NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.4% → 99.9%

Time: 13.0s

Alternatives: 14

Speedup: 9.7×

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

• could not determine a ground truth

  1. x = -7.074250416175752e+213
  2. eps = 5.361636106014531e-124

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: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(\varepsilon - 1\right) \cdot x}\\ \mathbf{if}\;\left(1 - \frac{-1}{\varepsilon}\right) \cdot t\_0 - e^{\left(-1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right) \leq 2.000000000002:\\ \;\;\;\;\left(\left(\left(2 + x\right) + x\right) \cdot e^{-x}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\left(-\varepsilon\right) \cdot x} + t\_0\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (* (- eps 1.0) x))))
   (if (<=
        (-
         (* (- 1.0 (/ -1.0 eps)) t_0)
         (* (exp (* (- -1.0 eps) x)) (- (/ 1.0 eps) 1.0)))
        2.000000000002)
     (* (* (+ (+ 2.0 x) x) (exp (- x))) 0.5)
     (* (+ (exp (* (- eps) x)) t_0) 0.5))))

C

double code(double x, double eps) {
	double t_0 = exp(((eps - 1.0) * x));
	double tmp;
	if ((((1.0 - (-1.0 / eps)) * t_0) - (exp(((-1.0 - eps) * x)) * ((1.0 / eps) - 1.0))) <= 2.000000000002) {
		tmp = (((2.0 + x) + x) * exp(-x)) * 0.5;
	} else {
		tmp = (exp((-eps * x)) + t_0) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(((eps - 1.0d0) * x))
    if ((((1.0d0 - ((-1.0d0) / eps)) * t_0) - (exp((((-1.0d0) - eps) * x)) * ((1.0d0 / eps) - 1.0d0))) <= 2.000000000002d0) then
        tmp = (((2.0d0 + x) + x) * exp(-x)) * 0.5d0
    else
        tmp = (exp((-eps * x)) + t_0) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.exp(((eps - 1.0) * x));
	double tmp;
	if ((((1.0 - (-1.0 / eps)) * t_0) - (Math.exp(((-1.0 - eps) * x)) * ((1.0 / eps) - 1.0))) <= 2.000000000002) {
		tmp = (((2.0 + x) + x) * Math.exp(-x)) * 0.5;
	} else {
		tmp = (Math.exp((-eps * x)) + t_0) * 0.5;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.exp(((eps - 1.0) * x))
	tmp = 0
	if (((1.0 - (-1.0 / eps)) * t_0) - (math.exp(((-1.0 - eps) * x)) * ((1.0 / eps) - 1.0))) <= 2.000000000002:
		tmp = (((2.0 + x) + x) * math.exp(-x)) * 0.5
	else:
		tmp = (math.exp((-eps * x)) + t_0) * 0.5
	return tmp

Julia

function code(x, eps)
	t_0 = exp(Float64(Float64(eps - 1.0) * x))
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 - Float64(-1.0 / eps)) * t_0) - Float64(exp(Float64(Float64(-1.0 - eps) * x)) * Float64(Float64(1.0 / eps) - 1.0))) <= 2.000000000002)
		tmp = Float64(Float64(Float64(Float64(2.0 + x) + x) * exp(Float64(-x))) * 0.5);
	else
		tmp = Float64(Float64(exp(Float64(Float64(-eps) * x)) + t_0) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = exp(((eps - 1.0) * x));
	tmp = 0.0;
	if ((((1.0 - (-1.0 / eps)) * t_0) - (exp(((-1.0 - eps) * x)) * ((1.0 / eps) - 1.0))) <= 2.000000000002)
		tmp = (((2.0 + x) + x) * exp(-x)) * 0.5;
	else
		tmp = (exp((-eps * x)) + t_0) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Exp[N[(N[(eps - 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[(1.0 - N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.000000000002], N[(N[(N[(N[(2.0 + x), $MachinePrecision] + x), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Exp[N[((-eps) * x), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision] * 0.5), $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))))) < 2.0000000000020002

