NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.6% → 99.0%

Time: 21.8s

Alternatives: 14

Speedup: 1.1×

• 38.5% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.0%

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

   1. sub-neg 72.0%

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

   2. neg-sub0 72.0%

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

   3. associate-+r- 72.0%

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

3. Simplified 72.0%

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

4. Taylor expanded in eps around inf 99.9%

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

   1. associate-*r* 99.9%

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

   2. sub-neg 99.9%

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

   3. mul-1-neg 99.9%

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

   4. associate-*r* 99.9%

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

   5. associate-*r* 99.9%

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

   6. neg-mul-1 99.9%

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

   7. mul-1-neg 99.9%

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

   8. sub-neg 99.9%

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

   9. mul-1-neg 99.9%

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

   10. neg-mul-1 99.9%

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

   11. distribute-lft-neg-in 99.9%

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

   12. +-commutative 99.9%

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

6. Simplified 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 85.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.0%

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

   1. sub-neg 72.0%

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

   2. neg-sub0 72.0%

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

   3. associate-+r- 72.0%

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

3. Simplified 72.0%

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

4. Taylor expanded in eps around inf 99.9%

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

   1. associate-*r* 99.9%

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

   2. sub-neg 99.9%

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

   3. mul-1-neg 99.9%

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

   4. associate-*r* 99.9%

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

   5. associate-*r* 99.9%

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

   6. neg-mul-1 99.9%

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

   7. mul-1-neg 99.9%

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

   8. sub-neg 99.9%

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

   9. mul-1-neg 99.9%

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

   10. neg-mul-1 99.9%

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

   11. distribute-lft-neg-in 99.9%

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

   12. +-commutative 99.9%

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

6. Simplified 99.9%

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

7. Taylor expanded in eps around inf 90.4%

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

   1. *-commutative 90.4%

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

9. Simplified 90.4%

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

10. Taylor expanded in eps around inf 87.4%

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

    1. mul-1-neg 87.4%

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

    2. distribute-rgt-neg-in 87.4%

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

12. Simplified 87.4%

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

13. Final simplification 87.4%

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

Alternative 3: 67.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-210}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+21}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} - -1}{2}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+31}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+133}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} - -1}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -9.5e-210)
   (/ (+ 1.0 (exp (* x (- eps)))) 2.0)
   (if (<= x 4.4e+21)
     (/ (- (exp (* x eps)) -1.0) 2.0)
     (if (<= x 2.5e+31)
       0.0
       (if (<= x 1.2e+133) (/ (- (exp (* x (+ eps -1.0))) -1.0) 2.0) 0.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -9.5e-210) {
		tmp = (1.0 + exp((x * -eps))) / 2.0;
	} else if (x <= 4.4e+21) {
		tmp = (exp((x * eps)) - -1.0) / 2.0;
	} else if (x <= 2.5e+31) {
		tmp = 0.0;
	} else if (x <= 1.2e+133) {
		tmp = (exp((x * (eps + -1.0))) - -1.0) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-9.5d-210)) then
        tmp = (1.0d0 + exp((x * -eps))) / 2.0d0
    else if (x <= 4.4d+21) then
        tmp = (exp((x * eps)) - (-1.0d0)) / 2.0d0
    else if (x <= 2.5d+31) then
        tmp = 0.0d0
    else if (x <= 1.2d+133) then
        tmp = (exp((x * (eps + (-1.0d0)))) - (-1.0d0)) / 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 <= -9.5e-210) {
		tmp = (1.0 + Math.exp((x * -eps))) / 2.0;
	} else if (x <= 4.4e+21) {
		tmp = (Math.exp((x * eps)) - -1.0) / 2.0;
	} else if (x <= 2.5e+31) {
		tmp = 0.0;
	} else if (x <= 1.2e+133) {
		tmp = (Math.exp((x * (eps + -1.0))) - -1.0) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -9.5e-210:
		tmp = (1.0 + math.exp((x * -eps))) / 2.0
	elif x <= 4.4e+21:
		tmp = (math.exp((x * eps)) - -1.0) / 2.0
	elif x <= 2.5e+31:
		tmp = 0.0
	elif x <= 1.2e+133:
		tmp = (math.exp((x * (eps + -1.0))) - -1.0) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -9.5e-210)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-eps)))) / 2.0);
	elseif (x <= 4.4e+21)
		tmp = Float64(Float64(exp(Float64(x * eps)) - -1.0) / 2.0);
	elseif (x <= 2.5e+31)
		tmp = 0.0;
	elseif (x <= 1.2e+133)
		tmp = Float64(Float64(exp(Float64(x * Float64(eps + -1.0))) - -1.0) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -9.5e-210)
		tmp = (1.0 + exp((x * -eps))) / 2.0;
	elseif (x <= 4.4e+21)
		tmp = (exp((x * eps)) - -1.0) / 2.0;
	elseif (x <= 2.5e+31)
		tmp = 0.0;
	elseif (x <= 1.2e+133)
		tmp = (exp((x * (eps + -1.0))) - -1.0) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -9.5e-210], N[(N[(1.0 + N[Exp[N[(x * (-eps)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 4.4e+21], N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.5e+31], 0.0, If[LessEqual[x, 1.2e+133], N[(N[(N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if x < -9.4999999999999997e-210

   1. Initial program 69.7%

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

      1. sub-neg 69.7%

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

      2. neg-sub0 69.7%

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

      3. associate-+r- 69.7%

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

   3. Simplified 69.7%

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

   4. Taylor expanded in eps around inf 99.9%

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

      1. associate-*r* 99.9%

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

      2. sub-neg 99.9%

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

      3. mul-1-neg 99.9%

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

      4. associate-*r* 99.9%

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

      5. associate-*r* 99.9%

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

      6. neg-mul-1 99.9%

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

      7. mul-1-neg 99.9%

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

      8. sub-neg 99.9%

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

      9. mul-1-neg 99.9%

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

      10. neg-mul-1 99.9%

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

      11. distribute-lft-neg-in 99.9%

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

      12. +-commutative 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in eps around inf 99.9%

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

      1. *-commutative 99.9%

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

   9. Simplified 99.9%

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

   10. Taylor expanded in eps around inf 100.0%

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

       1. mul-1-neg 100.0%

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

       2. distribute-rgt-neg-in 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in x around 0 73.7%

