NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.3% → 98.9%

Time: 12.2s

Alternatives: 12

Speedup: 1.7×

• 37.6% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.3% 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: 98.9% 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 72.0%

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

3. Simplified 72.0%

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

5. Taylor expanded in eps around inf 99.1%

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

6. Taylor expanded in eps around -inf 99.1%

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

8. Simplified 99.1%

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

9. Taylor expanded in eps around inf 99.1%

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

    1. +-commutative 99.1%

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

    2. mul-1-neg 99.1%

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

    3. distribute-rgt-neg-in 99.1%

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

    4. sub-neg 99.1%

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

    5. mul-1-neg 99.1%

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

    6. distribute-neg-in 99.1%

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

    7. metadata-eval 99.1%

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

    8. mul-1-neg 99.1%

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

    9. remove-double-neg 99.1%

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

    10. exp-prod 99.1%

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

    11. *-lft-identity 99.1%

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

    12. metadata-eval 99.1%

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

    13. cancel-sign-sub-inv 99.1%

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

    14. exp-prod 99.1%

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

    15. mul-1-neg 99.1%

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

    16. distribute-rgt-neg-in 99.1%

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

    17. sub-neg 99.1%

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

    18. mul-1-neg 99.1%

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

    19. remove-double-neg 99.1%

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

11. Simplified 99.1%

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

12. Final simplification 99.1%

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

Alternative 2: 74.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -3.1 \cdot 10^{-260}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)} + t_0}{2}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+24}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + 1}{2}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+139} \lor \neg \left(x \leq 1.35 \cdot 10^{+243}\right):\\ \;\;\;\;\frac{2 \cdot t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -3.1e-260)
     (/ (+ (exp (* x (- -1.0 eps))) t_0) 2.0)
     (if (<= x 9e+24)
       (/ (+ (exp (* x eps)) 1.0) 2.0)
       (if (or (<= x 9e+139) (not (<= x 1.35e+243)))
         (/ (* 2.0 t_0) 2.0)
         (/ (+ 1.0 (exp (* x (+ -1.0 eps)))) 2.0))))))

C

double code(double x, double eps) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -3.1e-260) {
		tmp = (exp((x * (-1.0 - eps))) + t_0) / 2.0;
	} else if (x <= 9e+24) {
		tmp = (exp((x * eps)) + 1.0) / 2.0;
	} else if ((x <= 9e+139) || !(x <= 1.35e+243)) {
		tmp = (2.0 * t_0) / 2.0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (x <= (-3.1d-260)) then
        tmp = (exp((x * ((-1.0d0) - eps))) + t_0) / 2.0d0
    else if (x <= 9d+24) then
        tmp = (exp((x * eps)) + 1.0d0) / 2.0d0
    else if ((x <= 9d+139) .or. (.not. (x <= 1.35d+243))) then
        tmp = (2.0d0 * t_0) / 2.0d0
    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 t_0 = Math.exp(-x);
	double tmp;
	if (x <= -3.1e-260) {
		tmp = (Math.exp((x * (-1.0 - eps))) + t_0) / 2.0;
	} else if (x <= 9e+24) {
		tmp = (Math.exp((x * eps)) + 1.0) / 2.0;
	} else if ((x <= 9e+139) || !(x <= 1.35e+243)) {
		tmp = (2.0 * t_0) / 2.0;
	} else {
		tmp = (1.0 + Math.exp((x * (-1.0 + eps)))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.exp(-x)
	tmp = 0
	if x <= -3.1e-260:
		tmp = (math.exp((x * (-1.0 - eps))) + t_0) / 2.0
	elif x <= 9e+24:
		tmp = (math.exp((x * eps)) + 1.0) / 2.0
	elif (x <= 9e+139) or not (x <= 1.35e+243):
		tmp = (2.0 * t_0) / 2.0
	else:
		tmp = (1.0 + math.exp((x * (-1.0 + eps)))) / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -3.1e-260)
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 - eps))) + t_0) / 2.0);
	elseif (x <= 9e+24)
		tmp = Float64(Float64(exp(Float64(x * eps)) + 1.0) / 2.0);
	elseif ((x <= 9e+139) || !(x <= 1.35e+243))
		tmp = Float64(Float64(2.0 * t_0) / 2.0);
	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)
	t_0 = exp(-x);
	tmp = 0.0;
	if (x <= -3.1e-260)
		tmp = (exp((x * (-1.0 - eps))) + t_0) / 2.0;
	elseif (x <= 9e+24)
		tmp = (exp((x * eps)) + 1.0) / 2.0;
	elseif ((x <= 9e+139) || ~((x <= 1.35e+243)))
		tmp = (2.0 * t_0) / 2.0;
	else
		tmp = (1.0 + exp((x * (-1.0 + eps)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -3.1e-260], N[(N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 9e+24], N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 9e+139], N[Not[LessEqual[x, 1.35e+243]], $MachinePrecision]], N[(N[(2.0 * t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], 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 < -3.09999999999999983e-260

