NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.4% → 99.9%

Time: 13.7s

Alternatives: 15

Speedup: 7.2×

• 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 = -9.373209203485e+297
  2. eps = -1.4413362485554466e-37

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, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 1.4 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(\left(\left(2 + x\right) + x\right) \cdot e^{-x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(eps\_m - 1\right) \cdot x} \cdot \left(\frac{1}{eps\_m} + 1\right) - e^{\left(-1 - eps\_m\right) \cdot x} \cdot \left(\frac{1}{eps\_m} - 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 1.4e-8)
   (* 0.5 (* (+ (+ 2.0 x) x) (exp (- x))))
   (/
    (-
     (* (exp (* (- eps_m 1.0) x)) (+ (/ 1.0 eps_m) 1.0))
     (* (exp (* (- -1.0 eps_m) x)) (- (/ 1.0 eps_m) 1.0)))
    2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.4e-8) {
		tmp = 0.5 * (((2.0 + x) + x) * exp(-x));
	} else {
		tmp = ((exp(((eps_m - 1.0) * x)) * ((1.0 / eps_m) + 1.0)) - (exp(((-1.0 - eps_m) * x)) * ((1.0 / eps_m) - 1.0))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 1.4d-8) then
        tmp = 0.5d0 * (((2.0d0 + x) + x) * exp(-x))
    else
        tmp = ((exp(((eps_m - 1.0d0) * x)) * ((1.0d0 / eps_m) + 1.0d0)) - (exp((((-1.0d0) - eps_m) * x)) * ((1.0d0 / eps_m) - 1.0d0))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.4e-8) {
		tmp = 0.5 * (((2.0 + x) + x) * Math.exp(-x));
	} else {
		tmp = ((Math.exp(((eps_m - 1.0) * x)) * ((1.0 / eps_m) + 1.0)) - (Math.exp(((-1.0 - eps_m) * x)) * ((1.0 / eps_m) - 1.0))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 1.4e-8:
		tmp = 0.5 * (((2.0 + x) + x) * math.exp(-x))
	else:
		tmp = ((math.exp(((eps_m - 1.0) * x)) * ((1.0 / eps_m) + 1.0)) - (math.exp(((-1.0 - eps_m) * x)) * ((1.0 / eps_m) - 1.0))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 1.4e-8)
		tmp = Float64(0.5 * Float64(Float64(Float64(2.0 + x) + x) * exp(Float64(-x))));
	else
		tmp = Float64(Float64(Float64(exp(Float64(Float64(eps_m - 1.0) * x)) * Float64(Float64(1.0 / eps_m) + 1.0)) - Float64(exp(Float64(Float64(-1.0 - eps_m) * x)) * Float64(Float64(1.0 / eps_m) - 1.0))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 1.4e-8)
		tmp = 0.5 * (((2.0 + x) + x) * exp(-x));
	else
		tmp = ((exp(((eps_m - 1.0) * x)) * ((1.0 / eps_m) + 1.0)) - (exp(((-1.0 - eps_m) * x)) * ((1.0 / eps_m) - 1.0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 1.4e-8], N[(0.5 * N[(N[(N[(2.0 + x), $MachinePrecision] + x), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Exp[N[(N[(eps$95$m - 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Exp[N[(N[(-1.0 - eps$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.4e-8

