NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.4% → 99.6%

Time: 11.6s

Alternatives: 12

Speedup: 0.7×

• 38.2% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -1.233054505946495e+63
  2. eps = -2.9835340827367445e-276

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.4% 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: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(\varepsilon - 1\right) \cdot x} - e^{\left(-1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right)\\ \mathbf{if}\;t\_0 \leq 5:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0
         (-
          (* (+ (/ 1.0 eps) 1.0) (exp (* (- eps 1.0) x)))
          (* (exp (* (- -1.0 eps) x)) (- (/ 1.0 eps) 1.0)))))
   (if (<= t_0 5.0) (* (exp (- x)) (+ x 1.0)) (/ t_0 2.0))))

C

double code(double x, double eps) {
	double t_0 = (((1.0 / eps) + 1.0) * exp(((eps - 1.0) * x))) - (exp(((-1.0 - eps) * x)) * ((1.0 / eps) - 1.0));
	double tmp;
	if (t_0 <= 5.0) {
		tmp = exp(-x) * (x + 1.0);
	} else {
		tmp = t_0 / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((1.0d0 / eps) + 1.0d0) * exp(((eps - 1.0d0) * x))) - (exp((((-1.0d0) - eps) * x)) * ((1.0d0 / eps) - 1.0d0))
    if (t_0 <= 5.0d0) then
        tmp = exp(-x) * (x + 1.0d0)
    else
        tmp = t_0 / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = (((1.0 / eps) + 1.0) * Math.exp(((eps - 1.0) * x))) - (Math.exp(((-1.0 - eps) * x)) * ((1.0 / eps) - 1.0));
	double tmp;
	if (t_0 <= 5.0) {
		tmp = Math.exp(-x) * (x + 1.0);
	} else {
		tmp = t_0 / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (((1.0 / eps) + 1.0) * math.exp(((eps - 1.0) * x))) - (math.exp(((-1.0 - eps) * x)) * ((1.0 / eps) - 1.0))
	tmp = 0
	if t_0 <= 5.0:
		tmp = math.exp(-x) * (x + 1.0)
	else:
		tmp = t_0 / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(Float64(Float64(1.0 / eps) + 1.0) * exp(Float64(Float64(eps - 1.0) * x))) - Float64(exp(Float64(Float64(-1.0 - eps) * x)) * Float64(Float64(1.0 / eps) - 1.0)))
	tmp = 0.0
	if (t_0 <= 5.0)
		tmp = Float64(exp(Float64(-x)) * Float64(x + 1.0));
	else
		tmp = Float64(t_0 / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (((1.0 / eps) + 1.0) * exp(((eps - 1.0) * x))) - (exp(((-1.0 - eps) * x)) * ((1.0 / eps) - 1.0));
	tmp = 0.0;
	if (t_0 <= 5.0)
		tmp = exp(-x) * (x + 1.0);
	else
		tmp = t_0 / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(N[(1.0 / eps), $MachinePrecision] + 1.0), $MachinePrecision] * N[Exp[N[(N[(eps - 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5.0], N[(N[Exp[(-x)], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / 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))))) < 5

   1. Initial program 53.5%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

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

   if 5 < (-.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. Final simplification 100.0%

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

Alternative 2: 78.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\varepsilon} - 1\\ t_1 := e^{\left(\varepsilon - 1\right) \cdot x}\\ \mathbf{if}\;\left(\frac{1}{\varepsilon} + 1\right) \cdot t\_1 - e^{\left(-1 - \varepsilon\right) \cdot x} \cdot t\_0 \leq 5:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot t\_1 - t\_0}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 eps) 1.0)) (t_1 (exp (* (- eps 1.0) x))))
   (if (<=
        (- (* (+ (/ 1.0 eps) 1.0) t_1) (* (exp (* (- -1.0 eps) x)) t_0))
        5.0)
     (* (exp (- x)) (+ x 1.0))
     (/ (- (* 1.0 t_1) t_0) 2.0))))