   1. Initial program 71.7%

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

   3. Simplified 71.7%

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

   5. Taylor expanded in eps around inf 98.7%

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

   6. Taylor expanded in eps around 0 88.8%

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

   7. Taylor expanded in eps around -inf 88.8%

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

      1. sub-neg 88.8%

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

      2. mul-1-neg 88.8%

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

      3. mul-1-neg 88.8%

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

      4. remove-double-neg 88.8%

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

      5. mul-1-neg 88.8%

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

      6. distribute-rgt-neg-in 88.8%

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

      7. sub-neg 88.8%

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

      8. mul-1-neg 88.8%

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

      9. remove-double-neg 88.8%

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

      10. distribute-neg-in 88.8%

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

      11. metadata-eval 88.8%

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

      12. unsub-neg 88.8%

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

   9. Simplified 88.8%

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

   if -3.09999999999999983e-260 < x < 9.00000000000000039e24

   1. Initial program 53.1%

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

   3. Simplified 43.1%

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

   5. Taylor expanded in eps around inf 98.9%

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

   6. Taylor expanded in x around 0 83.8%

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

   7. Taylor expanded in eps around inf 84.6%

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

      1. *-commutative 84.3%

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

   9. Simplified 84.6%

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

   if 9.00000000000000039e24 < x < 8.9999999999999999e139 or 1.3500000000000001e243 < 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%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around 0 75.4%

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

      1. neg-mul-1 75.4%

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

      2. rec-exp 75.4%

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

      3. count-2 75.4%

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

      4. rec-exp 75.4%

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

   8. Simplified 75.4%

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

   if 8.9999999999999999e139 < x < 1.3500000000000001e243

   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%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 40.0%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-260}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)} + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+24}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + 1}{2}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+139} \lor \neg \left(x \leq 1.35 \cdot 10^{+243}\right):\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2 \cdot e^{-x}}{2}\\ \mathbf{if}\;x \leq -700:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-211}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(-1 + \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) + \frac{-1 + \frac{1}{\varepsilon}}{\frac{1 - \varepsilon}{1 - {\varepsilon}^{2}}}\right)}{2}\\ \mathbf{elif}\;x \leq 10^{+22}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + 1}{2}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+139} \lor \neg \left(x \leq 8.5 \cdot 10^{+243}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (* 2.0 (exp (- x))) 2.0)))
   (if (<= x -700.0)
     t_0
     (if (<= x -1.3e-211)
       (/
        (+
         2.0
         (*
          x
          (+
           (* (+ -1.0 eps) (+ 1.0 (/ 1.0 eps)))
           (/ (+ -1.0 (/ 1.0 eps)) (/ (- 1.0 eps) (- 1.0 (pow eps 2.0)))))))
        2.0)
       (if (<= x 1e+22)
         (/ (+ (exp (* x eps)) 1.0) 2.0)
         (if (or (<= x 3.4e+139) (not (<= x 8.5e+243)))
           t_0
           (/ (+ 1.0 (exp (* x (+ -1.0 eps)))) 2.0)))))))

