NMSE Section 6.1 mentioned, A

Percentage Accurate: 72.4% → 98.8%

Time: 12.9s

Alternatives: 13

Speedup: 1.1×

• 37.6% 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: 72.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return (exp(((eps + -1.0) * 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 + (-1.0d0)) * x)) + exp((x * ((-1.0d0) - eps)))) / 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.8%

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

   1. div-sub 72.8%

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

   2. +-rgt-identity 72.8%

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

   3. div-sub 72.8%

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

3. Simplified 72.8%

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

4. Taylor expanded in eps around inf 99.5%

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

   1. mul-1-neg 99.5%

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

   2. distribute-lft-neg-in 99.5%

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

   3. mul-1-neg 99.5%

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

   4. exp-prod 99.5%

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

   5. +-commutative 99.5%

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

   6. *-commutative 99.5%

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

   7. exp-prod 99.5%

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

   8. neg-mul-1 99.5%

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

   9. +-commutative 99.5%

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

6. Simplified 99.5%

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

7. Final simplification 99.5%

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

Alternative 2: 88.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.8%

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

   1. div-sub 72.8%

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

   2. +-rgt-identity 72.8%

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

   3. div-sub 72.8%

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

3. Simplified 72.8%

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

4. Taylor expanded in eps around inf 99.5%

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

   1. mul-1-neg 99.5%

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

   2. distribute-lft-neg-in 99.5%

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

   3. mul-1-neg 99.5%

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

   4. exp-prod 99.5%

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

   5. +-commutative 99.5%

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

   6. *-commutative 99.5%

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

   7. exp-prod 99.5%

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

   8. neg-mul-1 99.5%

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

   9. +-commutative 99.5%

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

6. Simplified 99.5%

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

7. Taylor expanded in eps around inf 87.4%

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

8. Taylor expanded in eps around inf 87.4%

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

   1. distribute-lft-neg-in 87.4%

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

   2. *-commutative 87.4%

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

   3. sub-neg 87.4%

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

   4. metadata-eval 87.4%

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

10. Simplified 87.4%

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

11. Final simplification 87.4%

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

Alternative 3: 68.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-254}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+24}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(1 + \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+68}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 10^{+97}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(\varepsilon + -1\right) \cdot x} + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+214}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+302}:\\ \;\;\;\;\frac{\left(1 + \left(\varepsilon \cdot x + \left(x - x\right)\right)\right) + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -6.6e-254)
   (/ (+ 1.0 (exp (* eps (- x)))) 2.0)
   (if (<= x 5e+24)
     (/ (+ 1.0 (exp (* x (+ 1.0 eps)))) 2.0)
     (if (<= x 3.8e+68)
       0.0
       (if (<= x 1e+97)
         (/
          (+
           (* (+ 1.0 (/ 1.0 eps)) (exp (* (+ eps -1.0) x)))
           (+ 1.0 (/ -1.0 eps)))
          2.0)
         (if (<= x 6.2e+214)
           0.0
           (if (<= x 3.1e+302)
             (/ (+ (+ 1.0 (+ (* eps x) (- x x))) (exp (* x (- -1.0 eps)))) 2.0)
             0.0)))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -6.6e-254) {
		tmp = (1.0 + exp((eps * -x))) / 2.0;
	} else if (x <= 5e+24) {
		tmp = (1.0 + exp((x * (1.0 + eps)))) / 2.0;
	} else if (x <= 3.8e+68) {
		tmp = 0.0;
	} else if (x <= 1e+97) {
		tmp = (((1.0 + (1.0 / eps)) * exp(((eps + -1.0) * x))) + (1.0 + (-1.0 / eps))) / 2.0;
	} else if (x <= 6.2e+214) {
		tmp = 0.0;
	} else if (x <= 3.1e+302) {
		tmp = ((1.0 + ((eps * x) + (x - x))) + exp((x * (-1.0 - eps)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-6.6d-254)) then
        tmp = (1.0d0 + exp((eps * -x))) / 2.0d0
    else if (x <= 5d+24) then
        tmp = (1.0d0 + exp((x * (1.0d0 + eps)))) / 2.0d0
    else if (x <= 3.8d+68) then
        tmp = 0.0d0
    else if (x <= 1d+97) then
        tmp = (((1.0d0 + (1.0d0 / eps)) * exp(((eps + (-1.0d0)) * x))) + (1.0d0 + ((-1.0d0) / eps))) / 2.0d0
    else if (x <= 6.2d+214) then
        tmp = 0.0d0
    else if (x <= 3.1d+302) then
        tmp = ((1.0d0 + ((eps * x) + (x - x))) + exp((x * ((-1.0d0) - eps)))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -6.6e-254) {
		tmp = (1.0 + Math.exp((eps * -x))) / 2.0;
	} else if (x <= 5e+24) {
		tmp = (1.0 + Math.exp((x * (1.0 + eps)))) / 2.0;
	} else if (x <= 3.8e+68) {
		tmp = 0.0;
	} else if (x <= 1e+97) {
		tmp = (((1.0 + (1.0 / eps)) * Math.exp(((eps + -1.0) * x))) + (1.0 + (-1.0 / eps))) / 2.0;
	} else if (x <= 6.2e+214) {
		tmp = 0.0;
	} else if (x <= 3.1e+302) {
		tmp = ((1.0 + ((eps * x) + (x - x))) + Math.exp((x * (-1.0 - eps)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -6.6e-254:
		tmp = (1.0 + math.exp((eps * -x))) / 2.0
	elif x <= 5e+24:
		tmp = (1.0 + math.exp((x * (1.0 + eps)))) / 2.0
	elif x <= 3.8e+68:
		tmp = 0.0
	elif x <= 1e+97:
		tmp = (((1.0 + (1.0 / eps)) * math.exp(((eps + -1.0) * x))) + (1.0 + (-1.0 / eps))) / 2.0
	elif x <= 6.2e+214:
		tmp = 0.0
	elif x <= 3.1e+302:
		tmp = ((1.0 + ((eps * x) + (x - x))) + math.exp((x * (-1.0 - eps)))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -6.6e-254)
		tmp = Float64(Float64(1.0 + exp(Float64(eps * Float64(-x)))) / 2.0);
	elseif (x <= 5e+24)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(1.0 + eps)))) / 2.0);
	elseif (x <= 3.8e+68)
		tmp = 0.0;
	elseif (x <= 1e+97)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(Float64(eps + -1.0) * x))) + Float64(1.0 + Float64(-1.0 / eps))) / 2.0);
	elseif (x <= 6.2e+214)
		tmp = 0.0;
	elseif (x <= 3.1e+302)
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(eps * x) + Float64(x - x))) + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -6.6e-254)
		tmp = (1.0 + exp((eps * -x))) / 2.0;
	elseif (x <= 5e+24)
		tmp = (1.0 + exp((x * (1.0 + eps)))) / 2.0;
	elseif (x <= 3.8e+68)
		tmp = 0.0;
	elseif (x <= 1e+97)
		tmp = (((1.0 + (1.0 / eps)) * exp(((eps + -1.0) * x))) + (1.0 + (-1.0 / eps))) / 2.0;
	elseif (x <= 6.2e+214)
		tmp = 0.0;
	elseif (x <= 3.1e+302)
		tmp = ((1.0 + ((eps * x) + (x - x))) + exp((x * (-1.0 - eps)))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -6.6e-254], N[(N[(1.0 + N[Exp[N[(eps * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5e+24], N[(N[(1.0 + N[Exp[N[(x * N[(1.0 + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3.8e+68], 0.0, If[LessEqual[x, 1e+97], N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(eps + -1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 6.2e+214], 0.0, If[LessEqual[x, 3.1e+302], N[(N[(N[(1.0 + N[(N[(eps * x), $MachinePrecision] + N[(x - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -6.60000000000000033e-254