C

double code(double x, double eps) {
	double t_0 = (1.0 / eps) - 1.0;
	double t_1 = exp(((eps - 1.0) * x));
	double tmp;
	if (((((1.0 / eps) + 1.0) * t_1) - (exp(((-1.0 - eps) * x)) * t_0)) <= 5.0) {
		tmp = exp(-x) * (x + 1.0);
	} else {
		tmp = ((1.0 * t_1) - t_0) / 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 = (1.0d0 / eps) - 1.0d0
    t_1 = exp(((eps - 1.0d0) * x))
    if (((((1.0d0 / eps) + 1.0d0) * t_1) - (exp((((-1.0d0) - eps) * x)) * t_0)) <= 5.0d0) then
        tmp = exp(-x) * (x + 1.0d0)
    else
        tmp = ((1.0d0 * t_1) - t_0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = (1.0 / eps) - 1.0;
	double t_1 = Math.exp(((eps - 1.0) * x));
	double tmp;
	if (((((1.0 / eps) + 1.0) * t_1) - (Math.exp(((-1.0 - eps) * x)) * t_0)) <= 5.0) {
		tmp = Math.exp(-x) * (x + 1.0);
	} else {
		tmp = ((1.0 * t_1) - t_0) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (1.0 / eps) - 1.0
	t_1 = math.exp(((eps - 1.0) * x))
	tmp = 0
	if ((((1.0 / eps) + 1.0) * t_1) - (math.exp(((-1.0 - eps) * x)) * t_0)) <= 5.0:
		tmp = math.exp(-x) * (x + 1.0)
	else:
		tmp = ((1.0 * t_1) - t_0) / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(1.0 / eps) - 1.0)
	t_1 = exp(Float64(Float64(eps - 1.0) * x))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(1.0 / eps) + 1.0) * t_1) - Float64(exp(Float64(Float64(-1.0 - eps) * x)) * t_0)) <= 5.0)
		tmp = Float64(exp(Float64(-x)) * Float64(x + 1.0));
	else
		tmp = Float64(Float64(Float64(1.0 * t_1) - t_0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (1.0 / eps) - 1.0;
	t_1 = exp(((eps - 1.0) * x));
	tmp = 0.0;
	if (((((1.0 / eps) + 1.0) * t_1) - (exp(((-1.0 - eps) * x)) * t_0)) <= 5.0)
		tmp = exp(-x) * (x + 1.0);
	else
		tmp = ((1.0 * t_1) - t_0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[(eps - 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(1.0 / eps), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision] - N[(N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 5.0], N[(N[Exp[(-x)], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 * t$95$1), $MachinePrecision] - t$95$0), $MachinePrecision] / 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))))) < 5

   1. Initial program 53.5%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

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

   if 5 < (-.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

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 60.5

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

   5. Applied rewrites 60.5%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 60.5%

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

4. Final simplification 80.7%

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

Alternative 3: 66.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(\varepsilon - 1\right) \cdot x} - e^{\left(-1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right) \leq 0:\\ \;\;\;\;\frac{-1}{\left(x - 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, 0.5, -1\right), x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (-
       (* (+ (/ 1.0 eps) 1.0) (exp (* (- eps 1.0) x)))
       (* (exp (* (- -1.0 eps) x)) (- (/ 1.0 eps) 1.0)))
      0.0)
   (/ -1.0 (* (- x 1.0) (fma (fma 0.5 x 1.0) x 1.0)))
   (* 1.0 (fma (fma x 0.5 -1.0) x 1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (((((1.0 / eps) + 1.0) * exp(((eps - 1.0) * x))) - (exp(((-1.0 - eps) * x)) * ((1.0 / eps) - 1.0))) <= 0.0) {
		tmp = -1.0 / ((x - 1.0) * fma(fma(0.5, x, 1.0), x, 1.0));
	} else {
		tmp = 1.0 * fma(fma(x, 0.5, -1.0), x, 1.0);
	}
	return tmp;
}

Julia

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

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[(N[(N[(1.0 / eps), $MachinePrecision] + 1.0), $MachinePrecision] * N[Exp[N[(N[(eps - 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(-1.0 / N[(N[(x - 1.0), $MachinePrecision] * N[(N[(0.5 * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(N[(x * 0.5 + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $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 38.1%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 83.4%

      \[\leadsto \left(2 \cdot \frac{1 + x}{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.5, 1\right), x, 1\right)}\right) \cdot 0.5 \]
   9. Step-by-step derivation

   10. Applied rewrites 64.6%

       \[\leadsto \frac{\mathsf{fma}\left(x, x, -1\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right) \cdot \left(x - 1\right)}} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, x, 1\right), x, 1\right)} \cdot \left(x - 1\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 86.8%