   1. Initial program 68.5%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 69.5%

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

   if 1.4e-8 < 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. Recombined 2 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.4 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(\left(\left(2 + x\right) + x\right) \cdot e^{-x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(\varepsilon - 1\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right) - e^{\left(-1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{1}{eps\_m} - 1\\ t_1 := \frac{eps\_m - 1}{eps\_m} - \left(1 - eps\_m\right)\\ \mathbf{if}\;e^{\left(eps\_m - 1\right) \cdot x} \cdot \left(\frac{1}{eps\_m} + 1\right) - e^{\left(-1 - eps\_m\right) \cdot x} \cdot t\_0 \leq 10^{+39}:\\ \;\;\;\;0.5 \cdot \left(\left(\left(2 + x\right) + x\right) \cdot e^{-x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(-1 - eps\_m, \left(\frac{1 + eps\_m}{eps\_m} + -1\right) - eps\_m, t\_1 \cdot \left(eps\_m - 1\right)\right) \cdot 0.5, x, \mathsf{fma}\left(t\_0, 1 + eps\_m, t\_1\right)\right), 1\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 eps_m) 1.0))
        (t_1 (- (/ (- eps_m 1.0) eps_m) (- 1.0 eps_m))))
   (if (<=
        (-
         (* (exp (* (- eps_m 1.0) x)) (+ (/ 1.0 eps_m) 1.0))
         (* (exp (* (- -1.0 eps_m) x)) t_0))
        1e+39)
     (* 0.5 (* (+ (+ 2.0 x) x) (exp (- x))))
     (fma
      (* 0.5 x)
      (fma
       (*
        (fma
         (- -1.0 eps_m)
         (- (+ (/ (+ 1.0 eps_m) eps_m) -1.0) eps_m)
         (* t_1 (- eps_m 1.0)))
        0.5)
       x
       (fma t_0 (+ 1.0 eps_m) t_1))
      1.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (1.0 / eps_m) - 1.0;
	double t_1 = ((eps_m - 1.0) / eps_m) - (1.0 - eps_m);
	double tmp;
	if (((exp(((eps_m - 1.0) * x)) * ((1.0 / eps_m) + 1.0)) - (exp(((-1.0 - eps_m) * x)) * t_0)) <= 1e+39) {
		tmp = 0.5 * (((2.0 + x) + x) * exp(-x));
	} else {
		tmp = fma((0.5 * x), fma((fma((-1.0 - eps_m), ((((1.0 + eps_m) / eps_m) + -1.0) - eps_m), (t_1 * (eps_m - 1.0))) * 0.5), x, fma(t_0, (1.0 + eps_m), t_1)), 1.0);
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(1.0 / eps_m) - 1.0)
	t_1 = Float64(Float64(Float64(eps_m - 1.0) / eps_m) - Float64(1.0 - eps_m))
	tmp = 0.0
	if (Float64(Float64(exp(Float64(Float64(eps_m - 1.0) * x)) * Float64(Float64(1.0 / eps_m) + 1.0)) - Float64(exp(Float64(Float64(-1.0 - eps_m) * x)) * t_0)) <= 1e+39)
		tmp = Float64(0.5 * Float64(Float64(Float64(2.0 + x) + x) * exp(Float64(-x))));
	else
		tmp = fma(Float64(0.5 * x), fma(Float64(fma(Float64(-1.0 - eps_m), Float64(Float64(Float64(Float64(1.0 + eps_m) / eps_m) + -1.0) - eps_m), Float64(t_1 * Float64(eps_m - 1.0))) * 0.5), x, fma(t_0, Float64(1.0 + eps_m), t_1)), 1.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(eps$95$m - 1.0), $MachinePrecision] / eps$95$m), $MachinePrecision] - N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[Exp[N[(N[(eps$95$m - 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Exp[N[(N[(-1.0 - eps$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 1e+39], N[(0.5 * N[(N[(N[(2.0 + x), $MachinePrecision] + x), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * x), $MachinePrecision] * N[(N[(N[(N[(-1.0 - eps$95$m), $MachinePrecision] * N[(N[(N[(N[(1.0 + eps$95$m), $MachinePrecision] / eps$95$m), $MachinePrecision] + -1.0), $MachinePrecision] - eps$95$m), $MachinePrecision] + N[(t$95$1 * N[(eps$95$m - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * x + N[(t$95$0 * N[(1.0 + eps$95$m), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision] + 1.0), $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))))) < 9.9999999999999994e38

   1. Initial program 59.6%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 98.6%

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

   if 9.9999999999999994e38 < (-.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 99.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 83.8%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\left(\varepsilon - 1\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right) - e^{\left(-1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right) \leq 10^{+39}:\\ \;\;\;\;0.5 \cdot \left(\left(\left(2 + x\right) + x\right) \cdot e^{-x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(-1 - \varepsilon, \left(\frac{1 + \varepsilon}{\varepsilon} + -1\right) - \varepsilon, \left(\frac{\varepsilon - 1}{\varepsilon} - \left(1 - \varepsilon\right)\right) \cdot \left(\varepsilon - 1\right)\right) \cdot 0.5, x, \mathsf{fma}\left(\frac{1}{\varepsilon} - 1, 1 + \varepsilon, \frac{\varepsilon - 1}{\varepsilon} - \left(1 - \varepsilon\right)\right)\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;e^{\left(eps\_m - 1\right) \cdot x} \cdot \left(\frac{1}{eps\_m} + 1\right) - e^{\left(-1 - eps\_m\right) \cdot x} \cdot \left(\frac{1}{eps\_m} - 1\right) \leq 10^{+39}:\\ \;\;\;\;0.5 \cdot \left(\left(\left(2 + x\right) + x\right) \cdot e^{-x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(eps\_m - 1, eps\_m - 1, \left(-1 - eps\_m\right) \cdot \left(-1 - eps\_m\right)\right), 0.5 \cdot x, -2\right), x, 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<=
      (-
       (* (exp (* (- eps_m 1.0) x)) (+ (/ 1.0 eps_m) 1.0))
       (* (exp (* (- -1.0 eps_m) x)) (- (/ 1.0 eps_m) 1.0)))
      1e+39)
   (* 0.5 (* (+ (+ 2.0 x) x) (exp (- x))))
   (*
    (fma
     (fma
      (fma (- eps_m 1.0) (- eps_m 1.0) (* (- -1.0 eps_m) (- -1.0 eps_m)))
      (* 0.5 x)
      -2.0)
     x
     2.0)
    0.5)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (((exp(((eps_m - 1.0) * x)) * ((1.0 / eps_m) + 1.0)) - (exp(((-1.0 - eps_m) * x)) * ((1.0 / eps_m) - 1.0))) <= 1e+39) {
		tmp = 0.5 * (((2.0 + x) + x) * exp(-x));
	} else {
		tmp = fma(fma(fma((eps_m - 1.0), (eps_m - 1.0), ((-1.0 - eps_m) * (-1.0 - eps_m))), (0.5 * x), -2.0), x, 2.0) * 0.5;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (Float64(Float64(exp(Float64(Float64(eps_m - 1.0) * x)) * Float64(Float64(1.0 / eps_m) + 1.0)) - Float64(exp(Float64(Float64(-1.0 - eps_m) * x)) * Float64(Float64(1.0 / eps_m) - 1.0))) <= 1e+39)
		tmp = Float64(0.5 * Float64(Float64(Float64(2.0 + x) + x) * exp(Float64(-x))));
	else
		tmp = Float64(fma(fma(fma(Float64(eps_m - 1.0), Float64(eps_m - 1.0), Float64(Float64(-1.0 - eps_m) * Float64(-1.0 - eps_m))), Float64(0.5 * x), -2.0), x, 2.0) * 0.5);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[N[(N[(N[Exp[N[(N[(eps$95$m - 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Exp[N[(N[(-1.0 - eps$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+39], N[(0.5 * N[(N[(N[(2.0 + x), $MachinePrecision] + x), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(eps$95$m - 1.0), $MachinePrecision] * N[(eps$95$m - 1.0), $MachinePrecision] + N[(N[(-1.0 - eps$95$m), $MachinePrecision] * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 * x), $MachinePrecision] + -2.0), $MachinePrecision] * x + 2.0), $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))))) < 9.9999999999999994e38