   1. Initial program 68.1%

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

      1. div-sub 68.1%

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

      2. +-rgt-identity 68.1%

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

      3. div-sub 68.1%

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

   3. Simplified 68.1%

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

   4. Taylor expanded in x around 0 39.0%

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

   5. Taylor expanded in eps around inf 69.8%

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

   6. Taylor expanded in eps around inf 70.8%

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

      1. *-commutative 70.8%

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

   8. Simplified 70.8%

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

   if -6.60000000000000033e-254 < x < 5.00000000000000045e24

   1. Initial program 55.1%

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

      1. div-sub 55.1%

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

      2. +-rgt-identity 55.1%

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

      3. div-sub 55.1%

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

   3. Simplified 55.1%

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

   4. Taylor expanded in x around 0 39.4%

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

      1. add-log-exp 39.4%

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

      2. *-un-lft-identity 39.4%

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

      3. log-prod 39.4%

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

      4. metadata-eval 39.4%

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

      5. add-log-exp 39.4%

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

      6. *-commutative 39.4%

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

      7. add-sqr-sqrt 7.5%

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

      8. sqrt-unprod 34.8%

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

      9. sqr-neg 34.8%

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

      10. sqrt-unprod 30.9%

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

      11. add-sqr-sqrt 38.4%

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

   6. Applied egg-rr 38.4%

      \[\leadsto \frac{\left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\color{blue}{0 + x \cdot \left(1 + \varepsilon\right)}}}{2} \]
   7. Step-by-step derivation

      1. +-lft-identity 38.4%

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

      2. +-commutative 38.4%

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

   8. Simplified 38.4%

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

   9. Taylor expanded in eps around inf 83.1%

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

   if 5.00000000000000045e24 < x < 3.8000000000000001e68 or 1.0000000000000001e97 < x < 6.19999999999999957e214 or 3.0999999999999998e302 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 71.6%

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

      1. div-sub 71.6%

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

      2. rec-exp 71.6%

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

      3. mul-1-neg 71.6%

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

      4. +-inverses 71.6%

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

   6. Simplified 71.6%

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

   if 3.8000000000000001e68 < x < 1.0000000000000001e97

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 46.1%

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

   if 6.19999999999999957e214 < x < 3.0999999999999998e302

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 35.4%

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

   5. Taylor expanded in eps around inf 35.8%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-254}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+24}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(1 + \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+68}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 10^{+97}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(\varepsilon + -1\right) \cdot x} + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+214}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+302}:\\ \;\;\;\;\frac{\left(1 + \left(\varepsilon \cdot x + \left(x - x\right)\right)\right) + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4: 67.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-254}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 10^{+24}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(1 + \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+67}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+97}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(\varepsilon + -1\right) \cdot x} + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 6.9 \cdot 10^{+214}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+302}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -6.6e-254)
   (/ (+ 1.0 (exp (* eps (- x)))) 2.0)
   (if (<= x 1e+24)
     (/ (+ 1.0 (exp (* x (+ 1.0 eps)))) 2.0)
     (if (<= x 1.6e+67)
       0.0
       (if (<= x 1.4e+97)
         (/
          (+
           (* (+ 1.0 (/ 1.0 eps)) (exp (* (+ eps -1.0) x)))
           (+ 1.0 (/ -1.0 eps)))
          2.0)
         (if (<= x 6.9e+214)
           0.0
           (if (<= x 7.5e+302) (/ (+ 2.0 (* eps x)) 2.0) 0.0)))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -6.6e-254) {
		tmp = (1.0 + exp((eps * -x))) / 2.0;
	} else if (x <= 1e+24) {
		tmp = (1.0 + exp((x * (1.0 + eps)))) / 2.0;
	} else if (x <= 1.6e+67) {
		tmp = 0.0;
	} else if (x <= 1.4e+97) {
		tmp = (((1.0 + (1.0 / eps)) * exp(((eps + -1.0) * x))) + (1.0 + (-1.0 / eps))) / 2.0;
	} else if (x <= 6.9e+214) {
		tmp = 0.0;
	} else if (x <= 7.5e+302) {
		tmp = (2.0 + (eps * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-6.6d-254)) then
        tmp = (1.0d0 + exp((eps * -x))) / 2.0d0
    else if (x <= 1d+24) then
        tmp = (1.0d0 + exp((x * (1.0d0 + eps)))) / 2.0d0
    else if (x <= 1.6d+67) then
        tmp = 0.0d0
    else if (x <= 1.4d+97) then
        tmp = (((1.0d0 + (1.0d0 / eps)) * exp(((eps + (-1.0d0)) * x))) + (1.0d0 + ((-1.0d0) / eps))) / 2.0d0
    else if (x <= 6.9d+214) then
        tmp = 0.0d0
    else if (x <= 7.5d+302) then
        tmp = (2.0d0 + (eps * x)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -6.6e-254) {
		tmp = (1.0 + Math.exp((eps * -x))) / 2.0;
	} else if (x <= 1e+24) {
		tmp = (1.0 + Math.exp((x * (1.0 + eps)))) / 2.0;
	} else if (x <= 1.6e+67) {
		tmp = 0.0;
	} else if (x <= 1.4e+97) {
		tmp = (((1.0 + (1.0 / eps)) * Math.exp(((eps + -1.0) * x))) + (1.0 + (-1.0 / eps))) / 2.0;
	} else if (x <= 6.9e+214) {
		tmp = 0.0;
	} else if (x <= 7.5e+302) {
		tmp = (2.0 + (eps * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -6.6e-254:
		tmp = (1.0 + math.exp((eps * -x))) / 2.0
	elif x <= 1e+24:
		tmp = (1.0 + math.exp((x * (1.0 + eps)))) / 2.0
	elif x <= 1.6e+67:
		tmp = 0.0
	elif x <= 1.4e+97:
		tmp = (((1.0 + (1.0 / eps)) * math.exp(((eps + -1.0) * x))) + (1.0 + (-1.0 / eps))) / 2.0
	elif x <= 6.9e+214:
		tmp = 0.0
	elif x <= 7.5e+302:
		tmp = (2.0 + (eps * x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -6.6e-254)
		tmp = Float64(Float64(1.0 + exp(Float64(eps * Float64(-x)))) / 2.0);
	elseif (x <= 1e+24)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(1.0 + eps)))) / 2.0);
	elseif (x <= 1.6e+67)
		tmp = 0.0;
	elseif (x <= 1.4e+97)
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(Float64(eps + -1.0) * x))) + Float64(1.0 + Float64(-1.0 / eps))) / 2.0);
	elseif (x <= 6.9e+214)
		tmp = 0.0;
	elseif (x <= 7.5e+302)
		tmp = Float64(Float64(2.0 + Float64(eps * x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -6.6e-254)
		tmp = (1.0 + exp((eps * -x))) / 2.0;
	elseif (x <= 1e+24)
		tmp = (1.0 + exp((x * (1.0 + eps)))) / 2.0;
	elseif (x <= 1.6e+67)
		tmp = 0.0;
	elseif (x <= 1.4e+97)
		tmp = (((1.0 + (1.0 / eps)) * exp(((eps + -1.0) * x))) + (1.0 + (-1.0 / eps))) / 2.0;
	elseif (x <= 6.9e+214)
		tmp = 0.0;
	elseif (x <= 7.5e+302)
		tmp = (2.0 + (eps * x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -6.6e-254], N[(N[(1.0 + N[Exp[N[(eps * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1e+24], N[(N[(1.0 + N[Exp[N[(x * N[(1.0 + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.6e+67], 0.0, If[LessEqual[x, 1.4e+97], N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(eps + -1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 6.9e+214], 0.0, If[LessEqual[x, 7.5e+302], N[(N[(2.0 + N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -6.60000000000000033e-254