C

double code(double x, double eps) {
	double t_0 = (2.0 * exp(-x)) / 2.0;
	double tmp;
	if (x <= -700.0) {
		tmp = t_0;
	} else if (x <= -1.3e-211) {
		tmp = (2.0 + (x * (((-1.0 + eps) * (1.0 + (1.0 / eps))) + ((-1.0 + (1.0 / eps)) / ((1.0 - eps) / (1.0 - pow(eps, 2.0))))))) / 2.0;
	} else if (x <= 1e+22) {
		tmp = (exp((x * eps)) + 1.0) / 2.0;
	} else if ((x <= 3.4e+139) || !(x <= 8.5e+243)) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (2.0d0 * exp(-x)) / 2.0d0
    if (x <= (-700.0d0)) then
        tmp = t_0
    else if (x <= (-1.3d-211)) then
        tmp = (2.0d0 + (x * ((((-1.0d0) + eps) * (1.0d0 + (1.0d0 / eps))) + (((-1.0d0) + (1.0d0 / eps)) / ((1.0d0 - eps) / (1.0d0 - (eps ** 2.0d0))))))) / 2.0d0
    else if (x <= 1d+22) then
        tmp = (exp((x * eps)) + 1.0d0) / 2.0d0
    else if ((x <= 3.4d+139) .or. (.not. (x <= 8.5d+243))) then
        tmp = t_0
    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 t_0 = (2.0 * Math.exp(-x)) / 2.0;
	double tmp;
	if (x <= -700.0) {
		tmp = t_0;
	} else if (x <= -1.3e-211) {
		tmp = (2.0 + (x * (((-1.0 + eps) * (1.0 + (1.0 / eps))) + ((-1.0 + (1.0 / eps)) / ((1.0 - eps) / (1.0 - Math.pow(eps, 2.0))))))) / 2.0;
	} else if (x <= 1e+22) {
		tmp = (Math.exp((x * eps)) + 1.0) / 2.0;
	} else if ((x <= 3.4e+139) || !(x <= 8.5e+243)) {
		tmp = t_0;
	} else {
		tmp = (1.0 + Math.exp((x * (-1.0 + eps)))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (2.0 * math.exp(-x)) / 2.0
	tmp = 0
	if x <= -700.0:
		tmp = t_0
	elif x <= -1.3e-211:
		tmp = (2.0 + (x * (((-1.0 + eps) * (1.0 + (1.0 / eps))) + ((-1.0 + (1.0 / eps)) / ((1.0 - eps) / (1.0 - math.pow(eps, 2.0))))))) / 2.0
	elif x <= 1e+22:
		tmp = (math.exp((x * eps)) + 1.0) / 2.0
	elif (x <= 3.4e+139) or not (x <= 8.5e+243):
		tmp = t_0
	else:
		tmp = (1.0 + math.exp((x * (-1.0 + eps)))) / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0)
	tmp = 0.0
	if (x <= -700.0)
		tmp = t_0;
	elseif (x <= -1.3e-211)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(-1.0 + eps) * Float64(1.0 + Float64(1.0 / eps))) + Float64(Float64(-1.0 + Float64(1.0 / eps)) / Float64(Float64(1.0 - eps) / Float64(1.0 - (eps ^ 2.0))))))) / 2.0);
	elseif (x <= 1e+22)
		tmp = Float64(Float64(exp(Float64(x * eps)) + 1.0) / 2.0);
	elseif ((x <= 3.4e+139) || !(x <= 8.5e+243))
		tmp = t_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)
	t_0 = (2.0 * exp(-x)) / 2.0;
	tmp = 0.0;
	if (x <= -700.0)
		tmp = t_0;
	elseif (x <= -1.3e-211)
		tmp = (2.0 + (x * (((-1.0 + eps) * (1.0 + (1.0 / eps))) + ((-1.0 + (1.0 / eps)) / ((1.0 - eps) / (1.0 - (eps ^ 2.0))))))) / 2.0;
	elseif (x <= 1e+22)
		tmp = (exp((x * eps)) + 1.0) / 2.0;
	elseif ((x <= 3.4e+139) || ~((x <= 8.5e+243)))
		tmp = t_0;
	else
		tmp = (1.0 + exp((x * (-1.0 + eps)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -700.0], t$95$0, If[LessEqual[x, -1.3e-211], N[(N[(2.0 + N[(x * N[(N[(N[(-1.0 + eps), $MachinePrecision] * N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - eps), $MachinePrecision] / N[(1.0 - N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1e+22], N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 3.4e+139], N[Not[LessEqual[x, 8.5e+243]], $MachinePrecision]], t$95$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 < -700 or 1e22 < x < 3.4000000000000002e139 or 8.50000000000000026e243 < 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%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around 0 85.6%

