NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.6% → 99.8%

Time: 13.1s

Alternatives: 18

Speedup: 0.4×

• 38.3% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -3.885617184918427e+280
  2. eps = -3.6463083903105903e-14

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: 73.6% 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: 99.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 57.4%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 100.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in eps around inf

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

      1. lower-*.f64 100.0

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

   10. Applied rewrites 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 77.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x} \leq 4:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{{\varepsilon}^{-1} - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (-
       (* (+ 1.0 (pow eps -1.0)) (exp (* (+ -1.0 eps) x)))
       (* (- (pow eps -1.0) 1.0) (exp (* (- -1.0 eps) x))))
      4.0)
   (* (* 2.0 (/ (+ 1.0 x) (exp x))) 0.5)
   (/ (- (pow eps -1.0) (/ -1.0 (exp (fma x eps x)))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if ((((1.0 + pow(eps, -1.0)) * exp(((-1.0 + eps) * x))) - ((pow(eps, -1.0) - 1.0) * exp(((-1.0 - eps) * x)))) <= 4.0) {
		tmp = (2.0 * ((1.0 + x) / exp(x))) * 0.5;
	} else {
		tmp = (pow(eps, -1.0) - (-1.0 / exp(fma(x, eps, x)))) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 + (eps ^ -1.0)) * exp(Float64(Float64(-1.0 + eps) * x))) - Float64(Float64((eps ^ -1.0) - 1.0) * exp(Float64(Float64(-1.0 - eps) * x)))) <= 4.0)
		tmp = Float64(Float64(2.0 * Float64(Float64(1.0 + x) / exp(x))) * 0.5);
	else
		tmp = Float64(Float64((eps ^ -1.0) - Float64(-1.0 / exp(fma(x, eps, x)))) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[(N[(1.0 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4.0], N[(N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[Power[eps, -1.0], $MachinePrecision] - N[(-1.0 / N[Exp[N[(x * eps + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 57.9%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 71.0

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

   5. Applied rewrites 71.0%

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

   6. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

      2. neg-mul-1 N/A

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

      3. lower-exp.f64 N/A

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

      4. neg-mul-1 N/A

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

      5. lower-neg.f64 11.6

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

   8. Applied rewrites 11.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 3.1%

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

   12. Taylor expanded in eps around inf

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

       1. exp-neg N/A

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

       2. associate-*r/ N/A

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

       3. metadata-eval N/A

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

       4. lower-/.f64 N/A

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

       5. lower-exp.f64 N/A

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

       6. +-commutative N/A

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

       7. distribute-lft-in N/A

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

       8. *-rgt-identity N/A

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

       9. lower-fma.f64 31.1

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

   14. Applied rewrites 31.1%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x} \leq 4:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{{\varepsilon}^{-1} - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x}\\ \mathbf{if}\;t\_0 - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x} \leq 4:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - -1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (+ 1.0 (pow eps -1.0)) (exp (* (+ -1.0 eps) x)))))
   (if (<= (- t_0 (* (- (pow eps -1.0) 1.0) (exp (* (- -1.0 eps) x)))) 4.0)
     (* (* 2.0 (/ (+ 1.0 x) (exp x))) 0.5)
     (/ (- t_0 -1.0) 2.0))))

C

double code(double x, double eps) {
	double t_0 = (1.0 + pow(eps, -1.0)) * exp(((-1.0 + eps) * x));
	double tmp;
	if ((t_0 - ((pow(eps, -1.0) - 1.0) * exp(((-1.0 - eps) * x)))) <= 4.0) {
		tmp = (2.0 * ((1.0 + x) / exp(x))) * 0.5;
	} else {
		tmp = (t_0 - -1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + (eps ** (-1.0d0))) * exp((((-1.0d0) + eps) * x))
    if ((t_0 - (((eps ** (-1.0d0)) - 1.0d0) * exp((((-1.0d0) - eps) * x)))) <= 4.0d0) then
        tmp = (2.0d0 * ((1.0d0 + x) / exp(x))) * 0.5d0
    else
        tmp = (t_0 - (-1.0d0)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = (1.0 + Math.pow(eps, -1.0)) * Math.exp(((-1.0 + eps) * x));
	double tmp;
	if ((t_0 - ((Math.pow(eps, -1.0) - 1.0) * Math.exp(((-1.0 - eps) * x)))) <= 4.0) {
		tmp = (2.0 * ((1.0 + x) / Math.exp(x))) * 0.5;
	} else {
		tmp = (t_0 - -1.0) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (1.0 + math.pow(eps, -1.0)) * math.exp(((-1.0 + eps) * x))
	tmp = 0
	if (t_0 - ((math.pow(eps, -1.0) - 1.0) * math.exp(((-1.0 - eps) * x)))) <= 4.0:
		tmp = (2.0 * ((1.0 + x) / math.exp(x))) * 0.5
	else:
		tmp = (t_0 - -1.0) / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(1.0 + (eps ^ -1.0)) * exp(Float64(Float64(-1.0 + eps) * x)))
	tmp = 0.0
	if (Float64(t_0 - Float64(Float64((eps ^ -1.0) - 1.0) * exp(Float64(Float64(-1.0 - eps) * x)))) <= 4.0)
		tmp = Float64(Float64(2.0 * Float64(Float64(1.0 + x) / exp(x))) * 0.5);
	else
		tmp = Float64(Float64(t_0 - -1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (1.0 + (eps ^ -1.0)) * exp(((-1.0 + eps) * x));
	tmp = 0.0;
	if ((t_0 - (((eps ^ -1.0) - 1.0) * exp(((-1.0 - eps) * x)))) <= 4.0)
		tmp = (2.0 * ((1.0 + x) / exp(x))) * 0.5;
	else
		tmp = (t_0 - -1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(1.0 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[(N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4.0], N[(N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(t$95$0 - -1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 57.9%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 71.0%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x} \leq 4:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - -1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x} \leq 4:\\ \;\;\;\;\frac{1 + x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, 0.1111111111111111, -0.25\right) \cdot \left(x \cdot x\right)}{\mathsf{fma}\left(0.3333333333333333, x, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (-
       (* (+ 1.0 (pow eps -1.0)) (exp (* (+ -1.0 eps) x)))
       (* (- (pow eps -1.0) 1.0) (exp (* (- -1.0 eps) x))))
      4.0)
   (/ (+ 1.0 x) (fma (fma (fma 0.16666666666666666 x 0.5) x 1.0) x 1.0))
   (/
    (* (fma (* x x) 0.1111111111111111 -0.25) (* x x))
    (fma 0.3333333333333333 x 0.5))))