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

   if -9.4999999999999997e-210 < x < 4.4e21

   1. Initial program 57.3%

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

      1. sub-neg 57.3%

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

      2. neg-sub0 57.3%

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

      3. associate-+r- 57.3%

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

   3. Simplified 57.3%

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

   4. Taylor expanded in eps around inf 99.8%

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

      1. associate-*r* 99.8%

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

      2. sub-neg 99.8%

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

      3. mul-1-neg 99.8%

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

      4. associate-*r* 99.8%

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

      5. associate-*r* 99.8%

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

      6. neg-mul-1 99.8%

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

      7. mul-1-neg 99.8%

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

      8. sub-neg 99.8%

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

      9. mul-1-neg 99.8%

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

      10. neg-mul-1 99.8%

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

      11. distribute-lft-neg-in 99.8%

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

      12. +-commutative 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in eps around inf 98.1%

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

      1. *-commutative 98.1%

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

   9. Simplified 98.1%

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

   10. Taylor expanded in x around 0 90.9%

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

   if 4.4e21 < x < 2.50000000000000013e31 or 1.1999999999999999e133 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 64.7%

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

      1. rec-exp 64.7%

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

      2. div-sub 64.7%

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

      3. +-inverses 64.7%

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

   6. Simplified 64.7%

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

   if 2.50000000000000013e31 < x < 1.1999999999999999e133

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. sub-neg 100.0%

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

      3. mul-1-neg 100.0%

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

      4. associate-*r* 100.0%

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

      5. associate-*r* 100.0%

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

      6. neg-mul-1 100.0%

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

      7. mul-1-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. mul-1-neg 100.0%

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

      10. neg-mul-1 100.0%

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

      11. distribute-lft-neg-in 100.0%

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

      12. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 45.8%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-210}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+21}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} - -1}{2}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+31}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+133}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} - -1}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4: 59.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot 0.25}{\varepsilon}\\ \mathbf{elif}\;x \leq -2.15 \cdot 10^{-10}:\\ \;\;\;\;\frac{x \cdot \frac{-1 + \frac{1}{\varepsilon}}{\frac{1 - \varepsilon}{1 - \varepsilon \cdot \varepsilon}}}{2}\\ \mathbf{elif}\;x \leq 550:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+27}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+132}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.35e+154)
   (/ (* (* x x) 0.25) eps)
   (if (<= x -2.15e-10)
     (/ (* x (/ (+ -1.0 (/ 1.0 eps)) (/ (- 1.0 eps) (- 1.0 (* eps eps))))) 2.0)
     (if (<= x 550.0)
       1.0
       (if (<= x 5e+27)
         0.0
         (if (<= x 5.8e+132) (/ (/ (expm1 x) eps) 2.0) 0.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.35e+154) {
		tmp = ((x * x) * 0.25) / eps;
	} else if (x <= -2.15e-10) {
		tmp = (x * ((-1.0 + (1.0 / eps)) / ((1.0 - eps) / (1.0 - (eps * eps))))) / 2.0;
	} else if (x <= 550.0) {
		tmp = 1.0;
	} else if (x <= 5e+27) {
		tmp = 0.0;
	} else if (x <= 5.8e+132) {
		tmp = (expm1(x) / eps) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.35e+154) {
		tmp = ((x * x) * 0.25) / eps;
	} else if (x <= -2.15e-10) {
		tmp = (x * ((-1.0 + (1.0 / eps)) / ((1.0 - eps) / (1.0 - (eps * eps))))) / 2.0;
	} else if (x <= 550.0) {
		tmp = 1.0;
	} else if (x <= 5e+27) {
		tmp = 0.0;
	} else if (x <= 5.8e+132) {
		tmp = (Math.expm1(x) / eps) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.35e+154:
		tmp = ((x * x) * 0.25) / eps
	elif x <= -2.15e-10:
		tmp = (x * ((-1.0 + (1.0 / eps)) / ((1.0 - eps) / (1.0 - (eps * eps))))) / 2.0
	elif x <= 550.0:
		tmp = 1.0
	elif x <= 5e+27:
		tmp = 0.0
	elif x <= 5.8e+132:
		tmp = (math.expm1(x) / eps) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.35e+154)
		tmp = Float64(Float64(Float64(x * x) * 0.25) / eps);
	elseif (x <= -2.15e-10)
		tmp = Float64(Float64(x * Float64(Float64(-1.0 + Float64(1.0 / eps)) / Float64(Float64(1.0 - eps) / Float64(1.0 - Float64(eps * eps))))) / 2.0);
	elseif (x <= 550.0)
		tmp = 1.0;
	elseif (x <= 5e+27)
		tmp = 0.0;
	elseif (x <= 5.8e+132)
		tmp = Float64(Float64(expm1(x) / eps) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.35e+154], N[(N[(N[(x * x), $MachinePrecision] * 0.25), $MachinePrecision] / eps), $MachinePrecision], If[LessEqual[x, -2.15e-10], N[(N[(x * N[(N[(-1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - eps), $MachinePrecision] / N[(1.0 - N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 550.0], 1.0, If[LessEqual[x, 5e+27], 0.0, If[LessEqual[x, 5.8e+132], N[(N[(N[(Exp[x] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1.35000000000000003e154

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 35.1%

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

      1. neg-mul-1 35.1%

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

      2. distribute-lft-neg-in 35.1%

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

      3. +-commutative 35.1%

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

   6. Simplified 35.1%

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

   7. Taylor expanded in eps around 0 50.0%

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

      1. associate--r+ 50.0%

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

      2. cancel-sign-sub-inv 50.0%

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

      3. expm1-def 50.0%

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

      4. mul-1-neg 50.0%

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

      5. metadata-eval 50.0%

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

      6. *-lft-identity 50.0%

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

   9. Simplified 50.0%

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

   10. Taylor expanded in x around 0 50.0%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{x}^{2}}{\varepsilon}} \]
   11. Step-by-step derivation

       1. *-commutative 50.0%

          \[\leadsto \color{blue}{\frac{{x}^{2}}{\varepsilon} \cdot 0.25} \]

       2. associate-*l/ 50.0%

          \[\leadsto \color{blue}{\frac{{x}^{2} \cdot 0.25}{\varepsilon}} \]

       3. unpow2 50.0%

          \[\leadsto \frac{\color{blue}{\left(x \cdot x\right)} \cdot 0.25}{\varepsilon} \]