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

      1. neg-mul-1 85.6%

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

      2. rec-exp 85.6%

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

      3. count-2 85.6%

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

      4. rec-exp 85.6%

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

   8. Simplified 85.6%

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

   if -700 < x < -1.3e-211

   1. Initial program 57.3%

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

   3. Simplified 41.7%

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

   5. Taylor expanded in x around 0 61.9%

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

      1. *-commutative 61.9%

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

      2. flip-+ 75.1%

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

      3. associate-*r/ 75.1%

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

      4. sub-neg 75.1%

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

      5. distribute-neg-frac 75.1%

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

      6. metadata-eval 75.1%

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

      7. metadata-eval 75.1%

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

      8. pow2 75.1%

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

   7. Applied egg-rr 75.1%

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

      1. associate-/l* 75.1%

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

   9. Simplified 75.1%

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

   if -1.3e-211 < x < 1e22

   1. Initial program 54.0%

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

   3. Simplified 44.3%

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

   5. Taylor expanded in eps around inf 99.0%

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

   6. Taylor expanded in x around 0 84.5%

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

   7. Taylor expanded in eps around inf 85.2%

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

      1. *-commutative 85.2%

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

   9. Simplified 85.2%

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

   if 3.4000000000000002e139 < x < 8.50000000000000026e243

   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%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 40.0%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -700:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-211}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(-1 + \varepsilon\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) + \frac{-1 + \frac{1}{\varepsilon}}{\frac{1 - \varepsilon}{1 - {\varepsilon}^{2}}}\right)}{2}\\ \mathbf{elif}\;x \leq 10^{+22}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + 1}{2}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+139} \lor \neg \left(x \leq 8.5 \cdot 10^{+243}\right):\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2 \cdot e^{-x}}{2}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-259}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+20}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + 1}{2}\\ \mathbf{elif}\;x \leq 1.42 \cdot 10^{+141} \lor \neg \left(x \leq 10^{+244}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (* 2.0 (exp (- x))) 2.0)))
   (if (<= x -2e-259)
     t_0
     (if (<= x 1.15e+20)
       (/ (+ (exp (* x eps)) 1.0) 2.0)
       (if (or (<= x 1.42e+141) (not (<= x 1e+244)))
         t_0
         (/ (+ 1.0 (exp (* x (+ -1.0 eps)))) 2.0))))))