C

double code(double x, double eps) {
	double tmp;
	if ((((1.0 + pow(eps, -1.0)) * exp(((-1.0 + eps) * x))) - ((pow(eps, -1.0) - 1.0) * exp(((-1.0 - eps) * x)))) <= 4.0) {
		tmp = (1.0 + x) / fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0);
	} else {
		tmp = (fma((x * x), 0.1111111111111111, -0.25) * (x * x)) / fma(0.3333333333333333, x, 0.5);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 + (eps ^ -1.0)) * exp(Float64(Float64(-1.0 + eps) * x))) - Float64(Float64((eps ^ -1.0) - 1.0) * exp(Float64(Float64(-1.0 - eps) * x)))) <= 4.0)
		tmp = Float64(Float64(1.0 + x) / fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0));
	else
		tmp = Float64(Float64(fma(Float64(x * x), 0.1111111111111111, -0.25) * Float64(x * x)) / fma(0.3333333333333333, x, 0.5));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[(N[(1.0 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4.0], N[(N[(1.0 + x), $MachinePrecision] / N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.1111111111111111 + -0.25), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(0.3333333333333333 * x + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 57.9%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 89.4%

      \[\leadsto \left(2 \cdot \frac{1 + x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}\right) \cdot 0.5 \]
   9. Step-by-step derivation

   10. Applied rewrites 89.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 1.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 27.9%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 27.3%

       \[\leadsto \left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right) \cdot x\right) \cdot x \]
   12. Step-by-step derivation

   13. Applied rewrites 31.6%

       \[\leadsto \frac{\mathsf{fma}\left(x \cdot x, 0.1111111111111111, -0.25\right) \cdot \left(x \cdot x\right)}{\mathsf{fma}\left(0.3333333333333333, x, 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x} \leq 4:\\ \;\;\;\;\frac{1 + x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x \cdot x, 0.1111111111111111, -0.25\right) \cdot \left(x \cdot x\right)}{\mathsf{fma}\left(0.3333333333333333, x, 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x} \leq 0.2:\\ \;\;\;\;\frac{1 + x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x + 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (-
       (* (+ 1.0 (pow eps -1.0)) (exp (* (+ -1.0 eps) x)))
       (* (- (pow eps -1.0) 1.0) (exp (* (- -1.0 eps) x))))
      0.2)
   (/ (+ 1.0 x) (fma (fma (fma 0.16666666666666666 x 0.5) x 1.0) x 1.0))
   (* (+ x 1.0) (fma (fma 0.5 x -1.0) x 1.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((((1.0 + pow(eps, -1.0)) * exp(((-1.0 + eps) * x))) - ((pow(eps, -1.0) - 1.0) * exp(((-1.0 - eps) * x)))) <= 0.2) {
		tmp = (1.0 + x) / fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0);
	} else {
		tmp = (x + 1.0) * fma(fma(0.5, x, -1.0), x, 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 + (eps ^ -1.0)) * exp(Float64(Float64(-1.0 + eps) * x))) - Float64(Float64((eps ^ -1.0) - 1.0) * exp(Float64(Float64(-1.0 - eps) * x)))) <= 0.2)
		tmp = Float64(Float64(1.0 + x) / fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0));
	else
		tmp = Float64(Float64(x + 1.0) * fma(fma(0.5, x, -1.0), x, 1.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[(N[(1.0 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.2], N[(N[(1.0 + x), $MachinePrecision] / N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x + 1.0), $MachinePrecision] * N[(N[(0.5 * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 0.20000000000000001

   1. Initial program 42.5%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 86.0%

      \[\leadsto \left(2 \cdot \frac{1 + x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}\right) \cdot 0.5 \]
   9. Step-by-step derivation

   10. Applied rewrites 86.0%

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

   if 0.20000000000000001 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 28.1%

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

   7. Applied rewrites 28.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 47.2%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right) \cdot e^{\left(-1 - \varepsilon\right) \cdot x} \leq 0.2:\\ \;\;\;\;\frac{1 + x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x + 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.95 \cdot 10^{-13}:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{elif}\;\varepsilon \leq 1.72 \cdot 10^{+103} \lor \neg \left(\varepsilon \leq 8 \cdot 10^{+184}\right):\\ \;\;\;\;\frac{\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left({\varepsilon}^{-1} + 1\right) - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 1.95e-13)
   (* (* 2.0 (/ (+ 1.0 x) (exp x))) 0.5)
   (if (or (<= eps 1.72e+103) (not (<= eps 8e+184)))
     (/
      (-
       (* (+ 1.0 (pow eps -1.0)) (exp (* (+ -1.0 eps) x)))
       (- (pow eps -1.0) 1.0))
      2.0)
     (/ (- (+ (pow eps -1.0) 1.0) (/ -1.0 (exp (fma x eps x)))) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 1.95e-13) {
		tmp = (2.0 * ((1.0 + x) / exp(x))) * 0.5;
	} else if ((eps <= 1.72e+103) || !(eps <= 8e+184)) {
		tmp = (((1.0 + pow(eps, -1.0)) * exp(((-1.0 + eps) * x))) - (pow(eps, -1.0) - 1.0)) / 2.0;
	} else {
		tmp = ((pow(eps, -1.0) + 1.0) - (-1.0 / exp(fma(x, eps, x)))) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 1.95e-13)
		tmp = Float64(Float64(2.0 * Float64(Float64(1.0 + x) / exp(x))) * 0.5);
	elseif ((eps <= 1.72e+103) || !(eps <= 8e+184))
		tmp = Float64(Float64(Float64(Float64(1.0 + (eps ^ -1.0)) * exp(Float64(Float64(-1.0 + eps) * x))) - Float64((eps ^ -1.0) - 1.0)) / 2.0);
	else
		tmp = Float64(Float64(Float64((eps ^ -1.0) + 1.0) - Float64(-1.0 / exp(fma(x, eps, x)))) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 1.95e-13], N[(N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[Or[LessEqual[eps, 1.72e+103], N[Not[LessEqual[eps, 8e+184]], $MachinePrecision]], N[(N[(N[(N[(1.0 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[Power[eps, -1.0], $MachinePrecision] + 1.0), $MachinePrecision] - N[(-1.0 / N[Exp[N[(x * eps + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 1.95000000000000002e-13