   1. Initial program 68.1%

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

      1. div-sub 68.1%

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

      2. +-rgt-identity 68.1%

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

      3. div-sub 68.1%

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

   3. Simplified 68.1%

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

   4. Taylor expanded in x around 0 39.0%

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

   5. Taylor expanded in eps around inf 69.8%

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

   6. Taylor expanded in eps around inf 70.8%

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

      1. *-commutative 70.8%

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

   8. Simplified 70.8%

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

   if -6.60000000000000033e-254 < x < 9.9999999999999998e23

   1. Initial program 55.1%

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

      1. div-sub 55.1%

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

      2. +-rgt-identity 55.1%

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

      3. div-sub 55.1%

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

   3. Simplified 55.1%

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

   4. Taylor expanded in x around 0 39.4%

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

      1. add-log-exp 39.4%

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

      2. *-un-lft-identity 39.4%

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

      3. log-prod 39.4%

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

      4. metadata-eval 39.4%

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

      5. add-log-exp 39.4%

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

      6. *-commutative 39.4%

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

      7. add-sqr-sqrt 7.5%

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

      8. sqrt-unprod 34.8%

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

      9. sqr-neg 34.8%

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

      10. sqrt-unprod 30.9%

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

      11. add-sqr-sqrt 38.4%

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

   6. Applied egg-rr 38.4%

      \[\leadsto \frac{\left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\color{blue}{0 + x \cdot \left(1 + \varepsilon\right)}}}{2} \]
   7. Step-by-step derivation

      1. +-lft-identity 38.4%

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

      2. +-commutative 38.4%

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

   8. Simplified 38.4%

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

   9. Taylor expanded in eps around inf 83.1%

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

   if 9.9999999999999998e23 < x < 1.59999999999999991e67 or 1.4e97 < x < 6.89999999999999976e214 or 7.49999999999999924e302 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 71.6%

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

      1. div-sub 71.6%

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

      2. rec-exp 71.6%

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

      3. mul-1-neg 71.6%

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

      4. +-inverses 71.6%

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

   6. Simplified 71.6%

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

   if 1.59999999999999991e67 < x < 1.4e97

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 46.1%

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

   if 6.89999999999999976e214 < x < 7.49999999999999924e302

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 35.4%

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

   5. Taylor expanded in x around 0 35.4%

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

   6. Taylor expanded in eps around inf 35.8%

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

      1. associate-+r+ 35.8%

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

      2. distribute-lft1-in 35.8%

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

      3. metadata-eval 35.8%

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

      4. mul0-lft 35.8%

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

      5. metadata-eval 35.8%

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

      6. metadata-eval 35.8%

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

      7. *-commutative 35.8%

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

   8. Simplified 35.8%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-254}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 10^{+24}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(1 + \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+67}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+97}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(\varepsilon + -1\right) \cdot x} + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 6.9 \cdot 10^{+214}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+302}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5: 58.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + e^{x}}{2}\\ \mathbf{if}\;x \leq -1.1 \cdot 10^{-242}:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 10^{+24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+67}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 10^{+97}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+214}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.26 \cdot 10^{+302}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (exp x)) 2.0)))
   (if (<= x -1.1e-242)
     (/ (- 2.0 (* eps x)) 2.0)
     (if (<= x 1e+24)
       t_0
       (if (<= x 2e+67)
         0.0
         (if (<= x 1e+97)
           t_0
           (if (<= x 5.8e+214)
             0.0
             (if (<= x 1.26e+302) (/ (+ 2.0 (* eps x)) 2.0) 0.0))))))))