   1. Initial program 59.6%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 98.6%

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

   if 9.9999999999999994e38 < (-.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 99.1%

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

   3. Taylor expanded in eps around 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 99.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 19.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \varepsilon, 1 - \varepsilon, \left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right), 0.5, \mathsf{fma}\left(0.16666666666666666 \cdot \left(\varepsilon - 1\right), \left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right), \left(-0.16666666666666666 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right) \cdot x\right), x, \left(\varepsilon - 1\right) - \varepsilon\right) - 1, x, 2\right) \cdot 0.5 \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 83.8%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\left(\varepsilon - 1\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right) - e^{\left(-1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right) \leq 10^{+39}:\\ \;\;\;\;0.5 \cdot \left(\left(\left(2 + x\right) + x\right) \cdot e^{-x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon - 1, \varepsilon - 1, \left(-1 - \varepsilon\right) \cdot \left(-1 - \varepsilon\right)\right), 0.5 \cdot x, -2\right), x, 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;e^{\left(eps\_m - 1\right) \cdot x} \cdot \left(\frac{1}{eps\_m} + 1\right) - e^{\left(-1 - eps\_m\right) \cdot x} \cdot \left(\frac{1}{eps\_m} - 1\right) \leq 10^{+39}:\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(eps\_m - 1, eps\_m - 1, \left(-1 - eps\_m\right) \cdot \left(-1 - eps\_m\right)\right), 0.5 \cdot x, -2\right), x, 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<=
      (-
       (* (exp (* (- eps_m 1.0) x)) (+ (/ 1.0 eps_m) 1.0))
       (* (exp (* (- -1.0 eps_m) x)) (- (/ 1.0 eps_m) 1.0)))
      1e+39)
   (exp (- x))
   (*
    (fma
     (fma
      (fma (- eps_m 1.0) (- eps_m 1.0) (* (- -1.0 eps_m) (- -1.0 eps_m)))
      (* 0.5 x)
      -2.0)
     x
     2.0)
    0.5)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (((exp(((eps_m - 1.0) * x)) * ((1.0 / eps_m) + 1.0)) - (exp(((-1.0 - eps_m) * x)) * ((1.0 / eps_m) - 1.0))) <= 1e+39) {
		tmp = exp(-x);
	} else {
		tmp = fma(fma(fma((eps_m - 1.0), (eps_m - 1.0), ((-1.0 - eps_m) * (-1.0 - eps_m))), (0.5 * x), -2.0), x, 2.0) * 0.5;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (Float64(Float64(exp(Float64(Float64(eps_m - 1.0) * x)) * Float64(Float64(1.0 / eps_m) + 1.0)) - Float64(exp(Float64(Float64(-1.0 - eps_m) * x)) * Float64(Float64(1.0 / eps_m) - 1.0))) <= 1e+39)
		tmp = exp(Float64(-x));
	else
		tmp = Float64(fma(fma(fma(Float64(eps_m - 1.0), Float64(eps_m - 1.0), Float64(Float64(-1.0 - eps_m) * Float64(-1.0 - eps_m))), Float64(0.5 * x), -2.0), x, 2.0) * 0.5);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[N[(N[(N[Exp[N[(N[(eps$95$m - 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Exp[N[(N[(-1.0 - eps$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+39], N[Exp[(-x)], $MachinePrecision], N[(N[(N[(N[(N[(eps$95$m - 1.0), $MachinePrecision] * N[(eps$95$m - 1.0), $MachinePrecision] + N[(N[(-1.0 - eps$95$m), $MachinePrecision] * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 * x), $MachinePrecision] + -2.0), $MachinePrecision] * x + 2.0), $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))))) < 9.9999999999999994e38