C

double code(double x, double eps) {
	double t_0 = (2.0 * exp(-x)) / 2.0;
	double tmp;
	if (x <= -2e-259) {
		tmp = t_0;
	} else if (x <= 1.15e+20) {
		tmp = (exp((x * eps)) + 1.0) / 2.0;
	} else if ((x <= 1.42e+141) || !(x <= 1e+244)) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (2.0d0 * exp(-x)) / 2.0d0
    if (x <= (-2d-259)) then
        tmp = t_0
    else if (x <= 1.15d+20) then
        tmp = (exp((x * eps)) + 1.0d0) / 2.0d0
    else if ((x <= 1.42d+141) .or. (.not. (x <= 1d+244))) then
        tmp = t_0
    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 t_0 = (2.0 * Math.exp(-x)) / 2.0;
	double tmp;
	if (x <= -2e-259) {
		tmp = t_0;
	} else if (x <= 1.15e+20) {
		tmp = (Math.exp((x * eps)) + 1.0) / 2.0;
	} else if ((x <= 1.42e+141) || !(x <= 1e+244)) {
		tmp = t_0;
	} else {
		tmp = (1.0 + Math.exp((x * (-1.0 + eps)))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (2.0 * math.exp(-x)) / 2.0
	tmp = 0
	if x <= -2e-259:
		tmp = t_0
	elif x <= 1.15e+20:
		tmp = (math.exp((x * eps)) + 1.0) / 2.0
	elif (x <= 1.42e+141) or not (x <= 1e+244):
		tmp = t_0
	else:
		tmp = (1.0 + math.exp((x * (-1.0 + eps)))) / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0)
	tmp = 0.0
	if (x <= -2e-259)
		tmp = t_0;
	elseif (x <= 1.15e+20)
		tmp = Float64(Float64(exp(Float64(x * eps)) + 1.0) / 2.0);
	elseif ((x <= 1.42e+141) || !(x <= 1e+244))
		tmp = t_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)
	t_0 = (2.0 * exp(-x)) / 2.0;
	tmp = 0.0;
	if (x <= -2e-259)
		tmp = t_0;
	elseif (x <= 1.15e+20)
		tmp = (exp((x * eps)) + 1.0) / 2.0;
	elseif ((x <= 1.42e+141) || ~((x <= 1e+244)))
		tmp = t_0;
	else
		tmp = (1.0 + exp((x * (-1.0 + eps)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -2e-259], t$95$0, If[LessEqual[x, 1.15e+20], N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 1.42e+141], N[Not[LessEqual[x, 1e+244]], $MachinePrecision]], t$95$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 3 regimes

2. if x < -2.0000000000000001e-259 or 1.15e20 < x < 1.42000000000000005e141 or 1.00000000000000007e244 < x