   12. Simplified 50.0%

       \[\leadsto \color{blue}{\frac{\left(x \cdot x\right) \cdot 0.25}{\varepsilon}} \]

   if -1.35000000000000003e154 < x < -2.15000000000000007e-10

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 49.8%

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

      1. neg-mul-1 49.8%

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

      2. distribute-lft-neg-in 49.8%

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

      3. +-commutative 49.8%

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

   6. Simplified 49.8%

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

   7. Taylor expanded in x around inf 11.8%

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

      1. *-commutative 11.8%

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

      2. flip-+ 25.0%

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

      3. associate-*r/ 25.0%

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

      4. sub-neg 25.0%

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

      5. metadata-eval 25.0%

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

      6. metadata-eval 25.0%

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

   9. Applied egg-rr 25.0%

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

       1. associate-/l* 25.0%

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

       2. +-commutative 25.0%

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

   11. Simplified 25.0%

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

   if -2.15000000000000007e-10 < x < 550

   1. Initial program 51.9%

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

      1. sub-neg 51.9%

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

      2. neg-sub0 51.9%

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

      3. associate-+r- 51.9%

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

   3. Simplified 51.9%

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

   4. Taylor expanded in x around 0 78.4%

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

   if 550 < x < 4.99999999999999979e27 or 5.7999999999999997e132 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 63.4%

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

      1. rec-exp 63.4%

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

      2. div-sub 63.4%

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

      3. +-inverses 63.4%

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

   6. Simplified 63.4%

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

   if 4.99999999999999979e27 < x < 5.7999999999999997e132

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 80.4%

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

      1. associate-*r* 80.4%

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

      2. mul-1-neg 80.4%

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

   6. Simplified 80.4%

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

   7. Taylor expanded in eps around 0 1.8%

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

      1. expm1-def 1.8%

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

      2. mul-1-neg 1.8%

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

   9. Simplified 1.8%

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

       1. expm1-log1p-u 1.6%

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

       2. expm1-udef 1.5%

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

       3. add-sqr-sqrt 0.0%

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

       4. sqrt-unprod 44.1%

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

       5. sqr-neg 44.1%

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

       6. sqrt-unprod 44.1%

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

       7. add-sqr-sqrt 44.1%

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

   11. Applied egg-rr 44.1%

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

       1. expm1-def 44.1%

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

       2. expm1-log1p 44.4%

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

   13. Simplified 44.4%

       \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot 0.25}{\varepsilon}\\ \mathbf{elif}\;x \leq -2.15 \cdot 10^{-10}:\\ \;\;\;\;\frac{x \cdot \frac{-1 + \frac{1}{\varepsilon}}{\frac{1 - \varepsilon}{1 - \varepsilon \cdot \varepsilon}}}{2}\\ \mathbf{elif}\;x \leq 550:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+27}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+132}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5: 71.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{-294}:\\ \;\;\;\;\frac{2 + {\left(x \cdot \varepsilon\right)}^{2}}{2}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+21} \lor \neg \left(x \leq 5.8 \cdot 10^{+27}\right) \land x \leq 2.25 \cdot 10^{+132}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} - -1}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 1.6e-294)
   (/ (+ 2.0 (pow (* x eps) 2.0)) 2.0)
   (if (or (<= x 3.8e+21) (and (not (<= x 5.8e+27)) (<= x 2.25e+132)))
     (/ (- (exp (* x eps)) -1.0) 2.0)
     0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 1.6e-294) {
		tmp = (2.0 + pow((x * eps), 2.0)) / 2.0;
	} else if ((x <= 3.8e+21) || (!(x <= 5.8e+27) && (x <= 2.25e+132))) {
		tmp = (exp((x * eps)) - -1.0) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 1.6d-294) then
        tmp = (2.0d0 + ((x * eps) ** 2.0d0)) / 2.0d0
    else if ((x <= 3.8d+21) .or. (.not. (x <= 5.8d+27)) .and. (x <= 2.25d+132)) then
        tmp = (exp((x * eps)) - (-1.0d0)) / 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 <= 1.6e-294) {
		tmp = (2.0 + Math.pow((x * eps), 2.0)) / 2.0;
	} else if ((x <= 3.8e+21) || (!(x <= 5.8e+27) && (x <= 2.25e+132))) {
		tmp = (Math.exp((x * eps)) - -1.0) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 1.6e-294:
		tmp = (2.0 + math.pow((x * eps), 2.0)) / 2.0
	elif (x <= 3.8e+21) or (not (x <= 5.8e+27) and (x <= 2.25e+132)):
		tmp = (math.exp((x * eps)) - -1.0) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 1.6e-294)
		tmp = Float64(Float64(2.0 + (Float64(x * eps) ^ 2.0)) / 2.0);
	elseif ((x <= 3.8e+21) || (!(x <= 5.8e+27) && (x <= 2.25e+132)))
		tmp = Float64(Float64(exp(Float64(x * eps)) - -1.0) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 1.6e-294)
		tmp = (2.0 + ((x * eps) ^ 2.0)) / 2.0;
	elseif ((x <= 3.8e+21) || (~((x <= 5.8e+27)) && (x <= 2.25e+132)))
		tmp = (exp((x * eps)) - -1.0) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 1.6e-294], N[(N[(2.0 + N[Power[N[(x * eps), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 3.8e+21], And[N[Not[LessEqual[x, 5.8e+27]], $MachinePrecision], LessEqual[x, 2.25e+132]]], N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 1.6000000000000001e-294

   1. Initial program 65.4%

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

      1. sub-neg 65.4%

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

      2. neg-sub0 65.4%

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

      3. associate-+r- 65.4%

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

   3. Simplified 65.4%

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

   4. Taylor expanded in eps around inf 99.9%

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

      1. associate-*r* 99.9%

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

      2. sub-neg 99.9%

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

      3. mul-1-neg 99.9%

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

      4. associate-*r* 99.9%

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

      5. associate-*r* 99.9%

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

      6. neg-mul-1 99.9%

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

      7. mul-1-neg 99.9%

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

      8. sub-neg 99.9%

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

      9. mul-1-neg 99.9%

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

      10. neg-mul-1 99.9%

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

      11. distribute-lft-neg-in 99.9%

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

      12. +-commutative 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in eps around inf 99.9%

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

      1. *-commutative 99.9%

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

   9. Simplified 99.9%

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

   10. Taylor expanded in eps around inf 100.0%

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

       1. mul-1-neg 100.0%

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

       2. distribute-rgt-neg-in 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in x around 0 87.8%

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

       1. associate-+r+ 87.8%

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

       2. *-commutative 87.8%

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

       3. distribute-rgt1-in 87.8%

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

       4. metadata-eval 87.8%

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

       5. mul0-lft 87.8%

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

       6. mul0-lft 87.8%

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

       7. metadata-eval 87.8%

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

       8. unpow2 87.8%

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

       9. unpow2 87.8%

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

       10. swap-sqr 89.0%

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

       11. unpow2 89.0%

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

   15. Simplified 89.0%

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

   if 1.6000000000000001e-294 < x < 3.8e21 or 5.8000000000000002e27 < x < 2.24999999999999986e132

