NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.9% → 99.1%

Time: 13.6s

Alternatives: 13

Speedup: 2.1×

• 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 = -1.5440548540638612e+285
  2. eps = -2.661536568658366e-26

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: 73.9% 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: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.6%

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

3. Simplified 78.6%

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

5. Taylor expanded in eps around inf 99.4%

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

6. Final simplification 99.4%

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

Alternative 2: 77.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-298}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{e}^{\left(x \cdot \left(-1 + \varepsilon\right)\right)} + e^{-x}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1e-298)
   (/ (+ 1.0 (exp (* x (- -1.0 eps)))) 2.0)
   (/ (+ (pow E (* x (+ -1.0 eps))) (exp (- x))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1e-298) {
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	} else {
		tmp = (pow(((double) M_E), (x * (-1.0 + eps))) + exp(-x)) / 2.0;
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1e-298) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps)))) / 2.0;
	} else {
		tmp = (Math.pow(Math.E, (x * (-1.0 + eps))) + Math.exp(-x)) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1e-298:
		tmp = (1.0 + math.exp((x * (-1.0 - eps)))) / 2.0
	else:
		tmp = (math.pow(math.e, (x * (-1.0 + eps))) + math.exp(-x)) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1e-298)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	else
		tmp = Float64(Float64((exp(1) ^ Float64(x * Float64(-1.0 + eps))) + exp(Float64(-x))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1e-298)
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	else
		tmp = ((2.71828182845904523536 ^ (x * (-1.0 + eps))) + exp(-x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1e-298], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Power[E, N[(x * N[(-1.0 + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.99999999999999912e-299

   1. Initial program 77.0%

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

   3. Simplified 77.0%

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

   5. Taylor expanded in x around 0 48.7%

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

   6. Taylor expanded in eps around inf 70.7%

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

      1. mul-1-neg 70.7%

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

      2. neg-mul-1 70.7%

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

      3. distribute-lft-neg-in 70.7%

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

      4. *-commutative 70.7%

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

      5. +-commutative 70.7%

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

   8. Simplified 70.7%

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

   if -9.99999999999999912e-299 < x

   1. Initial program 79.7%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in eps around inf 99.6%

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

   6. Taylor expanded in eps around 0 77.6%

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

      1. neg-mul-1 77.6%

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

   8. Simplified 77.6%

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

      1. *-un-lft-identity 77.6%

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

      2. exp-prod 77.6%

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

      3. associate-*r* 77.6%

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

      4. neg-mul-1 77.6%

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

   10. Applied egg-rr 77.6%

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

       1. exp-1-e 77.6%

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

       2. distribute-lft-neg-in 77.6%

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

       3. distribute-rgt-neg-in 77.6%

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

       4. sub-neg 77.6%

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

       5. neg-mul-1 77.6%

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

       6. distribute-neg-in 77.6%

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

       7. metadata-eval 77.6%

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

       8. neg-mul-1 77.6%

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

       9. remove-double-neg 77.6%

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

   12. Simplified 77.6%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-298}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{e}^{\left(x \cdot \left(-1 + \varepsilon\right)\right)} + e^{-x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.6%

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

3. Simplified 68.4%

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

5. Taylor expanded in eps around inf 99.4%

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

6. Final simplification 99.4%

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

Alternative 4: 81.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.85:\\ \;\;\;\;\frac{e^{\log 2 - x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \left(\frac{1}{\varepsilon} - x \cdot \left(\frac{1}{\varepsilon} + \left(x \cdot \left(0.5 - \left(0.5 \cdot \frac{1}{\varepsilon} - \varepsilon \cdot \left(0.5 - \varepsilon \cdot 0.5\right)\right)\right) - \varepsilon\right)\right)\right)\right) + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 0.85)
   (/ (exp (- (log 2.0) x)) 2.0)
   (/
    (+
     (+
      1.0
      (-
       (/ 1.0 eps)
       (*
        x
        (+
         (/ 1.0 eps)
         (-
          (* x (- 0.5 (- (* 0.5 (/ 1.0 eps)) (* eps (- 0.5 (* eps 0.5))))))
          eps)))))
     (* (exp (* x (- -1.0 eps))) (+ 1.0 (/ -1.0 eps))))
    2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 0.85) {
		tmp = exp((log(2.0) - x)) / 2.0;
	} else {
		tmp = ((1.0 + ((1.0 / eps) - (x * ((1.0 / eps) + ((x * (0.5 - ((0.5 * (1.0 / eps)) - (eps * (0.5 - (eps * 0.5)))))) - eps))))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 0.85d0) then
        tmp = exp((log(2.0d0) - x)) / 2.0d0
    else
        tmp = ((1.0d0 + ((1.0d0 / eps) - (x * ((1.0d0 / eps) + ((x * (0.5d0 - ((0.5d0 * (1.0d0 / eps)) - (eps * (0.5d0 - (eps * 0.5d0)))))) - eps))))) + (exp((x * ((-1.0d0) - eps))) * (1.0d0 + ((-1.0d0) / eps)))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= 0.85) {
		tmp = Math.exp((Math.log(2.0) - x)) / 2.0;
	} else {
		tmp = ((1.0 + ((1.0 / eps) - (x * ((1.0 / eps) + ((x * (0.5 - ((0.5 * (1.0 / eps)) - (eps * (0.5 - (eps * 0.5)))))) - eps))))) + (Math.exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 0.85:
		tmp = math.exp((math.log(2.0) - x)) / 2.0
	else:
		tmp = ((1.0 + ((1.0 / eps) - (x * ((1.0 / eps) + ((x * (0.5 - ((0.5 * (1.0 / eps)) - (eps * (0.5 - (eps * 0.5)))))) - eps))))) + (math.exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 0.85)
		tmp = Float64(exp(Float64(log(2.0) - x)) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(1.0 / eps) - Float64(x * Float64(Float64(1.0 / eps) + Float64(Float64(x * Float64(0.5 - Float64(Float64(0.5 * Float64(1.0 / eps)) - Float64(eps * Float64(0.5 - Float64(eps * 0.5)))))) - eps))))) + Float64(exp(Float64(x * Float64(-1.0 - eps))) * Float64(1.0 + Float64(-1.0 / eps)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 0.85)
		tmp = exp((log(2.0) - x)) / 2.0;
	else
		tmp = ((1.0 + ((1.0 / eps) - (x * ((1.0 / eps) + ((x * (0.5 - ((0.5 * (1.0 / eps)) - (eps * (0.5 - (eps * 0.5)))))) - eps))))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 0.85], N[(N[Exp[N[(N[Log[2.0], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + N[(N[(1.0 / eps), $MachinePrecision] - N[(x * N[(N[(1.0 / eps), $MachinePrecision] + N[(N[(x * N[(0.5 - N[(N[(0.5 * N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] - N[(eps * N[(0.5 - N[(eps * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 0.849999999999999978

