NMSE Section 6.1 mentioned, A

Percentage Accurate: 74.0% → 98.7%

Time: 12.5s

Alternatives: 12

Speedup: 1.1×

• 39.2% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 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 77.3%

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

3. Simplified 63.7%

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

5. Taylor expanded in eps around inf 99.3%

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

6. Final simplification 99.3%

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

Alternative 2: 88.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 77.3%

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

3. Simplified 63.7%

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

5. Taylor expanded in eps around inf 99.3%

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

6. Taylor expanded in eps around inf 88.7%

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

   1. neg-mul-1 88.7%

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

   2. distribute-lft-neg-in 88.7%

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

   3. *-commutative 88.7%

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

8. Simplified 88.7%

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

9. Final simplification 88.7%

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

Alternative 3: 63.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-21}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{-1}{\varepsilon} - -1\right)}{2}\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{+110}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+184}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -5.2e-21)
   (/
    (+ (+ 1.0 (/ 1.0 eps)) (* (exp (* x (- -1.0 eps))) (- (/ -1.0 eps) -1.0)))
    2.0)
   (if (<= x 9.2e+110)
     (/ (+ 1.0 (exp (* x eps))) 2.0)
     (if (<= x 2.2e+184) 0.0 (/ (+ 1.0 (exp (* x (+ -1.0 eps)))) 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -5.2e-21) {
		tmp = ((1.0 + (1.0 / eps)) + (exp((x * (-1.0 - eps))) * ((-1.0 / eps) - -1.0))) / 2.0;
	} else if (x <= 9.2e+110) {
		tmp = (1.0 + exp((x * eps))) / 2.0;
	} else if (x <= 2.2e+184) {
		tmp = 0.0;
	} else {
		tmp = (1.0 + exp((x * (-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 (x <= (-5.2d-21)) then
        tmp = ((1.0d0 + (1.0d0 / eps)) + (exp((x * ((-1.0d0) - eps))) * (((-1.0d0) / eps) - (-1.0d0)))) / 2.0d0
    else if (x <= 9.2d+110) then
        tmp = (1.0d0 + exp((x * eps))) / 2.0d0
    else if (x <= 2.2d+184) then
        tmp = 0.0d0
    else
        tmp = (1.0d0 + exp((x * ((-1.0d0) + eps)))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -5.2e-21) {
		tmp = ((1.0 + (1.0 / eps)) + (Math.exp((x * (-1.0 - eps))) * ((-1.0 / eps) - -1.0))) / 2.0;
	} else if (x <= 9.2e+110) {
		tmp = (1.0 + Math.exp((x * eps))) / 2.0;
	} else if (x <= 2.2e+184) {
		tmp = 0.0;
	} else {
		tmp = (1.0 + Math.exp((x * (-1.0 + eps)))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -5.2e-21:
		tmp = ((1.0 + (1.0 / eps)) + (math.exp((x * (-1.0 - eps))) * ((-1.0 / eps) - -1.0))) / 2.0
	elif x <= 9.2e+110:
		tmp = (1.0 + math.exp((x * eps))) / 2.0
	elif x <= 2.2e+184:
		tmp = 0.0
	else:
		tmp = (1.0 + math.exp((x * (-1.0 + eps)))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -5.2e-21)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) + Float64(exp(Float64(x * Float64(-1.0 - eps))) * Float64(Float64(-1.0 / eps) - -1.0))) / 2.0);
	elseif (x <= 9.2e+110)
		tmp = Float64(Float64(1.0 + exp(Float64(x * eps))) / 2.0);
	elseif (x <= 2.2e+184)
		tmp = 0.0;
	else
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 + eps)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -5.2e-21)
		tmp = ((1.0 + (1.0 / eps)) + (exp((x * (-1.0 - eps))) * ((-1.0 / eps) - -1.0))) / 2.0;
	elseif (x <= 9.2e+110)
		tmp = (1.0 + exp((x * eps))) / 2.0;
	elseif (x <= 2.2e+184)
		tmp = 0.0;
	else
		tmp = (1.0 + exp((x * (-1.0 + eps)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -5.2e-21], N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(-1.0 / eps), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 9.2e+110], N[(N[(1.0 + N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.2e+184], 0.0, N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.20000000000000035e-21