   1. Initial program 59.6%

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

   3. Taylor expanded in eps around 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 97.1%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 95.8%

      \[\leadsto e^{-x} \]

   if 9.9999999999999994e38 < (-.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 99.1%

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

   3. Taylor expanded in eps around 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 99.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 19.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \varepsilon, 1 - \varepsilon, \left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right), 0.5, \mathsf{fma}\left(0.16666666666666666 \cdot \left(\varepsilon - 1\right), \left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right), \left(-0.16666666666666666 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right) \cdot x\right), x, \left(\varepsilon - 1\right) - \varepsilon\right) - 1, x, 2\right) \cdot 0.5 \]

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 83.8%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\left(\varepsilon - 1\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right) - e^{\left(-1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right) \leq 10^{+39}:\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon - 1, \varepsilon - 1, \left(-1 - \varepsilon\right) \cdot \left(-1 - \varepsilon\right)\right), 0.5 \cdot x, -2\right), x, 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 1.4 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(\left(\left(2 + x\right) + x\right) \cdot e^{-x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\left(eps\_m - 1\right) \cdot x} + e^{\left(-1 - eps\_m\right) \cdot x}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 1.4e-8)
   (* 0.5 (* (+ (+ 2.0 x) x) (exp (- x))))
   (* (+ (exp (* (- eps_m 1.0) x)) (exp (* (- -1.0 eps_m) x))) 0.5)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.4e-8) {
		tmp = 0.5 * (((2.0 + x) + x) * exp(-x));
	} else {
		tmp = (exp(((eps_m - 1.0) * x)) + exp(((-1.0 - eps_m) * x))) * 0.5;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 1.4d-8) then
        tmp = 0.5d0 * (((2.0d0 + x) + x) * exp(-x))
    else
        tmp = (exp(((eps_m - 1.0d0) * x)) + exp((((-1.0d0) - eps_m) * x))) * 0.5d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.4e-8) {
		tmp = 0.5 * (((2.0 + x) + x) * Math.exp(-x));
	} else {
		tmp = (Math.exp(((eps_m - 1.0) * x)) + Math.exp(((-1.0 - eps_m) * x))) * 0.5;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 1.4e-8:
		tmp = 0.5 * (((2.0 + x) + x) * math.exp(-x))
	else:
		tmp = (math.exp(((eps_m - 1.0) * x)) + math.exp(((-1.0 - eps_m) * x))) * 0.5
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 1.4e-8)
		tmp = Float64(0.5 * Float64(Float64(Float64(2.0 + x) + x) * exp(Float64(-x))));
	else
		tmp = Float64(Float64(exp(Float64(Float64(eps_m - 1.0) * x)) + exp(Float64(Float64(-1.0 - eps_m) * x))) * 0.5);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 1.4e-8)
		tmp = 0.5 * (((2.0 + x) + x) * exp(-x));
	else
		tmp = (exp(((eps_m - 1.0) * x)) + exp(((-1.0 - eps_m) * x))) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 1.4e-8], N[(0.5 * N[(N[(N[(2.0 + x), $MachinePrecision] + x), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[N[(N[(eps$95$m - 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(N[(-1.0 - eps$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.4e-8

   1. Initial program 68.5%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 69.5%

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

   if 1.4e-8 < 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 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 100.0%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.4 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(\left(\left(2 + x\right) + x\right) \cdot e^{-x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\left(\varepsilon - 1\right) \cdot x} + e^{\left(-1 - \varepsilon\right) \cdot x}\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 1.4 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(\left(\left(2 + x\right) + x\right) \cdot e^{-x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\left(-x\right) \cdot eps\_m} + e^{\left(eps\_m - 1\right) \cdot x}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 1.4e-8)
   (* 0.5 (* (+ (+ 2.0 x) x) (exp (- x))))
   (* (+ (exp (* (- x) eps_m)) (exp (* (- eps_m 1.0) x))) 0.5)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.4e-8) {
		tmp = 0.5 * (((2.0 + x) + x) * exp(-x));
	} else {
		tmp = (exp((-x * eps_m)) + exp(((eps_m - 1.0) * x))) * 0.5;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 1.4d-8) then
        tmp = 0.5d0 * (((2.0d0 + x) + x) * exp(-x))
    else
        tmp = (exp((-x * eps_m)) + exp(((eps_m - 1.0d0) * x))) * 0.5d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.4e-8) {
		tmp = 0.5 * (((2.0 + x) + x) * Math.exp(-x));
	} else {
		tmp = (Math.exp((-x * eps_m)) + Math.exp(((eps_m - 1.0) * x))) * 0.5;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 1.4e-8:
		tmp = 0.5 * (((2.0 + x) + x) * math.exp(-x))
	else:
		tmp = (math.exp((-x * eps_m)) + math.exp(((eps_m - 1.0) * x))) * 0.5
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 1.4e-8)
		tmp = Float64(0.5 * Float64(Float64(Float64(2.0 + x) + x) * exp(Float64(-x))));
	else
		tmp = Float64(Float64(exp(Float64(Float64(-x) * eps_m)) + exp(Float64(Float64(eps_m - 1.0) * x))) * 0.5);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 1.4e-8)
		tmp = 0.5 * (((2.0 + x) + x) * exp(-x));
	else
		tmp = (exp((-x * eps_m)) + exp(((eps_m - 1.0) * x))) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 1.4e-8], N[(0.5 * N[(N[(N[(2.0 + x), $MachinePrecision] + x), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[N[((-x) * eps$95$m), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(N[(eps$95$m - 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.4e-8