   1. Initial program 70.0%

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

   3. Simplified 70.0%

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

   5. Taylor expanded in eps around inf 99.1%

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

   6. Taylor expanded in eps around 0 81.0%

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

      1. neg-mul-1 81.0%

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

   8. Simplified 81.0%

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

   9. Taylor expanded in eps around 0 74.7%

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

       1. cancel-sign-sub-inv 74.7%

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

       2. neg-mul-1 74.7%

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

       3. metadata-eval 74.7%

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

       4. *-lft-identity 74.7%

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

       5. count-2 74.7%

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

   11. Simplified 74.7%

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

       1. add-exp-log 74.7%

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

       2. *-commutative 74.7%

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

       3. log-prod 74.7%

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

       4. add-log-exp 74.7%

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

   13. Applied egg-rr 74.7%

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

   if 0.849999999999999978 < eps

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 48.1%

      \[\leadsto \frac{\color{blue}{\left(1 + \left(x \cdot \left(-1 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right) + x \cdot \left(-0.16666666666666666 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(1 - \varepsilon\right)}^{3}\right)\right) + 0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(1 - \varepsilon\right)}^{2}\right)\right)\right) + \frac{1}{\varepsilon}\right)\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   6. Taylor expanded in eps around 0 56.3%

      \[\leadsto \frac{\left(1 + \left(x \cdot \color{blue}{\frac{\left(\varepsilon \cdot \left(\varepsilon \cdot \left(1 + \left(-0.5 \cdot x + \varepsilon \cdot \left(x \cdot \left(0.5 + -0.3333333333333333 \cdot x\right)\right)\right)\right) + x \cdot \left(0.3333333333333333 \cdot x - 0.5\right)\right) + x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right)\right) - 1}{\varepsilon}} + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   7. Taylor expanded in x around 0 92.2%