   1. Initial program 65.6%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 70.3%

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

   if 1.95000000000000002e-13 < eps < 1.72e103 or 8.00000000000000014e184 < 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

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 83.1

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

   5. Applied rewrites 83.1%

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

   if 1.72e103 < eps < 8.00000000000000014e184

   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

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 76.8

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

   8. Applied rewrites 76.8%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.95 \cdot 10^{-13}:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{elif}\;\varepsilon \leq 1.72 \cdot 10^{+103} \lor \neg \left(\varepsilon \leq 8 \cdot 10^{+184}\right):\\ \;\;\;\;\frac{\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - \left({\varepsilon}^{-1} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left({\varepsilon}^{-1} + 1\right) - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\varepsilon}^{-1} + 1\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{+146}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, \mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), \frac{-1}{\varepsilon}\right), x, {\varepsilon}^{-1}\right) - \left({\varepsilon}^{-1} - 1\right)}{2}\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-176}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, t\_0, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+246}:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, t\_0, {\varepsilon}^{-1}\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (pow eps -1.0) 1.0)))
   (if (<= x -7.2e+146)
     (/
      (-
       (fma
        (fma (/ x eps) (fma -0.16666666666666666 x 0.5) (/ -1.0 eps))
        x
        (pow eps -1.0))
       (- (pow eps -1.0) 1.0))
      2.0)
     (if (<= x -1.4e-176)
       (fma (* 0.5 x) (fma (- eps 1.0) t_0 (/ (- 1.0 (* eps eps)) eps)) 1.0)
       (if (<= x 5.4e+246)
         (* (* 2.0 (/ (+ 1.0 x) (exp x))) 0.5)
         (fma (* 0.5 x) (fma (- eps 1.0) t_0 (pow eps -1.0)) 1.0))))))

C

double code(double x, double eps) {
	double t_0 = pow(eps, -1.0) + 1.0;
	double tmp;
	if (x <= -7.2e+146) {
		tmp = (fma(fma((x / eps), fma(-0.16666666666666666, x, 0.5), (-1.0 / eps)), x, pow(eps, -1.0)) - (pow(eps, -1.0) - 1.0)) / 2.0;
	} else if (x <= -1.4e-176) {
		tmp = fma((0.5 * x), fma((eps - 1.0), t_0, ((1.0 - (eps * eps)) / eps)), 1.0);
	} else if (x <= 5.4e+246) {
		tmp = (2.0 * ((1.0 + x) / exp(x))) * 0.5;
	} else {
		tmp = fma((0.5 * x), fma((eps - 1.0), t_0, pow(eps, -1.0)), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((eps ^ -1.0) + 1.0)
	tmp = 0.0
	if (x <= -7.2e+146)
		tmp = Float64(Float64(fma(fma(Float64(x / eps), fma(-0.16666666666666666, x, 0.5), Float64(-1.0 / eps)), x, (eps ^ -1.0)) - Float64((eps ^ -1.0) - 1.0)) / 2.0);
	elseif (x <= -1.4e-176)
		tmp = fma(Float64(0.5 * x), fma(Float64(eps - 1.0), t_0, Float64(Float64(1.0 - Float64(eps * eps)) / eps)), 1.0);
	elseif (x <= 5.4e+246)
		tmp = Float64(Float64(2.0 * Float64(Float64(1.0 + x) / exp(x))) * 0.5);
	else
		tmp = fma(Float64(0.5 * x), fma(Float64(eps - 1.0), t_0, (eps ^ -1.0)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[eps, -1.0], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -7.2e+146], N[(N[(N[(N[(N[(x / eps), $MachinePrecision] * N[(-0.16666666666666666 * x + 0.5), $MachinePrecision] + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision] * x + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision] - N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -1.4e-176], N[(N[(0.5 * x), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * t$95$0 + N[(N[(1.0 - N[(eps * eps), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 5.4e+246], N[(N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(0.5 * x), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * t$95$0 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.1999999999999997e146

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 76.9

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

   5. Applied rewrites 76.9%

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

   6. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

      2. neg-mul-1 N/A

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

      3. lower-exp.f64 N/A

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

      4. neg-mul-1 N/A

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

      5. lower-neg.f64 23.8

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

   8. Applied rewrites 23.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 23.8%

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

   if -7.1999999999999997e146 < x < -1.4000000000000001e-176

   1. Initial program 71.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 45.3%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 51.8%

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

   if -1.4000000000000001e-176 < x < 5.4e246

   1. Initial program 72.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 72.9%

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

   if 5.4e246 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 55.1%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+146}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, \mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), \frac{-1}{\varepsilon}\right), x, {\varepsilon}^{-1}\right) - \left({\varepsilon}^{-1} - 1\right)}{2}\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-176}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, {\varepsilon}^{-1} + 1, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+246}:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, {\varepsilon}^{-1} + 1, {\varepsilon}^{-1}\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\varepsilon}^{-1} + 1\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{+146}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, \mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), \frac{-1}{\varepsilon}\right), x, {\varepsilon}^{-1}\right) - \left({\varepsilon}^{-1} - 1\right)}{2}\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-176}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, t\_0, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+246}:\\ \;\;\;\;\left(x + 1\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, t\_0, {\varepsilon}^{-1}\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (pow eps -1.0) 1.0)))
   (if (<= x -7.2e+146)
     (/
      (-
       (fma
        (fma (/ x eps) (fma -0.16666666666666666 x 0.5) (/ -1.0 eps))
        x
        (pow eps -1.0))
       (- (pow eps -1.0) 1.0))
      2.0)
     (if (<= x -1.4e-176)
       (fma (* 0.5 x) (fma (- eps 1.0) t_0 (/ (- 1.0 (* eps eps)) eps)) 1.0)
       (if (<= x 5.4e+246)
         (* (+ x 1.0) (exp (- x)))
         (fma (* 0.5 x) (fma (- eps 1.0) t_0 (pow eps -1.0)) 1.0))))))