   1. Initial program 68.6%

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

      1. sub-neg 68.6%

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

      2. neg-sub0 68.6%

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

      3. associate-+r- 68.6%

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

   3. Simplified 68.6%

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

   4. Taylor expanded in eps around inf 99.8%

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

      1. associate-*r* 99.8%

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

      2. sub-neg 99.8%

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

      3. mul-1-neg 99.8%

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

      4. associate-*r* 99.8%

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

      5. associate-*r* 99.8%

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

      6. neg-mul-1 99.8%

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

      7. mul-1-neg 99.8%

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

      8. sub-neg 99.8%

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

      9. mul-1-neg 99.8%

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

      10. neg-mul-1 99.8%

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

      11. distribute-lft-neg-in 99.8%

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

      12. +-commutative 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in eps around inf 92.4%

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

      1. *-commutative 92.4%

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

   9. Simplified 92.4%

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

   10. Taylor expanded in x around 0 77.7%

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

   if 3.8e21 < x < 5.8000000000000002e27 or 2.24999999999999986e132 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 64.7%

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

      1. rec-exp 64.7%

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

      2. div-sub 64.7%

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

      3. +-inverses 64.7%

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

   6. Simplified 64.7%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{-294}:\\ \;\;\;\;\frac{2 + {\left(x \cdot \varepsilon\right)}^{2}}{2}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+21} \lor \neg \left(x \leq 5.8 \cdot 10^{+27}\right) \land x \leq 2.25 \cdot 10^{+132}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} - -1}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6: 67.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-209}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+21} \lor \neg \left(x \leq 5 \cdot 10^{+27}\right) \land x \leq 7.2 \cdot 10^{+133}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} - -1}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2e-209)
   (/ (+ 1.0 (exp (* x (- eps)))) 2.0)
   (if (or (<= x 4.4e+21) (and (not (<= x 5e+27)) (<= x 7.2e+133)))
     (/ (- (exp (* x eps)) -1.0) 2.0)
     0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2e-209) {
		tmp = (1.0 + exp((x * -eps))) / 2.0;
	} else if ((x <= 4.4e+21) || (!(x <= 5e+27) && (x <= 7.2e+133))) {
		tmp = (exp((x * eps)) - -1.0) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2d-209)) then
        tmp = (1.0d0 + exp((x * -eps))) / 2.0d0
    else if ((x <= 4.4d+21) .or. (.not. (x <= 5d+27)) .and. (x <= 7.2d+133)) then
        tmp = (exp((x * eps)) - (-1.0d0)) / 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 <= -2e-209) {
		tmp = (1.0 + Math.exp((x * -eps))) / 2.0;
	} else if ((x <= 4.4e+21) || (!(x <= 5e+27) && (x <= 7.2e+133))) {
		tmp = (Math.exp((x * eps)) - -1.0) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2e-209:
		tmp = (1.0 + math.exp((x * -eps))) / 2.0
	elif (x <= 4.4e+21) or (not (x <= 5e+27) and (x <= 7.2e+133)):
		tmp = (math.exp((x * eps)) - -1.0) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2e-209)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-eps)))) / 2.0);
	elseif ((x <= 4.4e+21) || (!(x <= 5e+27) && (x <= 7.2e+133)))
		tmp = Float64(Float64(exp(Float64(x * eps)) - -1.0) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2e-209)
		tmp = (1.0 + exp((x * -eps))) / 2.0;
	elseif ((x <= 4.4e+21) || (~((x <= 5e+27)) && (x <= 7.2e+133)))
		tmp = (exp((x * eps)) - -1.0) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2e-209], N[(N[(1.0 + N[Exp[N[(x * (-eps)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 4.4e+21], And[N[Not[LessEqual[x, 5e+27]], $MachinePrecision], LessEqual[x, 7.2e+133]]], N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -2.0000000000000001e-209

   1. Initial program 69.7%

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

      1. sub-neg 69.7%

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

      2. neg-sub0 69.7%

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

      3. associate-+r- 69.7%

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

   3. Simplified 69.7%

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

   4. Taylor expanded in eps around inf 99.9%

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

      1. associate-*r* 99.9%

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

      2. sub-neg 99.9%

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

      3. mul-1-neg 99.9%

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

      4. associate-*r* 99.9%

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

      5. associate-*r* 99.9%

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

      6. neg-mul-1 99.9%

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

      7. mul-1-neg 99.9%

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

      8. sub-neg 99.9%

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

      9. mul-1-neg 99.9%

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

      10. neg-mul-1 99.9%

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

      11. distribute-lft-neg-in 99.9%

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

      12. +-commutative 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in eps around inf 99.9%

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

      1. *-commutative 99.9%

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

   9. Simplified 99.9%

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

   10. Taylor expanded in eps around inf 100.0%

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

       1. mul-1-neg 100.0%

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

       2. distribute-rgt-neg-in 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in x around 0 73.7%

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

   if -2.0000000000000001e-209 < x < 4.4e21 or 4.99999999999999979e27 < x < 7.19999999999999956e133

   1. Initial program 65.3%

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

      1. sub-neg 65.3%

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

      2. neg-sub0 65.3%

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

      3. associate-+r- 65.3%

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

   3. Simplified 65.3%

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

   4. Taylor expanded in eps around inf 99.9%

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

      1. associate-*r* 99.9%

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

      2. sub-neg 99.9%

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

      3. mul-1-neg 99.9%

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

      4. associate-*r* 99.9%

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

      5. associate-*r* 99.9%

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

      6. neg-mul-1 99.9%

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

      7. mul-1-neg 99.9%

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

      8. sub-neg 99.9%

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

      9. mul-1-neg 99.9%

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

      10. neg-mul-1 99.9%

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

      11. distribute-lft-neg-in 99.9%

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

      12. +-commutative 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in eps around inf 94.1%

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

      1. *-commutative 94.1%

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

   9. Simplified 94.1%

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

   10. Taylor expanded in x around 0 82.4%

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

   if 4.4e21 < x < 4.99999999999999979e27 or 7.19999999999999956e133 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 64.7%