      \[\leadsto \frac{\left(1 + \left(x \cdot \color{blue}{\left(\left(\varepsilon + x \cdot \left(\left(0.5 \cdot \frac{1}{\varepsilon} + \varepsilon \cdot \left(0.5 \cdot \varepsilon - 0.5\right)\right) - 0.5\right)\right) - \frac{1}{\varepsilon}\right)} + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.85:\\ \;\;\;\;\frac{e^{\log 2 - x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \left(\frac{1}{\varepsilon} - x \cdot \left(\frac{1}{\varepsilon} + \left(x \cdot \left(0.5 - \left(0.5 \cdot \frac{1}{\varepsilon} - \varepsilon \cdot \left(0.5 - \varepsilon \cdot 0.5\right)\right)\right) - \varepsilon\right)\right)\right)\right) + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.85:\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \left(\frac{1}{\varepsilon} - x \cdot \left(\frac{1}{\varepsilon} + \left(x \cdot \left(0.5 - \left(0.5 \cdot \frac{1}{\varepsilon} - \varepsilon \cdot \left(0.5 - \varepsilon \cdot 0.5\right)\right)\right) - \varepsilon\right)\right)\right)\right) + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 0.85)
   (exp (- x))
   (/
    (+
     (+
      1.0
      (-
       (/ 1.0 eps)
       (*
        x
        (+
         (/ 1.0 eps)
         (-
          (* x (- 0.5 (- (* 0.5 (/ 1.0 eps)) (* eps (- 0.5 (* eps 0.5))))))
          eps)))))
     (* (exp (* x (- -1.0 eps))) (+ 1.0 (/ -1.0 eps))))
    2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 0.85) {
		tmp = exp(-x);
	} else {
		tmp = ((1.0 + ((1.0 / eps) - (x * ((1.0 / eps) + ((x * (0.5 - ((0.5 * (1.0 / eps)) - (eps * (0.5 - (eps * 0.5)))))) - eps))))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 0.85d0) then
        tmp = exp(-x)
    else
        tmp = ((1.0d0 + ((1.0d0 / eps) - (x * ((1.0d0 / eps) + ((x * (0.5d0 - ((0.5d0 * (1.0d0 / eps)) - (eps * (0.5d0 - (eps * 0.5d0)))))) - eps))))) + (exp((x * ((-1.0d0) - eps))) * (1.0d0 + ((-1.0d0) / eps)))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= 0.85) {
		tmp = Math.exp(-x);
	} else {
		tmp = ((1.0 + ((1.0 / eps) - (x * ((1.0 / eps) + ((x * (0.5 - ((0.5 * (1.0 / eps)) - (eps * (0.5 - (eps * 0.5)))))) - eps))))) + (Math.exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 0.85:
		tmp = math.exp(-x)
	else:
		tmp = ((1.0 + ((1.0 / eps) - (x * ((1.0 / eps) + ((x * (0.5 - ((0.5 * (1.0 / eps)) - (eps * (0.5 - (eps * 0.5)))))) - eps))))) + (math.exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 0.85)
		tmp = exp(Float64(-x));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(1.0 / eps) - Float64(x * Float64(Float64(1.0 / eps) + Float64(Float64(x * Float64(0.5 - Float64(Float64(0.5 * Float64(1.0 / eps)) - Float64(eps * Float64(0.5 - Float64(eps * 0.5)))))) - eps))))) + Float64(exp(Float64(x * Float64(-1.0 - eps))) * Float64(1.0 + Float64(-1.0 / eps)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 0.85)
		tmp = exp(-x);
	else
		tmp = ((1.0 + ((1.0 / eps) - (x * ((1.0 / eps) + ((x * (0.5 - ((0.5 * (1.0 / eps)) - (eps * (0.5 - (eps * 0.5)))))) - eps))))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 0.85], N[Exp[(-x)], $MachinePrecision], N[(N[(N[(1.0 + N[(N[(1.0 / eps), $MachinePrecision] - N[(x * N[(N[(1.0 / eps), $MachinePrecision] + N[(N[(x * N[(0.5 - N[(N[(0.5 * N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] - N[(eps * N[(0.5 - N[(eps * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 0.849999999999999978

   1. Initial program 70.0%

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

   3. Simplified 70.0%

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

   5. Taylor expanded in eps around inf 99.1%

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

   6. Taylor expanded in eps around 0 81.0%

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

      1. neg-mul-1 81.0%

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

   8. Simplified 81.0%

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

      1. *-un-lft-identity 81.0%

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

      2. exp-prod 81.0%

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

      3. associate-*r* 81.0%

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

      4. neg-mul-1 81.0%

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

   10. Applied egg-rr 81.0%

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

       1. exp-1-e 81.0%

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

       2. distribute-lft-neg-in 81.0%

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

       3. distribute-rgt-neg-in 81.0%

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

       4. sub-neg 81.0%

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

       5. neg-mul-1 81.0%

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

       6. distribute-neg-in 81.0%

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

       7. metadata-eval 81.0%

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

       8. neg-mul-1 81.0%

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

       9. remove-double-neg 81.0%

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

   12. Simplified 81.0%

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

   13. Taylor expanded in eps around 0 74.7%

       \[\leadsto \color{blue}{0.5 \cdot \left(e^{-1 \cdot \left(x \cdot \log e\right)} - -1 \cdot e^{-x}\right)} \]
   14. Step-by-step derivation

       1. mul-1-neg 74.7%

          \[\leadsto 0.5 \cdot \left(e^{\color{blue}{-x \cdot \log e}} - -1 \cdot e^{-x}\right) \]

       2. log-E 74.7%

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

       3. distribute-lft-neg-out 74.7%

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

       4. *-rgt-identity 74.7%

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

       5. *-lft-identity 74.7%

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

       6. distribute-rgt-out-- 74.7%

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

       7. metadata-eval 74.7%

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

       8. *-commutative 74.7%

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

       9. associate-*l* 74.7%

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

       10. metadata-eval 74.7%

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

       11. *-rgt-identity 74.7%

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

   15. Simplified 74.7%

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

   if 0.849999999999999978 < eps

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 48.1%

      \[\leadsto \frac{\color{blue}{\left(1 + \left(x \cdot \left(-1 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right) + x \cdot \left(-0.16666666666666666 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(1 - \varepsilon\right)}^{3}\right)\right) + 0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(1 - \varepsilon\right)}^{2}\right)\right)\right) + \frac{1}{\varepsilon}\right)\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   6. Taylor expanded in eps around 0 56.3%

      \[\leadsto \frac{\left(1 + \left(x \cdot \color{blue}{\frac{\left(\varepsilon \cdot \left(\varepsilon \cdot \left(1 + \left(-0.5 \cdot x + \varepsilon \cdot \left(x \cdot \left(0.5 + -0.3333333333333333 \cdot x\right)\right)\right)\right) + x \cdot \left(0.3333333333333333 \cdot x - 0.5\right)\right) + x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right)\right) - 1}{\varepsilon}} + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   7. Taylor expanded in x around 0 92.2%