C

double code(double x, double eps) {
	double t_0 = pow(eps, -1.0) + 1.0;
	double tmp;
	if (x <= -7.2e+146) {
		tmp = (fma(fma((x / eps), fma(-0.16666666666666666, x, 0.5), (-1.0 / eps)), x, pow(eps, -1.0)) - (pow(eps, -1.0) - 1.0)) / 2.0;
	} else if (x <= -1.4e-176) {
		tmp = fma((0.5 * x), fma((eps - 1.0), t_0, ((1.0 - (eps * eps)) / eps)), 1.0);
	} else if (x <= 5.4e+246) {
		tmp = (x + 1.0) * exp(-x);
	} else {
		tmp = fma((0.5 * x), fma((eps - 1.0), t_0, pow(eps, -1.0)), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((eps ^ -1.0) + 1.0)
	tmp = 0.0
	if (x <= -7.2e+146)
		tmp = Float64(Float64(fma(fma(Float64(x / eps), fma(-0.16666666666666666, x, 0.5), Float64(-1.0 / eps)), x, (eps ^ -1.0)) - Float64((eps ^ -1.0) - 1.0)) / 2.0);
	elseif (x <= -1.4e-176)
		tmp = fma(Float64(0.5 * x), fma(Float64(eps - 1.0), t_0, Float64(Float64(1.0 - Float64(eps * eps)) / eps)), 1.0);
	elseif (x <= 5.4e+246)
		tmp = Float64(Float64(x + 1.0) * exp(Float64(-x)));
	else
		tmp = fma(Float64(0.5 * x), fma(Float64(eps - 1.0), t_0, (eps ^ -1.0)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[eps, -1.0], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -7.2e+146], N[(N[(N[(N[(N[(x / eps), $MachinePrecision] * N[(-0.16666666666666666 * x + 0.5), $MachinePrecision] + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision] * x + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision] - N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -1.4e-176], N[(N[(0.5 * x), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * t$95$0 + N[(N[(1.0 - N[(eps * eps), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 5.4e+246], N[(N[(x + 1.0), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * x), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * t$95$0 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.1999999999999997e146

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 76.9

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

   5. Applied rewrites 76.9%

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

   6. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

      2. neg-mul-1 N/A

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

      3. lower-exp.f64 N/A

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

      4. neg-mul-1 N/A

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

      5. lower-neg.f64 23.8

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

   8. Applied rewrites 23.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 23.8%

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

   if -7.1999999999999997e146 < x < -1.4000000000000001e-176

   1. Initial program 71.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 45.3%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 51.8%

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

   if -1.4000000000000001e-176 < x < 5.4e246

   1. Initial program 72.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 72.9%

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

   7. Applied rewrites 72.9%

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

   if 5.4e246 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 55.1%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+146}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, \mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), \frac{-1}{\varepsilon}\right), x, {\varepsilon}^{-1}\right) - \left({\varepsilon}^{-1} - 1\right)}{2}\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-176}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, {\varepsilon}^{-1} + 1, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+246}:\\ \;\;\;\;\left(x + 1\right) \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, {\varepsilon}^{-1} + 1, {\varepsilon}^{-1}\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\varepsilon}^{-1} - 1\\ t_1 := {\varepsilon}^{-1} + 1\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{+146}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, \mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), \frac{-1}{\varepsilon}\right), x, {\varepsilon}^{-1}\right) - t\_0}{2}\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-176}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, t\_1, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{elif}\;x \leq 370:\\ \;\;\;\;\frac{1 + x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+246}:\\ \;\;\;\;\frac{t\_1 - t\_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, t\_1, {\varepsilon}^{-1}\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow eps -1.0) 1.0)) (t_1 (+ (pow eps -1.0) 1.0)))
   (if (<= x -7.2e+146)
     (/
      (-
       (fma
        (fma (/ x eps) (fma -0.16666666666666666 x 0.5) (/ -1.0 eps))
        x
        (pow eps -1.0))
       t_0)
      2.0)
     (if (<= x -1.4e-176)
       (fma (* 0.5 x) (fma (- eps 1.0) t_1 (/ (- 1.0 (* eps eps)) eps)) 1.0)
       (if (<= x 370.0)
         (/ (+ 1.0 x) (fma (fma (fma 0.16666666666666666 x 0.5) x 1.0) x 1.0))
         (if (<= x 5.4e+246)
           (/ (- t_1 t_0) 2.0)
           (fma (* 0.5 x) (fma (- eps 1.0) t_1 (pow eps -1.0)) 1.0)))))))

C

double code(double x, double eps) {
	double t_0 = pow(eps, -1.0) - 1.0;
	double t_1 = pow(eps, -1.0) + 1.0;
	double tmp;
	if (x <= -7.2e+146) {
		tmp = (fma(fma((x / eps), fma(-0.16666666666666666, x, 0.5), (-1.0 / eps)), x, pow(eps, -1.0)) - t_0) / 2.0;
	} else if (x <= -1.4e-176) {
		tmp = fma((0.5 * x), fma((eps - 1.0), t_1, ((1.0 - (eps * eps)) / eps)), 1.0);
	} else if (x <= 370.0) {
		tmp = (1.0 + x) / fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0);
	} else if (x <= 5.4e+246) {
		tmp = (t_1 - t_0) / 2.0;
	} else {
		tmp = fma((0.5 * x), fma((eps - 1.0), t_1, pow(eps, -1.0)), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((eps ^ -1.0) - 1.0)
	t_1 = Float64((eps ^ -1.0) + 1.0)
	tmp = 0.0
	if (x <= -7.2e+146)
		tmp = Float64(Float64(fma(fma(Float64(x / eps), fma(-0.16666666666666666, x, 0.5), Float64(-1.0 / eps)), x, (eps ^ -1.0)) - t_0) / 2.0);
	elseif (x <= -1.4e-176)
		tmp = fma(Float64(0.5 * x), fma(Float64(eps - 1.0), t_1, Float64(Float64(1.0 - Float64(eps * eps)) / eps)), 1.0);
	elseif (x <= 370.0)
		tmp = Float64(Float64(1.0 + x) / fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0));
	elseif (x <= 5.4e+246)
		tmp = Float64(Float64(t_1 - t_0) / 2.0);
	else
		tmp = fma(Float64(0.5 * x), fma(Float64(eps - 1.0), t_1, (eps ^ -1.0)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[eps, -1.0], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -7.2e+146], N[(N[(N[(N[(N[(x / eps), $MachinePrecision] * N[(-0.16666666666666666 * x + 0.5), $MachinePrecision] + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision] * x + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -1.4e-176], N[(N[(0.5 * x), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * t$95$1 + N[(N[(1.0 - N[(eps * eps), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 370.0], N[(N[(1.0 + x), $MachinePrecision] / N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.4e+246], N[(N[(t$95$1 - t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(0.5 * x), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * t$95$1 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -7.1999999999999997e146