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 53.9%

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

   if -5.20000000000000035e-21 < x < 9.2000000000000001e110

   1. Initial program 64.6%

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

   3. Simplified 64.6%

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

   5. Taylor expanded in x around 0 42.9%

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

   6. Taylor expanded in eps around inf 77.8%

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

      1. *-commutative 77.8%

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

      2. sub-neg 77.8%

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

      3. neg-mul-1 77.8%

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

      4. *-commutative 77.8%

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

      5. associate-*r* 77.8%

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

      6. neg-mul-1 77.8%

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

      7. sub-neg 77.8%

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

      8. *-lft-identity 77.8%

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

      9. *-commutative 77.8%

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

      10. *-lft-identity 77.8%

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

      11. associate-*r* 77.8%

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

      12. neg-mul-1 77.8%

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

      13. neg-sub0 77.8%

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

      14. associate--r- 77.8%

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

      15. metadata-eval 77.8%

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

      16. +-commutative 77.8%

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

   8. Simplified 77.8%

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

   9. Taylor expanded in eps around inf 78.1%

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

       1. *-commutative 78.1%

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

   11. Simplified 78.1%

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

   if 9.2000000000000001e110 < x < 2.2e184

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 82.6%

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

      1. associate-/l/ 82.6%

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

      2. distribute-rgt1-in 82.6%

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

      3. metadata-eval 82.6%

         \[\leadsto \frac{\color{blue}{0} \cdot e^{-1 \cdot x}}{2 \cdot \varepsilon} \]

      4. mul0-lft 82.6%

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

      5. associate-/r* 82.6%

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

      6. metadata-eval 82.6%

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

      7. metadata-eval 82.6%

         \[\leadsto \frac{\color{blue}{-0}}{\varepsilon} \]

      8. distribute-neg-frac 82.6%

         \[\leadsto \color{blue}{-\frac{0}{\varepsilon}} \]

   7. Applied egg-rr 82.6%

      \[\leadsto \color{blue}{-\frac{0}{\varepsilon}} \]

   8. Taylor expanded in eps around 0 82.6%

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

   if 2.2e184 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 37.4%

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

   6. Taylor expanded in eps around inf 37.7%

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

      1. *-commutative 37.7%

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

      2. sub-neg 37.7%

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

      3. neg-mul-1 37.7%

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

      4. *-commutative 37.7%

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

      5. associate-*r* 37.7%

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

      6. neg-mul-1 37.7%

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

      7. sub-neg 37.7%

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

      8. *-lft-identity 37.7%

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

      9. *-commutative 37.7%

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

      10. *-lft-identity 37.7%

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

      11. associate-*r* 37.7%

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

      12. neg-mul-1 37.7%

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

      13. neg-sub0 37.7%

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

      14. associate--r- 37.7%

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

      15. metadata-eval 37.7%

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

      16. +-commutative 37.7%

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

   8. Simplified 37.7%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-21}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(\frac{-1}{\varepsilon} - -1\right)}{2}\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{+110}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+184}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-19}:\\ \;\;\;\;{\left(\varepsilon \cdot 0\right)}^{-1}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+116}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+184}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.05e-19)
   (pow (* eps 0.0) -1.0)
   (if (<= x 1.45e+116)
     (/ (+ 1.0 (exp (* x eps))) 2.0)
     (if (<= x 2.1e+184) 0.0 (/ (+ 1.0 (exp (* x (+ -1.0 eps)))) 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.05e-19) {
		tmp = pow((eps * 0.0), -1.0);
	} else if (x <= 1.45e+116) {
		tmp = (1.0 + exp((x * eps))) / 2.0;
	} else if (x <= 2.1e+184) {
		tmp = 0.0;
	} else {
		tmp = (1.0 + exp((x * (-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 (x <= (-1.05d-19)) then
        tmp = (eps * 0.0d0) ** (-1.0d0)
    else if (x <= 1.45d+116) then
        tmp = (1.0d0 + exp((x * eps))) / 2.0d0
    else if (x <= 2.1d+184) then
        tmp = 0.0d0
    else
        tmp = (1.0d0 + exp((x * ((-1.0d0) + eps)))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.05e-19) {
		tmp = Math.pow((eps * 0.0), -1.0);
	} else if (x <= 1.45e+116) {
		tmp = (1.0 + Math.exp((x * eps))) / 2.0;
	} else if (x <= 2.1e+184) {
		tmp = 0.0;
	} else {
		tmp = (1.0 + Math.exp((x * (-1.0 + eps)))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.05e-19:
		tmp = math.pow((eps * 0.0), -1.0)
	elif x <= 1.45e+116:
		tmp = (1.0 + math.exp((x * eps))) / 2.0
	elif x <= 2.1e+184:
		tmp = 0.0
	else:
		tmp = (1.0 + math.exp((x * (-1.0 + eps)))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.05e-19)
		tmp = Float64(eps * 0.0) ^ -1.0;
	elseif (x <= 1.45e+116)
		tmp = Float64(Float64(1.0 + exp(Float64(x * eps))) / 2.0);
	elseif (x <= 2.1e+184)
		tmp = 0.0;
	else
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 + eps)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.05e-19)
		tmp = (eps * 0.0) ^ -1.0;
	elseif (x <= 1.45e+116)
		tmp = (1.0 + exp((x * eps))) / 2.0;
	elseif (x <= 2.1e+184)
		tmp = 0.0;
	else
		tmp = (1.0 + exp((x * (-1.0 + eps)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.05e-19], N[Power[N[(eps * 0.0), $MachinePrecision], -1.0], $MachinePrecision], If[LessEqual[x, 1.45e+116], N[(N[(1.0 + N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.1e+184], 0.0, N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.0499999999999999e-19