      \[\leadsto \frac{\left(1 + \left(x \cdot \color{blue}{\left(\left(\varepsilon + x \cdot \left(\left(0.5 \cdot \frac{1}{\varepsilon} + \varepsilon \cdot \left(0.5 \cdot \varepsilon - 0.5\right)\right) - 0.5\right)\right) - \frac{1}{\varepsilon}\right)} + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.85:\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \left(\frac{1}{\varepsilon} - x \cdot \left(\frac{1}{\varepsilon} + \left(x \cdot \left(0.5 - \left(0.5 \cdot \frac{1}{\varepsilon} - \varepsilon \cdot \left(0.5 - \varepsilon \cdot 0.5\right)\right)\right) - \varepsilon\right)\right)\right)\right) + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1000000000:\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)} + \left(1 + \left(\frac{1}{\varepsilon} - x \cdot \left(\frac{1}{\varepsilon} + \left(x \cdot \left(0.5 - \left(0.5 \cdot \frac{1}{\varepsilon} - \varepsilon \cdot \left(0.5 - \varepsilon \cdot 0.5\right)\right)\right) - \varepsilon\right)\right)\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 1000000000.0)
   (exp (- x))
   (/
    (+
     (exp (* x (- -1.0 eps)))
     (+
      1.0
      (-
       (/ 1.0 eps)
       (*
        x
        (+
         (/ 1.0 eps)
         (-
          (* x (- 0.5 (- (* 0.5 (/ 1.0 eps)) (* eps (- 0.5 (* eps 0.5))))))
          eps))))))
    2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 1000000000.0) {
		tmp = exp(-x);
	} else {
		tmp = (exp((x * (-1.0 - eps))) + (1.0 + ((1.0 / eps) - (x * ((1.0 / eps) + ((x * (0.5 - ((0.5 * (1.0 / eps)) - (eps * (0.5 - (eps * 0.5)))))) - eps)))))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 1000000000.0d0) then
        tmp = exp(-x)
    else
        tmp = (exp((x * ((-1.0d0) - eps))) + (1.0d0 + ((1.0d0 / eps) - (x * ((1.0d0 / eps) + ((x * (0.5d0 - ((0.5d0 * (1.0d0 / eps)) - (eps * (0.5d0 - (eps * 0.5d0)))))) - eps)))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= 1000000000.0) {
		tmp = Math.exp(-x);
	} else {
		tmp = (Math.exp((x * (-1.0 - eps))) + (1.0 + ((1.0 / eps) - (x * ((1.0 / eps) + ((x * (0.5 - ((0.5 * (1.0 / eps)) - (eps * (0.5 - (eps * 0.5)))))) - eps)))))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 1000000000.0:
		tmp = math.exp(-x)
	else:
		tmp = (math.exp((x * (-1.0 - eps))) + (1.0 + ((1.0 / eps) - (x * ((1.0 / eps) + ((x * (0.5 - ((0.5 * (1.0 / eps)) - (eps * (0.5 - (eps * 0.5)))))) - eps)))))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 1000000000.0)
		tmp = exp(Float64(-x));
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 - eps))) + Float64(1.0 + Float64(Float64(1.0 / eps) - Float64(x * Float64(Float64(1.0 / eps) + Float64(Float64(x * Float64(0.5 - Float64(Float64(0.5 * Float64(1.0 / eps)) - Float64(eps * Float64(0.5 - Float64(eps * 0.5)))))) - eps)))))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 1000000000.0)
		tmp = exp(-x);
	else
		tmp = (exp((x * (-1.0 - eps))) + (1.0 + ((1.0 / eps) - (x * ((1.0 / eps) + ((x * (0.5 - ((0.5 * (1.0 / eps)) - (eps * (0.5 - (eps * 0.5)))))) - eps)))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 1000000000.0], N[Exp[(-x)], $MachinePrecision], N[(N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 + N[(N[(1.0 / eps), $MachinePrecision] - N[(x * N[(N[(1.0 / eps), $MachinePrecision] + N[(N[(x * N[(0.5 - N[(N[(0.5 * N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] - N[(eps * N[(0.5 - N[(eps * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1e9

   1. Initial program 71.0%

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

   3. Simplified 71.0%

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

   5. Taylor expanded in eps around inf 99.2%

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

   6. Taylor expanded in eps around 0 81.6%

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

      1. neg-mul-1 81.6%

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

   8. Simplified 81.6%

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

      1. *-un-lft-identity 81.6%

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

      2. exp-prod 81.6%

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

      3. associate-*r* 81.6%

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

      4. neg-mul-1 81.6%

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

   10. Applied egg-rr 81.6%

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

       1. exp-1-e 81.6%

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

       2. distribute-lft-neg-in 81.6%

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

       3. distribute-rgt-neg-in 81.6%

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

       4. sub-neg 81.6%

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

       5. neg-mul-1 81.6%

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

       6. distribute-neg-in 81.6%

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

       7. metadata-eval 81.6%

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

       8. neg-mul-1 81.6%

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

       9. remove-double-neg 81.6%

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

   12. Simplified 81.6%

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

   13. Taylor expanded in eps around 0 75.0%

       \[\leadsto \color{blue}{0.5 \cdot \left(e^{-1 \cdot \left(x \cdot \log e\right)} - -1 \cdot e^{-x}\right)} \]
   14. Step-by-step derivation

       1. mul-1-neg 75.0%

          \[\leadsto 0.5 \cdot \left(e^{\color{blue}{-x \cdot \log e}} - -1 \cdot e^{-x}\right) \]