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

      1. rec-exp 64.7%

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

      2. div-sub 64.7%

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

      3. +-inverses 64.7%

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

   6. Simplified 64.7%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-209}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+21} \lor \neg \left(x \leq 5 \cdot 10^{+27}\right) \land x \leq 7.2 \cdot 10^{+133}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} - -1}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7: 61.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -460:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 480:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+27}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+133}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -460.0)
   (/ (/ (expm1 (- x)) eps) 2.0)
   (if (<= x 480.0)
     1.0
     (if (<= x 5e+27) 0.0 (if (<= x 2e+133) (/ (/ (expm1 x) eps) 2.0) 0.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -460.0) {
		tmp = (expm1(-x) / eps) / 2.0;
	} else if (x <= 480.0) {
		tmp = 1.0;
	} else if (x <= 5e+27) {
		tmp = 0.0;
	} else if (x <= 2e+133) {
		tmp = (expm1(x) / eps) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -460.0) {
		tmp = (Math.expm1(-x) / eps) / 2.0;
	} else if (x <= 480.0) {
		tmp = 1.0;
	} else if (x <= 5e+27) {
		tmp = 0.0;
	} else if (x <= 2e+133) {
		tmp = (Math.expm1(x) / eps) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -460.0:
		tmp = (math.expm1(-x) / eps) / 2.0
	elif x <= 480.0:
		tmp = 1.0
	elif x <= 5e+27:
		tmp = 0.0
	elif x <= 2e+133:
		tmp = (math.expm1(x) / eps) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -460.0)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps) / 2.0);
	elseif (x <= 480.0)
		tmp = 1.0;
	elseif (x <= 5e+27)
		tmp = 0.0;
	elseif (x <= 2e+133)
		tmp = Float64(Float64(expm1(x) / eps) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -460.0], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 480.0], 1.0, If[LessEqual[x, 5e+27], 0.0, If[LessEqual[x, 2e+133], N[(N[(N[(Exp[x] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if x < -460

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in eps around 0 47.2%

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

      1. expm1-def 47.2%

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

      2. mul-1-neg 47.2%

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

   9. Simplified 47.2%

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

   if -460 < x < 480

   1. Initial program 52.8%

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

      1. sub-neg 52.8%

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

      2. neg-sub0 52.8%

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

      3. associate-+r- 52.8%

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

   3. Simplified 52.8%

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

   4. Taylor expanded in x around 0 76.9%

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

   if 480 < x < 4.99999999999999979e27 or 2e133 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 63.4%

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

      1. rec-exp 63.4%

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

      2. div-sub 63.4%

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

      3. +-inverses 63.4%

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

   6. Simplified 63.4%

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

   if 4.99999999999999979e27 < x < 2e133

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 80.4%

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

      1. associate-*r* 80.4%

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

      2. mul-1-neg 80.4%

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

   6. Simplified 80.4%

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

   7. Taylor expanded in eps around 0 1.8%

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

      1. expm1-def 1.8%

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

      2. mul-1-neg 1.8%

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

   9. Simplified 1.8%

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

       1. expm1-log1p-u 1.6%

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

       2. expm1-udef 1.5%

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

       3. add-sqr-sqrt 0.0%

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

       4. sqrt-unprod 44.1%

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

       5. sqr-neg 44.1%

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

       6. sqrt-unprod 44.1%

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

       7. add-sqr-sqrt 44.1%

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

   11. Applied egg-rr 44.1%

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

       1. expm1-def 44.1%

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

       2. expm1-log1p 44.4%

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

   13. Simplified 44.4%

       \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -460:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 480:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+27}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+133}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8: 70.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 720:\\ \;\;\;\;\frac{2 + {\left(x \cdot \varepsilon\right)}^{2}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+27}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 10^{+131}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 720.0)
   (/ (+ 2.0 (pow (* x eps) 2.0)) 2.0)
   (if (<= x 5e+27) 0.0 (if (<= x 1e+131) (/ (/ (expm1 x) eps) 2.0) 0.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 720.0) {
		tmp = (2.0 + pow((x * eps), 2.0)) / 2.0;
	} else if (x <= 5e+27) {
		tmp = 0.0;
	} else if (x <= 1e+131) {
		tmp = (expm1(x) / eps) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 720.0) {
		tmp = (2.0 + Math.pow((x * eps), 2.0)) / 2.0;
	} else if (x <= 5e+27) {
		tmp = 0.0;
	} else if (x <= 1e+131) {
		tmp = (Math.expm1(x) / eps) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 720.0:
		tmp = (2.0 + math.pow((x * eps), 2.0)) / 2.0
	elif x <= 5e+27:
		tmp = 0.0
	elif x <= 1e+131:
		tmp = (math.expm1(x) / eps) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 720.0)
		tmp = Float64(Float64(2.0 + (Float64(x * eps) ^ 2.0)) / 2.0);
	elseif (x <= 5e+27)
		tmp = 0.0;
	elseif (x <= 1e+131)
		tmp = Float64(Float64(expm1(x) / eps) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 720.0], N[(N[(2.0 + N[Power[N[(x * eps), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5e+27], 0.0, If[LessEqual[x, 1e+131], N[(N[(N[(Exp[x] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 720

   1. Initial program 61.8%

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

      1. sub-neg 61.8%

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

      2. neg-sub0 61.8%

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

      3. associate-+r- 61.8%

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

   3. Simplified 61.8%

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

   4. Taylor expanded in eps around inf 99.9%

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

      1. associate-*r* 99.9%

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

      2. sub-neg 99.9%

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

      3. mul-1-neg 99.9%

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

      4. associate-*r* 99.9%

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

      5. associate-*r* 99.9%

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

      6. neg-mul-1 99.9%

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

      7. mul-1-neg 99.9%

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

      8. sub-neg 99.9%

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

      9. mul-1-neg 99.9%

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

      10. neg-mul-1 99.9%

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

      11. distribute-lft-neg-in 99.9%

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

      12. +-commutative 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in eps around inf 99.9%

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

      1. *-commutative 99.9%

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

   9. Simplified 99.9%

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

   10. Taylor expanded in eps around inf 100.0%

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

       1. mul-1-neg 100.0%

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

       2. distribute-rgt-neg-in 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in x around 0 86.5%

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

       1. associate-+r+ 86.5%

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

       2. *-commutative 86.5%

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

       3. distribute-rgt1-in 86.5%

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

       4. metadata-eval 86.5%

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

       5. mul0-lft 86.5%

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

       6. mul0-lft 86.5%

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

       7. metadata-eval 86.5%

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

       8. unpow2 86.5%

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

       9. unpow2 86.5%

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

       10. swap-sqr 85.4%

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

       11. unpow2 85.4%

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

   15. Simplified 85.4%

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

   if 720 < x < 4.99999999999999979e27 or 9.9999999999999991e130 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 63.4%