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 0.1%

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

   7. Applied egg-rr 52.2%

      \[\leadsto \color{blue}{{\left(\varepsilon \cdot 0\right)}^{-1}} \]

   if -1.0499999999999999e-19 < x < 1.4500000000000001e116

   1. Initial program 64.8%

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

   3. Simplified 64.8%

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

   5. Taylor expanded in x around 0 43.1%

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

   6. Taylor expanded in eps around inf 77.7%

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

      1. *-commutative 77.7%

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

      2. sub-neg 77.7%

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

      3. neg-mul-1 77.7%

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

      4. *-commutative 77.7%

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

      5. associate-*r* 77.7%

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

      6. neg-mul-1 77.7%

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

      7. sub-neg 77.7%

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

      8. *-lft-identity 77.7%

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

      9. *-commutative 77.7%

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

      10. *-lft-identity 77.7%

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

      11. associate-*r* 77.7%

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

      12. neg-mul-1 77.7%

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

      13. neg-sub0 77.7%

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

      14. associate--r- 77.7%

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

      15. metadata-eval 77.7%

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

      16. +-commutative 77.7%

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

   8. Simplified 77.7%

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

   9. Taylor expanded in eps around inf 78.0%

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

       1. *-commutative 78.0%

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

   11. Simplified 78.0%

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

   if 1.4500000000000001e116 < x < 2.1e184

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 82.6%

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

      1. associate-/l/ 82.6%

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

      2. distribute-rgt1-in 82.6%

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

      3. metadata-eval 82.6%

         \[\leadsto \frac{\color{blue}{0} \cdot e^{-1 \cdot x}}{2 \cdot \varepsilon} \]

      4. mul0-lft 82.6%

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

      5. associate-/r* 82.6%

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

      6. metadata-eval 82.6%

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

      7. metadata-eval 82.6%

         \[\leadsto \frac{\color{blue}{-0}}{\varepsilon} \]

      8. distribute-neg-frac 82.6%

         \[\leadsto \color{blue}{-\frac{0}{\varepsilon}} \]

   7. Applied egg-rr 82.6%

      \[\leadsto \color{blue}{-\frac{0}{\varepsilon}} \]

   8. Taylor expanded in eps around 0 82.6%

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

   if 2.1e184 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 37.4%

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

   6. Taylor expanded in eps around inf 37.7%

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

      1. *-commutative 37.7%

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

      2. sub-neg 37.7%

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

      3. neg-mul-1 37.7%

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

      4. *-commutative 37.7%

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

      5. associate-*r* 37.7%

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

      6. neg-mul-1 37.7%

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

      7. sub-neg 37.7%

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

      8. *-lft-identity 37.7%

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

      9. *-commutative 37.7%

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

      10. *-lft-identity 37.7%

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

      11. associate-*r* 37.7%

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

      12. neg-mul-1 37.7%

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

      13. neg-sub0 37.7%

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

      14. associate--r- 37.7%

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

      15. metadata-eval 37.7%

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

      16. +-commutative 37.7%

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

   8. Simplified 37.7%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-19}:\\ \;\;\;\;{\left(\varepsilon \cdot 0\right)}^{-1}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+116}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+184}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-19}:\\ \;\;\;\;{\left(\varepsilon \cdot 0\right)}^{-1}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+111} \lor \neg \left(x \leq 2.3 \cdot 10^{+184}\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.1e-19)
   (pow (* eps 0.0) -1.0)
   (if (or (<= x 1.35e+111) (not (<= x 2.3e+184)))
     (/ (+ 1.0 (exp (* x eps))) 2.0)
     0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.1e-19) {
		tmp = pow((eps * 0.0), -1.0);
	} else if ((x <= 1.35e+111) || !(x <= 2.3e+184)) {
		tmp = (1.0 + exp((x * eps))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2.1d-19)) then
        tmp = (eps * 0.0d0) ** (-1.0d0)
    else if ((x <= 1.35d+111) .or. (.not. (x <= 2.3d+184))) then
        tmp = (1.0d0 + exp((x * eps))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.1e-19) {
		tmp = Math.pow((eps * 0.0), -1.0);
	} else if ((x <= 1.35e+111) || !(x <= 2.3e+184)) {
		tmp = (1.0 + Math.exp((x * eps))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.1e-19:
		tmp = math.pow((eps * 0.0), -1.0)
	elif (x <= 1.35e+111) or not (x <= 2.3e+184):
		tmp = (1.0 + math.exp((x * eps))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.1e-19)
		tmp = Float64(eps * 0.0) ^ -1.0;
	elseif ((x <= 1.35e+111) || !(x <= 2.3e+184))
		tmp = Float64(Float64(1.0 + exp(Float64(x * eps))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.1e-19)
		tmp = (eps * 0.0) ^ -1.0;
	elseif ((x <= 1.35e+111) || ~((x <= 2.3e+184)))
		tmp = (1.0 + exp((x * eps))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.1e-19], N[Power[N[(eps * 0.0), $MachinePrecision], -1.0], $MachinePrecision], If[Or[LessEqual[x, 1.35e+111], N[Not[LessEqual[x, 2.3e+184]], $MachinePrecision]], N[(N[(1.0 + N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -2.0999999999999999e-19