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 76.9

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

   5. Applied rewrites 76.9%

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

   6. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

      2. neg-mul-1 N/A

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

      3. lower-exp.f64 N/A

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

      4. neg-mul-1 N/A

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

      5. lower-neg.f64 23.8

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

   8. Applied rewrites 23.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 23.8%

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

   if -7.1999999999999997e146 < x < -1.4000000000000001e-176

   1. Initial program 71.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 45.3%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 51.8%

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

   if -1.4000000000000001e-176 < x < 370

   1. Initial program 50.8%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 84.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 84.0%

      \[\leadsto \left(2 \cdot \frac{1 + x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}\right) \cdot 0.5 \]
   9. Step-by-step derivation

   10. Applied rewrites 84.0%

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

   if 370 < x < 5.4e246

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 34.1

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

   5. Applied rewrites 34.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 58.1

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

   8. Applied rewrites 58.1%

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

   if 5.4e246 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 55.1%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+146}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{x}{\varepsilon}, \mathsf{fma}\left(-0.16666666666666666, x, 0.5\right), \frac{-1}{\varepsilon}\right), x, {\varepsilon}^{-1}\right) - \left({\varepsilon}^{-1} - 1\right)}{2}\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-176}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, {\varepsilon}^{-1} + 1, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{elif}\;x \leq 370:\\ \;\;\;\;\frac{1 + x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+246}:\\ \;\;\;\;\frac{\left({\varepsilon}^{-1} + 1\right) - \left({\varepsilon}^{-1} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, {\varepsilon}^{-1} + 1, {\varepsilon}^{-1}\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\varepsilon}^{-1} - 1\\ t_1 := {\varepsilon}^{-1} + 1\\ \mathbf{if}\;x \leq -1.1 \cdot 10^{+183}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{\varepsilon}, x, \frac{-1}{\varepsilon}\right), x, {\varepsilon}^{-1}\right) - t\_0}{2}\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-176}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, t\_1, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{elif}\;x \leq 370:\\ \;\;\;\;\frac{1 + x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+246}:\\ \;\;\;\;\frac{t\_1 - t\_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, t\_1, {\varepsilon}^{-1}\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow eps -1.0) 1.0)) (t_1 (+ (pow eps -1.0) 1.0)))
   (if (<= x -1.1e+183)
     (/ (- (fma (fma (/ 0.5 eps) x (/ -1.0 eps)) x (pow eps -1.0)) t_0) 2.0)
     (if (<= x -1.4e-176)
       (fma (* 0.5 x) (fma (- eps 1.0) t_1 (/ (- 1.0 (* eps eps)) eps)) 1.0)
       (if (<= x 370.0)
         (/ (+ 1.0 x) (fma (fma (fma 0.16666666666666666 x 0.5) x 1.0) x 1.0))
         (if (<= x 5.4e+246)
           (/ (- t_1 t_0) 2.0)
           (fma (* 0.5 x) (fma (- eps 1.0) t_1 (pow eps -1.0)) 1.0)))))))

C

double code(double x, double eps) {
	double t_0 = pow(eps, -1.0) - 1.0;
	double t_1 = pow(eps, -1.0) + 1.0;
	double tmp;
	if (x <= -1.1e+183) {
		tmp = (fma(fma((0.5 / eps), x, (-1.0 / eps)), x, pow(eps, -1.0)) - t_0) / 2.0;
	} else if (x <= -1.4e-176) {
		tmp = fma((0.5 * x), fma((eps - 1.0), t_1, ((1.0 - (eps * eps)) / eps)), 1.0);
	} else if (x <= 370.0) {
		tmp = (1.0 + x) / fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0);
	} else if (x <= 5.4e+246) {
		tmp = (t_1 - t_0) / 2.0;
	} else {
		tmp = fma((0.5 * x), fma((eps - 1.0), t_1, pow(eps, -1.0)), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((eps ^ -1.0) - 1.0)
	t_1 = Float64((eps ^ -1.0) + 1.0)
	tmp = 0.0
	if (x <= -1.1e+183)
		tmp = Float64(Float64(fma(fma(Float64(0.5 / eps), x, Float64(-1.0 / eps)), x, (eps ^ -1.0)) - t_0) / 2.0);
	elseif (x <= -1.4e-176)
		tmp = fma(Float64(0.5 * x), fma(Float64(eps - 1.0), t_1, Float64(Float64(1.0 - Float64(eps * eps)) / eps)), 1.0);
	elseif (x <= 370.0)
		tmp = Float64(Float64(1.0 + x) / fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0));
	elseif (x <= 5.4e+246)
		tmp = Float64(Float64(t_1 - t_0) / 2.0);
	else
		tmp = fma(Float64(0.5 * x), fma(Float64(eps - 1.0), t_1, (eps ^ -1.0)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[eps, -1.0], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.1e+183], N[(N[(N[(N[(N[(0.5 / eps), $MachinePrecision] * x + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision] * x + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -1.4e-176], N[(N[(0.5 * x), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * t$95$1 + N[(N[(1.0 - N[(eps * eps), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 370.0], N[(N[(1.0 + x), $MachinePrecision] / N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.4e+246], N[(N[(t$95$1 - t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(0.5 * x), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * t$95$1 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1.09999999999999995e183