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

      1. rec-exp 63.4%

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

      2. div-sub 63.4%

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

      3. +-inverses 63.4%

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

   6. Simplified 63.4%

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

   if 4.99999999999999979e27 < x < 9.9999999999999991e130

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around inf 80.4%

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

      1. associate-*r* 80.4%

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

      2. mul-1-neg 80.4%

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

   6. Simplified 80.4%

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

   7. Taylor expanded in eps around 0 1.8%

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

      1. expm1-def 1.8%

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

      2. mul-1-neg 1.8%

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

   9. Simplified 1.8%

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

       1. expm1-log1p-u 1.6%

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

       2. expm1-udef 1.5%

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

       3. add-sqr-sqrt 0.0%

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

       4. sqrt-unprod 44.1%

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

       5. sqr-neg 44.1%

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

       6. sqrt-unprod 44.1%

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

       7. add-sqr-sqrt 44.1%

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

   11. Applied egg-rr 44.1%

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

       1. expm1-def 44.1%

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

       2. expm1-log1p 44.4%

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

   13. Simplified 44.4%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 720:\\ \;\;\;\;\frac{2 + {\left(x \cdot \varepsilon\right)}^{2}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+27}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 10^{+131}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9: 62.2% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot 0.25}{\varepsilon}\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-12}:\\ \;\;\;\;\frac{x \cdot \frac{-1 + \frac{1}{\varepsilon}}{\frac{1 - \varepsilon}{1 - \varepsilon \cdot \varepsilon}}}{2}\\ \mathbf{elif}\;x \leq 600:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.35e+154)
   (/ (* (* x x) 0.25) eps)
   (if (<= x -1.1e-12)
     (/ (* x (/ (+ -1.0 (/ 1.0 eps)) (/ (- 1.0 eps) (- 1.0 (* eps eps))))) 2.0)
     (if (<= x 600.0) 1.0 0.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.35e+154) {
		tmp = ((x * x) * 0.25) / eps;
	} else if (x <= -1.1e-12) {
		tmp = (x * ((-1.0 + (1.0 / eps)) / ((1.0 - eps) / (1.0 - (eps * eps))))) / 2.0;
	} else if (x <= 600.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-1.35d+154)) then
        tmp = ((x * x) * 0.25d0) / eps
    else if (x <= (-1.1d-12)) then
        tmp = (x * (((-1.0d0) + (1.0d0 / eps)) / ((1.0d0 - eps) / (1.0d0 - (eps * eps))))) / 2.0d0
    else if (x <= 600.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.35e+154) {
		tmp = ((x * x) * 0.25) / eps;
	} else if (x <= -1.1e-12) {
		tmp = (x * ((-1.0 + (1.0 / eps)) / ((1.0 - eps) / (1.0 - (eps * eps))))) / 2.0;
	} else if (x <= 600.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.35e+154:
		tmp = ((x * x) * 0.25) / eps
	elif x <= -1.1e-12:
		tmp = (x * ((-1.0 + (1.0 / eps)) / ((1.0 - eps) / (1.0 - (eps * eps))))) / 2.0
	elif x <= 600.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.35e+154)
		tmp = Float64(Float64(Float64(x * x) * 0.25) / eps);
	elseif (x <= -1.1e-12)
		tmp = Float64(Float64(x * Float64(Float64(-1.0 + Float64(1.0 / eps)) / Float64(Float64(1.0 - eps) / Float64(1.0 - Float64(eps * eps))))) / 2.0);
	elseif (x <= 600.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.35e+154)
		tmp = ((x * x) * 0.25) / eps;
	elseif (x <= -1.1e-12)
		tmp = (x * ((-1.0 + (1.0 / eps)) / ((1.0 - eps) / (1.0 - (eps * eps))))) / 2.0;
	elseif (x <= 600.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.35e+154], N[(N[(N[(x * x), $MachinePrecision] * 0.25), $MachinePrecision] / eps), $MachinePrecision], If[LessEqual[x, -1.1e-12], N[(N[(x * N[(N[(-1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - eps), $MachinePrecision] / N[(1.0 - N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 600.0], 1.0, 0.0]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.35000000000000003e154

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 35.1%

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

      1. neg-mul-1 35.1%

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

      2. distribute-lft-neg-in 35.1%

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

      3. +-commutative 35.1%

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

   6. Simplified 35.1%

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

   7. Taylor expanded in eps around 0 50.0%

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

      1. associate--r+ 50.0%

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

      2. cancel-sign-sub-inv 50.0%

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

      3. expm1-def 50.0%

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

      4. mul-1-neg 50.0%

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

      5. metadata-eval 50.0%

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

      6. *-lft-identity 50.0%

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

   9. Simplified 50.0%

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

   10. Taylor expanded in x around 0 50.0%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{x}^{2}}{\varepsilon}} \]
   11. Step-by-step derivation

       1. *-commutative 50.0%

          \[\leadsto \color{blue}{\frac{{x}^{2}}{\varepsilon} \cdot 0.25} \]

       2. associate-*l/ 50.0%

          \[\leadsto \color{blue}{\frac{{x}^{2} \cdot 0.25}{\varepsilon}} \]

       3. unpow2 50.0%

          \[\leadsto \frac{\color{blue}{\left(x \cdot x\right)} \cdot 0.25}{\varepsilon} \]

   12. Simplified 50.0%

       \[\leadsto \color{blue}{\frac{\left(x \cdot x\right) \cdot 0.25}{\varepsilon}} \]