C

double code(double x, double eps) {
	double t_0 = (1.0 + exp(x)) / 2.0;
	double tmp;
	if (x <= -1.1e-242) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else if (x <= 1e+24) {
		tmp = t_0;
	} else if (x <= 2e+67) {
		tmp = 0.0;
	} else if (x <= 1e+97) {
		tmp = t_0;
	} else if (x <= 5.8e+214) {
		tmp = 0.0;
	} else if (x <= 1.26e+302) {
		tmp = (2.0 + (eps * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + exp(x)) / 2.0d0
    if (x <= (-1.1d-242)) then
        tmp = (2.0d0 - (eps * x)) / 2.0d0
    else if (x <= 1d+24) then
        tmp = t_0
    else if (x <= 2d+67) then
        tmp = 0.0d0
    else if (x <= 1d+97) then
        tmp = t_0
    else if (x <= 5.8d+214) then
        tmp = 0.0d0
    else if (x <= 1.26d+302) then
        tmp = (2.0d0 + (eps * x)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = (1.0 + Math.exp(x)) / 2.0;
	double tmp;
	if (x <= -1.1e-242) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else if (x <= 1e+24) {
		tmp = t_0;
	} else if (x <= 2e+67) {
		tmp = 0.0;
	} else if (x <= 1e+97) {
		tmp = t_0;
	} else if (x <= 5.8e+214) {
		tmp = 0.0;
	} else if (x <= 1.26e+302) {
		tmp = (2.0 + (eps * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (1.0 + math.exp(x)) / 2.0
	tmp = 0
	if x <= -1.1e-242:
		tmp = (2.0 - (eps * x)) / 2.0
	elif x <= 1e+24:
		tmp = t_0
	elif x <= 2e+67:
		tmp = 0.0
	elif x <= 1e+97:
		tmp = t_0
	elif x <= 5.8e+214:
		tmp = 0.0
	elif x <= 1.26e+302:
		tmp = (2.0 + (eps * x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(1.0 + exp(x)) / 2.0)
	tmp = 0.0
	if (x <= -1.1e-242)
		tmp = Float64(Float64(2.0 - Float64(eps * x)) / 2.0);
	elseif (x <= 1e+24)
		tmp = t_0;
	elseif (x <= 2e+67)
		tmp = 0.0;
	elseif (x <= 1e+97)
		tmp = t_0;
	elseif (x <= 5.8e+214)
		tmp = 0.0;
	elseif (x <= 1.26e+302)
		tmp = Float64(Float64(2.0 + Float64(eps * x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (1.0 + exp(x)) / 2.0;
	tmp = 0.0;
	if (x <= -1.1e-242)
		tmp = (2.0 - (eps * x)) / 2.0;
	elseif (x <= 1e+24)
		tmp = t_0;
	elseif (x <= 2e+67)
		tmp = 0.0;
	elseif (x <= 1e+97)
		tmp = t_0;
	elseif (x <= 5.8e+214)
		tmp = 0.0;
	elseif (x <= 1.26e+302)
		tmp = (2.0 + (eps * x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -1.1e-242], N[(N[(2.0 - N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1e+24], t$95$0, If[LessEqual[x, 2e+67], 0.0, If[LessEqual[x, 1e+97], t$95$0, If[LessEqual[x, 5.8e+214], 0.0, If[LessEqual[x, 1.26e+302], N[(N[(2.0 + N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.10000000000000001e-242

   1. Initial program 67.3%

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

      1. div-sub 67.3%

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

      2. +-rgt-identity 67.3%

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

      3. div-sub 67.3%

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

   3. Simplified 67.3%

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

   4. Taylor expanded in x around 0 38.3%

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

   5. Taylor expanded in x around 0 35.3%

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

   6. Taylor expanded in eps around inf 53.4%

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

      1. neg-mul-1 53.4%

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

      2. distribute-lft-neg-in 53.4%

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

   8. Simplified 53.4%

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

   if -1.10000000000000001e-242 < x < 9.9999999999999998e23 or 1.99999999999999997e67 < x < 1.0000000000000001e97

   1. Initial program 59.6%

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

      1. div-sub 59.6%

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

      2. +-rgt-identity 59.6%

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

      3. div-sub 59.6%

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

   3. Simplified 59.6%

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

   4. Taylor expanded in eps around inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. distribute-lft-neg-in 99.8%

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

      3. mul-1-neg 99.8%

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

      4. exp-prod 99.8%

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

      5. +-commutative 99.8%

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

      6. *-commutative 99.8%

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

      7. exp-prod 99.8%

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

      8. neg-mul-1 99.8%

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

      9. +-commutative 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in eps around inf 96.2%

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

   8. Taylor expanded in eps around 0 65.1%

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

      1. mul-1-neg 65.1%

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

   10. Simplified 65.1%

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

       1. expm1-log1p-u 63.6%

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

       2. expm1-udef 63.6%

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

       3. add-sqr-sqrt 14.9%

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

       4. sqrt-unprod 75.2%

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

       5. sqr-neg 75.2%

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

       6. sqrt-unprod 60.4%

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

       7. add-sqr-sqrt 75.2%

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

   12. Applied egg-rr 75.2%

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

       1. expm1-def 75.2%

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

       2. expm1-log1p 76.7%

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

   14. Simplified 76.7%

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

   if 9.9999999999999998e23 < x < 1.99999999999999997e67 or 1.0000000000000001e97 < x < 5.7999999999999999e214 or 1.25999999999999996e302 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 71.6%

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

      1. div-sub 71.6%

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

      2. rec-exp 71.6%

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

      3. mul-1-neg 71.6%

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

      4. +-inverses 71.6%

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

   6. Simplified 71.6%

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

   if 5.7999999999999999e214 < x < 1.25999999999999996e302

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 35.4%

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

   5. Taylor expanded in x around 0 35.4%

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

   6. Taylor expanded in eps around inf 35.8%

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

      1. associate-+r+ 35.8%

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

      2. distribute-lft1-in 35.8%

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

      3. metadata-eval 35.8%

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

      4. mul0-lft 35.8%

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

      5. metadata-eval 35.8%

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

      6. metadata-eval 35.8%

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

      7. *-commutative 35.8%

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

   8. Simplified 35.8%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-242}:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 10^{+24}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+67}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 10^{+97}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+214}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.26 \cdot 10^{+302}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6: 65.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + e^{x}}{2}\\ \mathbf{if}\;x \leq -1 \cdot 10^{-242}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+67}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+96}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+214}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+302}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (exp x)) 2.0)))
   (if (<= x -1e-242)
     (/ (+ 1.0 (exp (* eps (- x)))) 2.0)
     (if (<= x 5e+24)
       t_0
       (if (<= x 5e+67)
         0.0
         (if (<= x 9.5e+96)
           t_0
           (if (<= x 6.2e+214)
             0.0
             (if (<= x 2.3e+302) (/ (+ 2.0 (* eps x)) 2.0) 0.0))))))))