       \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)} \cdot \left(x - 1\right)} \]

   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 99.5%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 22.6%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 22.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 45.3%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 47.8%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(\varepsilon - 1\right) \cdot x} - e^{\left(-1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right) \leq 0:\\ \;\;\;\;\frac{-1}{\left(x - 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, 0.5, -1\right), x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(\varepsilon - 1\right) \cdot x} - e^{\left(-1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right) \leq 0:\\ \;\;\;\;\frac{x + 1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, 0.5, -1\right), x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (-
       (* (+ (/ 1.0 eps) 1.0) (exp (* (- eps 1.0) x)))
       (* (exp (* (- -1.0 eps) x)) (- (/ 1.0 eps) 1.0)))
      0.0)
   (/ (+ x 1.0) (fma (fma 0.5 x 1.0) x 1.0))
   (* 1.0 (fma (fma x 0.5 -1.0) x 1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (((((1.0 / eps) + 1.0) * exp(((eps - 1.0) * x))) - (exp(((-1.0 - eps) * x)) * ((1.0 / eps) - 1.0))) <= 0.0) {
		tmp = (x + 1.0) / fma(fma(0.5, x, 1.0), x, 1.0);
	} else {
		tmp = 1.0 * fma(fma(x, 0.5, -1.0), x, 1.0);
	}
	return tmp;
}

Julia

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

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[(N[(N[(1.0 / eps), $MachinePrecision] + 1.0), $MachinePrecision] * N[Exp[N[(N[(eps - 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(x + 1.0), $MachinePrecision] / N[(N[(0.5 * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(N[(x * 0.5 + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $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 38.1%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 83.4%

      \[\leadsto \left(2 \cdot \frac{1 + x}{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.5, 1\right), x, 1\right)}\right) \cdot 0.5 \]
   9. Step-by-step derivation

   10. Applied rewrites 83.4%

       \[\leadsto \frac{1 + x}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)}} \]

   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 99.5%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 22.6%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 22.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 45.3%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 47.8%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(\varepsilon - 1\right) \cdot x} - e^{\left(-1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right) \leq 0:\\ \;\;\;\;\frac{x + 1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, 0.5, -1\right), x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{elif}\;\varepsilon \leq 6 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, 0.5, -1\right), x, 1\right) \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - e^{\left(-1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 1.0)
   (* (exp (- x)) (+ x 1.0))
   (if (<= eps 6e+25)
     (* (fma (fma x 0.5 -1.0) x 1.0) (+ x 1.0))
     (/ (- 1.0 (* (exp (* (- -1.0 eps) x)) (- (/ 1.0 eps) 1.0))) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 1.0) {
		tmp = exp(-x) * (x + 1.0);
	} else if (eps <= 6e+25) {
		tmp = fma(fma(x, 0.5, -1.0), x, 1.0) * (x + 1.0);
	} else {
		tmp = (1.0 - (exp(((-1.0 - eps) * x)) * ((1.0 / eps) - 1.0))) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 1.0)
		tmp = Float64(exp(Float64(-x)) * Float64(x + 1.0));
	elseif (eps <= 6e+25)
		tmp = Float64(fma(fma(x, 0.5, -1.0), x, 1.0) * Float64(x + 1.0));
	else
		tmp = Float64(Float64(1.0 - Float64(exp(Float64(Float64(-1.0 - eps) * x)) * Float64(Float64(1.0 / eps) - 1.0))) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 1.0], N[(N[Exp[(-x)], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 6e+25], N[(N[(N[(x * 0.5 + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 1

   1. Initial program 66.9%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 62.0%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 62.0%

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

   if 1 < eps < 6.00000000000000011e25

   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 eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 23.3%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 23.3%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 79.1%

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

   if 6.00000000000000011e25 < 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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 53.8

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

   5. Applied rewrites 53.8%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 53.8%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{elif}\;\varepsilon \leq 6 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, 0.5, -1\right), x, 1\right) \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - e^{\left(-1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;x \leq -720:\\ \;\;\;\;1 \cdot t\_0\\ \mathbf{elif}\;x \leq -9.2 \cdot 10^{-229}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, \frac{1}{\varepsilon}, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+113}:\\ \;\;\;\;t\_0 \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= x -720.0)
     (* 1.0 t_0)
     (if (<= x -9.2e-229)
       (fma
        (* 0.5 x)
        (fma (- eps 1.0) (/ 1.0 eps) (/ (- 1.0 (* eps eps)) eps))
        1.0)
       (if (<= x 9.5e+113)
         (* t_0 (+ x 1.0))
         (fma (fma 0.3333333333333333 x -0.5) (* x x) 1.0))))))