   1. Initial program 68.5%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 69.5%

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

   if 1.4e-8 < 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 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 100.0%

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

   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 100.0%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.4 \cdot 10^{-8}:\\ \;\;\;\;0.5 \cdot \left(\left(\left(2 + x\right) + x\right) \cdot e^{-x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\left(-x\right) \cdot \varepsilon} + e^{\left(\varepsilon - 1\right) \cdot x}\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 0.0152:\\ \;\;\;\;0.5 \cdot \left(\left(\left(2 + x\right) + x\right) \cdot e^{-x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(eps\_m - 1\right) \cdot x} \cdot \left(\frac{1}{eps\_m} + 1\right) - \mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot x, 1 + eps\_m, -1\right) \cdot \left(1 + eps\_m\right), x, 1\right) \cdot \left(\frac{1}{eps\_m} - 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 0.0152)
   (* 0.5 (* (+ (+ 2.0 x) x) (exp (- x))))
   (/
    (-
     (* (exp (* (- eps_m 1.0) x)) (+ (/ 1.0 eps_m) 1.0))
     (*
      (fma (* (fma (* 0.5 x) (+ 1.0 eps_m) -1.0) (+ 1.0 eps_m)) x 1.0)
      (- (/ 1.0 eps_m) 1.0)))
    2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.0152) {
		tmp = 0.5 * (((2.0 + x) + x) * exp(-x));
	} else {
		tmp = ((exp(((eps_m - 1.0) * x)) * ((1.0 / eps_m) + 1.0)) - (fma((fma((0.5 * x), (1.0 + eps_m), -1.0) * (1.0 + eps_m)), x, 1.0) * ((1.0 / eps_m) - 1.0))) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 0.0152)
		tmp = Float64(0.5 * Float64(Float64(Float64(2.0 + x) + x) * exp(Float64(-x))));
	else
		tmp = Float64(Float64(Float64(exp(Float64(Float64(eps_m - 1.0) * x)) * Float64(Float64(1.0 / eps_m) + 1.0)) - Float64(fma(Float64(fma(Float64(0.5 * x), Float64(1.0 + eps_m), -1.0) * Float64(1.0 + eps_m)), x, 1.0) * Float64(Float64(1.0 / eps_m) - 1.0))) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 0.0152], N[(0.5 * N[(N[(N[(2.0 + x), $MachinePrecision] + x), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Exp[N[(N[(eps$95$m - 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(N[(0.5 * x), $MachinePrecision] * N[(1.0 + eps$95$m), $MachinePrecision] + -1.0), $MachinePrecision] * N[(1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] * N[(N[(1.0 / eps$95$m), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 0.0152

   1. Initial program 68.8%

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

   3. Taylor expanded in 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 69.8%

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

   if 0.0152 < 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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 90.3