C

double code(double x, double eps) {
	double t_0 = (1.0 + exp(x)) / 2.0;
	double tmp;
	if (x <= -1e-242) {
		tmp = (1.0 + exp((eps * -x))) / 2.0;
	} else if (x <= 5e+24) {
		tmp = t_0;
	} else if (x <= 5e+67) {
		tmp = 0.0;
	} else if (x <= 9.5e+96) {
		tmp = t_0;
	} else if (x <= 6.2e+214) {
		tmp = 0.0;
	} else if (x <= 2.3e+302) {
		tmp = (2.0 + (eps * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + exp(x)) / 2.0d0
    if (x <= (-1d-242)) then
        tmp = (1.0d0 + exp((eps * -x))) / 2.0d0
    else if (x <= 5d+24) then
        tmp = t_0
    else if (x <= 5d+67) then
        tmp = 0.0d0
    else if (x <= 9.5d+96) then
        tmp = t_0
    else if (x <= 6.2d+214) then
        tmp = 0.0d0
    else if (x <= 2.3d+302) then
        tmp = (2.0d0 + (eps * x)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = (1.0 + Math.exp(x)) / 2.0;
	double tmp;
	if (x <= -1e-242) {
		tmp = (1.0 + Math.exp((eps * -x))) / 2.0;
	} else if (x <= 5e+24) {
		tmp = t_0;
	} else if (x <= 5e+67) {
		tmp = 0.0;
	} else if (x <= 9.5e+96) {
		tmp = t_0;
	} else if (x <= 6.2e+214) {
		tmp = 0.0;
	} else if (x <= 2.3e+302) {
		tmp = (2.0 + (eps * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (1.0 + math.exp(x)) / 2.0
	tmp = 0
	if x <= -1e-242:
		tmp = (1.0 + math.exp((eps * -x))) / 2.0
	elif x <= 5e+24:
		tmp = t_0
	elif x <= 5e+67:
		tmp = 0.0
	elif x <= 9.5e+96:
		tmp = t_0
	elif x <= 6.2e+214:
		tmp = 0.0
	elif x <= 2.3e+302:
		tmp = (2.0 + (eps * x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(1.0 + exp(x)) / 2.0)
	tmp = 0.0
	if (x <= -1e-242)
		tmp = Float64(Float64(1.0 + exp(Float64(eps * Float64(-x)))) / 2.0);
	elseif (x <= 5e+24)
		tmp = t_0;
	elseif (x <= 5e+67)
		tmp = 0.0;
	elseif (x <= 9.5e+96)
		tmp = t_0;
	elseif (x <= 6.2e+214)
		tmp = 0.0;
	elseif (x <= 2.3e+302)
		tmp = Float64(Float64(2.0 + Float64(eps * x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (1.0 + exp(x)) / 2.0;
	tmp = 0.0;
	if (x <= -1e-242)
		tmp = (1.0 + exp((eps * -x))) / 2.0;
	elseif (x <= 5e+24)
		tmp = t_0;
	elseif (x <= 5e+67)
		tmp = 0.0;
	elseif (x <= 9.5e+96)
		tmp = t_0;
	elseif (x <= 6.2e+214)
		tmp = 0.0;
	elseif (x <= 2.3e+302)
		tmp = (2.0 + (eps * x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -1e-242], N[(N[(1.0 + N[Exp[N[(eps * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5e+24], t$95$0, If[LessEqual[x, 5e+67], 0.0, If[LessEqual[x, 9.5e+96], t$95$0, If[LessEqual[x, 6.2e+214], 0.0, If[LessEqual[x, 2.3e+302], N[(N[(2.0 + N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1e-242

   1. Initial program 67.3%

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

      1. div-sub 67.3%

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

      2. +-rgt-identity 67.3%

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

      3. div-sub 67.3%

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

   3. Simplified 67.3%

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

   4. Taylor expanded in x around 0 38.3%

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

   5. Taylor expanded in eps around inf 69.9%

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

   6. Taylor expanded in eps around inf 70.9%

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

      1. *-commutative 70.9%

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

   8. Simplified 70.9%

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

   if -1e-242 < x < 5.00000000000000045e24 or 4.99999999999999976e67 < x < 9.50000000000000089e96

   1. Initial program 59.6%

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

      1. div-sub 59.6%

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

      2. +-rgt-identity 59.6%

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

      3. div-sub 59.6%

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

   3. Simplified 59.6%

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

   4. Taylor expanded in eps around inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. distribute-lft-neg-in 99.8%

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

      3. mul-1-neg 99.8%

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

      4. exp-prod 99.8%

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

      5. +-commutative 99.8%

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

      6. *-commutative 99.8%

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

      7. exp-prod 99.8%

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

      8. neg-mul-1 99.8%

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

      9. +-commutative 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in eps around inf 96.2%

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

   8. Taylor expanded in eps around 0 65.1%

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

      1. mul-1-neg 65.1%

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

   10. Simplified 65.1%

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

       1. expm1-log1p-u 63.6%

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

       2. expm1-udef 63.6%

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

       3. add-sqr-sqrt 14.9%

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

       4. sqrt-unprod 75.2%

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

       5. sqr-neg 75.2%

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

       6. sqrt-unprod 60.4%

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

       7. add-sqr-sqrt 75.2%

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

   12. Applied egg-rr 75.2%

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

       1. expm1-def 75.2%

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

       2. expm1-log1p 76.7%

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

   14. Simplified 76.7%

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

   if 5.00000000000000045e24 < x < 4.99999999999999976e67 or 9.50000000000000089e96 < x < 6.19999999999999957e214 or 2.3000000000000001e302 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 71.6%

