NMSE Section 6.1 mentioned, A

Percentage Accurate: 72.7% → 98.9%

Time: 11.5s

Alternatives: 9

Speedup: 2.1×

• 37.9% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.7% 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 76.9%

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

3. Simplified 64.7%

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

5. Taylor expanded in eps around inf 99.1%

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

6. Final simplification 99.1%

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

Alternative 2: 88.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.9%

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

3. Simplified 64.7%

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

5. Taylor expanded in eps around inf 99.1%

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

6. Taylor expanded in eps around inf 88.9%

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

   1. mul-1-neg 88.9%

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

   2. *-commutative 88.9%

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

   3. distribute-rgt-neg-in 88.9%

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

8. Simplified 88.9%

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

9. Final simplification 88.9%

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

Alternative 3: 85.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{2 \cdot \cosh \left(x \cdot \varepsilon\right)}{2} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (/ (* 2.0 (cosh (* x eps))) 2.0))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.9%

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

3. Simplified 64.7%

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

5. Taylor expanded in eps around inf 99.1%

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

6. Taylor expanded in eps around inf 88.9%

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

   1. mul-1-neg 88.9%

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

   2. *-commutative 88.9%

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

   3. distribute-rgt-neg-in 88.9%

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

8. Simplified 88.9%

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

9. Taylor expanded in eps around inf 85.5%

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

    1. *-commutative 85.5%

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

11. Simplified 85.5%

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

    1. +-commutative 85.5%

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

    2. distribute-rgt-neg-out 85.5%

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

    3. cosh-undef 85.5%

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

13. Applied egg-rr 85.5%

    \[\leadsto \frac{\color{blue}{2 \cdot \cosh \left(x \cdot \varepsilon\right)}}{2} \]
14. Add Preprocessing

Alternative 4: 58.5% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(2 - x \cdot \varepsilon\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 550:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 10^{+164}:\\ \;\;\;\;\frac{\frac{\left(1 - x\right) + \left(-1 + x\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-1 + x\right) + \left(1 + x \cdot \left(-1 + x \cdot 0.5\right)\right)}{\varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -6e-37)
   (/ (/ (- (* eps (- 2.0 (* x eps))) x) eps) 2.0)
   (if (<= x 550.0)
     1.0
     (if (<= x 1e+164)
       (/ (/ (+ (- 1.0 x) (+ -1.0 x)) eps) 2.0)
       (/ (/ (+ (+ -1.0 x) (+ 1.0 (* x (+ -1.0 (* x 0.5))))) eps) 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -6e-37) {
		tmp = (((eps * (2.0 - (x * eps))) - x) / eps) / 2.0;
	} else if (x <= 550.0) {
		tmp = 1.0;
	} else if (x <= 1e+164) {
		tmp = (((1.0 - x) + (-1.0 + x)) / eps) / 2.0;
	} else {
		tmp = (((-1.0 + x) + (1.0 + (x * (-1.0 + (x * 0.5))))) / eps) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-6d-37)) then
        tmp = (((eps * (2.0d0 - (x * eps))) - x) / eps) / 2.0d0
    else if (x <= 550.0d0) then
        tmp = 1.0d0
    else if (x <= 1d+164) then
        tmp = (((1.0d0 - x) + ((-1.0d0) + x)) / eps) / 2.0d0
    else
        tmp = ((((-1.0d0) + x) + (1.0d0 + (x * ((-1.0d0) + (x * 0.5d0))))) / eps) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -6e-37) {
		tmp = (((eps * (2.0 - (x * eps))) - x) / eps) / 2.0;
	} else if (x <= 550.0) {
		tmp = 1.0;
	} else if (x <= 1e+164) {
		tmp = (((1.0 - x) + (-1.0 + x)) / eps) / 2.0;
	} else {
		tmp = (((-1.0 + x) + (1.0 + (x * (-1.0 + (x * 0.5))))) / eps) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -6e-37:
		tmp = (((eps * (2.0 - (x * eps))) - x) / eps) / 2.0
	elif x <= 550.0:
		tmp = 1.0
	elif x <= 1e+164:
		tmp = (((1.0 - x) + (-1.0 + x)) / eps) / 2.0
	else:
		tmp = (((-1.0 + x) + (1.0 + (x * (-1.0 + (x * 0.5))))) / eps) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -6e-37)
		tmp = Float64(Float64(Float64(Float64(eps * Float64(2.0 - Float64(x * eps))) - x) / eps) / 2.0);
	elseif (x <= 550.0)
		tmp = 1.0;
	elseif (x <= 1e+164)
		tmp = Float64(Float64(Float64(Float64(1.0 - x) + Float64(-1.0 + x)) / eps) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(-1.0 + x) + Float64(1.0 + Float64(x * Float64(-1.0 + Float64(x * 0.5))))) / eps) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -6e-37)
		tmp = (((eps * (2.0 - (x * eps))) - x) / eps) / 2.0;
	elseif (x <= 550.0)
		tmp = 1.0;
	elseif (x <= 1e+164)
		tmp = (((1.0 - x) + (-1.0 + x)) / eps) / 2.0;
	else
		tmp = (((-1.0 + x) + (1.0 + (x * (-1.0 + (x * 0.5))))) / eps) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -6e-37], N[(N[(N[(N[(eps * N[(2.0 - N[(x * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 550.0], 1.0, If[LessEqual[x, 1e+164], N[(N[(N[(N[(1.0 - x), $MachinePrecision] + N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(-1.0 + x), $MachinePrecision] + N[(1.0 + N[(x * N[(-1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6e-37