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 0.1%

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

   7. Applied egg-rr 52.2%

      \[\leadsto \color{blue}{{\left(\varepsilon \cdot 0\right)}^{-1}} \]

   if -2.0999999999999999e-19 < x < 1.3499999999999999e111 or 2.3e184 < x

   1. Initial program 69.9%

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

   3. Simplified 69.9%

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

   5. Taylor expanded in x around 0 42.3%

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

   6. Taylor expanded in eps around inf 71.9%

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

      1. *-commutative 71.9%

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

      2. sub-neg 71.9%

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

      3. neg-mul-1 71.9%

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

      4. *-commutative 71.9%

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

      5. associate-*r* 71.9%

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

      6. neg-mul-1 71.9%

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

      7. sub-neg 71.9%

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

      8. *-lft-identity 71.9%

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

      9. *-commutative 71.9%

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

      10. *-lft-identity 71.9%

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

      11. associate-*r* 71.9%

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

      12. neg-mul-1 71.9%

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

      13. neg-sub0 71.9%

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

      14. associate--r- 71.9%

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

      15. metadata-eval 71.9%

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

      16. +-commutative 71.9%

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

   8. Simplified 71.9%

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

   9. Taylor expanded in eps around inf 72.1%

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

       1. *-commutative 72.1%

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

   11. Simplified 72.1%

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

   if 1.3499999999999999e111 < x < 2.3e184

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 82.6%

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

      1. associate-/l/ 82.6%

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

      2. distribute-rgt1-in 82.6%

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

      3. metadata-eval 82.6%

         \[\leadsto \frac{\color{blue}{0} \cdot e^{-1 \cdot x}}{2 \cdot \varepsilon} \]

      4. mul0-lft 82.6%

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

      5. associate-/r* 82.6%

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

      6. metadata-eval 82.6%

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

      7. metadata-eval 82.6%

         \[\leadsto \frac{\color{blue}{-0}}{\varepsilon} \]

      8. distribute-neg-frac 82.6%

         \[\leadsto \color{blue}{-\frac{0}{\varepsilon}} \]

   7. Applied egg-rr 82.6%

      \[\leadsto \color{blue}{-\frac{0}{\varepsilon}} \]

   8. Taylor expanded in eps around 0 82.6%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-19}:\\ \;\;\;\;{\left(\varepsilon \cdot 0\right)}^{-1}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+111} \lor \neg \left(x \leq 2.3 \cdot 10^{+184}\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\varepsilon \cdot 0\right)}^{-1}\\ \mathbf{if}\;x \leq -5 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 14500000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+184}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (* eps 0.0) -1.0)))
   (if (<= x -5e-18)
     t_0
     (if (<= x 14500000.0) 1.0 (if (<= x 2.2e+184) 0.0 t_0)))))

C

double code(double x, double eps) {
	double t_0 = pow((eps * 0.0), -1.0);
	double tmp;
	if (x <= -5e-18) {
		tmp = t_0;
	} else if (x <= 14500000.0) {
		tmp = 1.0;
	} else if (x <= 2.2e+184) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (eps * 0.0d0) ** (-1.0d0)
    if (x <= (-5d-18)) then
        tmp = t_0
    else if (x <= 14500000.0d0) then
        tmp = 1.0d0
    else if (x <= 2.2d+184) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((eps * 0.0), -1.0);
	double tmp;
	if (x <= -5e-18) {
		tmp = t_0;
	} else if (x <= 14500000.0) {
		tmp = 1.0;
	} else if (x <= 2.2e+184) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((eps * 0.0), -1.0)
	tmp = 0
	if x <= -5e-18:
		tmp = t_0
	elif x <= 14500000.0:
		tmp = 1.0
	elif x <= 2.2e+184:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(eps * 0.0) ^ -1.0
	tmp = 0.0
	if (x <= -5e-18)
		tmp = t_0;
	elseif (x <= 14500000.0)
		tmp = 1.0;
	elseif (x <= 2.2e+184)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (eps * 0.0) ^ -1.0;
	tmp = 0.0;
	if (x <= -5e-18)
		tmp = t_0;
	elseif (x <= 14500000.0)
		tmp = 1.0;
	elseif (x <= 2.2e+184)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[(eps * 0.0), $MachinePrecision], -1.0], $MachinePrecision]}, If[LessEqual[x, -5e-18], t$95$0, If[LessEqual[x, 14500000.0], 1.0, If[LessEqual[x, 2.2e+184], 0.0, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.00000000000000036e-18 or 2.2e184 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 13.9%