   if -1.35000000000000003e154 < x < -1.09999999999999996e-12

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 49.8%

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

      1. neg-mul-1 49.8%

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

      2. distribute-lft-neg-in 49.8%

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

      3. +-commutative 49.8%

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

   6. Simplified 49.8%

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

   7. Taylor expanded in x around inf 11.8%

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

      1. *-commutative 11.8%

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

      2. flip-+ 25.0%

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

      3. associate-*r/ 25.0%

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

      4. sub-neg 25.0%

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

      5. metadata-eval 25.0%

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

      6. metadata-eval 25.0%

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

   9. Applied egg-rr 25.0%

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

       1. associate-/l* 25.0%

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

       2. +-commutative 25.0%

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

   11. Simplified 25.0%

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

   if -1.09999999999999996e-12 < x < 600

   1. Initial program 51.9%

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

      1. sub-neg 51.9%

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

      2. neg-sub0 51.9%

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

      3. associate-+r- 51.9%

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

   3. Simplified 51.9%

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

   4. Taylor expanded in x around 0 78.4%

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

   if 600 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 49.3%

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

      1. rec-exp 49.3%

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

      2. div-sub 49.3%

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

      3. +-inverses 49.3%

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

   6. Simplified 49.3%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot 0.25}{\varepsilon}\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-12}:\\ \;\;\;\;\frac{x \cdot \frac{-1 + \frac{1}{\varepsilon}}{\frac{1 - \varepsilon}{1 - \varepsilon \cdot \varepsilon}}}{2}\\ \mathbf{elif}\;x \leq 600:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10: 60.7% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot 0.25}{\varepsilon}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-10}:\\ \;\;\;\;\frac{1 + \left(1 - x \cdot \left(\varepsilon - -1\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.35e+154)
   (/ (* (* x x) 0.25) eps)
   (if (<= x 3.3e-10) (/ (+ 1.0 (- 1.0 (* x (- eps -1.0)))) 2.0) 0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.35e+154) {
		tmp = ((x * x) * 0.25) / eps;
	} else if (x <= 3.3e-10) {
		tmp = (1.0 + (1.0 - (x * (eps - -1.0)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-1.35d+154)) then
        tmp = ((x * x) * 0.25d0) / eps
    else if (x <= 3.3d-10) then
        tmp = (1.0d0 + (1.0d0 - (x * (eps - (-1.0d0))))) / 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 <= -1.35e+154) {
		tmp = ((x * x) * 0.25) / eps;
	} else if (x <= 3.3e-10) {
		tmp = (1.0 + (1.0 - (x * (eps - -1.0)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.35e+154:
		tmp = ((x * x) * 0.25) / eps
	elif x <= 3.3e-10:
		tmp = (1.0 + (1.0 - (x * (eps - -1.0)))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.35e+154)
		tmp = Float64(Float64(Float64(x * x) * 0.25) / eps);
	elseif (x <= 3.3e-10)
		tmp = Float64(Float64(1.0 + Float64(1.0 - Float64(x * Float64(eps - -1.0)))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.35e+154)
		tmp = ((x * x) * 0.25) / eps;
	elseif (x <= 3.3e-10)
		tmp = (1.0 + (1.0 - (x * (eps - -1.0)))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.35e+154], N[(N[(N[(x * x), $MachinePrecision] * 0.25), $MachinePrecision] / eps), $MachinePrecision], If[LessEqual[x, 3.3e-10], N[(N[(1.0 + N[(1.0 - N[(x * N[(eps - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -1.35000000000000003e154

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 35.1%

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

      1. neg-mul-1 35.1%

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

      2. distribute-lft-neg-in 35.1%

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

      3. +-commutative 35.1%

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

   6. Simplified 35.1%

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

   7. Taylor expanded in eps around 0 50.0%

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

      1. associate--r+ 50.0%

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

      2. cancel-sign-sub-inv 50.0%

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

      3. expm1-def 50.0%

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

      4. mul-1-neg 50.0%

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

      5. metadata-eval 50.0%

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

      6. *-lft-identity 50.0%

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

   9. Simplified 50.0%

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

   10. Taylor expanded in x around 0 50.0%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{x}^{2}}{\varepsilon}} \]
   11. Step-by-step derivation

       1. *-commutative 50.0%

          \[\leadsto \color{blue}{\frac{{x}^{2}}{\varepsilon} \cdot 0.25} \]

       2. associate-*l/ 50.0%

          \[\leadsto \color{blue}{\frac{{x}^{2} \cdot 0.25}{\varepsilon}} \]

       3. unpow2 50.0%

          \[\leadsto \frac{\color{blue}{\left(x \cdot x\right)} \cdot 0.25}{\varepsilon} \]

   12. Simplified 50.0%

       \[\leadsto \color{blue}{\frac{\left(x \cdot x\right) \cdot 0.25}{\varepsilon}} \]

   if -1.35000000000000003e154 < x < 3.3e-10

   1. Initial program 57.0%

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

      1. sub-neg 57.0%

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

      2. neg-sub0 57.0%

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

      3. associate-+r- 57.0%

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

   3. Simplified 57.0%

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

   4. Taylor expanded in eps around inf 99.8%

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

      1. associate-*r* 99.8%

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

      2. sub-neg 99.8%

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

      3. mul-1-neg 99.8%

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

      4. associate-*r* 99.8%

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

      5. associate-*r* 99.8%

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

      6. neg-mul-1 99.8%

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

      7. mul-1-neg 99.8%

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

      8. sub-neg 99.8%

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

      9. mul-1-neg 99.8%

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

      10. neg-mul-1 99.8%

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

      11. distribute-lft-neg-in 99.8%

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

      12. +-commutative 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in eps around inf 99.9%

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

      1. *-commutative 99.9%

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

   9. Simplified 99.9%

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

   10. Taylor expanded in x around 0 85.6%

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

   11. Taylor expanded in x around 0 70.9%

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

   if 3.3e-10 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 47.3%

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

      1. rec-exp 47.3%

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

      2. div-sub 47.3%

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

      3. +-inverses 47.3%

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

   6. Simplified 47.3%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot 0.25}{\varepsilon}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-10}:\\ \;\;\;\;\frac{1 + \left(1 - x \cdot \left(\varepsilon - -1\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11: 61.1% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot 0.25}{\varepsilon}\\ \mathbf{elif}\;x \leq -2.15 \cdot 10^{-10}:\\ \;\;\;\;\frac{x \cdot \left(-\varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 520:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.35e+154)
   (/ (* (* x x) 0.25) eps)
   (if (<= x -2.15e-10) (/ (* x (- eps)) 2.0) (if (<= x 520.0) 1.0 0.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.35e+154) {
		tmp = ((x * x) * 0.25) / eps;
	} else if (x <= -2.15e-10) {
		tmp = (x * -eps) / 2.0;
	} else if (x <= 520.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-1.35d+154)) then
        tmp = ((x * x) * 0.25d0) / eps
    else if (x <= (-2.15d-10)) then
        tmp = (x * -eps) / 2.0d0
    else if (x <= 520.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.35e+154) {
		tmp = ((x * x) * 0.25) / eps;
	} else if (x <= -2.15e-10) {
		tmp = (x * -eps) / 2.0;
	} else if (x <= 520.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.35e+154:
		tmp = ((x * x) * 0.25) / eps
	elif x <= -2.15e-10:
		tmp = (x * -eps) / 2.0
	elif x <= 520.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.35e+154)
		tmp = Float64(Float64(Float64(x * x) * 0.25) / eps);
	elseif (x <= -2.15e-10)
		tmp = Float64(Float64(x * Float64(-eps)) / 2.0);
	elseif (x <= 520.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.35e+154)
		tmp = ((x * x) * 0.25) / eps;
	elseif (x <= -2.15e-10)
		tmp = (x * -eps) / 2.0;
	elseif (x <= 520.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.35e+154], N[(N[(N[(x * x), $MachinePrecision] * 0.25), $MachinePrecision] / eps), $MachinePrecision], If[LessEqual[x, -2.15e-10], N[(N[(x * (-eps)), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 520.0], 1.0, 0.0]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.35000000000000003e154