       2. log-E 75.0%

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

       3. distribute-lft-neg-out 75.0%

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

       4. *-rgt-identity 75.0%

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

       5. *-lft-identity 75.0%

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

       6. distribute-rgt-out-- 75.0%

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

       7. metadata-eval 75.0%

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

       8. *-commutative 75.0%

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

       9. associate-*l* 75.0%

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

       10. metadata-eval 75.0%

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

       11. *-rgt-identity 75.0%

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

   15. Simplified 75.0%

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

   if 1e9 < eps

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 46.5%

      \[\leadsto \frac{\color{blue}{\left(1 + \left(x \cdot \left(-1 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right) + x \cdot \left(-0.16666666666666666 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(1 - \varepsilon\right)}^{3}\right)\right) + 0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(1 - \varepsilon\right)}^{2}\right)\right)\right) + \frac{1}{\varepsilon}\right)\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   6. Taylor expanded in eps around 0 56.9%

      \[\leadsto \frac{\left(1 + \left(x \cdot \color{blue}{\frac{\left(\varepsilon \cdot \left(\varepsilon \cdot \left(1 + \left(-0.5 \cdot x + \varepsilon \cdot \left(x \cdot \left(0.5 + -0.3333333333333333 \cdot x\right)\right)\right)\right) + x \cdot \left(0.3333333333333333 \cdot x - 0.5\right)\right) + x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right)\right) - 1}{\varepsilon}} + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   7. Taylor expanded in x around 0 91.5%

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

   8. Taylor expanded in eps around inf 91.5%

      \[\leadsto \frac{\left(1 + \left(x \cdot \left(\left(\varepsilon + x \cdot \left(\left(0.5 \cdot \frac{1}{\varepsilon} + \varepsilon \cdot \left(0.5 \cdot \varepsilon - 0.5\right)\right) - 0.5\right)\right) - \frac{1}{\varepsilon}\right) + \frac{1}{\varepsilon}\right)\right) - \color{blue}{-1 \cdot e^{-1 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1000000000:\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)} + \left(1 + \left(\frac{1}{\varepsilon} - x \cdot \left(\frac{1}{\varepsilon} + \left(x \cdot \left(0.5 - \left(0.5 \cdot \frac{1}{\varepsilon} - \varepsilon \cdot \left(0.5 - \varepsilon \cdot 0.5\right)\right)\right) - \varepsilon\right)\right)\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.2 \cdot 10^{+23}:\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \left(\frac{1}{\varepsilon} - x \cdot \left(\frac{1}{\varepsilon} + \left(x \cdot \left(0.5 - \left(0.5 \cdot \frac{1}{\varepsilon} - \varepsilon \cdot \left(0.5 - \varepsilon \cdot 0.5\right)\right)\right) - \varepsilon\right)\right)\right)\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 1.2e+23)
   (exp (- x))
   (/
    (+
     (+
      1.0
      (-
       (/ 1.0 eps)
       (*
        x
        (+
         (/ 1.0 eps)
         (-
          (* x (- 0.5 (- (* 0.5 (/ 1.0 eps)) (* eps (- 0.5 (* eps 0.5))))))
          eps)))))
     (+ 1.0 (/ -1.0 eps)))
    2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 1.2e+23) {
		tmp = exp(-x);
	} else {
		tmp = ((1.0 + ((1.0 / eps) - (x * ((1.0 / eps) + ((x * (0.5 - ((0.5 * (1.0 / eps)) - (eps * (0.5 - (eps * 0.5)))))) - eps))))) + (1.0 + (-1.0 / eps))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 1.2d+23) then
        tmp = exp(-x)
    else
        tmp = ((1.0d0 + ((1.0d0 / eps) - (x * ((1.0d0 / eps) + ((x * (0.5d0 - ((0.5d0 * (1.0d0 / eps)) - (eps * (0.5d0 - (eps * 0.5d0)))))) - eps))))) + (1.0d0 + ((-1.0d0) / eps))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= 1.2e+23) {
		tmp = Math.exp(-x);
	} else {
		tmp = ((1.0 + ((1.0 / eps) - (x * ((1.0 / eps) + ((x * (0.5 - ((0.5 * (1.0 / eps)) - (eps * (0.5 - (eps * 0.5)))))) - eps))))) + (1.0 + (-1.0 / eps))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 1.2e+23:
		tmp = math.exp(-x)
	else:
		tmp = ((1.0 + ((1.0 / eps) - (x * ((1.0 / eps) + ((x * (0.5 - ((0.5 * (1.0 / eps)) - (eps * (0.5 - (eps * 0.5)))))) - eps))))) + (1.0 + (-1.0 / eps))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 1.2e+23)
		tmp = exp(Float64(-x));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(1.0 / eps) - Float64(x * Float64(Float64(1.0 / eps) + Float64(Float64(x * Float64(0.5 - Float64(Float64(0.5 * Float64(1.0 / eps)) - Float64(eps * Float64(0.5 - Float64(eps * 0.5)))))) - eps))))) + Float64(1.0 + Float64(-1.0 / eps))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 1.2e+23)
		tmp = exp(-x);
	else
		tmp = ((1.0 + ((1.0 / eps) - (x * ((1.0 / eps) + ((x * (0.5 - ((0.5 * (1.0 / eps)) - (eps * (0.5 - (eps * 0.5)))))) - eps))))) + (1.0 + (-1.0 / eps))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 1.2e+23], N[Exp[(-x)], $MachinePrecision], N[(N[(N[(1.0 + N[(N[(1.0 / eps), $MachinePrecision] - N[(x * N[(N[(1.0 / eps), $MachinePrecision] + N[(N[(x * N[(0.5 - N[(N[(0.5 * N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] - N[(eps * N[(0.5 - N[(eps * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.2e23

