NMSE Section 6.1 mentioned, A

Percentage Accurate: 74.0% → 99.1%

Time: 3.0s

Alternatives: 8

Speedup: 2.5×

• 38.0% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -2.1247062122418665e+61
  2. eps = 1.2563524472448189e-136

Specification

\[\mathsf{TRUE}\left(\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\left(1 + \varepsilon\right) \cdot x}\\ t_1 := \left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot t\_0\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (- (* (+ 1.0 eps) x))))
        (t_1
         (-
          (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x))))
          (* (- (/ 1.0 eps) 1.0) t_0))))
   (if (<= t_1 0.0) t_0 (/ t_1 2.0))))

C

double code(double x, double eps) {
	double t_0 = exp(-((1.0 + eps) * x));
	double t_1 = ((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * t_0);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else {
		tmp = t_1 / 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) :: t_1
    real(8) :: tmp
    t_0 = exp(-((1.0d0 + eps) * x))
    t_1 = ((1.0d0 + (1.0d0 / eps)) * exp(-((1.0d0 - eps) * x))) - (((1.0d0 / eps) - 1.0d0) * t_0)
    if (t_1 <= 0.0d0) then
        tmp = t_0
    else
        tmp = t_1 / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.exp(-((1.0 + eps) * x));
	double t_1 = ((1.0 + (1.0 / eps)) * Math.exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * t_0);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else {
		tmp = t_1 / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.exp(-((1.0 + eps) * x))
	t_1 = ((1.0 + (1.0 / eps)) * math.exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * t_0)
	tmp = 0
	if t_1 <= 0.0:
		tmp = t_0
	else:
		tmp = t_1 / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = exp(Float64(-Float64(Float64(1.0 + eps) * x)))
	t_1 = 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) * t_0))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_0;
	else
		tmp = Float64(t_1 / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = exp(-((1.0 + eps) * x));
	t_1 = ((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * t_0);
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = t_0;
	else
		tmp = t_1 / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Exp[(-N[(N[(1.0 + eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]}, Block[{t$95$1 = 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] * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], t$95$0, N[(t$95$1 / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 0.0

   1. Initial program 41.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 41.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 41.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 19.8%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 97.1%

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

   if 0.0 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) 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. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 48.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + \varepsilon\right) \cdot x\\ t_1 := 1 + \frac{1}{\varepsilon}\\ t_2 := t\_1 \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-t\_0}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;1 + \varepsilon\\ \mathbf{elif}\;t\_2 \leq 4:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (+ 1.0 eps) x))
        (t_1 (+ 1.0 (/ 1.0 eps)))
        (t_2
         (-
          (* t_1 (exp (- (* (- 1.0 eps) x))))
          (* (- (/ 1.0 eps) 1.0) (exp (- t_0))))))
   (if (<= t_2 0.0) (+ 1.0 eps) (if (<= t_2 4.0) t_1 t_0))))

C

double code(double x, double eps) {
	double t_0 = (1.0 + eps) * x;
	double t_1 = 1.0 + (1.0 / eps);
	double t_2 = (t_1 * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-t_0));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = 1.0 + eps;
	} else if (t_2 <= 4.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (1.0d0 + eps) * x
    t_1 = 1.0d0 + (1.0d0 / eps)
    t_2 = (t_1 * exp(-((1.0d0 - eps) * x))) - (((1.0d0 / eps) - 1.0d0) * exp(-t_0))
    if (t_2 <= 0.0d0) then
        tmp = 1.0d0 + eps
    else if (t_2 <= 4.0d0) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = (1.0 + eps) * x;
	double t_1 = 1.0 + (1.0 / eps);
	double t_2 = (t_1 * Math.exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * Math.exp(-t_0));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = 1.0 + eps;
	} else if (t_2 <= 4.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (1.0 + eps) * x
	t_1 = 1.0 + (1.0 / eps)
	t_2 = (t_1 * math.exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * math.exp(-t_0))
	tmp = 0
	if t_2 <= 0.0:
		tmp = 1.0 + eps
	elif t_2 <= 4.0:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(1.0 + eps) * x)
	t_1 = Float64(1.0 + Float64(1.0 / eps))
	t_2 = Float64(Float64(t_1 * exp(Float64(-Float64(Float64(1.0 - eps) * x)))) - Float64(Float64(Float64(1.0 / eps) - 1.0) * exp(Float64(-t_0))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(1.0 + eps);
	elseif (t_2 <= 4.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (1.0 + eps) * x;
	t_1 = 1.0 + (1.0 / eps);
	t_2 = (t_1 * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-t_0));
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = 1.0 + eps;
	elseif (t_2 <= 4.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(1.0 + eps), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * N[Exp[(-N[(N[(1.0 - eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[(-t$95$0)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(1.0 + eps), $MachinePrecision], If[LessEqual[t$95$2, 4.0], t$95$1, t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 0.0