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

      1. div-sub 71.6%

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

      2. rec-exp 71.6%

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

      3. mul-1-neg 71.6%

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

      4. +-inverses 71.6%

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

   6. Simplified 71.6%

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

   if 6.19999999999999957e214 < x < 2.3000000000000001e302

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 35.4%

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

   5. Taylor expanded in x around 0 35.4%

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

   6. Taylor expanded in eps around inf 35.8%

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

      1. associate-+r+ 35.8%

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

      2. distribute-lft1-in 35.8%

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

      3. metadata-eval 35.8%

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

      4. mul0-lft 35.8%

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

      5. metadata-eval 35.8%

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

      6. metadata-eval 35.8%

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

      7. *-commutative 35.8%

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

   8. Simplified 35.8%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-242}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+24}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+67}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+96}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+214}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+302}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7: 68.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-254}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+28}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(1 + \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+68}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+96}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+214}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+302}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -6.6e-254)
   (/ (+ 1.0 (exp (* eps (- x)))) 2.0)
   (if (<= x 5e+28)
     (/ (+ 1.0 (exp (* x (+ 1.0 eps)))) 2.0)
     (if (<= x 2e+68)
       0.0
       (if (<= x 5.4e+96)
         (/ (+ 1.0 (exp x)) 2.0)
         (if (<= x 5.5e+214)
           0.0
           (if (<= x 6.5e+302) (/ (+ 2.0 (* eps x)) 2.0) 0.0)))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -6.6e-254) {
		tmp = (1.0 + exp((eps * -x))) / 2.0;
	} else if (x <= 5e+28) {
		tmp = (1.0 + exp((x * (1.0 + eps)))) / 2.0;
	} else if (x <= 2e+68) {
		tmp = 0.0;
	} else if (x <= 5.4e+96) {
		tmp = (1.0 + exp(x)) / 2.0;
	} else if (x <= 5.5e+214) {
		tmp = 0.0;
	} else if (x <= 6.5e+302) {
		tmp = (2.0 + (eps * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-6.6d-254)) then
        tmp = (1.0d0 + exp((eps * -x))) / 2.0d0
    else if (x <= 5d+28) then
        tmp = (1.0d0 + exp((x * (1.0d0 + eps)))) / 2.0d0
    else if (x <= 2d+68) then
        tmp = 0.0d0
    else if (x <= 5.4d+96) then
        tmp = (1.0d0 + exp(x)) / 2.0d0
    else if (x <= 5.5d+214) then
        tmp = 0.0d0
    else if (x <= 6.5d+302) then
        tmp = (2.0d0 + (eps * x)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -6.6e-254) {
		tmp = (1.0 + Math.exp((eps * -x))) / 2.0;
	} else if (x <= 5e+28) {
		tmp = (1.0 + Math.exp((x * (1.0 + eps)))) / 2.0;
	} else if (x <= 2e+68) {
		tmp = 0.0;
	} else if (x <= 5.4e+96) {
		tmp = (1.0 + Math.exp(x)) / 2.0;
	} else if (x <= 5.5e+214) {
		tmp = 0.0;
	} else if (x <= 6.5e+302) {
		tmp = (2.0 + (eps * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -6.6e-254:
		tmp = (1.0 + math.exp((eps * -x))) / 2.0
	elif x <= 5e+28:
		tmp = (1.0 + math.exp((x * (1.0 + eps)))) / 2.0
	elif x <= 2e+68:
		tmp = 0.0
	elif x <= 5.4e+96:
		tmp = (1.0 + math.exp(x)) / 2.0
	elif x <= 5.5e+214:
		tmp = 0.0
	elif x <= 6.5e+302:
		tmp = (2.0 + (eps * x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -6.6e-254)
		tmp = Float64(Float64(1.0 + exp(Float64(eps * Float64(-x)))) / 2.0);
	elseif (x <= 5e+28)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(1.0 + eps)))) / 2.0);
	elseif (x <= 2e+68)
		tmp = 0.0;
	elseif (x <= 5.4e+96)
		tmp = Float64(Float64(1.0 + exp(x)) / 2.0);
	elseif (x <= 5.5e+214)
		tmp = 0.0;
	elseif (x <= 6.5e+302)
		tmp = Float64(Float64(2.0 + Float64(eps * x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -6.6e-254)
		tmp = (1.0 + exp((eps * -x))) / 2.0;
	elseif (x <= 5e+28)
		tmp = (1.0 + exp((x * (1.0 + eps)))) / 2.0;
	elseif (x <= 2e+68)
		tmp = 0.0;
	elseif (x <= 5.4e+96)
		tmp = (1.0 + exp(x)) / 2.0;
	elseif (x <= 5.5e+214)
		tmp = 0.0;
	elseif (x <= 6.5e+302)
		tmp = (2.0 + (eps * x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -6.6e-254], N[(N[(1.0 + N[Exp[N[(eps * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5e+28], N[(N[(1.0 + N[Exp[N[(x * N[(1.0 + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2e+68], 0.0, If[LessEqual[x, 5.4e+96], N[(N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5.5e+214], 0.0, If[LessEqual[x, 6.5e+302], N[(N[(2.0 + N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -6.60000000000000033e-254

   1. Initial program 68.1%

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

      1. div-sub 68.1%

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

      2. +-rgt-identity 68.1%

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

      3. div-sub 68.1%

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

   3. Simplified 68.1%

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

   4. Taylor expanded in x around 0 39.0%

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

   5. Taylor expanded in eps around inf 69.8%

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

   6. Taylor expanded in eps around inf 70.8%

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

      1. *-commutative 70.8%

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

   8. Simplified 70.8%

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

   if -6.60000000000000033e-254 < x < 4.99999999999999957e28

   1. Initial program 55.1%

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

      1. div-sub 55.1%

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

      2. +-rgt-identity 55.1%

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

      3. div-sub 55.1%

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

   3. Simplified 55.1%

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

   4. Taylor expanded in x around 0 39.4%

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

      1. add-log-exp 39.4%

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

      2. *-un-lft-identity 39.4%

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

      3. log-prod 39.4%

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

      4. metadata-eval 39.4%

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

      5. add-log-exp 39.4%

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

      6. *-commutative 39.4%

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

      7. add-sqr-sqrt 7.5%

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

      8. sqrt-unprod 34.8%

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

      9. sqr-neg 34.8%

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

      10. sqrt-unprod 30.9%

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

      11. add-sqr-sqrt 38.4%

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

   6. Applied egg-rr 38.4%

      \[\leadsto \frac{\left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\color{blue}{0 + x \cdot \left(1 + \varepsilon\right)}}}{2} \]
   7. Step-by-step derivation

      1. +-lft-identity 38.4%

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

      2. +-commutative 38.4%

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

   8. Simplified 38.4%

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

   9. Taylor expanded in eps around inf 83.1%

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

   if 4.99999999999999957e28 < x < 1.99999999999999991e68 or 5.40000000000000044e96 < x < 5.5000000000000003e214 or 6.50000000000000031e302 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 71.6%

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

      1. div-sub 71.6%

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

      2. rec-exp 71.6%

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

      3. mul-1-neg 71.6%

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

      4. +-inverses 71.6%

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

   6. Simplified 71.6%

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

   if 1.99999999999999991e68 < x < 5.40000000000000044e96

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-lft-neg-in 100.0%

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

      3. mul-1-neg 100.0%

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

      4. exp-prod 100.0%

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

      5. +-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. exp-prod 100.0%

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

      8. neg-mul-1 100.0%

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

      9. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in eps around inf 78.3%

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

   8. Taylor expanded in eps around 0 3.1%

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

      1. mul-1-neg 3.1%

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

   10. Simplified 3.1%

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

       1. expm1-log1p-u 3.1%

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

       2. expm1-udef 3.1%

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

       3. add-sqr-sqrt 0.0%

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

       4. sqrt-unprod 78.1%

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

       5. sqr-neg 78.1%

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

       6. sqrt-unprod 78.1%

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

       7. add-sqr-sqrt 78.1%

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

   12. Applied egg-rr 78.1%

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

       1. expm1-def 78.1%

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

       2. expm1-log1p 78.1%

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

   14. Simplified 78.1%

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

   if 5.5000000000000003e214 < x < 6.50000000000000031e302

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 35.4%

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

   5. Taylor expanded in x around 0 35.4%

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

   6. Taylor expanded in eps around inf 35.8%

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

      1. associate-+r+ 35.8%

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

      2. distribute-lft1-in 35.8%

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

      3. metadata-eval 35.8%

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

      4. mul0-lft 35.8%

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

      5. metadata-eval 35.8%

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

      6. metadata-eval 35.8%

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

      7. *-commutative 35.8%

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

   8. Simplified 35.8%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.6 \cdot 10^{-254}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+28}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(1 + \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+68}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+96}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+214}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{+302}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8: 68.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 520:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 5.9 \cdot 10^{+214}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+302}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 520.0)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (<= x 5.9e+214)
     0.0
     (if (<= x 2.7e+302) (/ (+ 2.0 (* eps x)) 2.0) 0.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 520.0) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 5.9e+214) {
		tmp = 0.0;
	} else if (x <= 2.7e+302) {
		tmp = (2.0 + (eps * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 520.0d0) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if (x <= 5.9d+214) then
        tmp = 0.0d0
    else if (x <= 2.7d+302) then
        tmp = (2.0d0 + (eps * x)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 520.0) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 5.9e+214) {
		tmp = 0.0;
	} else if (x <= 2.7e+302) {
		tmp = (2.0 + (eps * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 520.0:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 5.9e+214:
		tmp = 0.0
	elif x <= 2.7e+302:
		tmp = (2.0 + (eps * x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 520.0)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 5.9e+214)
		tmp = 0.0;
	elseif (x <= 2.7e+302)
		tmp = Float64(Float64(2.0 + Float64(eps * x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 520.0)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif (x <= 5.9e+214)
		tmp = 0.0;
	elseif (x <= 2.7e+302)
		tmp = (2.0 + (eps * x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 520.0], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5.9e+214], 0.0, If[LessEqual[x, 2.7e+302], N[(N[(2.0 + N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 520