   1. Initial program 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in x around 0 5.3%

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

   6. Taylor expanded in eps around 0 28.0%

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

   7. Taylor expanded in eps around inf 28.1%

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

      1. mul-1-neg 28.1%

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

   9. Simplified 28.1%

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

   10. Taylor expanded in eps around 0 45.7%

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

   if -6e-37 < x < 550

   1. Initial program 58.9%

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

   3. Simplified 36.4%

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

   5. Taylor expanded in x around 0 72.6%

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

      1. add-sqr-sqrt 32.6%

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

      2. sqrt-unprod 45.9%

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

      3. frac-times 45.2%

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

      4. metadata-eval 45.2%

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

      5. metadata-eval 45.2%

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

      6. frac-times 45.9%

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

      7. sqrt-unprod 24.7%

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

      8. add-sqr-sqrt 49.2%

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

      9. div-inv 49.2%

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

      10. mul-1-neg 49.2%

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

   7. Applied egg-rr 49.2%

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

   8. Taylor expanded in eps around 0 49.2%

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

   9. Taylor expanded in x around 0 72.6%

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

   if 550 < x < 1e164

   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 55.3%

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

   6. Taylor expanded in x around 0 1.7%

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

   7. Taylor expanded in x around 0 55.3%

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

      1. neg-mul-1 55.3%

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

      2. unsub-neg 55.3%

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

   9. Simplified 55.3%

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

   if 1e164 < 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 43.8%

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

   6. Taylor expanded in x around 0 2.1%

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

   7. Taylor expanded in x around 0 29.2%

      \[\leadsto \frac{\frac{\color{blue}{\left(1 + x \cdot \left(0.5 \cdot x - 1\right)\right)} + \left(x - 1\right)}{\varepsilon}}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(2 - x \cdot \varepsilon\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 550:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 10^{+164}:\\ \;\;\;\;\frac{\frac{\left(1 - x\right) + \left(-1 + x\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(-1 + x\right) + \left(1 + x \cdot \left(-1 + x \cdot 0.5\right)\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.0% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-34}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(2 - x \cdot \varepsilon\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 520:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 - x\right) + \left(-1 + x\right)}{\varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -5.5e-34)
   (/ (/ (- (* eps (- 2.0 (* x eps))) x) eps) 2.0)
   (if (<= x 520.0) 1.0 (/ (/ (+ (- 1.0 x) (+ -1.0 x)) eps) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -5.5e-34) {
		tmp = (((eps * (2.0 - (x * eps))) - x) / eps) / 2.0;
	} else if (x <= 520.0) {
		tmp = 1.0;
	} else {
		tmp = (((1.0 - x) + (-1.0 + 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 <= (-5.5d-34)) then
        tmp = (((eps * (2.0d0 - (x * eps))) - x) / eps) / 2.0d0
    else if (x <= 520.0d0) then
        tmp = 1.0d0
    else
        tmp = (((1.0d0 - x) + ((-1.0d0) + x)) / eps) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -5.5e-34) {
		tmp = (((eps * (2.0 - (x * eps))) - x) / eps) / 2.0;
	} else if (x <= 520.0) {
		tmp = 1.0;
	} else {
		tmp = (((1.0 - x) + (-1.0 + x)) / eps) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -5.5e-34:
		tmp = (((eps * (2.0 - (x * eps))) - x) / eps) / 2.0
	elif x <= 520.0:
		tmp = 1.0
	else:
		tmp = (((1.0 - x) + (-1.0 + x)) / eps) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -5.5e-34)
		tmp = Float64(Float64(Float64(Float64(eps * Float64(2.0 - Float64(x * eps))) - x) / eps) / 2.0);
	elseif (x <= 520.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - x) + Float64(-1.0 + x)) / eps) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -5.5e-34)
		tmp = (((eps * (2.0 - (x * eps))) - x) / eps) / 2.0;
	elseif (x <= 520.0)
		tmp = 1.0;
	else
		tmp = (((1.0 - x) + (-1.0 + x)) / eps) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -5.5e-34], N[(N[(N[(N[(eps * N[(2.0 - N[(x * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 520.0], 1.0, N[(N[(N[(N[(1.0 - x), $MachinePrecision] + N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.50000000000000014e-34