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

   5. Applied rewrites 90.3%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.0152:\\ \;\;\;\;0.5 \cdot \left(\left(\left(2 + x\right) + x\right) \cdot e^{-x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(\varepsilon - 1\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \mathsf{fma}\left(\mathsf{fma}\left(0.5 \cdot x, 1 + \varepsilon, -1\right) \cdot \left(1 + \varepsilon\right), x, 1\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 0.0152:\\ \;\;\;\;0.5 \cdot \left(\left(\left(2 + x\right) + x\right) \cdot e^{-x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(eps\_m, x, x\right) \cdot \left(1 + eps\_m\right), 0.5, -1 - eps\_m\right), x, 1\right) + e^{\left(eps\_m - 1\right) \cdot x}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 0.0152)
   (* 0.5 (* (+ (+ 2.0 x) x) (exp (- x))))
   (*
    (+
     (fma (fma (* (fma eps_m x x) (+ 1.0 eps_m)) 0.5 (- -1.0 eps_m)) x 1.0)
     (exp (* (- eps_m 1.0) x)))
    0.5)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.0152) {
		tmp = 0.5 * (((2.0 + x) + x) * exp(-x));
	} else {
		tmp = (fma(fma((fma(eps_m, x, x) * (1.0 + eps_m)), 0.5, (-1.0 - eps_m)), x, 1.0) + exp(((eps_m - 1.0) * x))) * 0.5;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 0.0152)
		tmp = Float64(0.5 * Float64(Float64(Float64(2.0 + x) + x) * exp(Float64(-x))));
	else
		tmp = Float64(Float64(fma(fma(Float64(fma(eps_m, x, x) * Float64(1.0 + eps_m)), 0.5, Float64(-1.0 - eps_m)), x, 1.0) + exp(Float64(Float64(eps_m - 1.0) * x))) * 0.5);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 0.0152], N[(0.5 * N[(N[(N[(2.0 + x), $MachinePrecision] + x), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[(eps$95$m * x + x), $MachinePrecision] * N[(1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] + N[Exp[N[(N[(eps$95$m - 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 0.0152

   1. Initial program 68.8%

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

   3. Taylor expanded in 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 69.8%

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

   if 0.0152 < 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 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 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 90.3%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.0152:\\ \;\;\;\;0.5 \cdot \left(\left(\left(2 + x\right) + x\right) \cdot e^{-x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, x, x\right) \cdot \left(1 + \varepsilon\right), 0.5, -1 - \varepsilon\right), x, 1\right) + e^{\left(\varepsilon - 1\right) \cdot x}\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 9.4 \cdot 10^{-172}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(1 - x\right) \cdot x\right) \cdot eps\_m\right) \cdot eps\_m, x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1 - eps\_m, 1 - eps\_m, \left(-1 - eps\_m\right) \cdot \left(-1 - eps\_m\right)\right), 0.5, \mathsf{fma}\left(0.16666666666666666 \cdot \left(eps\_m - 1\right), \left(eps\_m - 1\right) \cdot \left(eps\_m - 1\right), 1 \cdot \left(-0.16666666666666666 \cdot \left(1 + eps\_m\right)\right)\right) \cdot x\right), x, \left(eps\_m - 1\right) - eps\_m\right) - 1, x, 2\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 9.4e-172)
   (* (fma (* (* (* (- 1.0 x) x) eps_m) eps_m) x 2.0) 0.5)
   (*
    (fma
     (-
      (fma
       (fma
        (fma (- 1.0 eps_m) (- 1.0 eps_m) (* (- -1.0 eps_m) (- -1.0 eps_m)))
        0.5
        (*
         (fma
          (* 0.16666666666666666 (- eps_m 1.0))
          (* (- eps_m 1.0) (- eps_m 1.0))
          (* 1.0 (* -0.16666666666666666 (+ 1.0 eps_m))))
         x))
       x
       (- (- eps_m 1.0) eps_m))
      1.0)
     x
     2.0)
    0.5)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 9.4e-172) {
		tmp = fma(((((1.0 - x) * x) * eps_m) * eps_m), x, 2.0) * 0.5;
	} else {
		tmp = fma((fma(fma(fma((1.0 - eps_m), (1.0 - eps_m), ((-1.0 - eps_m) * (-1.0 - eps_m))), 0.5, (fma((0.16666666666666666 * (eps_m - 1.0)), ((eps_m - 1.0) * (eps_m - 1.0)), (1.0 * (-0.16666666666666666 * (1.0 + eps_m)))) * x)), x, ((eps_m - 1.0) - eps_m)) - 1.0), x, 2.0) * 0.5;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 9.4e-172)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(1.0 - x) * x) * eps_m) * eps_m), x, 2.0) * 0.5);
	else
		tmp = Float64(fma(Float64(fma(fma(fma(Float64(1.0 - eps_m), Float64(1.0 - eps_m), Float64(Float64(-1.0 - eps_m) * Float64(-1.0 - eps_m))), 0.5, Float64(fma(Float64(0.16666666666666666 * Float64(eps_m - 1.0)), Float64(Float64(eps_m - 1.0) * Float64(eps_m - 1.0)), Float64(1.0 * Float64(-0.16666666666666666 * Float64(1.0 + eps_m)))) * x)), x, Float64(Float64(eps_m - 1.0) - eps_m)) - 1.0), x, 2.0) * 0.5);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 9.4e-172], N[(N[(N[(N[(N[(N[(1.0 - x), $MachinePrecision] * x), $MachinePrecision] * eps$95$m), $MachinePrecision] * eps$95$m), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(1.0 - eps$95$m), $MachinePrecision] * N[(1.0 - eps$95$m), $MachinePrecision] + N[(N[(-1.0 - eps$95$m), $MachinePrecision] * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(N[(N[(0.16666666666666666 * N[(eps$95$m - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(eps$95$m - 1.0), $MachinePrecision] * N[(eps$95$m - 1.0), $MachinePrecision]), $MachinePrecision] + N[(1.0 * N[(-0.16666666666666666 * N[(1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x + N[(N[(eps$95$m - 1.0), $MachinePrecision] - eps$95$m), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.39999999999999952e-172