   1. Initial program 59.2%

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

      1. div-sub 59.2%

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

      2. +-rgt-identity 59.2%

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

      3. div-sub 59.2%

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

   3. Simplified 59.2%

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

   4. Taylor expanded in eps around inf 99.3%

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

      1. mul-1-neg 99.3%

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

      2. distribute-lft-neg-in 99.3%

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

      3. mul-1-neg 99.3%

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

      4. exp-prod 99.3%

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

      5. +-commutative 99.3%

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

      6. *-commutative 99.3%

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

      7. exp-prod 99.3%

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

      8. neg-mul-1 99.3%

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

      9. +-commutative 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in eps around inf 99.4%

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

   8. Taylor expanded in eps around 0 80.9%

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

      1. mul-1-neg 80.9%

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

   10. Simplified 80.9%

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

   if 520 < x < 5.90000000000000004e214 or 2.6999999999999999e302 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 58.7%

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

      1. div-sub 58.7%

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

      2. rec-exp 58.7%

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

      3. mul-1-neg 58.7%

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

      4. +-inverses 58.7%

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

   6. Simplified 58.7%

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

   if 5.90000000000000004e214 < x < 2.6999999999999999e302

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 35.4%

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

   5. Taylor expanded in x around 0 35.4%

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

   6. Taylor expanded in eps around inf 35.8%

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

      1. associate-+r+ 35.8%

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

      2. distribute-lft1-in 35.8%

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

      3. metadata-eval 35.8%

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

      4. mul0-lft 35.8%

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

      5. metadata-eval 35.8%

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

      6. metadata-eval 35.8%

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

      7. *-commutative 35.8%

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

   8. Simplified 35.8%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 520:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 5.9 \cdot 10^{+214}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+302}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9: 55.2% accurate, 17.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 470:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5.9 \cdot 10^{+214}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+302}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 470.0)
   1.0
   (if (<= x 5.9e+214) 0.0 (if (<= x 7e+302) (/ (+ 2.0 (* eps x)) 2.0) 0.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 470.0) {
		tmp = 1.0;
	} else if (x <= 5.9e+214) {
		tmp = 0.0;
	} else if (x <= 7e+302) {
		tmp = (2.0 + (eps * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 470.0d0) then
        tmp = 1.0d0
    else if (x <= 5.9d+214) then
        tmp = 0.0d0
    else if (x <= 7d+302) then
        tmp = (2.0d0 + (eps * x)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 470.0) {
		tmp = 1.0;
	} else if (x <= 5.9e+214) {
		tmp = 0.0;
	} else if (x <= 7e+302) {
		tmp = (2.0 + (eps * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 470.0:
		tmp = 1.0
	elif x <= 5.9e+214:
		tmp = 0.0
	elif x <= 7e+302:
		tmp = (2.0 + (eps * x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 470.0)
		tmp = 1.0;
	elseif (x <= 5.9e+214)
		tmp = 0.0;
	elseif (x <= 7e+302)
		tmp = Float64(Float64(2.0 + Float64(eps * x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 470.0)
		tmp = 1.0;
	elseif (x <= 5.9e+214)
		tmp = 0.0;
	elseif (x <= 7e+302)
		tmp = (2.0 + (eps * x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 470.0], 1.0, If[LessEqual[x, 5.9e+214], 0.0, If[LessEqual[x, 7e+302], N[(N[(2.0 + N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 470

   1. Initial program 59.2%

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

      1. div-sub 59.2%

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

      2. +-rgt-identity 59.2%

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

      3. div-sub 59.2%

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

   3. Simplified 59.2%

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

   4. Taylor expanded in x around 0 63.9%

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

   if 470 < x < 5.90000000000000004e214 or 6.99999999999999939e302 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 58.7%

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

      1. div-sub 58.7%

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

      2. rec-exp 58.7%

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

      3. mul-1-neg 58.7%

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

      4. +-inverses 58.7%

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

   6. Simplified 58.7%

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

   if 5.90000000000000004e214 < x < 6.99999999999999939e302

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 35.4%

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

   5. Taylor expanded in x around 0 35.4%

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

   6. Taylor expanded in eps around inf 35.8%

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

      1. associate-+r+ 35.8%

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

      2. distribute-lft1-in 35.8%

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

      3. metadata-eval 35.8%

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

      4. mul0-lft 35.8%

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

      5. metadata-eval 35.8%

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

      6. metadata-eval 35.8%

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

      7. *-commutative 35.8%

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

   8. Simplified 35.8%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 470:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5.9 \cdot 10^{+214}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+302}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10: 58.0% accurate, 17.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{-12}:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+214}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+302}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 5.8e-12)
   (/ (- 2.0 (* eps x)) 2.0)
   (if (<= x 5.8e+214)
     0.0
     (if (<= x 7.5e+302) (/ (+ 2.0 (* eps x)) 2.0) 0.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 5.8e-12) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else if (x <= 5.8e+214) {
		tmp = 0.0;
	} else if (x <= 7.5e+302) {
		tmp = (2.0 + (eps * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 5.8d-12) then
        tmp = (2.0d0 - (eps * x)) / 2.0d0
    else if (x <= 5.8d+214) then
        tmp = 0.0d0
    else if (x <= 7.5d+302) then
        tmp = (2.0d0 + (eps * x)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 5.8e-12) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else if (x <= 5.8e+214) {
		tmp = 0.0;
	} else if (x <= 7.5e+302) {
		tmp = (2.0 + (eps * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 5.8e-12:
		tmp = (2.0 - (eps * x)) / 2.0
	elif x <= 5.8e+214:
		tmp = 0.0
	elif x <= 7.5e+302:
		tmp = (2.0 + (eps * x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 5.8e-12)
		tmp = Float64(Float64(2.0 - Float64(eps * x)) / 2.0);
	elseif (x <= 5.8e+214)
		tmp = 0.0;
	elseif (x <= 7.5e+302)
		tmp = Float64(Float64(2.0 + Float64(eps * x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 5.8e-12)
		tmp = (2.0 - (eps * x)) / 2.0;
	elseif (x <= 5.8e+214)
		tmp = 0.0;
	elseif (x <= 7.5e+302)
		tmp = (2.0 + (eps * x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 5.8e-12], N[(N[(2.0 - N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5.8e+214], 0.0, If[LessEqual[x, 7.5e+302], N[(N[(2.0 + N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 5.8000000000000003e-12