   1. Initial program 80.6%

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

   3. Simplified 73.8%

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

   5. Taylor expanded in eps around inf 99.1%

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

   6. Taylor expanded in eps around 0 75.8%

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

      1. neg-mul-1 75.8%

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

      2. rec-exp 75.8%

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

      3. count-2 75.8%

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

      4. rec-exp 75.8%

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

   8. Simplified 75.8%

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

   if -2.0000000000000001e-259 < x < 1.15e20

   1. Initial program 53.1%

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

   3. Simplified 43.1%

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

   5. Taylor expanded in eps around inf 98.9%

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

   6. Taylor expanded in x around 0 83.8%

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

   7. Taylor expanded in eps around inf 84.6%

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

      1. *-commutative 84.3%

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

   9. Simplified 84.6%

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

   if 1.42000000000000005e141 < x < 1.00000000000000007e244

   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%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 40.0%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-259}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+20}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + 1}{2}\\ \mathbf{elif}\;x \leq 1.42 \cdot 10^{+141} \lor \neg \left(x \leq 10^{+244}\right):\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 + \varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-244} \lor \neg \left(x \leq 2.3 \cdot 10^{+17}\right) \land \left(x \leq 4.4 \cdot 10^{+144} \lor \neg \left(x \leq 1.08 \cdot 10^{+242}\right)\right):\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + 1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -5e-244)
         (and (not (<= x 2.3e+17))
              (or (<= x 4.4e+144) (not (<= x 1.08e+242)))))
   (/ (* 2.0 (exp (- x))) 2.0)
   (/ (+ (exp (* x eps)) 1.0) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -5e-244) || (!(x <= 2.3e+17) && ((x <= 4.4e+144) || !(x <= 1.08e+242)))) {
		tmp = (2.0 * exp(-x)) / 2.0;
	} else {
		tmp = (exp((x * eps)) + 1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-5d-244)) .or. (.not. (x <= 2.3d+17)) .and. (x <= 4.4d+144) .or. (.not. (x <= 1.08d+242))) then
        tmp = (2.0d0 * exp(-x)) / 2.0d0
    else
        tmp = (exp((x * eps)) + 1.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -5e-244) || (!(x <= 2.3e+17) && ((x <= 4.4e+144) || !(x <= 1.08e+242)))) {
		tmp = (2.0 * Math.exp(-x)) / 2.0;
	} else {
		tmp = (Math.exp((x * eps)) + 1.0) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -5e-244) or (not (x <= 2.3e+17) and ((x <= 4.4e+144) or not (x <= 1.08e+242))):
		tmp = (2.0 * math.exp(-x)) / 2.0
	else:
		tmp = (math.exp((x * eps)) + 1.0) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -5e-244) || (!(x <= 2.3e+17) && ((x <= 4.4e+144) || !(x <= 1.08e+242))))
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * eps)) + 1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -5e-244) || (~((x <= 2.3e+17)) && ((x <= 4.4e+144) || ~((x <= 1.08e+242)))))
		tmp = (2.0 * exp(-x)) / 2.0;
	else
		tmp = (exp((x * eps)) + 1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -5e-244], And[N[Not[LessEqual[x, 2.3e+17]], $MachinePrecision], Or[LessEqual[x, 4.4e+144], N[Not[LessEqual[x, 1.08e+242]], $MachinePrecision]]]], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.99999999999999998e-244 or 2.3e17 < x < 4.39999999999999976e144 or 1.07999999999999989e242 < x

   1. Initial program 80.6%

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

   3. Simplified 73.8%

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

   5. Taylor expanded in eps around inf 99.1%

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

   6. Taylor expanded in eps around 0 75.8%

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

      1. neg-mul-1 75.8%

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

      2. rec-exp 75.8%

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

      3. count-2 75.8%

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

      4. rec-exp 75.8%

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

   8. Simplified 75.8%

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

   if -4.99999999999999998e-244 < x < 2.3e17 or 4.39999999999999976e144 < x < 1.07999999999999989e242

   1. Initial program 61.6%

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

   3. Simplified 53.4%

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

   5. Taylor expanded in eps around inf 99.1%

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

   6. Taylor expanded in x around 0 75.9%

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

   7. Taylor expanded in eps around inf 76.5%

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

      1. *-commutative 76.1%

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

   9. Simplified 76.5%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-244} \lor \neg \left(x \leq 2.3 \cdot 10^{+17}\right) \land \left(x \leq 4.4 \cdot 10^{+144} \lor \neg \left(x \leq 1.08 \cdot 10^{+242}\right)\right):\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + 1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{2 \cdot e^{-x}}{2} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.0%

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

3. Simplified 64.6%

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

5. Taylor expanded in eps around inf 99.1%

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

6. Taylor expanded in eps around 0 71.7%

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

   1. neg-mul-1 71.7%

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

   2. rec-exp 71.7%

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

   3. count-2 71.7%

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

   4. rec-exp 71.7%

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

8. Simplified 71.7%

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

9. Final simplification 71.7%

   \[\leadsto \frac{2 \cdot e^{-x}}{2} \]
10. Add Preprocessing

Alternative 7: 60.5% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.16 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{-1}{\varepsilon} + \left(\varepsilon + 1\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 1.16e-5)
   (/ (+ 2.0 (* x (+ (/ -1.0 eps) (* (+ eps 1.0) (+ -1.0 (/ 1.0 eps)))))) 2.0)
   0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 1.16e-5) {
		tmp = (2.0 + (x * ((-1.0 / eps) + ((eps + 1.0) * (-1.0 + (1.0 / eps)))))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 1.16d-5) then
        tmp = (2.0d0 + (x * (((-1.0d0) / eps) + ((eps + 1.0d0) * ((-1.0d0) + (1.0d0 / eps)))))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 1.16e-5) {
		tmp = (2.0 + (x * ((-1.0 / eps) + ((eps + 1.0) * (-1.0 + (1.0 / eps)))))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 1.16e-5:
		tmp = (2.0 + (x * ((-1.0 / eps) + ((eps + 1.0) * (-1.0 + (1.0 / eps)))))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 1.16e-5)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(-1.0 / eps) + Float64(Float64(eps + 1.0) * Float64(-1.0 + Float64(1.0 / eps)))))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 1.16e-5)
		tmp = (2.0 + (x * ((-1.0 / eps) + ((eps + 1.0) * (-1.0 + (1.0 / eps)))))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 1.16e-5], N[(N[(2.0 + N[(x * N[(N[(-1.0 / eps), $MachinePrecision] + N[(N[(eps + 1.0), $MachinePrecision] * N[(-1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.1600000000000001e-5