   1. Initial program 70.1%

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

   3. Taylor expanded in eps around 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 97.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 53.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \varepsilon, 1 - \varepsilon, \left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right), 0.5, \mathsf{fma}\left(0.16666666666666666 \cdot \left(\varepsilon - 1\right), \left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right), \left(-0.16666666666666666 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right) \cdot x\right), x, \left(\varepsilon - 1\right) - \varepsilon\right) - 1, x, 2\right) \cdot 0.5 \]

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 89.1%

       \[\leadsto \mathsf{fma}\left(\left(\left(\left(1 - x\right) \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon, x, 2\right) \cdot 0.5 \]

   if 9.39999999999999952e-172 < x

   1. Initial program 85.9%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 98.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 27.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \varepsilon, 1 - \varepsilon, \left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right), 0.5, \mathsf{fma}\left(0.16666666666666666 \cdot \left(\varepsilon - 1\right), \left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right), \left(-0.16666666666666666 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right) \cdot x\right), x, \left(\varepsilon - 1\right) - \varepsilon\right) - 1, x, 2\right) \cdot 0.5 \]

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 43.4%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \varepsilon, 1 - \varepsilon, \left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right), 0.5, \mathsf{fma}\left(0.16666666666666666 \cdot \left(\varepsilon - 1\right), \left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right), \left(-0.16666666666666666 \cdot \left(1 + \varepsilon\right)\right) \cdot 1\right) \cdot x\right), x, \left(\varepsilon - 1\right) - \varepsilon\right) - 1, x, 2\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.4 \cdot 10^{-172}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(1 - x\right) \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon, x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \varepsilon, 1 - \varepsilon, \left(-1 - \varepsilon\right) \cdot \left(-1 - \varepsilon\right)\right), 0.5, \mathsf{fma}\left(0.16666666666666666 \cdot \left(\varepsilon - 1\right), \left(\varepsilon - 1\right) \cdot \left(\varepsilon - 1\right), 1 \cdot \left(-0.16666666666666666 \cdot \left(1 + \varepsilon\right)\right)\right) \cdot x\right), x, \left(\varepsilon - 1\right) - \varepsilon\right) - 1, x, 2\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.5% accurate, 7.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \left(1 + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(eps\_m, x, x\right) \cdot \left(1 + eps\_m\right), 0.5, -1 - eps\_m\right), x, 1\right)\right) \cdot 0.5 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (*
  (+
   1.0
   (fma (fma (* (fma eps_m x x) (+ 1.0 eps_m)) 0.5 (- -1.0 eps_m)) x 1.0))
  0.5))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return (1.0 + fma(fma((fma(eps_m, x, x) * (1.0 + eps_m)), 0.5, (-1.0 - eps_m)), x, 1.0)) * 0.5;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return Float64(Float64(1.0 + fma(fma(Float64(fma(eps_m, x, x) * Float64(1.0 + eps_m)), 0.5, Float64(-1.0 - eps_m)), x, 1.0)) * 0.5)
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := N[(N[(1.0 + N[(N[(N[(N[(eps$95$m * x + x), $MachinePrecision] * N[(1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 76.9%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
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 98.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 60.4%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 75.1%

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

12. Final simplification 75.1%

    \[\leadsto \left(1 + \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, x, x\right) \cdot \left(1 + \varepsilon\right), 0.5, -1 - \varepsilon\right), x, 1\right)\right) \cdot 0.5 \]
13. Add Preprocessing

Alternative 11: 72.9% accurate, 7.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.32 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(1 - x\right) \cdot x\right) \cdot eps\_m\right) \cdot eps\_m, x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 0.5, -1\right), x, 1\right) \cdot \left(\left(2 + x\right) + x\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1.32e+20)
   (* (fma (* (* (* (- 1.0 x) x) eps_m) eps_m) x 2.0) 0.5)
   (* (* (fma (fma x 0.5 -1.0) x 1.0) (+ (+ 2.0 x) x)) 0.5)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.32e+20) {
		tmp = fma(((((1.0 - x) * x) * eps_m) * eps_m), x, 2.0) * 0.5;
	} else {
		tmp = (fma(fma(x, 0.5, -1.0), x, 1.0) * ((2.0 + x) + x)) * 0.5;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 1.32e+20)
		tmp = Float64(fma(Float64(Float64(Float64(Float64(1.0 - x) * x) * eps_m) * eps_m), x, 2.0) * 0.5);
	else
		tmp = Float64(Float64(fma(fma(x, 0.5, -1.0), x, 1.0) * Float64(Float64(2.0 + x) + x)) * 0.5);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 1.32e+20], N[(N[(N[(N[(N[(N[(1.0 - x), $MachinePrecision] * x), $MachinePrecision] * eps$95$m), $MachinePrecision] * eps$95$m), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(x * 0.5 + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] * N[(N[(2.0 + x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.32e20