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 35.1%

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

      1. neg-mul-1 35.1%

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

      2. distribute-lft-neg-in 35.1%

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

      3. +-commutative 35.1%

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

   6. Simplified 35.1%

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

   7. Taylor expanded in eps around 0 50.0%

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

      1. associate--r+ 50.0%

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

      2. cancel-sign-sub-inv 50.0%

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

      3. expm1-def 50.0%

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

      4. mul-1-neg 50.0%

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

      5. metadata-eval 50.0%

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

      6. *-lft-identity 50.0%

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

   9. Simplified 50.0%

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

   10. Taylor expanded in x around 0 50.0%

       \[\leadsto \color{blue}{0.25 \cdot \frac{{x}^{2}}{\varepsilon}} \]
   11. Step-by-step derivation

       1. *-commutative 50.0%

          \[\leadsto \color{blue}{\frac{{x}^{2}}{\varepsilon} \cdot 0.25} \]

       2. associate-*l/ 50.0%

          \[\leadsto \color{blue}{\frac{{x}^{2} \cdot 0.25}{\varepsilon}} \]

       3. unpow2 50.0%

          \[\leadsto \frac{\color{blue}{\left(x \cdot x\right)} \cdot 0.25}{\varepsilon} \]

   12. Simplified 50.0%

       \[\leadsto \color{blue}{\frac{\left(x \cdot x\right) \cdot 0.25}{\varepsilon}} \]

   if -1.35000000000000003e154 < x < -2.15000000000000007e-10

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 49.8%

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

      1. neg-mul-1 49.8%

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

      2. distribute-lft-neg-in 49.8%

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

      3. +-commutative 49.8%

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

   6. Simplified 49.8%

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

   7. Taylor expanded in x around inf 11.8%

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

   8. Taylor expanded in eps around inf 11.8%

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

      1. associate-*r* 11.8%

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

      2. neg-mul-1 11.8%

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

   10. Simplified 11.8%

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

   if -2.15000000000000007e-10 < x < 520

   1. Initial program 51.9%

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

      1. sub-neg 51.9%

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

      2. neg-sub0 51.9%

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

      3. associate-+r- 51.9%

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

   3. Simplified 51.9%

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

   4. Taylor expanded in x around 0 78.4%

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

   if 520 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 49.3%

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

      1. rec-exp 49.3%

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

      2. div-sub 49.3%

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

      3. +-inverses 49.3%

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

   6. Simplified 49.3%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(x \cdot x\right) \cdot 0.25}{\varepsilon}\\ \mathbf{elif}\;x \leq -2.15 \cdot 10^{-10}:\\ \;\;\;\;\frac{x \cdot \left(-\varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 520:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 12: 60.2% accurate, 28.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-10}:\\ \;\;\;\;\frac{x \cdot \left(-\varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 540:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.15e-10) (/ (* x (- eps)) 2.0) (if (<= x 540.0) 1.0 0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.15e-10) {
		tmp = (x * -eps) / 2.0;
	} else if (x <= 540.0) {
		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 <= (-2.15d-10)) then
        tmp = (x * -eps) / 2.0d0
    else if (x <= 540.0d0) 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 <= -2.15e-10) {
		tmp = (x * -eps) / 2.0;
	} else if (x <= 540.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.15e-10:
		tmp = (x * -eps) / 2.0
	elif x <= 540.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.15e-10)
		tmp = Float64(Float64(x * Float64(-eps)) / 2.0);
	elseif (x <= 540.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.15e-10)
		tmp = (x * -eps) / 2.0;
	elseif (x <= 540.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.15e-10], N[(N[(x * (-eps)), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 540.0], 1.0, 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -2.15000000000000007e-10

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. neg-sub0 100.0%

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

      3. associate-+r- 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 43.0%

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

      1. neg-mul-1 43.0%

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

      2. distribute-lft-neg-in 43.0%

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

      3. +-commutative 43.0%

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

   6. Simplified 43.0%

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

   7. Taylor expanded in x around inf 20.0%

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

   8. Taylor expanded in eps around inf 20.0%

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

      1. associate-*r* 20.0%

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

      2. neg-mul-1 20.0%

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

   10. Simplified 20.0%

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

   if -2.15000000000000007e-10 < x < 540

   1. Initial program 51.9%

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

      1. sub-neg 51.9%

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

      2. neg-sub0 51.9%

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

      3. associate-+r- 51.9%

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

   3. Simplified 51.9%

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

   4. Taylor expanded in x around 0 78.4%

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

   if 540 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 49.3%

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

      1. rec-exp 49.3%

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

      2. div-sub 49.3%

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

      3. +-inverses 49.3%

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

   6. Simplified 49.3%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-10}:\\ \;\;\;\;\frac{x \cdot \left(-\varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 540:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 13: 57.0% accurate, 74.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 600:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (if (<= x 600.0) 1.0 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 600.0) {
		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 <= 600.0d0) 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 <= 600.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 600.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 600.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 600.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 600.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 600

   1. Initial program 61.8%

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

      1. sub-neg 61.8%

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

      2. neg-sub0 61.8%

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

      3. associate-+r- 61.8%

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

   3. Simplified 61.8%

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

   4. Taylor expanded in x around 0 62.8%

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

   if 600 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 49.3%

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

      1. rec-exp 49.3%

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

      2. div-sub 49.3%

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

      3. +-inverses 49.3%

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

   6. Simplified 49.3%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 600:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 14: 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 72.0%

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

3. Simplified 72.0%

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

4. Taylor expanded in eps around 0 14.7%

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

   1. rec-exp 14.7%

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

   2. div-sub 14.7%

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

   3. +-inverses 15.0%

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

6. Simplified 15.0%

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

7. Final simplification 15.0%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023283 
(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))