   1. Initial program 61.7%

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

   3. Simplified 51.5%

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

   5. Taylor expanded in x around 0 61.1%

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

   6. Taylor expanded in eps around 0 63.1%

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

   if 1.1600000000000001e-5 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around -inf 100.0%

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

   8. Simplified 100.0%

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

   10. Applied egg-rr 0.3%

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

       1. +-inverses 0.3%

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

       2. mul0-rgt 58.6%

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

   12. Simplified 58.6%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.16 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{-1}{\varepsilon} + \left(\varepsilon + 1\right) \cdot \left(-1 + \frac{1}{\varepsilon}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.2% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.16 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 - x \cdot \left(\varepsilon + 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 1.16e-5) (/ (- 2.0 (* x (+ eps 2.0))) 2.0) 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 1.16e-5) {
		tmp = (2.0 - (x * (eps + 2.0))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 1.16d-5) then
        tmp = (2.0d0 - (x * (eps + 2.0d0))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 1.16e-5) {
		tmp = (2.0 - (x * (eps + 2.0))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 1.16e-5:
		tmp = (2.0 - (x * (eps + 2.0))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 1.16e-5)
		tmp = Float64(Float64(2.0 - Float64(x * Float64(eps + 2.0))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 1.16e-5)
		tmp = (2.0 - (x * (eps + 2.0))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 1.16e-5], N[(N[(2.0 - N[(x * N[(eps + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.1600000000000001e-5

   1. Initial program 61.7%

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

   3. Simplified 61.7%

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

   5. Taylor expanded in eps around inf 98.8%

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

   6. Taylor expanded in eps around 0 88.6%

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

   7. Taylor expanded in x around 0 62.4%

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

      1. mul-1-neg 62.4%

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

      2. unsub-neg 62.4%

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

   9. Simplified 62.4%

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

   if 1.1600000000000001e-5 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around -inf 100.0%

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

   8. Simplified 100.0%

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

   10. Applied egg-rr 0.3%

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

       1. +-inverses 0.3%

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

       2. mul0-rgt 58.6%

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

   12. Simplified 58.6%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.16 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 - x \cdot \left(\varepsilon + 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.4% accurate, 25.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.39:\\ \;\;\;\;\frac{x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+15}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -0.39) (/ (* x (- -1.0 eps)) 2.0) (if (<= x 2.6e+15) 1.0 0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -0.39) {
		tmp = (x * (-1.0 - eps)) / 2.0;
	} else if (x <= 2.6e+15) {
		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 <= (-0.39d0)) then
        tmp = (x * ((-1.0d0) - eps)) / 2.0d0
    else if (x <= 2.6d+15) 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 <= -0.39) {
		tmp = (x * (-1.0 - eps)) / 2.0;
	} else if (x <= 2.6e+15) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -0.39:
		tmp = (x * (-1.0 - eps)) / 2.0
	elif x <= 2.6e+15:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -0.39)
		tmp = Float64(Float64(x * Float64(-1.0 - eps)) / 2.0);
	elseif (x <= 2.6e+15)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -0.39)
		tmp = (x * (-1.0 - eps)) / 2.0;
	elseif (x <= 2.6e+15)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -0.39], N[(N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.6e+15], 1.0, 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -0.39000000000000001

   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%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 49.8%