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 73.1

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

   5. Applied rewrites 73.1%

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

   6. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

      2. neg-mul-1 N/A

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

      3. lower-exp.f64 N/A

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

      4. neg-mul-1 N/A

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

      5. lower-neg.f64 27.8

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

   8. Applied rewrites 27.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 12.0%

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

   if -1.09999999999999995e183 < x < -1.4000000000000001e-176

   1. Initial program 72.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 43.1%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 49.1%

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

   if -1.4000000000000001e-176 < x < 370

   1. Initial program 50.8%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 84.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 84.0%

      \[\leadsto \left(2 \cdot \frac{1 + x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}\right) \cdot 0.5 \]
   9. Step-by-step derivation

   10. Applied rewrites 84.0%

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

   if 370 < x < 5.4e246

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 34.1

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

   5. Applied rewrites 34.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 58.1

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

   8. Applied rewrites 58.1%

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

   if 5.4e246 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 55.1%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+183}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{\varepsilon}, x, \frac{-1}{\varepsilon}\right), x, {\varepsilon}^{-1}\right) - \left({\varepsilon}^{-1} - 1\right)}{2}\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-176}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, {\varepsilon}^{-1} + 1, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{elif}\;x \leq 370:\\ \;\;\;\;\frac{1 + x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+246}:\\ \;\;\;\;\frac{\left({\varepsilon}^{-1} + 1\right) - \left({\varepsilon}^{-1} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, {\varepsilon}^{-1} + 1, {\varepsilon}^{-1}\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.65:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{elif}\;\varepsilon \leq 1.72 \cdot 10^{+103} \lor \neg \left(\varepsilon \leq 8 \cdot 10^{+184}\right):\\ \;\;\;\;\frac{\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - -1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left({\varepsilon}^{-1} + 1\right) - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 1.65)
   (* (* 2.0 (/ (+ 1.0 x) (exp x))) 0.5)
   (if (or (<= eps 1.72e+103) (not (<= eps 8e+184)))
     (/ (- (* (+ 1.0 (pow eps -1.0)) (exp (* (+ -1.0 eps) x))) -1.0) 2.0)
     (/ (- (+ (pow eps -1.0) 1.0) (/ -1.0 (exp (fma x eps x)))) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 1.65) {
		tmp = (2.0 * ((1.0 + x) / exp(x))) * 0.5;
	} else if ((eps <= 1.72e+103) || !(eps <= 8e+184)) {
		tmp = (((1.0 + pow(eps, -1.0)) * exp(((-1.0 + eps) * x))) - -1.0) / 2.0;
	} else {
		tmp = ((pow(eps, -1.0) + 1.0) - (-1.0 / exp(fma(x, eps, x)))) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 1.65)
		tmp = Float64(Float64(2.0 * Float64(Float64(1.0 + x) / exp(x))) * 0.5);
	elseif ((eps <= 1.72e+103) || !(eps <= 8e+184))
		tmp = Float64(Float64(Float64(Float64(1.0 + (eps ^ -1.0)) * exp(Float64(Float64(-1.0 + eps) * x))) - -1.0) / 2.0);
	else
		tmp = Float64(Float64(Float64((eps ^ -1.0) + 1.0) - Float64(-1.0 / exp(fma(x, eps, x)))) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 1.65], N[(N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[Or[LessEqual[eps, 1.72e+103], N[Not[LessEqual[eps, 8e+184]], $MachinePrecision]], N[(N[(N[(N[(1.0 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[Power[eps, -1.0], $MachinePrecision] + 1.0), $MachinePrecision] - N[(-1.0 / N[Exp[N[(x * eps + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 1.6499999999999999

   1. Initial program 65.7%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 70.4%

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

   if 1.6499999999999999 < eps < 1.72e103 or 8.00000000000000014e184 < 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

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 97.9

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 80.6%

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

   if 1.72e103 < eps < 8.00000000000000014e184

   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

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 76.8

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

   8. Applied rewrites 76.8%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.65:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{elif}\;\varepsilon \leq 1.72 \cdot 10^{+103} \lor \neg \left(\varepsilon \leq 8 \cdot 10^{+184}\right):\\ \;\;\;\;\frac{\left(1 + {\varepsilon}^{-1}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} - -1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left({\varepsilon}^{-1} + 1\right) - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\varepsilon}^{-1} + 1\\ t_1 := {\varepsilon}^{-1} - 1\\ \mathbf{if}\;x \leq -9.8 \cdot 10^{+42}:\\ \;\;\;\;\frac{{\varepsilon}^{-1} - \mathsf{fma}\left(-1 - \varepsilon, x, 1\right) \cdot t\_1}{2}\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-176}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, t\_0, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{elif}\;x \leq 370:\\ \;\;\;\;\frac{1 + x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+246}:\\ \;\;\;\;\frac{t\_0 - t\_1}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, t\_0, {\varepsilon}^{-1}\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (pow eps -1.0) 1.0)) (t_1 (- (pow eps -1.0) 1.0)))
   (if (<= x -9.8e+42)
     (/ (- (pow eps -1.0) (* (fma (- -1.0 eps) x 1.0) t_1)) 2.0)
     (if (<= x -1.4e-176)
       (fma (* 0.5 x) (fma (- eps 1.0) t_0 (/ (- 1.0 (* eps eps)) eps)) 1.0)
       (if (<= x 370.0)
         (/ (+ 1.0 x) (fma (fma (fma 0.16666666666666666 x 0.5) x 1.0) x 1.0))
         (if (<= x 5.4e+246)
           (/ (- t_0 t_1) 2.0)
           (fma (* 0.5 x) (fma (- eps 1.0) t_0 (pow eps -1.0)) 1.0)))))))