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

   7. Applied egg-rr 46.2%

      \[\leadsto \color{blue}{{\left(\varepsilon \cdot 0\right)}^{-1}} \]

   if -5.00000000000000036e-18 < x < 1.45e7

   1. Initial program 59.3%

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

   3. Simplified 59.3%

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

   5. Taylor expanded in x around 0 75.2%

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

   if 1.45e7 < x < 2.2e184

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 64.7%

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

      1. associate-/l/ 64.7%

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

      2. distribute-rgt1-in 64.7%

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

      3. metadata-eval 64.7%

         \[\leadsto \frac{\color{blue}{0} \cdot e^{-1 \cdot x}}{2 \cdot \varepsilon} \]

      4. mul0-lft 64.7%

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

      5. associate-/r* 64.7%

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

      6. metadata-eval 64.7%

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

      7. metadata-eval 64.7%

         \[\leadsto \frac{\color{blue}{-0}}{\varepsilon} \]

      8. distribute-neg-frac 64.7%

         \[\leadsto \color{blue}{-\frac{0}{\varepsilon}} \]

   7. Applied egg-rr 64.7%

      \[\leadsto \color{blue}{-\frac{0}{\varepsilon}} \]

   8. Taylor expanded in eps around 0 64.7%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-18}:\\ \;\;\;\;{\left(\varepsilon \cdot 0\right)}^{-1}\\ \mathbf{elif}\;x \leq 14500000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+184}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;{\left(\varepsilon \cdot 0\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.8% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 230:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+184}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(2 + x \cdot \varepsilon\right) - x}{\varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 230.0)
   (/ (- 2.0 (* x eps)) 2.0)
   (if (<= x 3.4e+184) 0.0 (/ (/ (- (* eps (+ 2.0 (* x eps))) x) eps) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 230.0) {
		tmp = (2.0 - (x * eps)) / 2.0;
	} else if (x <= 3.4e+184) {
		tmp = 0.0;
	} else {
		tmp = (((eps * (2.0 + (x * eps))) - x) / 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 (x <= 230.0d0) then
        tmp = (2.0d0 - (x * eps)) / 2.0d0
    else if (x <= 3.4d+184) then
        tmp = 0.0d0
    else
        tmp = (((eps * (2.0d0 + (x * eps))) - x) / eps) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 230.0) {
		tmp = (2.0 - (x * eps)) / 2.0;
	} else if (x <= 3.4e+184) {
		tmp = 0.0;
	} else {
		tmp = (((eps * (2.0 + (x * eps))) - x) / eps) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 230.0:
		tmp = (2.0 - (x * eps)) / 2.0
	elif x <= 3.4e+184:
		tmp = 0.0
	else:
		tmp = (((eps * (2.0 + (x * eps))) - x) / eps) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 230.0)
		tmp = Float64(Float64(2.0 - Float64(x * eps)) / 2.0);
	elseif (x <= 3.4e+184)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(Float64(eps * Float64(2.0 + Float64(x * eps))) - x) / eps) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 230.0)
		tmp = (2.0 - (x * eps)) / 2.0;
	elseif (x <= 3.4e+184)
		tmp = 0.0;
	else
		tmp = (((eps * (2.0 + (x * eps))) - x) / eps) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 230.0], N[(N[(2.0 - N[(x * eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3.4e+184], 0.0, N[(N[(N[(N[(eps * N[(2.0 + N[(x * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 230

   1. Initial program 68.9%

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

   3. Simplified 50.3%

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

   5. Taylor expanded in x around 0 58.3%

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

   6. Taylor expanded in eps around 0 63.8%

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

   7. Taylor expanded in eps around 0 63.8%

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

      1. *-commutative 63.8%

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

      2. neg-mul-1 63.8%

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

      3. unsub-neg 63.8%

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

   9. Simplified 63.8%

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

   if 230 < x < 3.4000000000000002e184

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 61.6%

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

      1. associate-/l/ 61.6%

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

      2. distribute-rgt1-in 61.6%

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

      3. metadata-eval 61.6%

         \[\leadsto \frac{\color{blue}{0} \cdot e^{-1 \cdot x}}{2 \cdot \varepsilon} \]