   1. Initial program 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in x around 0 5.3%

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

   6. Taylor expanded in eps around 0 28.0%

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

   7. Taylor expanded in eps around inf 28.1%

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

      1. mul-1-neg 28.1%

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

   9. Simplified 28.1%

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

   10. Taylor expanded in eps around 0 45.7%

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

   if -5.50000000000000014e-34 < x < 520

   1. Initial program 58.9%

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

   3. Simplified 36.4%

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

   5. Taylor expanded in x around 0 72.6%

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

      1. add-sqr-sqrt 32.6%

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

      2. sqrt-unprod 45.9%

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

      3. frac-times 45.2%

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

      4. metadata-eval 45.2%

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

      5. metadata-eval 45.2%

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

      6. frac-times 45.9%

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

      7. sqrt-unprod 24.7%

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

      8. add-sqr-sqrt 49.2%

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

      9. div-inv 49.2%

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

      10. mul-1-neg 49.2%

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

   7. Applied egg-rr 49.2%

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

   8. Taylor expanded in eps around 0 49.2%

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

   9. Taylor expanded in x around 0 72.6%

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

   if 520 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 48.8%

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

   6. Taylor expanded in x around 0 1.9%

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

   7. Taylor expanded in x around 0 48.8%

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

      1. neg-mul-1 48.8%

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

      2. unsub-neg 48.8%

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

   9. Simplified 48.8%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-34}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(2 - x \cdot \varepsilon\right) - x}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 520:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 - x\right) + \left(-1 + x\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.7% accurate, 14.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 - x\right) + \left(-1 + x\right)}{\varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 8.5e-10)
   (/ (- 2.0 (* x eps)) 2.0)
   (/ (/ (+ (- 1.0 x) (+ -1.0 x)) eps) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 8.5e-10) {
		tmp = (2.0 - (x * eps)) / 2.0;
	} else {
		tmp = (((1.0 - x) + (-1.0 + 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 <= 8.5d-10) then
        tmp = (2.0d0 - (x * eps)) / 2.0d0
    else
        tmp = (((1.0d0 - x) + ((-1.0d0) + x)) / eps) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 8.5e-10) {
		tmp = (2.0 - (x * eps)) / 2.0;
	} else {
		tmp = (((1.0 - x) + (-1.0 + x)) / eps) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 8.5e-10:
		tmp = (2.0 - (x * eps)) / 2.0
	else:
		tmp = (((1.0 - x) + (-1.0 + x)) / eps) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 8.5e-10)
		tmp = Float64(Float64(2.0 - Float64(x * eps)) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - x) + Float64(-1.0 + x)) / eps) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 8.5e-10)
		tmp = (2.0 - (x * eps)) / 2.0;
	else
		tmp = (((1.0 - x) + (-1.0 + x)) / eps) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 8.5e-10], N[(N[(2.0 - N[(x * eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(1.0 - x), $MachinePrecision] + N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.4999999999999996e-10

   1. Initial program 66.8%

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

   3. Simplified 49.2%

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

   5. Taylor expanded in x around 0 57.9%

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

   6. Taylor expanded in eps around 0 62.2%

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

   7. Taylor expanded in eps around inf 43.6%

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

      1. mul-1-neg 43.6%

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

   9. Simplified 43.6%

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

   10. Taylor expanded in eps around inf 62.2%

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

   if 8.4999999999999996e-10 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 47.0%