C

double code(double x, double eps) {
	double t_0 = pow(eps, -1.0) + 1.0;
	double t_1 = pow(eps, -1.0) - 1.0;
	double tmp;
	if (x <= -9.8e+42) {
		tmp = (pow(eps, -1.0) - (fma((-1.0 - eps), x, 1.0) * t_1)) / 2.0;
	} else if (x <= -1.4e-176) {
		tmp = fma((0.5 * x), fma((eps - 1.0), t_0, ((1.0 - (eps * eps)) / eps)), 1.0);
	} else if (x <= 370.0) {
		tmp = (1.0 + x) / fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0);
	} else if (x <= 5.4e+246) {
		tmp = (t_0 - t_1) / 2.0;
	} else {
		tmp = fma((0.5 * x), fma((eps - 1.0), t_0, pow(eps, -1.0)), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((eps ^ -1.0) + 1.0)
	t_1 = Float64((eps ^ -1.0) - 1.0)
	tmp = 0.0
	if (x <= -9.8e+42)
		tmp = Float64(Float64((eps ^ -1.0) - Float64(fma(Float64(-1.0 - eps), x, 1.0) * t_1)) / 2.0);
	elseif (x <= -1.4e-176)
		tmp = fma(Float64(0.5 * x), fma(Float64(eps - 1.0), t_0, Float64(Float64(1.0 - Float64(eps * eps)) / eps)), 1.0);
	elseif (x <= 370.0)
		tmp = Float64(Float64(1.0 + x) / fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0));
	elseif (x <= 5.4e+246)
		tmp = Float64(Float64(t_0 - t_1) / 2.0);
	else
		tmp = fma(Float64(0.5 * x), fma(Float64(eps - 1.0), t_0, (eps ^ -1.0)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[eps, -1.0], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -9.8e+42], N[(N[(N[Power[eps, -1.0], $MachinePrecision] - N[(N[(N[(-1.0 - eps), $MachinePrecision] * x + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -1.4e-176], N[(N[(0.5 * x), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * t$95$0 + N[(N[(1.0 - N[(eps * eps), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 370.0], N[(N[(1.0 + x), $MachinePrecision] / N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.4e+246], N[(N[(t$95$0 - t$95$1), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(0.5 * x), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * t$95$0 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -9.8000000000000004e42

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 69.9

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

   5. Applied rewrites 69.9%

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

   6. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

      2. neg-mul-1 N/A

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

      3. lower-exp.f64 N/A

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

      4. neg-mul-1 N/A

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

      5. lower-neg.f64 31.0

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

   8. Applied rewrites 31.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 3.1%

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

   12. Taylor expanded in x around 0

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

       1. associate--l+ N/A

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

       2. mul-1-neg N/A

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

       3. associate-*r* N/A

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

       4. distribute-lft-neg-in N/A

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

       5. distribute-lft1-in N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. distribute-lft-neg-in N/A

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

       9. lower-fma.f64 N/A

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

       10. neg-sub0 N/A

           \[\leadsto \frac{\frac{1}{\varepsilon} - \mathsf{fma}\left(\color{blue}{0 - \left(1 + \varepsilon\right)}, x, 1\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2} \]

       11. associate--r+ N/A

           \[\leadsto \frac{\frac{1}{\varepsilon} - \mathsf{fma}\left(\color{blue}{\left(0 - 1\right) - \varepsilon}, x, 1\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)}{2} \]

       12. metadata-eval N/A

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

       13. lower--.f64 N/A

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

       14. lower--.f64 N/A

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

       15. lower-/.f64 21.3

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

   14. Applied rewrites 21.3%

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

   if -9.8000000000000004e42 < x < -1.4000000000000001e-176

   1. Initial program 66.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 52.7%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 58.2%

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

   if -1.4000000000000001e-176 < x < 370

   1. Initial program 50.8%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 84.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 84.0%

      \[\leadsto \left(2 \cdot \frac{1 + x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}\right) \cdot 0.5 \]
   9. Step-by-step derivation

   10. Applied rewrites 84.0%

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

   if 370 < x < 5.4e246

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 34.1

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

   5. Applied rewrites 34.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 58.1

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

   8. Applied rewrites 58.1%

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

   if 5.4e246 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 55.1%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{+42}:\\ \;\;\;\;\frac{{\varepsilon}^{-1} - \mathsf{fma}\left(-1 - \varepsilon, x, 1\right) \cdot \left({\varepsilon}^{-1} - 1\right)}{2}\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-176}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, {\varepsilon}^{-1} + 1, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{elif}\;x \leq 370:\\ \;\;\;\;\frac{1 + x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+246}:\\ \;\;\;\;\frac{\left({\varepsilon}^{-1} + 1\right) - \left({\varepsilon}^{-1} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, {\varepsilon}^{-1} + 1, {\varepsilon}^{-1}\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\varepsilon}^{-1} + 1\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{+42}:\\ \;\;\;\;\frac{\frac{e^{-x}}{\varepsilon} - \left({\varepsilon}^{-1} - 1\right)}{2}\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-176}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, t\_0, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+246}:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, t\_0, {\varepsilon}^{-1}\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (pow eps -1.0) 1.0)))
   (if (<= x -1.5e+42)
     (/ (- (/ (exp (- x)) eps) (- (pow eps -1.0) 1.0)) 2.0)
     (if (<= x -1.4e-176)
       (fma (* 0.5 x) (fma (- eps 1.0) t_0 (/ (- 1.0 (* eps eps)) eps)) 1.0)
       (if (<= x 5.4e+246)
         (* (* 2.0 (/ (+ 1.0 x) (exp x))) 0.5)
         (fma (* 0.5 x) (fma (- eps 1.0) t_0 (pow eps -1.0)) 1.0))))))