C

double code(double x, double eps) {
	double t_0 = exp(-x);
	double tmp;
	if (x <= -720.0) {
		tmp = 1.0 * t_0;
	} else if (x <= -9.2e-229) {
		tmp = fma((0.5 * x), fma((eps - 1.0), (1.0 / eps), ((1.0 - (eps * eps)) / eps)), 1.0);
	} else if (x <= 9.5e+113) {
		tmp = t_0 * (x + 1.0);
	} else {
		tmp = fma(fma(0.3333333333333333, x, -0.5), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (x <= -720.0)
		tmp = Float64(1.0 * t_0);
	elseif (x <= -9.2e-229)
		tmp = fma(Float64(0.5 * x), fma(Float64(eps - 1.0), Float64(1.0 / eps), Float64(Float64(1.0 - Float64(eps * eps)) / eps)), 1.0);
	elseif (x <= 9.5e+113)
		tmp = Float64(t_0 * Float64(x + 1.0));
	else
		tmp = fma(fma(0.3333333333333333, x, -0.5), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[x, -720.0], N[(1.0 * t$95$0), $MachinePrecision], If[LessEqual[x, -9.2e-229], N[(N[(0.5 * x), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * N[(1.0 / eps), $MachinePrecision] + N[(N[(1.0 - N[(eps * eps), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 9.5e+113], N[(t$95$0 * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * x + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -720

   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 eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 0.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 100.0%

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

   if -720 < x < -9.19999999999999983e-229

   1. Initial program 48.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 56.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, \frac{1}{\varepsilon} + 1, \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(1 + \varepsilon\right)\right), 1\right)} \]

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 55.9%

      \[\leadsto \mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, \frac{1}{\color{blue}{\varepsilon}}, \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(1 + \varepsilon\right)\right), 1\right) \]

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 69.2%

       \[\leadsto \mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, \frac{1}{\varepsilon}, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right) \]

   if -9.19999999999999983e-229 < x < 9.5000000000000001e113

   1. Initial program 69.9%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 66.3%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 66.3%

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

   if 9.5000000000000001e113 < 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 eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 45.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), \color{blue}{x \cdot x}, 1\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -720:\\ \;\;\;\;1 \cdot e^{-x}\\ \mathbf{elif}\;x \leq -9.2 \cdot 10^{-229}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, \frac{1}{\varepsilon}, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+113}:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot e^{-x}\\ \mathbf{if}\;x \leq -720:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -9.2 \cdot 10^{-229}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, \frac{1}{\varepsilon}, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+113}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* 1.0 (exp (- x)))))
   (if (<= x -720.0)
     t_0
     (if (<= x -9.2e-229)
       (fma
        (* 0.5 x)
        (fma (- eps 1.0) (/ 1.0 eps) (/ (- 1.0 (* eps eps)) eps))
        1.0)
       (if (<= x 9.5e+113)
         t_0
         (fma (fma 0.3333333333333333 x -0.5) (* x x) 1.0))))))

C

double code(double x, double eps) {
	double t_0 = 1.0 * exp(-x);
	double tmp;
	if (x <= -720.0) {
		tmp = t_0;
	} else if (x <= -9.2e-229) {
		tmp = fma((0.5 * x), fma((eps - 1.0), (1.0 / eps), ((1.0 - (eps * eps)) / eps)), 1.0);
	} else if (x <= 9.5e+113) {
		tmp = t_0;
	} else {
		tmp = fma(fma(0.3333333333333333, x, -0.5), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(1.0 * exp(Float64(-x)))
	tmp = 0.0
	if (x <= -720.0)
		tmp = t_0;
	elseif (x <= -9.2e-229)
		tmp = fma(Float64(0.5 * x), fma(Float64(eps - 1.0), Float64(1.0 / eps), Float64(Float64(1.0 - Float64(eps * eps)) / eps)), 1.0);
	elseif (x <= 9.5e+113)
		tmp = t_0;
	else
		tmp = fma(fma(0.3333333333333333, x, -0.5), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -720.0], t$95$0, If[LessEqual[x, -9.2e-229], N[(N[(0.5 * x), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * N[(1.0 / eps), $MachinePrecision] + N[(N[(1.0 - N[(eps * eps), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 9.5e+113], t$95$0, N[(N[(0.3333333333333333 * x + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -720 or -9.19999999999999983e-229 < x < 9.5000000000000001e113