      4. mul0-lft 61.6%

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

      5. associate-/r* 61.6%

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

      6. metadata-eval 61.6%

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

      7. metadata-eval 61.6%

         \[\leadsto \frac{\color{blue}{-0}}{\varepsilon} \]

      8. distribute-neg-frac 61.6%

         \[\leadsto \color{blue}{-\frac{0}{\varepsilon}} \]

   7. Applied egg-rr 61.6%

      \[\leadsto \color{blue}{-\frac{0}{\varepsilon}} \]

   8. Taylor expanded in eps around 0 61.6%

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

   if 3.4000000000000002e184 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 37.4%

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

   6. Taylor expanded in x around 0 29.7%

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

      1. *-commutative 29.7%

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

      2. associate-*r* 29.7%

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

      3. mul-1-neg 29.7%

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

      4. distribute-lft-neg-in 29.7%

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

      5. associate-*l* 29.7%

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

      6. mul-1-neg 29.7%

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

      7. distribute-lft-in 29.7%

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

      8. metadata-eval 29.7%

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

      9. associate-*r/ 29.7%

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

      10. metadata-eval 29.7%

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

      11. *-commutative 29.7%

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

   8. Simplified 29.7%

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

   9. Taylor expanded in eps around 0 36.3%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 230:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+184}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(2 + x \cdot \varepsilon\right) - x}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.2% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{elif}\;x \leq 14500000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 8.4 \cdot 10^{+217}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \varepsilon}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.7e-12)
   (* x (* eps -0.5))
   (if (<= x 14500000.0)
     1.0
     (if (<= x 8.4e+217) 0.0 (/ (+ 2.0 (* x eps)) 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.7e-12) {
		tmp = x * (eps * -0.5);
	} else if (x <= 14500000.0) {
		tmp = 1.0;
	} else if (x <= 8.4e+217) {
		tmp = 0.0;
	} else {
		tmp = (2.0 + (x * 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 (x <= (-2.7d-12)) then
        tmp = x * (eps * (-0.5d0))
    else if (x <= 14500000.0d0) then
        tmp = 1.0d0
    else if (x <= 8.4d+217) then
        tmp = 0.0d0
    else
        tmp = (2.0d0 + (x * eps)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.7e-12) {
		tmp = x * (eps * -0.5);
	} else if (x <= 14500000.0) {
		tmp = 1.0;
	} else if (x <= 8.4e+217) {
		tmp = 0.0;
	} else {
		tmp = (2.0 + (x * eps)) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.7e-12:
		tmp = x * (eps * -0.5)
	elif x <= 14500000.0:
		tmp = 1.0
	elif x <= 8.4e+217:
		tmp = 0.0
	else:
		tmp = (2.0 + (x * eps)) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.7e-12)
		tmp = Float64(x * Float64(eps * -0.5));
	elseif (x <= 14500000.0)
		tmp = 1.0;
	elseif (x <= 8.4e+217)
		tmp = 0.0;
	else
		tmp = Float64(Float64(2.0 + Float64(x * eps)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.7e-12)
		tmp = x * (eps * -0.5);
	elseif (x <= 14500000.0)
		tmp = 1.0;
	elseif (x <= 8.4e+217)
		tmp = 0.0;
	else
		tmp = (2.0 + (x * eps)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.7e-12], N[(x * N[(eps * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 14500000.0], 1.0, If[LessEqual[x, 8.4e+217], 0.0, N[(N[(2.0 + N[(x * eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.6999999999999998e-12

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 3.2%

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

   6. Taylor expanded in eps around 0 27.8%

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

   7. Taylor expanded in eps around inf 27.8%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\varepsilon \cdot x\right)} \]
   8. Step-by-step derivation

      1. *-commutative 27.8%

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

      2. associate-*r* 27.8%

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

      3. *-commutative 27.8%

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

      4. associate-*l* 27.8%

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

   9. Simplified 27.8%

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

   if -2.6999999999999998e-12 < x < 1.45e7

   1. Initial program 59.3%

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

   3. Simplified 59.3%

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

   5. Taylor expanded in x around 0 75.2%

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

   if 1.45e7 < x < 8.4000000000000003e217

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 59.9%

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

      1. associate-/l/ 59.9%

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

      2. distribute-rgt1-in 59.9%

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

      3. metadata-eval 59.9%

         \[\leadsto \frac{\color{blue}{0} \cdot e^{-1 \cdot x}}{2 \cdot \varepsilon} \]

      4. mul0-lft 59.9%

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

      5. associate-/r* 59.9%

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

      6. metadata-eval 59.9%

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

      7. metadata-eval 59.9%

         \[\leadsto \frac{\color{blue}{-0}}{\varepsilon} \]

      8. distribute-neg-frac 59.9%

         \[\leadsto \color{blue}{-\frac{0}{\varepsilon}} \]