   1. Initial program 71.4%

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

   3. Simplified 71.4%

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

   5. Taylor expanded in eps around inf 99.2%

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

   6. Taylor expanded in eps around 0 81.8%

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

      1. neg-mul-1 81.8%

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

   8. Simplified 81.8%

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

      1. *-un-lft-identity 81.8%

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

      2. exp-prod 81.8%

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

      3. associate-*r* 81.8%

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

      4. neg-mul-1 81.8%

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

   10. Applied egg-rr 81.8%

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

       1. exp-1-e 81.8%

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

       2. distribute-lft-neg-in 81.8%

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

       3. distribute-rgt-neg-in 81.8%

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

       4. sub-neg 81.8%

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

       5. neg-mul-1 81.8%

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

       6. distribute-neg-in 81.8%

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

       7. metadata-eval 81.8%

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

       8. neg-mul-1 81.8%

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

       9. remove-double-neg 81.8%

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

   12. Simplified 81.8%

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

   13. Taylor expanded in eps around 0 75.4%

       \[\leadsto \color{blue}{0.5 \cdot \left(e^{-1 \cdot \left(x \cdot \log e\right)} - -1 \cdot e^{-x}\right)} \]
   14. Step-by-step derivation

       1. mul-1-neg 75.4%

          \[\leadsto 0.5 \cdot \left(e^{\color{blue}{-x \cdot \log e}} - -1 \cdot e^{-x}\right) \]

       2. log-E 75.4%

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

       3. distribute-lft-neg-out 75.4%

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

       4. *-rgt-identity 75.4%

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

       5. *-lft-identity 75.4%

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

       6. distribute-rgt-out-- 75.4%

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

       7. metadata-eval 75.4%

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

       8. *-commutative 75.4%

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

       9. associate-*l* 75.4%

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

       10. metadata-eval 75.4%

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

       11. *-rgt-identity 75.4%

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

   15. Simplified 75.4%

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

   if 1.2e23 < eps

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 47.1%

      \[\leadsto \frac{\color{blue}{\left(1 + \left(x \cdot \left(-1 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right) + x \cdot \left(-0.16666666666666666 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(1 - \varepsilon\right)}^{3}\right)\right) + 0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(1 - \varepsilon\right)}^{2}\right)\right)\right) + \frac{1}{\varepsilon}\right)\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   6. Taylor expanded in eps around 0 54.8%

      \[\leadsto \frac{\left(1 + \left(x \cdot \color{blue}{\frac{\left(\varepsilon \cdot \left(\varepsilon \cdot \left(1 + \left(-0.5 \cdot x + \varepsilon \cdot \left(x \cdot \left(0.5 + -0.3333333333333333 \cdot x\right)\right)\right)\right) + x \cdot \left(0.3333333333333333 \cdot x - 0.5\right)\right) + x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right)\right) - 1}{\varepsilon}} + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   7. Taylor expanded in x around 0 91.1%

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

   8. Taylor expanded in x around 0 83.7%

      \[\leadsto \frac{\left(1 + \left(x \cdot \left(\left(\varepsilon + x \cdot \left(\left(0.5 \cdot \frac{1}{\varepsilon} + \varepsilon \cdot \left(0.5 \cdot \varepsilon - 0.5\right)\right) - 0.5\right)\right) - \frac{1}{\varepsilon}\right) + \frac{1}{\varepsilon}\right)\right) - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.2 \cdot 10^{+23}:\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \left(\frac{1}{\varepsilon} - x \cdot \left(\frac{1}{\varepsilon} + \left(x \cdot \left(0.5 - \left(0.5 \cdot \frac{1}{\varepsilon} - \varepsilon \cdot \left(0.5 - \varepsilon \cdot 0.5\right)\right)\right) - \varepsilon\right)\right)\right)\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.4% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 4.6 \cdot 10^{+21}:\\ \;\;\;\;1 + x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \left(\frac{1}{\varepsilon} - x \cdot \left(\frac{1}{\varepsilon} + \left(x \cdot \left(0.5 - \left(0.5 \cdot \frac{1}{\varepsilon} - \varepsilon \cdot \left(0.5 - \varepsilon \cdot 0.5\right)\right)\right) - \varepsilon\right)\right)\right)\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 4.6e+21)
   (+ 1.0 (* x (+ -1.0 (* x (+ 0.5 (* x -0.16666666666666666))))))
   (/
    (+
     (+
      1.0
      (-
       (/ 1.0 eps)
       (*
        x
        (+
         (/ 1.0 eps)
         (-
          (* x (- 0.5 (- (* 0.5 (/ 1.0 eps)) (* eps (- 0.5 (* eps 0.5))))))
          eps)))))
     (+ 1.0 (/ -1.0 eps)))
    2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 4.6e+21) {
		tmp = 1.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))));
	} else {
		tmp = ((1.0 + ((1.0 / eps) - (x * ((1.0 / eps) + ((x * (0.5 - ((0.5 * (1.0 / eps)) - (eps * (0.5 - (eps * 0.5)))))) - eps))))) + (1.0 + (-1.0 / eps))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 4.6d+21) then
        tmp = 1.0d0 + (x * ((-1.0d0) + (x * (0.5d0 + (x * (-0.16666666666666666d0))))))
    else
        tmp = ((1.0d0 + ((1.0d0 / eps) - (x * ((1.0d0 / eps) + ((x * (0.5d0 - ((0.5d0 * (1.0d0 / eps)) - (eps * (0.5d0 - (eps * 0.5d0)))))) - eps))))) + (1.0d0 + ((-1.0d0) / eps))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= 4.6e+21) {
		tmp = 1.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))));
	} else {
		tmp = ((1.0 + ((1.0 / eps) - (x * ((1.0 / eps) + ((x * (0.5 - ((0.5 * (1.0 / eps)) - (eps * (0.5 - (eps * 0.5)))))) - eps))))) + (1.0 + (-1.0 / eps))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 4.6e+21:
		tmp = 1.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))
	else:
		tmp = ((1.0 + ((1.0 / eps) - (x * ((1.0 / eps) + ((x * (0.5 - ((0.5 * (1.0 / eps)) - (eps * (0.5 - (eps * 0.5)))))) - eps))))) + (1.0 + (-1.0 / eps))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 4.6e+21)
		tmp = Float64(1.0 + Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * -0.16666666666666666))))));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(Float64(1.0 / eps) - Float64(x * Float64(Float64(1.0 / eps) + Float64(Float64(x * Float64(0.5 - Float64(Float64(0.5 * Float64(1.0 / eps)) - Float64(eps * Float64(0.5 - Float64(eps * 0.5)))))) - eps))))) + Float64(1.0 + Float64(-1.0 / eps))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 4.6e+21)
		tmp = 1.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))));
	else
		tmp = ((1.0 + ((1.0 / eps) - (x * ((1.0 / eps) + ((x * (0.5 - ((0.5 * (1.0 / eps)) - (eps * (0.5 - (eps * 0.5)))))) - eps))))) + (1.0 + (-1.0 / eps))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 4.6e+21], N[(1.0 + N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(N[(1.0 / eps), $MachinePrecision] - N[(x * N[(N[(1.0 / eps), $MachinePrecision] + N[(N[(x * N[(0.5 - N[(N[(0.5 * N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] - N[(eps * N[(0.5 - N[(eps * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 4.6e21