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

   7. Taylor expanded in x around inf 19.5%

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

      1. mul-1-neg 19.5%

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

      2. distribute-rgt-neg-in 19.5%

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

      3. distribute-neg-in 19.5%

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

      4. metadata-eval 19.5%

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

      5. unsub-neg 19.5%

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

   9. Simplified 19.5%

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

   if -0.39000000000000001 < x < 2.6e15

   1. Initial program 54.6%

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

   3. Simplified 54.6%

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

   5. Taylor expanded in x around 0 71.7%

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

   if 2.6e15 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around -inf 100.0%

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

   8. Simplified 100.0%

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

   10. Applied egg-rr 0.3%

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

       1. +-inverses 0.3%

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

       2. mul0-rgt 62.1%

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

   12. Simplified 62.1%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.39:\\ \;\;\;\;\frac{x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+15}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.4% accurate, 28.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.39:\\ \;\;\;\;\frac{x \cdot \left(-\varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+15}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -0.39) (/ (* x (- eps)) 2.0) (if (<= x 2.6e+15) 1.0 0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -0.39) {
		tmp = (x * -eps) / 2.0;
	} else if (x <= 2.6e+15) {
		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 <= (-0.39d0)) then
        tmp = (x * -eps) / 2.0d0
    else if (x <= 2.6d+15) 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 <= -0.39) {
		tmp = (x * -eps) / 2.0;
	} else if (x <= 2.6e+15) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -0.39:
		tmp = (x * -eps) / 2.0
	elif x <= 2.6e+15:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -0.39)
		tmp = Float64(Float64(x * Float64(-eps)) / 2.0);
	elseif (x <= 2.6e+15)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -0.39)
		tmp = (x * -eps) / 2.0;
	elseif (x <= 2.6e+15)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -0.39], N[(N[(x * (-eps)), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.6e+15], 1.0, 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -0.39000000000000001

   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%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 49.8%

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

   7. Taylor expanded in eps around inf 49.8%

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

      1. *-commutative 49.8%

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

   9. Simplified 49.8%

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

   10. Taylor expanded in eps around inf 19.5%

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

       1. associate-*r* 19.5%

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

       2. mul-1-neg 19.5%

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

   12. Simplified 19.5%

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

   if -0.39000000000000001 < x < 2.6e15

   1. Initial program 54.6%

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

   3. Simplified 54.6%

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

   5. Taylor expanded in x around 0 71.7%

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

   if 2.6e15 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around -inf 100.0%

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

   8. Simplified 100.0%

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

   10. Applied egg-rr 0.3%

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

       1. +-inverses 0.3%

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

       2. mul0-rgt 62.1%

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

   12. Simplified 62.1%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.39:\\ \;\;\;\;\frac{x \cdot \left(-\varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+15}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.1% accurate, 74.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{+15}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (if (<= x 2.6e+15) 1.0 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 2.6e+15) {
		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 <= 2.6d+15) 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 <= 2.6e+15) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 2.6e+15:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 2.6e+15)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

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

Wolfram

code[x_, eps_] := If[LessEqual[x, 2.6e+15], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.6e15

   1. Initial program 62.5%

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

   3. Simplified 62.5%

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

   5. Taylor expanded in x around 0 59.9%

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

   if 2.6e15 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around -inf 100.0%

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

   8. Simplified 100.0%

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

   10. Applied egg-rr 0.3%

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

       1. +-inverses 0.3%

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

       2. mul0-rgt 62.1%

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

   12. Simplified 62.1%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{+15}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 16.2% accurate, 227.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 72.0%

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

3. Simplified 72.0%

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

5. Taylor expanded in eps around inf 99.1%

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

6. Taylor expanded in eps around -inf 99.1%

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

8. Simplified 99.1%

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

10. Applied egg-rr 1.8%

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

    1. +-inverses 1.8%

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

    2. mul0-rgt 17.6%

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

12. Simplified 17.6%

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

13. Final simplification 17.6%

    \[\leadsto 0 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(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))