   7. Applied egg-rr 59.9%

      \[\leadsto \color{blue}{-\frac{0}{\varepsilon}} \]

   8. Taylor expanded in eps around 0 59.9%

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

   if 8.4000000000000003e217 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 47.5%

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

   6. Taylor expanded in x around 0 46.5%

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

      1. *-commutative 46.5%

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

      2. associate-*r* 46.5%

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

      3. mul-1-neg 46.5%

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

      4. distribute-lft-neg-in 46.5%

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

      5. associate-*l* 46.5%

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

      6. mul-1-neg 46.5%

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

      7. distribute-lft-in 46.5%

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

      8. metadata-eval 46.5%

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

      9. associate-*r/ 46.5%

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

      10. metadata-eval 46.5%

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

      11. *-commutative 46.5%

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

   8. Simplified 46.5%

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

   9. Taylor expanded in eps around inf 46.8%

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

       1. *-commutative 46.8%

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

   11. Simplified 46.8%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{elif}\;x \leq 14500000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 8.4 \cdot 10^{+217}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.4% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 220:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+217}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \varepsilon}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 220.0)
   (/ (- 2.0 (* x eps)) 2.0)
   (if (<= x 4.6e+217) 0.0 (/ (+ 2.0 (* x eps)) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 220.0) {
		tmp = (2.0 - (x * eps)) / 2.0;
	} else if (x <= 4.6e+217) {
		tmp = 0.0;
	} else {
		tmp = (2.0 + (x * 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 (x <= 220.0d0) then
        tmp = (2.0d0 - (x * eps)) / 2.0d0
    else if (x <= 4.6d+217) then
        tmp = 0.0d0
    else
        tmp = (2.0d0 + (x * eps)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 220.0) {
		tmp = (2.0 - (x * eps)) / 2.0;
	} else if (x <= 4.6e+217) {
		tmp = 0.0;
	} else {
		tmp = (2.0 + (x * eps)) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 220.0:
		tmp = (2.0 - (x * eps)) / 2.0
	elif x <= 4.6e+217:
		tmp = 0.0
	else:
		tmp = (2.0 + (x * eps)) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 220.0)
		tmp = Float64(Float64(2.0 - Float64(x * eps)) / 2.0);
	elseif (x <= 4.6e+217)
		tmp = 0.0;
	else
		tmp = Float64(Float64(2.0 + Float64(x * eps)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 220.0)
		tmp = (2.0 - (x * eps)) / 2.0;
	elseif (x <= 4.6e+217)
		tmp = 0.0;
	else
		tmp = (2.0 + (x * eps)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 220.0], N[(N[(2.0 - N[(x * eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 4.6e+217], 0.0, N[(N[(2.0 + N[(x * eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 220

   1. Initial program 68.9%

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

   3. Simplified 50.3%

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

   5. Taylor expanded in x around 0 58.3%

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

   6. Taylor expanded in eps around 0 63.8%

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

   7. Taylor expanded in eps around 0 63.8%

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

      1. *-commutative 63.8%

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

      2. neg-mul-1 63.8%

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

      3. unsub-neg 63.8%

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

   9. Simplified 63.8%

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

   if 220 < x < 4.5999999999999998e217

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 57.8%

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

      1. associate-/l/ 57.8%

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

      2. distribute-rgt1-in 57.8%

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

      3. metadata-eval 57.8%

         \[\leadsto \frac{\color{blue}{0} \cdot e^{-1 \cdot x}}{2 \cdot \varepsilon} \]

      4. mul0-lft 57.8%

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

      5. associate-/r* 57.8%

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

      6. metadata-eval 57.8%

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

      7. metadata-eval 57.8%

         \[\leadsto \frac{\color{blue}{-0}}{\varepsilon} \]

      8. distribute-neg-frac 57.8%

         \[\leadsto \color{blue}{-\frac{0}{\varepsilon}} \]

   7. Applied egg-rr 57.8%

      \[\leadsto \color{blue}{-\frac{0}{\varepsilon}} \]

   8. Taylor expanded in eps around 0 57.8%

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

   if 4.5999999999999998e217 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 47.5%