   1. Initial program 71.4%

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

   3. Simplified 71.4%

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

   5. Taylor expanded in eps around inf 99.2%

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

   6. Taylor expanded in eps around 0 81.8%

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

      1. neg-mul-1 81.8%

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

   8. Simplified 81.8%

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

   9. Taylor expanded in eps around 0 75.4%

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

       1. cancel-sign-sub-inv 75.4%

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

       2. neg-mul-1 75.4%

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

       3. metadata-eval 75.4%

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

       4. *-lft-identity 75.4%

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

       5. count-2 75.4%

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

   11. Simplified 75.4%

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

   12. Taylor expanded in x around 0 53.6%

       \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right) - 1\right)} \]

   if 4.6e21 < eps

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 47.1%

      \[\leadsto \frac{\color{blue}{\left(1 + \left(x \cdot \left(-1 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right) + x \cdot \left(-0.16666666666666666 \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(1 - \varepsilon\right)}^{3}\right)\right) + 0.5 \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(1 - \varepsilon\right)}^{2}\right)\right)\right) + \frac{1}{\varepsilon}\right)\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   6. Taylor expanded in eps around 0 54.8%

      \[\leadsto \frac{\left(1 + \left(x \cdot \color{blue}{\frac{\left(\varepsilon \cdot \left(\varepsilon \cdot \left(1 + \left(-0.5 \cdot x + \varepsilon \cdot \left(x \cdot \left(0.5 + -0.3333333333333333 \cdot x\right)\right)\right)\right) + x \cdot \left(0.3333333333333333 \cdot x - 0.5\right)\right) + x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right)\right) - 1}{\varepsilon}} + \frac{1}{\varepsilon}\right)\right) - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}{2} \]

   7. Taylor expanded in x around 0 91.1%

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

   8. Taylor expanded in x around 0 83.7%

      \[\leadsto \frac{\left(1 + \left(x \cdot \left(\left(\varepsilon + x \cdot \left(\left(0.5 \cdot \frac{1}{\varepsilon} + \varepsilon \cdot \left(0.5 \cdot \varepsilon - 0.5\right)\right) - 0.5\right)\right) - \frac{1}{\varepsilon}\right) + \frac{1}{\varepsilon}\right)\right) - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 4.6 \cdot 10^{+21}:\\ \;\;\;\;1 + x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \left(\frac{1}{\varepsilon} - x \cdot \left(\frac{1}{\varepsilon} + \left(x \cdot \left(0.5 - \left(0.5 \cdot \frac{1}{\varepsilon} - \varepsilon \cdot \left(0.5 - \varepsilon \cdot 0.5\right)\right)\right) - \varepsilon\right)\right)\right)\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.0% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.56:\\ \;\;\;\;1 + x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 1.56)
   (+ 1.0 (* x (+ -1.0 (* x (+ 0.5 (* x -0.16666666666666666))))))
   0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 1.56) {
		tmp = 1.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))));
	} 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 <= 1.56d0) then
        tmp = 1.0d0 + (x * ((-1.0d0) + (x * (0.5d0 + (x * (-0.16666666666666666d0))))))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 1.56) {
		tmp = 1.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 1.56:
		tmp = 1.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 1.56)
		tmp = Float64(1.0 + Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * -0.16666666666666666))))));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 1.56)
		tmp = 1.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 1.56], N[(1.0 + N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.5600000000000001