   1. Initial program 77.1%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 73.5%

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

   if -720 < x < -9.19999999999999983e-229

   1. Initial program 48.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 56.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, \frac{1}{\varepsilon} + 1, \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(1 + \varepsilon\right)\right), 1\right)} \]

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 55.9%

      \[\leadsto \mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, \frac{1}{\color{blue}{\varepsilon}}, \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(1 + \varepsilon\right)\right), 1\right) \]

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 69.2%

       \[\leadsto \mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, \frac{1}{\varepsilon}, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right) \]

   if 9.5000000000000001e113 < 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 eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 45.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), \color{blue}{x \cdot x}, 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 68.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), x, 1\right) \cdot 1\\ \mathbf{elif}\;x \leq -9.2 \cdot 10^{-229}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, \frac{1}{\varepsilon}, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{elif}\;x \leq 1.8:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.3333333333333333\right), x, -0.5\right), x \cdot x, 1\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+113}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.5e+100)
   (* (fma (fma (fma -0.16666666666666666 x 0.5) x -1.0) x 1.0) 1.0)
   (if (<= x -9.2e-229)
     (fma
      (* 0.5 x)
      (fma (- eps 1.0) (/ 1.0 eps) (/ (- 1.0 (* eps eps)) eps))
      1.0)
     (if (<= x 1.8)
       (fma (fma (fma -0.125 x 0.3333333333333333) x -0.5) (* x x) 1.0)
       (if (<= x 9.5e+113)
         (/ (- (+ (/ 1.0 eps) 1.0) (- (/ 1.0 eps) 1.0)) 2.0)
         (fma (fma 0.3333333333333333 x -0.5) (* x x) 1.0))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.5e+100) {
		tmp = fma(fma(fma(-0.16666666666666666, x, 0.5), x, -1.0), x, 1.0) * 1.0;
	} else if (x <= -9.2e-229) {
		tmp = fma((0.5 * x), fma((eps - 1.0), (1.0 / eps), ((1.0 - (eps * eps)) / eps)), 1.0);
	} else if (x <= 1.8) {
		tmp = fma(fma(fma(-0.125, x, 0.3333333333333333), x, -0.5), (x * x), 1.0);
	} else if (x <= 9.5e+113) {
		tmp = (((1.0 / eps) + 1.0) - ((1.0 / eps) - 1.0)) / 2.0;
	} else {
		tmp = fma(fma(0.3333333333333333, x, -0.5), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.5e+100)
		tmp = Float64(fma(fma(fma(-0.16666666666666666, x, 0.5), x, -1.0), x, 1.0) * 1.0);
	elseif (x <= -9.2e-229)
		tmp = fma(Float64(0.5 * x), fma(Float64(eps - 1.0), Float64(1.0 / eps), Float64(Float64(1.0 - Float64(eps * eps)) / eps)), 1.0);
	elseif (x <= 1.8)
		tmp = fma(fma(fma(-0.125, x, 0.3333333333333333), x, -0.5), Float64(x * x), 1.0);
	elseif (x <= 9.5e+113)
		tmp = Float64(Float64(Float64(Float64(1.0 / eps) + 1.0) - Float64(Float64(1.0 / eps) - 1.0)) / 2.0);
	else
		tmp = fma(fma(0.3333333333333333, x, -0.5), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.5e+100], N[(N[(N[(N[(-0.16666666666666666 * x + 0.5), $MachinePrecision] * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[x, -9.2e-229], N[(N[(0.5 * x), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * N[(1.0 / eps), $MachinePrecision] + N[(N[(1.0 - N[(eps * eps), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 1.8], N[(N[(N[(-0.125 * x + 0.3333333333333333), $MachinePrecision] * x + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 9.5e+113], N[(N[(N[(N[(1.0 / eps), $MachinePrecision] + 1.0), $MachinePrecision] - N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(0.3333333333333333 * x + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -2.4999999999999999e100

   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 eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 0.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 96.0%

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

   if -2.4999999999999999e100 < x < -9.19999999999999983e-229

   1. Initial program 59.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 44.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, \frac{1}{\varepsilon} + 1, \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(1 + \varepsilon\right)\right), 1\right)} \]