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

   6. Taylor expanded in x around 0 46.5%

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

      1. *-commutative 46.5%

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

      2. associate-*r* 46.5%

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

      3. mul-1-neg 46.5%

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

      4. distribute-lft-neg-in 46.5%

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

      5. associate-*l* 46.5%

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

      6. mul-1-neg 46.5%

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

      7. distribute-lft-in 46.5%

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

      8. metadata-eval 46.5%

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

      9. associate-*r/ 46.5%

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

      10. metadata-eval 46.5%

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

      11. *-commutative 46.5%

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

   8. Simplified 46.5%

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

   9. Taylor expanded in eps around inf 46.8%

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

       1. *-commutative 46.8%

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

   11. Simplified 46.8%

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

Alternative 10: 59.0% accurate, 20.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{elif}\;x \leq 14500000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.7e-12) (* x (* eps -0.5)) (if (<= x 14500000.0) 1.0 0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.7e-12) {
		tmp = x * (eps * -0.5);
	} else if (x <= 14500000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2.7d-12)) then
        tmp = x * (eps * (-0.5d0))
    else if (x <= 14500000.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.7e-12) {
		tmp = x * (eps * -0.5);
	} else if (x <= 14500000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.7e-12:
		tmp = x * (eps * -0.5)
	elif x <= 14500000.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.7e-12)
		tmp = Float64(x * Float64(eps * -0.5));
	elseif (x <= 14500000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.7e-12)
		tmp = x * (eps * -0.5);
	elseif (x <= 14500000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.7e-12], N[(x * N[(eps * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 14500000.0], 1.0, 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -2.6999999999999998e-12

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 3.2%

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

   6. Taylor expanded in eps around 0 27.8%

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

   7. Taylor expanded in eps around inf 27.8%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\varepsilon \cdot x\right)} \]
   8. Step-by-step derivation

      1. *-commutative 27.8%

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

      2. associate-*r* 27.8%

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

      3. *-commutative 27.8%

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

      4. associate-*l* 27.8%

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

   9. Simplified 27.8%

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

   if -2.6999999999999998e-12 < x < 1.45e7

   1. Initial program 59.3%

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

   3. Simplified 59.3%

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

   5. Taylor expanded in x around 0 75.2%

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

   if 1.45e7 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 53.0%

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

      1. associate-/l/ 53.0%

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

      2. distribute-rgt1-in 53.0%

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

      3. metadata-eval 53.0%

         \[\leadsto \frac{\color{blue}{0} \cdot e^{-1 \cdot x}}{2 \cdot \varepsilon} \]

      4. mul0-lft 53.0%

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

      5. associate-/r* 53.0%

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

      6. metadata-eval 53.0%

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

      7. metadata-eval 53.0%

         \[\leadsto \frac{\color{blue}{-0}}{\varepsilon} \]

      8. distribute-neg-frac 53.0%

         \[\leadsto \color{blue}{-\frac{0}{\varepsilon}} \]

   7. Applied egg-rr 53.0%

      \[\leadsto \color{blue}{-\frac{0}{\varepsilon}} \]

   8. Taylor expanded in eps around 0 53.0%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{elif}\;x \leq 14500000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.1% accurate, 37.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 14500000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (if (<= x 14500000.0) 1.0 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 14500000.0) {
		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 <= 14500000.0d0) 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 <= 14500000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 14500000.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 14500000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 14500000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 14500000.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.45e7

   1. Initial program 69.2%

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

   3. Simplified 69.2%

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

   5. Taylor expanded in x around 0 57.7%

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

   if 1.45e7 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 53.0%

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

      1. associate-/l/ 53.0%

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

      2. distribute-rgt1-in 53.0%

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

      3. metadata-eval 53.0%

         \[\leadsto \frac{\color{blue}{0} \cdot e^{-1 \cdot x}}{2 \cdot \varepsilon} \]

      4. mul0-lft 53.0%

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

      5. associate-/r* 53.0%

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

      6. metadata-eval 53.0%

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

      7. metadata-eval 53.0%

         \[\leadsto \frac{\color{blue}{-0}}{\varepsilon} \]

      8. distribute-neg-frac 53.0%

         \[\leadsto \color{blue}{-\frac{0}{\varepsilon}} \]

   7. Applied egg-rr 53.0%

      \[\leadsto \color{blue}{-\frac{0}{\varepsilon}} \]

   8. Taylor expanded in eps around 0 53.0%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 14500000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 15.9% 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 77.3%

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

3. Simplified 63.7%

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

5. Taylor expanded in eps around 0 15.4%

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

   1. associate-/l/ 15.4%

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

   2. distribute-rgt1-in 15.4%

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

   3. metadata-eval 15.4%

      \[\leadsto \frac{\color{blue}{0} \cdot e^{-1 \cdot x}}{2 \cdot \varepsilon} \]

   4. mul0-lft 15.7%

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

   5. associate-/r* 15.7%

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

   6. metadata-eval 15.7%

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

   7. metadata-eval 15.7%

      \[\leadsto \frac{\color{blue}{-0}}{\varepsilon} \]

   8. distribute-neg-frac 15.7%

      \[\leadsto \color{blue}{-\frac{0}{\varepsilon}} \]

7. Applied egg-rr 15.7%

   \[\leadsto \color{blue}{-\frac{0}{\varepsilon}} \]

8. Taylor expanded in eps around 0 15.7%

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

Reproduce

herbie shell --seed 2024118 
(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))