   1. Initial program 70.2%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in eps around inf 99.4%

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

   6. Taylor expanded in eps around 0 88.3%

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

      1. neg-mul-1 88.3%

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

   8. Simplified 88.3%

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

   9. Taylor expanded in eps around 0 75.1%

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

       1. cancel-sign-sub-inv 75.1%

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

       2. neg-mul-1 75.1%

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

       3. metadata-eval 75.1%

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

       4. *-lft-identity 75.1%

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

       5. count-2 75.1%

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

   11. Simplified 75.1%

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

   12. Taylor expanded in x around 0 69.4%

       \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right) - 1\right)} \]

   if 1.5600000000000001 < x

   1. Initial program 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in eps around 0 50.9%

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

   6. Taylor expanded in x around 0 51.3%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.56:\\ \;\;\;\;1 + x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.9% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 550:\\ \;\;\;\;\frac{2 + x \cdot \left(x - 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 550.0) (/ (+ 2.0 (* x (- x 2.0))) 2.0) 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 550.0) {
		tmp = (2.0 + (x * (x - 2.0))) / 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 <= 550.0d0) then
        tmp = (2.0d0 + (x * (x - 2.0d0))) / 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 <= 550.0) {
		tmp = (2.0 + (x * (x - 2.0))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 550.0:
		tmp = (2.0 + (x * (x - 2.0))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 550.0)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(x - 2.0))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 550.0)
		tmp = (2.0 + (x * (x - 2.0))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 550.0], N[(N[(2.0 + N[(x * N[(x - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 550

   1. Initial program 70.3%

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

   3. Simplified 70.3%

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

   5. Taylor expanded in eps around inf 99.4%

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

   6. Taylor expanded in eps around 0 87.8%

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

      1. neg-mul-1 87.8%

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

   8. Simplified 87.8%

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

   9. Taylor expanded in eps around 0 74.7%

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

       1. cancel-sign-sub-inv 74.7%

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

       2. neg-mul-1 74.7%

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

       3. metadata-eval 74.7%

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

       4. *-lft-identity 74.7%

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

       5. count-2 74.7%

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

   11. Simplified 74.7%

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

   12. Taylor expanded in x around 0 62.5%

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

   if 550 < x

   1. Initial program 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in eps around 0 51.6%

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

   6. Taylor expanded in x around 0 52.0%

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

Alternative 11: 57.2% accurate, 28.3× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 - x;
	} 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 <= 1.0d0) then
        tmp = 1.0d0 - x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(1.0 - x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

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

Wolfram

code[x_, eps_] := If[LessEqual[x, 1.0], N[(1.0 - x), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 70.2%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in eps around inf 99.4%

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

   6. Taylor expanded in eps around 0 88.3%

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

      1. neg-mul-1 88.3%

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

   8. Simplified 88.3%

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

   9. Taylor expanded in eps around 0 75.1%

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

       1. cancel-sign-sub-inv 75.1%

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

       2. neg-mul-1 75.1%

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

       3. metadata-eval 75.1%

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

       4. *-lft-identity 75.1%

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

       5. count-2 75.1%

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

   11. Simplified 75.1%

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

   12. Taylor expanded in x around 0 52.2%

       \[\leadsto \color{blue}{1 + -1 \cdot x} \]
   13. Step-by-step derivation

       1. neg-mul-1 52.2%

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

       2. unsub-neg 52.2%

          \[\leadsto \color{blue}{1 - x} \]

   14. Simplified 52.2%

       \[\leadsto \color{blue}{1 - x} \]

   if 1 < x

   1. Initial program 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in eps around 0 50.9%

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

   6. Taylor expanded in x around 0 51.3%

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

Alternative 12: 57.2% accurate, 37.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 550:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (if (<= x 550.0) 1.0 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 550.0) {
		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 <= 550.0d0) 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 <= 550.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 550.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 550.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 550.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 550.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 550

   1. Initial program 70.3%

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

   3. Simplified 70.3%

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

   5. Taylor expanded in eps around inf 99.4%

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

   6. Taylor expanded in eps around 0 87.8%

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

      1. neg-mul-1 87.8%

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

   8. Simplified 87.8%

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

      1. *-un-lft-identity 87.8%

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

      2. exp-prod 87.8%

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

      3. associate-*r* 87.8%

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

      4. neg-mul-1 87.8%

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

   10. Applied egg-rr 87.8%

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

       1. exp-1-e 87.8%

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

       2. distribute-lft-neg-in 87.8%

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

       3. distribute-rgt-neg-in 87.8%

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

       4. sub-neg 87.8%

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

       5. neg-mul-1 87.8%

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

       6. distribute-neg-in 87.8%

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

       7. metadata-eval 87.8%

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

       8. neg-mul-1 87.8%

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

       9. remove-double-neg 87.8%

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

   12. Simplified 87.8%

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

   13. Taylor expanded in x around 0 51.4%

       \[\leadsto \color{blue}{1} \]

   if 550 < x

   1. Initial program 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in eps around 0 51.6%

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

   6. Taylor expanded in x around 0 52.0%

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

Alternative 13: 15.8% 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 78.6%

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

3. Simplified 68.4%

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

5. Taylor expanded in eps around 0 16.1%

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

6. Taylor expanded in x around 0 16.5%

   \[\leadsto \color{blue}{0} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024185 
(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))