   1. Initial program 67.6%

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

   3. Taylor expanded in eps around 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 97.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 55.3%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \varepsilon, 1 - \varepsilon, \left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right), 0.5, \mathsf{fma}\left(0.16666666666666666 \cdot \left(\varepsilon - 1\right), \left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right), \left(-0.16666666666666666 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right) \cdot x\right), x, \left(\varepsilon - 1\right) - \varepsilon\right) - 1, x, 2\right) \cdot 0.5 \]

   9. Taylor expanded in eps around inf

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

   11. Applied rewrites 86.6%

       \[\leadsto \mathsf{fma}\left(\left(\left(\left(1 - x\right) \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon, x, 2\right) \cdot 0.5 \]

   if 1.32e20 < 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 50.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 42.6%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.32 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(1 - x\right) \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon, x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 0.5, -1\right), x, 1\right) \cdot \left(\left(2 + x\right) + x\right)\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.1% accurate, 9.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -210000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 1\right), x, -2\right), x, 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -210000.0)
   (* (fma (fma (fma -0.3333333333333333 x 1.0) x -2.0) x 2.0) 0.5)
   (fma (fma 0.3333333333333333 x -0.5) (* x x) 1.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -210000.0) {
		tmp = fma(fma(fma(-0.3333333333333333, x, 1.0), x, -2.0), x, 2.0) * 0.5;
	} else {
		tmp = fma(fma(0.3333333333333333, x, -0.5), (x * x), 1.0);
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -210000.0)
		tmp = Float64(fma(fma(fma(-0.3333333333333333, x, 1.0), x, -2.0), x, 2.0) * 0.5);
	else
		tmp = fma(fma(0.3333333333333333, x, -0.5), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -210000.0], N[(N[(N[(N[(-0.3333333333333333 * x + 1.0), $MachinePrecision] * x + -2.0), $MachinePrecision] * x + 2.0), $MachinePrecision] * 0.5), $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 < -2.1e5

   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 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 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 31.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(1 - \varepsilon, 1 - \varepsilon, \left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right), 0.5, \mathsf{fma}\left(0.16666666666666666 \cdot \left(\varepsilon - 1\right), \left(1 - \varepsilon\right) \cdot \left(1 - \varepsilon\right), \left(-0.16666666666666666 \cdot \left(1 + \varepsilon\right)\right) \cdot \left(\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right)\right)\right) \cdot x\right), x, \left(\varepsilon - 1\right) - \varepsilon\right) - 1, x, 2\right) \cdot 0.5 \]

   9. Taylor expanded in eps around 0

      \[\leadsto \mathsf{fma}\left(x \cdot \left(1 + \frac{-1}{3} \cdot x\right) - 2, x, 2\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 67.3%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, x, 1\right), x, -2\right), x, 2\right) \cdot 0.5 \]

   if -2.1e5 < x

   1. Initial program 73.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 65.5%

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

   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 60.3%

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

Alternative 13: 59.8% accurate, 11.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -62:\\ \;\;\;\;\left(\mathsf{fma}\left(-1 - eps\_m, x, 1\right) + 1\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -62.0)
   (* (+ (fma (- -1.0 eps_m) x 1.0) 1.0) 0.5)
   (fma (fma 0.3333333333333333 x -0.5) (* x x) 1.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -62.0) {
		tmp = (fma((-1.0 - eps_m), x, 1.0) + 1.0) * 0.5;
	} else {
		tmp = fma(fma(0.3333333333333333, x, -0.5), (x * x), 1.0);
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -62.0)
		tmp = Float64(Float64(fma(Float64(-1.0 - eps_m), x, 1.0) + 1.0) * 0.5);
	else
		tmp = fma(fma(0.3333333333333333, x, -0.5), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -62.0], N[(N[(N[(N[(-1.0 - eps$95$m), $MachinePrecision] * x + 1.0), $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $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 < -62

   1. Initial program 97.3%

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

   3. Taylor expanded in eps around 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 97.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 44.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 20.4%

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

   if -62 < x

   1. Initial program 73.4%

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

   3. Taylor expanded in 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 65.6%

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

   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 60.9%

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

Alternative 14: 52.5% accurate, 15.2× speedup

Math

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

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (fma (fma 0.3333333333333333 x -0.5) (* x x) 1.0))

C

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

Julia

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := N[(N[(0.3333333333333333 * x + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 76.9%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
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 56.6%

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

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 52.1%

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

Alternative 15: 44.0% accurate, 273.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ 1 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 1.0)

C

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

Fortran

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

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return 1.0;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return 1.0

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return 1.0
end

MATLAB

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := 1.0

Derivation

1. Initial program 76.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 41.0%

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

Reproduce

herbie shell --seed 2024240 
(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))