   1. Initial program 41.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 41.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 59.7%

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

   if 0.0 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 4

   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. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + x \cdot \left(\frac{1}{2} \cdot \left(x \cdot \left(\frac{1}{6} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{3}\right) - \frac{-1}{6} \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) + \frac{1}{2} \cdot \left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right) - \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)\right)} \]

   5. Applied rewrites 83.8%

      \[\leadsto \color{blue}{1 + \frac{1}{\varepsilon}} \]

   if 4 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) 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. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto 2 + \color{blue}{x \cdot \left(\left(x \cdot \left(\left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right) + x \cdot \left(\frac{1}{6} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{3}\right) - \frac{-1}{6} \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) - \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)} \]

   8. Applied rewrites 17.5%

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

Alternative 3: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\left(1 + \varepsilon\right) \cdot x}\\ t_1 := \left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot t\_0\\ \mathbf{if}\;t\_1 \leq 4:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (- (* (+ 1.0 eps) x))))
        (t_1
         (-
          (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x))))
          (* (- (/ 1.0 eps) 1.0) t_0))))
   (if (<= t_1 4.0) t_0 t_1)))

C

double code(double x, double eps) {
	double t_0 = exp(-((1.0 + eps) * x));
	double t_1 = ((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * t_0);
	double tmp;
	if (t_1 <= 4.0) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = exp(-((1.0d0 + eps) * x))
    t_1 = ((1.0d0 + (1.0d0 / eps)) * exp(-((1.0d0 - eps) * x))) - (((1.0d0 / eps) - 1.0d0) * t_0)
    if (t_1 <= 4.0d0) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.exp(-((1.0 + eps) * x));
	double t_1 = ((1.0 + (1.0 / eps)) * Math.exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * t_0);
	double tmp;
	if (t_1 <= 4.0) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.exp(-((1.0 + eps) * x))
	t_1 = ((1.0 + (1.0 / eps)) * math.exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * t_0)
	tmp = 0
	if t_1 <= 4.0:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, eps)
	t_0 = exp(Float64(-Float64(Float64(1.0 + eps) * x)))
	t_1 = 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) * t_0))
	tmp = 0.0
	if (t_1 <= 4.0)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = exp(-((1.0 + eps) * x));
	t_1 = ((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * t_0);
	tmp = 0.0;
	if (t_1 <= 4.0)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Exp[(-N[(N[(1.0 + eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]}, Block[{t$95$1 = 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] * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 4.0], t$95$0, t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 4

   1. Initial program 56.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 51.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 51.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 36.8%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 97.0%

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

   if 4 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) 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. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

Alternative 4: 36.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + \varepsilon\right) \cdot x\\ \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-t\_0} \leq 2.000002:\\ \;\;\;\;1 + \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (+ 1.0 eps) x)))
   (if (<=
        (-
         (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x))))
         (* (- (/ 1.0 eps) 1.0) (exp (- t_0))))
        2.000002)
     (+ 1.0 eps)
     t_0)))

C

double code(double x, double eps) {
	double t_0 = (1.0 + eps) * x;
	double tmp;
	if ((((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-t_0))) <= 2.000002) {
		tmp = 1.0 + eps;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + eps) * x
    if ((((1.0d0 + (1.0d0 / eps)) * exp(-((1.0d0 - eps) * x))) - (((1.0d0 / eps) - 1.0d0) * exp(-t_0))) <= 2.000002d0) then
        tmp = 1.0d0 + eps
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = (1.0 + eps) * x;
	double tmp;
	if ((((1.0 + (1.0 / eps)) * Math.exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * Math.exp(-t_0))) <= 2.000002) {
		tmp = 1.0 + eps;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (1.0 + eps) * x
	tmp = 0
	if (((1.0 + (1.0 / eps)) * math.exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * math.exp(-t_0))) <= 2.000002:
		tmp = 1.0 + eps
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(1.0 + eps) * x)
	tmp = 0.0
	if (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(-t_0)))) <= 2.000002)
		tmp = Float64(1.0 + eps);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (1.0 + eps) * x;
	tmp = 0.0;
	if ((((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-t_0))) <= 2.000002)
		tmp = 1.0 + eps;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(1.0 + eps), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[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[(-t$95$0)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.000002], N[(1.0 + eps), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 2.00000199999999984

   1. Initial program 55.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 51.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 47.1%

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

   if 2.00000199999999984 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) 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. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto 2 + \color{blue}{x \cdot \left(\left(x \cdot \left(\left(\frac{1}{2} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{2}\right) + x \cdot \left(\frac{1}{6} \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot {\left(\varepsilon - 1\right)}^{3}\right) - \frac{-1}{6} \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) - \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)} \]

