. Taylor expanded in eps around inf 55.1%

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

      1. *-commutative 55.1%

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

   10. Simplified 55.1%

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

4. Final simplification 50.3%

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

Alternative 13: 52.1% accurate, 15.1× speedup

Math

\[\begin{array}{l} \\ \frac{2 + \left(x + \varepsilon \cdot \left(x + 0.5 \cdot \left(x \cdot x\right)\right)\right)}{2} \end{array} \]

FPCore

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

C

double code(double x, double eps) {
	return (2.0 + (x + (eps * (x + (0.5 * (x * x)))))) / 2.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (2.0d0 + (x + (eps * (x + (0.5d0 * (x * x)))))) / 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.6%

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

3. Simplified 74.6%

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

4. Taylor expanded in x around 0 39.8%

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

5. Taylor expanded in eps around inf 43.9%

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

   1. *-commutative 43.9%

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

7. Simplified 43.9%

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

8. Taylor expanded in eps around 0 54.5%

   \[\leadsto \frac{\color{blue}{2 + \left(x + \varepsilon \cdot \left(x + 0.5 \cdot {x}^{2}\right)\right)}}{2} \]
9. Step-by-step derivation

   1. unpow2 54.5%

      \[\leadsto \frac{2 + \left(x + \varepsilon \cdot \left(x + 0.5 \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)}{2} \]

10. Simplified 54.5%

    \[\leadsto \frac{\color{blue}{2 + \left(x + \varepsilon \cdot \left(x + 0.5 \cdot \left(x \cdot x\right)\right)\right)}}{2} \]

11. Final simplification 54.5%

    \[\leadsto \frac{2 + \left(x + \varepsilon \cdot \left(x + 0.5 \cdot \left(x \cdot x\right)\right)\right)}{2} \]

Alternative 14: 48.5% accurate, 25.0× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2e-18) (/ (* eps (- x)) 2.0) (/ (+ (* eps x) 2.0) 2.0)))

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	tmp = 0
	if x <= -2e-18:
		tmp = (eps * -x) / 2.0
	else:
		tmp = ((eps * x) + 2.0) / 2.0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := If[LessEqual[x, -2e-18], N[(N[(eps * (-x)), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(eps * x), $MachinePrecision] + 2.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.0000000000000001e-18

   1. Initial program 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in x around 0 49.4%

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

      1. associate-*r* 49.4%

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

      2. mul-1-neg 49.4%

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

      3. distribute-lft-out 49.4%

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

      4. *-rgt-identity 49.4%

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

      5. cancel-sign-sub-inv 49.4%

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

   6. Simplified 49.4%

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

   7. Taylor expanded in eps around inf 24.5%

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

      1. mul-1-neg 24.5%

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

      2. *-commutative 24.5%

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

   9. Simplified 24.5%

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

   if -2.0000000000000001e-18 < x

   1. Initial program 70.4%

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

   3. Simplified 70.4%

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

   4. Taylor expanded in x around 0 37.6%

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

   5. Taylor expanded in x around 0 40.0%

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

      1. associate-*r* 40.0%

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

      2. mul-1-neg 40.0%

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

      3. *-commutative 40.0%

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

   7. Simplified 40.0%

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

   8. Taylor expanded in eps around inf 55.1%

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

      1. *-commutative 55.1%

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

   10. Simplified 55.1%

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

4. Final simplification 50.3%

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

Alternative 15: 45.8% accurate, 28.2× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2e-18) (/ (* eps (- x)) 2.0) (/ (+ x 2.0) 2.0)))

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	tmp = 0
	if x <= -2e-18:
		tmp = (eps * -x) / 2.0
	else:
		tmp = (x + 2.0) / 2.0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := If[LessEqual[x, -2e-18], N[(N[(eps * (-x)), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x + 2.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.0000000000000001e-18

   1. Initial program 97.5%

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

   3. Simplified 97.5%

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

   4. Taylor expanded in x around 0 49.4%

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

      1. associate-*r* 49.4%

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

      2. mul-1-neg 49.4%

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

      3. distribute-lft-out 49.4%

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

      4. *-rgt-identity 49.4%

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

      5. cancel-sign-sub-inv 49.4%

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

   6. Simplified 49.4%

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

   7. Taylor expanded in eps around inf 24.5%

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

      1. mul-1-neg 24.5%

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

      2. *-commutative 24.5%

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

   9. Simplified 24.5%

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

   if -2.0000000000000001e-18 < x

   1. Initial program 70.4%

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

   3. Simplified 70.4%

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

   4. Taylor expanded in x around 0 37.6%

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

   5. Taylor expanded in eps around inf 43.0%

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

      1. *-commutative 43.0%

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

   7. Simplified 43.0%

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

   8. Taylor expanded in eps around 0 50.6%

      \[\leadsto \frac{\color{blue}{2 + x}}{2} \]
   9. Step-by-step derivation

      1. +-commutative 50.6%

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

   10. Simplified 50.6%

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

4. Final simplification 46.5%

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

Alternative 16: 42.7% accurate, 227.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 1.0

Julia

function code(x, eps)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 1.0

Derivation

1. Initial program 74.6%

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

3. Simplified 74.6%

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

4. Taylor expanded in x around 0 43.0%

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

5. Final simplification 43.0%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023297 
(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))