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 44.2%

      \[\leadsto \mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, \frac{1}{\color{blue}{\varepsilon}}, \left(\frac{1}{\varepsilon} - 1\right) \cdot \left(1 + \varepsilon\right)\right), 1\right) \]

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 60.4%

       \[\leadsto \mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, \frac{1}{\varepsilon}, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right) \]

   if -9.19999999999999983e-229 < x < 1.80000000000000004

   1. Initial program 58.5%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 78.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(x \cdot \left(\frac{1}{3} + \frac{-1}{8} \cdot x\right) - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 77.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.3333333333333333\right), x, -0.5\right), \color{blue}{x \cdot x}, 1\right) \]

   if 1.80000000000000004 < x < 9.5000000000000001e113

   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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 45.2

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

   5. Applied rewrites 45.2%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 36.4

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

   8. Applied rewrites 36.4%

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

   if 9.5000000000000001e113 < 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 eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 45.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), \color{blue}{x \cdot x}, 1\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), x, 1\right) \cdot 1\\ \mathbf{elif}\;x \leq -9.2 \cdot 10^{-229}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, \frac{1}{\varepsilon}, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{elif}\;x \leq 1.8:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.3333333333333333\right), x, -0.5\right), x \cdot x, 1\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+113}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.0% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5800:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), x, 1\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -5800.0)
   (* (fma (fma (fma -0.16666666666666666 x 0.5) x -1.0) x 1.0) 1.0)
   (fma (fma 0.3333333333333333 x -0.5) (* x x) 1.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -5800.0) {
		tmp = fma(fma(fma(-0.16666666666666666, x, 0.5), x, -1.0), x, 1.0) * 1.0;
	} else {
		tmp = fma(fma(0.3333333333333333, x, -0.5), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -5800.0)
		tmp = Float64(fma(fma(fma(-0.16666666666666666, x, 0.5), x, -1.0), x, 1.0) * 1.0);
	else
		tmp = fma(fma(0.3333333333333333, x, -0.5), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -5800.0], N[(N[(N[(N[(-0.16666666666666666 * x + 0.5), $MachinePrecision] * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] * 1.0), $MachinePrecision], N[(N[(0.3333333333333333 * x + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5800

   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 eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 0.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 60.6%

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

   if -5800 < x

   1. Initial program 72.3%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 60.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 57.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), \color{blue}{x \cdot x}, 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5800:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), x, -1\right), x, 1\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.8% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5800:\\ \;\;\;\;1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, 0.5, -1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -5800.0)
   (* 1.0 (fma (fma x 0.5 -1.0) x 1.0))
   (fma (fma 0.3333333333333333 x -0.5) (* x x) 1.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -5800.0) {
		tmp = 1.0 * fma(fma(x, 0.5, -1.0), x, 1.0);
	} else {
		tmp = fma(fma(0.3333333333333333, x, -0.5), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -5800.0)
		tmp = Float64(1.0 * fma(fma(x, 0.5, -1.0), x, 1.0));
	else
		tmp = fma(fma(0.3333333333333333, x, -0.5), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -5800.0], N[(1.0 * N[(N[(x * 0.5 + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * x + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5800

   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 eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5} \]
   6. Step-by-step derivation

   7. Applied rewrites 0.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 100.0%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 39.6%

       \[\leadsto 1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, 0.5, -1\right), \color{blue}{x}, 1\right) \]

   if -5800 < x

   1. Initial program 72.3%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 60.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 57.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), \color{blue}{x \cdot x}, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 52.9% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma (fma 0.3333333333333333 x -0.5) (* x x) 1.0))

C

double code(double x, double eps) {
	return fma(fma(0.3333333333333333, x, -0.5), (x * x), 1.0);
}

Julia

function code(x, eps)
	return fma(fma(0.3333333333333333, x, -0.5), Float64(x * x), 1.0)
end

Wolfram

code[x_, eps_] := N[(N[(0.3333333333333333 * x + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 76.2%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 51.9%

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

6. Taylor expanded in x around 0

   \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 49.6%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), \color{blue}{x \cdot x}, 1\right) \]
9. Add Preprocessing

Alternative 12: 43.9% accurate, 273.0× 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 76.2%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 37.8%

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

Reproduce

herbie shell --seed 2024284 
(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))