   8. Applied rewrites 17.5%

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

Alternative 5: 77.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\left(1 - \varepsilon\right) \cdot x\\ \mathbf{if}\;\varepsilon \leq 3300000:\\ \;\;\;\;e^{-\left(1 + \varepsilon\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{t\_0} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (* (- 1.0 eps) x))))
   (if (<= eps 3300000.0)
     (exp (- (* (+ 1.0 eps) x)))
     (- (* (+ 1.0 (/ 1.0 eps)) (exp t_0)) t_0))))

C

double code(double x, double eps) {
	double t_0 = -((1.0 - eps) * x);
	double tmp;
	if (eps <= 3300000.0) {
		tmp = exp(-((1.0 + eps) * x));
	} else {
		tmp = ((1.0 + (1.0 / eps)) * exp(t_0)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -((1.0d0 - eps) * x)
    if (eps <= 3300000.0d0) then
        tmp = exp(-((1.0d0 + eps) * x))
    else
        tmp = ((1.0d0 + (1.0d0 / eps)) * exp(t_0)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = -((1.0 - eps) * x);
	double tmp;
	if (eps <= 3300000.0) {
		tmp = Math.exp(-((1.0 + eps) * x));
	} else {
		tmp = ((1.0 + (1.0 / eps)) * Math.exp(t_0)) - t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = -((1.0 - eps) * x)
	tmp = 0
	if eps <= 3300000.0:
		tmp = math.exp(-((1.0 + eps) * x))
	else:
		tmp = ((1.0 + (1.0 / eps)) * math.exp(t_0)) - t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(-Float64(Float64(1.0 - eps) * x))
	tmp = 0.0
	if (eps <= 3300000.0)
		tmp = exp(Float64(-Float64(Float64(1.0 + eps) * x)));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(t_0)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = -((1.0 - eps) * x);
	tmp = 0.0;
	if (eps <= 3300000.0)
		tmp = exp(-((1.0 + eps) * x));
	else
		tmp = ((1.0 + (1.0 / eps)) * exp(t_0)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = (-N[(N[(1.0 - eps), $MachinePrecision] * x), $MachinePrecision])}, If[LessEqual[eps, 3300000.0], N[Exp[(-N[(N[(1.0 + eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision], N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[t$95$0], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 3.3e6

   1. Initial program 65.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 44.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 49.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 19.1%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 84.3%

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

   if 3.3e6 < eps

   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. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 76.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\left(x \cdot \left(-1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right) + x \cdot \left(\frac{-1}{6} \cdot \left(x \cdot \left({\left(1 + \varepsilon\right)}^{3} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) + \frac{1}{2} \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right) + \frac{1}{\varepsilon}\right) - \color{blue}{1}\right) \]

   8. Applied rewrites 74.1%

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

Alternative 6: 77.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2.8 \cdot 10^{+44}:\\ \;\;\;\;e^{-\left(1 + \varepsilon\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 2.8e+44)
   (exp (- (* (+ 1.0 eps) x)))
   (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x))))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 2.8e+44) {
		tmp = exp(-((1.0 + eps) * x));
	} else {
		tmp = (1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 2.8d+44) then
        tmp = exp(-((1.0d0 + eps) * x))
    else
        tmp = (1.0d0 + (1.0d0 / eps)) * exp(-((1.0d0 - eps) * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= 2.8e+44) {
		tmp = Math.exp(-((1.0 + eps) * x));
	} else {
		tmp = (1.0 + (1.0 / eps)) * Math.exp(-((1.0 - eps) * x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 2.8e+44:
		tmp = math.exp(-((1.0 + eps) * x))
	else:
		tmp = (1.0 + (1.0 / eps)) * math.exp(-((1.0 - eps) * x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 2.8e+44)
		tmp = exp(Float64(-Float64(Float64(1.0 + eps) * x)));
	else
		tmp = Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(-Float64(Float64(1.0 - eps) * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 2.8e+44)
		tmp = exp(-((1.0 + eps) * x));
	else
		tmp = (1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 2.8e+44], N[Exp[(-N[(N[(1.0 + eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision], N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[(1.0 - eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 2.8000000000000001e44

   1. Initial program 67.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 45.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 49.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 20.3%

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

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 83.6%

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

   if 2.8000000000000001e44 < eps

   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. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 67.5%

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

Alternative 7: 77.4% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 50.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 52.1%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 22.3%

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

12. Taylor expanded in x around inf

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

14. Applied rewrites 77.9%

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

Alternative 8: 29.2% accurate, 68.3× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (+ 1.0 eps))

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 1.0 + eps

Julia

function code(x, eps)
	return Float64(1.0 + eps)
end

MATLAB

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

Wolfram

code[x_, eps_] := N[(1.0 + eps), $MachinePrecision]

Derivation

1. Initial program 75.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 50.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 27.8%

   \[\leadsto 1 + \color{blue}{\varepsilon} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024321 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  :precision binary64
  :pre (TRUE)
  (/ (- (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x)))) (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x))))) 2.0))