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

   6. Taylor expanded in x around 0 1.9%

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

   7. Taylor expanded in x around 0 47.0%

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

      1. neg-mul-1 47.0%

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

      2. unsub-neg 47.0%

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

   9. Simplified 47.0%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1 - x\right) + \left(-1 + x\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.2% accurate, 14.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(x \cdot \varepsilon + \frac{x}{\varepsilon}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 8.5e-10)
   (/ (- 2.0 (* x eps)) 2.0)
   (/ (+ 2.0 (+ (* x eps) (/ x eps))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 8.5e-10) {
		tmp = (2.0 - (x * eps)) / 2.0;
	} else {
		tmp = (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 <= 8.5d-10) then
        tmp = (2.0d0 - (x * eps)) / 2.0d0
    else
        tmp = (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 <= 8.5e-10) {
		tmp = (2.0 - (x * eps)) / 2.0;
	} else {
		tmp = (2.0 + ((x * eps) + (x / eps))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 8.5e-10:
		tmp = (2.0 - (x * eps)) / 2.0
	else:
		tmp = (2.0 + ((x * eps) + (x / eps))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 8.5e-10)
		tmp = Float64(Float64(2.0 - Float64(x * eps)) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(x * eps) + Float64(x / eps))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 8.5e-10)
		tmp = (2.0 - (x * eps)) / 2.0;
	else
		tmp = (2.0 + ((x * eps) + (x / eps))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 8.5e-10], N[(N[(2.0 - N[(x * eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(N[(x * eps), $MachinePrecision] + N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.4999999999999996e-10

   1. Initial program 66.8%

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

   3. Simplified 49.2%

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

   5. Taylor expanded in x around 0 57.9%

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

   6. Taylor expanded in eps around 0 62.2%

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

   7. Taylor expanded in eps around inf 43.6%

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

      1. mul-1-neg 43.6%

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

   9. Simplified 43.6%

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

   10. Taylor expanded in eps around inf 62.2%

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

   if 8.4999999999999996e-10 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 3.1%

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

   6. Taylor expanded in eps around 0 20.0%

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

   7. Taylor expanded in eps around inf 19.4%

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

      1. mul-1-neg 19.4%

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

   9. Simplified 19.4%

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

       1. distribute-lft-in 19.4%

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

       2. add-sqr-sqrt 19.0%

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

       3. sqrt-unprod 35.4%

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

       4. sqr-neg 35.4%

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

       5. sqrt-unprod 15.3%

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

       6. add-sqr-sqrt 15.7%

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

       7. frac-2neg 15.7%

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

       8. metadata-eval 15.7%

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

       9. add-sqr-sqrt 0.5%

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

       10. sqrt-unprod 15.7%

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

       11. sqr-neg 15.7%

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

       12. sqrt-unprod 15.3%

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

       13. add-sqr-sqrt 15.7%

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

       14. div-inv 15.7%

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

   11. Applied egg-rr 15.7%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \left(x \cdot \varepsilon + \frac{x}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.0% accurate, 32.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.9%

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

3. Simplified 64.7%

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

5. Taylor expanded in x around 0 41.2%

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

6. Taylor expanded in eps around 0 49.3%

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

7. Taylor expanded in eps around inf 36.3%

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

   1. mul-1-neg 36.3%

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

9. Simplified 36.3%

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

10. Taylor expanded in eps around inf 49.3%

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

11. Final simplification 49.3%

    \[\leadsto \frac{2 - x \cdot \varepsilon}{2} \]
12. Add Preprocessing

Alternative 9: 44.2% accurate, 227.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 1.0)

C

double code(double x, double eps) {
	return 1.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 1.0d0
end function

Java

public static double code(double x, double eps) {
	return 1.0;
}

Python

def code(x, eps):
	return 1.0

Julia

function code(x, eps)
	return 1.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 1.0;
end

Wolfram

code[x_, eps_] := 1.0

Derivation

1. Initial program 76.9%

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

3. Simplified 64.7%

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

5. Taylor expanded in x around 0 41.2%

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

   1. add-sqr-sqrt 18.3%

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

   2. sqrt-unprod 26.5%

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

   3. frac-times 26.2%

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

   4. metadata-eval 26.2%

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

   5. metadata-eval 26.2%

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

   6. frac-times 26.5%

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

   7. sqrt-unprod 14.4%

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

   8. add-sqr-sqrt 28.3%

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

   9. div-inv 28.3%

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

   10. mul-1-neg 28.3%

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

7. Applied egg-rr 28.3%

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

8. Taylor expanded in eps around 0 28.2%

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

9. Taylor expanded in x around 0 41.2%

   \[\leadsto \color{blue}{1} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024163 
(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))