   1. Initial program 58.5%

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

      1. div-sub 58.5%

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

      2. +-rgt-identity 58.5%

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

      3. div-sub 58.5%

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

   3. Simplified 58.5%

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

   4. Taylor expanded in x around 0 37.5%

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

   5. Taylor expanded in x around 0 45.5%

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

   6. Taylor expanded in eps around inf 66.5%

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

      1. neg-mul-1 66.5%

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

      2. distribute-lft-neg-in 66.5%

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

   8. Simplified 66.5%

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

   if 5.8000000000000003e-12 < x < 5.7999999999999999e214 or 7.49999999999999924e302 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 56.1%

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

      1. div-sub 56.1%

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

      2. rec-exp 56.1%

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

      3. mul-1-neg 56.1%

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

      4. +-inverses 56.1%

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

   6. Simplified 56.1%

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

   if 5.7999999999999999e214 < x < 7.49999999999999924e302

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 35.4%

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

   5. Taylor expanded in x around 0 35.4%

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

   6. Taylor expanded in eps around inf 35.8%

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

      1. associate-+r+ 35.8%

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

      2. distribute-lft1-in 35.8%

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

      3. metadata-eval 35.8%

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

      4. mul0-lft 35.8%

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

      5. metadata-eval 35.8%

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

      6. metadata-eval 35.8%

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

      7. *-commutative 35.8%

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

   8. Simplified 35.8%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{-12}:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+214}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+302}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11: 61.9% accurate, 17.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 480:\\ \;\;\;\;\frac{\left(2 + 0.5 \cdot \left(x \cdot x\right)\right) - x}{2}\\ \mathbf{elif}\;x \leq 5.9 \cdot 10^{+214}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+302}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 480.0)
   (/ (- (+ 2.0 (* 0.5 (* x x))) x) 2.0)
   (if (<= x 5.9e+214)
     0.0
     (if (<= x 2.3e+302) (/ (+ 2.0 (* eps x)) 2.0) 0.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 480.0) {
		tmp = ((2.0 + (0.5 * (x * x))) - x) / 2.0;
	} else if (x <= 5.9e+214) {
		tmp = 0.0;
	} else if (x <= 2.3e+302) {
		tmp = (2.0 + (eps * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 480.0d0) then
        tmp = ((2.0d0 + (0.5d0 * (x * x))) - x) / 2.0d0
    else if (x <= 5.9d+214) then
        tmp = 0.0d0
    else if (x <= 2.3d+302) then
        tmp = (2.0d0 + (eps * x)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 480.0) {
		tmp = ((2.0 + (0.5 * (x * x))) - x) / 2.0;
	} else if (x <= 5.9e+214) {
		tmp = 0.0;
	} else if (x <= 2.3e+302) {
		tmp = (2.0 + (eps * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 480.0:
		tmp = ((2.0 + (0.5 * (x * x))) - x) / 2.0
	elif x <= 5.9e+214:
		tmp = 0.0
	elif x <= 2.3e+302:
		tmp = (2.0 + (eps * x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 480.0)
		tmp = Float64(Float64(Float64(2.0 + Float64(0.5 * Float64(x * x))) - x) / 2.0);
	elseif (x <= 5.9e+214)
		tmp = 0.0;
	elseif (x <= 2.3e+302)
		tmp = Float64(Float64(2.0 + Float64(eps * x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 480.0)
		tmp = ((2.0 + (0.5 * (x * x))) - x) / 2.0;
	elseif (x <= 5.9e+214)
		tmp = 0.0;
	elseif (x <= 2.3e+302)
		tmp = (2.0 + (eps * x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 480.0], N[(N[(N[(2.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5.9e+214], 0.0, If[LessEqual[x, 2.3e+302], N[(N[(2.0 + N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 480

   1. Initial program 59.2%

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

      1. div-sub 59.2%

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

      2. +-rgt-identity 59.2%

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

      3. div-sub 59.2%

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

   3. Simplified 59.2%

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

   4. Taylor expanded in eps around inf 99.3%

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

      1. mul-1-neg 99.3%

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

      2. distribute-lft-neg-in 99.3%

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

      3. mul-1-neg 99.3%

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

      4. exp-prod 99.3%

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

      5. +-commutative 99.3%

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

      6. *-commutative 99.3%

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

      7. exp-prod 99.3%

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

      8. neg-mul-1 99.3%

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

      9. +-commutative 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in eps around inf 99.4%

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

   8. Taylor expanded in eps around 0 80.9%

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

      1. mul-1-neg 80.9%

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

   10. Simplified 80.9%

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

   11. Taylor expanded in x around 0 72.6%

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

       1. associate-+r+ 72.6%

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

       2. mul-1-neg 72.6%

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

       3. unsub-neg 72.6%

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

       4. unpow2 72.6%

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

   13. Simplified 72.6%

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

   if 480 < x < 5.90000000000000004e214 or 2.3000000000000001e302 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 58.7%

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

      1. div-sub 58.7%

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

      2. rec-exp 58.7%

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

      3. mul-1-neg 58.7%

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

      4. +-inverses 58.7%

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

   6. Simplified 58.7%

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

   if 5.90000000000000004e214 < x < 2.3000000000000001e302

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 35.4%

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

   5. Taylor expanded in x around 0 35.4%

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

   6. Taylor expanded in eps around inf 35.8%

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

      1. associate-+r+ 35.8%

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

      2. distribute-lft1-in 35.8%

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

      3. metadata-eval 35.8%

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

      4. mul0-lft 35.8%

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

      5. metadata-eval 35.8%

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

      6. metadata-eval 35.8%

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

      7. *-commutative 35.8%

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

   8. Simplified 35.8%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 480:\\ \;\;\;\;\frac{\left(2 + 0.5 \cdot \left(x \cdot x\right)\right) - x}{2}\\ \mathbf{elif}\;x \leq 5.9 \cdot 10^{+214}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+302}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 12: 58.0% accurate, 74.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 460:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (if (<= x 460.0) 1.0 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 460.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 460.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 460.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 460.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 460.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 460.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 460.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 460

   1. Initial program 59.2%

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

      1. div-sub 59.2%

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

      2. +-rgt-identity 59.2%

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

      3. div-sub 59.2%

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

   3. Simplified 59.2%

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

   4. Taylor expanded in x around 0 63.9%

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

   if 460 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 53.7%

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

      1. div-sub 53.7%

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

      2. rec-exp 53.7%

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

      3. mul-1-neg 53.7%

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

      4. +-inverses 53.7%

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

   6. Simplified 53.7%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 460:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 13: 16.3% accurate, 227.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 72.8%

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

3. Simplified 61.7%

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

4. Taylor expanded in eps around 0 19.3%

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

   1. div-sub 19.3%

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

   2. rec-exp 19.3%

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

   3. mul-1-neg 19.3%

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

   4. +-inverses 19.5%

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

6. Simplified 19.5%

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

7. Final simplification 19.5%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023238 
(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))