   1. Initial program 49.6%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   2. 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 99.8%

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

   if 2.0000000000020002 < (-.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 99.9%

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

   3. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{-1}{\varepsilon}\right) \cdot e^{\left(\varepsilon - 1\right) \cdot x} - e^{\left(-1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right) \leq 2.000000000002:\\ \;\;\;\;\left(\left(\left(2 + x\right) + x\right) \cdot e^{-x}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\left(-\varepsilon\right) \cdot x} + e^{\left(\varepsilon - 1\right) \cdot x}\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{-1}{\varepsilon}\\ t_1 := e^{\left(-1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right)\\ \mathbf{if}\;t\_0 \cdot e^{\left(\varepsilon - 1\right) \cdot x} - t\_1 \leq 0:\\ \;\;\;\;\left(\left(\left(2 + x\right) + x\right) \cdot e^{-x}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot x, \varepsilon - 1, 1\right) \cdot \left(\varepsilon - 1\right), x, 1\right) \cdot t\_0 - t\_1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ -1.0 eps)))
        (t_1 (* (exp (* (- -1.0 eps) x)) (- (/ 1.0 eps) 1.0))))
   (if (<= (- (* t_0 (exp (* (- eps 1.0) x))) t_1) 0.0)
     (* (* (+ (+ 2.0 x) x) (exp (- x))) 0.5)
     (/
      (-
       (* (fma (* (fma (* 0.5 x) (- eps 1.0) 1.0) (- eps 1.0)) x 1.0) t_0)
       t_1)
      2.0))))

C

double code(double x, double eps) {
	double t_0 = 1.0 - (-1.0 / eps);
	double t_1 = exp(((-1.0 - eps) * x)) * ((1.0 / eps) - 1.0);
	double tmp;
	if (((t_0 * exp(((eps - 1.0) * x))) - t_1) <= 0.0) {
		tmp = (((2.0 + x) + x) * exp(-x)) * 0.5;
	} else {
		tmp = ((fma((fma((0.5 * x), (eps - 1.0), 1.0) * (eps - 1.0)), x, 1.0) * t_0) - t_1) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(1.0 - Float64(-1.0 / eps))
	t_1 = Float64(exp(Float64(Float64(-1.0 - eps) * x)) * Float64(Float64(1.0 / eps) - 1.0))
	tmp = 0.0
	if (Float64(Float64(t_0 * exp(Float64(Float64(eps - 1.0) * x))) - t_1) <= 0.0)
		tmp = Float64(Float64(Float64(Float64(2.0 + x) + x) * exp(Float64(-x))) * 0.5);
	else
		tmp = Float64(Float64(Float64(fma(Float64(fma(Float64(0.5 * x), Float64(eps - 1.0), 1.0) * Float64(eps - 1.0)), x, 1.0) * t_0) - t_1) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 - N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 * N[Exp[N[(N[(eps - 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], 0.0], N[(N[(N[(N[(2.0 + x), $MachinePrecision] + x), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(0.5 * x), $MachinePrecision] * N[(eps - 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(eps - 1.0), $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] - t$95$1), $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))))) < 0.0

   1. Initial program 36.9%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 99.9%

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

   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 99.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. associate-*r* N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. distribute-lft1-in N/A

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

      10. lower-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower--.f64 91.0

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

   5. Applied rewrites 91.0%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{-1}{\varepsilon}\right) \cdot e^{\left(\varepsilon - 1\right) \cdot x} - e^{\left(-1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right) \leq 0:\\ \;\;\;\;\left(\left(\left(2 + x\right) + x\right) \cdot e^{-x}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot x, \varepsilon - 1, 1\right) \cdot \left(\varepsilon - 1\right), x, 1\right) \cdot \left(1 - \frac{-1}{\varepsilon}\right) - e^{\left(-1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024230 
(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))