C

double code(double x, double eps) {
	double t_0 = pow(eps, -1.0) + 1.0;
	double tmp;
	if (x <= -1.5e+42) {
		tmp = ((exp(-x) / eps) - (pow(eps, -1.0) - 1.0)) / 2.0;
	} else if (x <= -1.4e-176) {
		tmp = fma((0.5 * x), fma((eps - 1.0), t_0, ((1.0 - (eps * eps)) / eps)), 1.0);
	} else if (x <= 5.4e+246) {
		tmp = (2.0 * ((1.0 + x) / exp(x))) * 0.5;
	} else {
		tmp = fma((0.5 * x), fma((eps - 1.0), t_0, pow(eps, -1.0)), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((eps ^ -1.0) + 1.0)
	tmp = 0.0
	if (x <= -1.5e+42)
		tmp = Float64(Float64(Float64(exp(Float64(-x)) / eps) - Float64((eps ^ -1.0) - 1.0)) / 2.0);
	elseif (x <= -1.4e-176)
		tmp = fma(Float64(0.5 * x), fma(Float64(eps - 1.0), t_0, Float64(Float64(1.0 - Float64(eps * eps)) / eps)), 1.0);
	elseif (x <= 5.4e+246)
		tmp = Float64(Float64(2.0 * Float64(Float64(1.0 + x) / exp(x))) * 0.5);
	else
		tmp = fma(Float64(0.5 * x), fma(Float64(eps - 1.0), t_0, (eps ^ -1.0)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[eps, -1.0], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.5e+42], N[(N[(N[(N[Exp[(-x)], $MachinePrecision] / eps), $MachinePrecision] - N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -1.4e-176], N[(N[(0.5 * x), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * t$95$0 + N[(N[(1.0 - N[(eps * eps), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 5.4e+246], N[(N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(0.5 * x), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * t$95$0 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.50000000000000014e42

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 69.9

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

   5. Applied rewrites 69.9%

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

   6. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

      2. neg-mul-1 N/A

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

      3. lower-exp.f64 N/A

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

      4. neg-mul-1 N/A

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

      5. lower-neg.f64 31.0

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

   8. Applied rewrites 31.0%

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

   if -1.50000000000000014e42 < x < -1.4000000000000001e-176

   1. Initial program 66.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 52.7%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 58.2%

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

   if -1.4000000000000001e-176 < x < 5.4e246

   1. Initial program 72.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 72.9%

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

   if 5.4e246 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 55.1%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+42}:\\ \;\;\;\;\frac{\frac{e^{-x}}{\varepsilon} - \left({\varepsilon}^{-1} - 1\right)}{2}\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-176}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, {\varepsilon}^{-1} + 1, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+246}:\\ \;\;\;\;\left(2 \cdot \frac{1 + x}{e^{x}}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, {\varepsilon}^{-1} + 1, {\varepsilon}^{-1}\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 60.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\varepsilon}^{-1} + 1\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{-176}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, t\_0, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{elif}\;x \leq 370:\\ \;\;\;\;\frac{1 + x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+246}:\\ \;\;\;\;\frac{t\_0 - \left({\varepsilon}^{-1} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, t\_0, {\varepsilon}^{-1}\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (pow eps -1.0) 1.0)))
   (if (<= x -1.4e-176)
     (fma (* 0.5 x) (fma (- eps 1.0) t_0 (/ (- 1.0 (* eps eps)) eps)) 1.0)
     (if (<= x 370.0)
       (/ (+ 1.0 x) (fma (fma (fma 0.16666666666666666 x 0.5) x 1.0) x 1.0))
       (if (<= x 5.4e+246)
         (/ (- t_0 (- (pow eps -1.0) 1.0)) 2.0)
         (fma (* 0.5 x) (fma (- eps 1.0) t_0 (pow eps -1.0)) 1.0))))))

C

double code(double x, double eps) {
	double t_0 = pow(eps, -1.0) + 1.0;
	double tmp;
	if (x <= -1.4e-176) {
		tmp = fma((0.5 * x), fma((eps - 1.0), t_0, ((1.0 - (eps * eps)) / eps)), 1.0);
	} else if (x <= 370.0) {
		tmp = (1.0 + x) / fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0);
	} else if (x <= 5.4e+246) {
		tmp = (t_0 - (pow(eps, -1.0) - 1.0)) / 2.0;
	} else {
		tmp = fma((0.5 * x), fma((eps - 1.0), t_0, pow(eps, -1.0)), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((eps ^ -1.0) + 1.0)
	tmp = 0.0
	if (x <= -1.4e-176)
		tmp = fma(Float64(0.5 * x), fma(Float64(eps - 1.0), t_0, Float64(Float64(1.0 - Float64(eps * eps)) / eps)), 1.0);
	elseif (x <= 370.0)
		tmp = Float64(Float64(1.0 + x) / fma(fma(fma(0.16666666666666666, x, 0.5), x, 1.0), x, 1.0));
	elseif (x <= 5.4e+246)
		tmp = Float64(Float64(t_0 - Float64((eps ^ -1.0) - 1.0)) / 2.0);
	else
		tmp = fma(Float64(0.5 * x), fma(Float64(eps - 1.0), t_0, (eps ^ -1.0)), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[eps, -1.0], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.4e-176], N[(N[(0.5 * x), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * t$95$0 + N[(N[(1.0 - N[(eps * eps), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 370.0], N[(N[(1.0 + x), $MachinePrecision] / N[(N[(N[(0.16666666666666666 * x + 0.5), $MachinePrecision] * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.4e+246], N[(N[(t$95$0 - N[(N[Power[eps, -1.0], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(0.5 * x), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * t$95$0 + N[Power[eps, -1.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.4000000000000001e-176

   1. Initial program 79.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 33.5%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 44.5%

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

   if -1.4000000000000001e-176 < x < 370

   1. Initial program 50.8%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 84.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 84.0%

      \[\leadsto \left(2 \cdot \frac{1 + x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}\right) \cdot 0.5 \]
   9. Step-by-step derivation

   10. Applied rewrites 84.0%

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

   if 370 < x < 5.4e246

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 34.1

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

   5. Applied rewrites 34.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 58.1

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

   8. Applied rewrites 58.1%

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

   if 5.4e246 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in eps around 0

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

   8. Applied rewrites 55.1%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-176}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, {\varepsilon}^{-1} + 1, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{elif}\;x \leq 370:\\ \;\;\;\;\frac{1 + x}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, x, 0.5\right), x, 1\right), x, 1\right)}\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+246}:\\ \;\;\;\;\frac{\left({\varepsilon}^{-1} + 1\right) - \left({\varepsilon}^{-1} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, {\varepsilon}^{-1} + 1, {\varepsilon}^{-1}\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 53.6% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.9 \cdot 10^{+32}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right) \cdot x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 2.9e+32) 1.0 (* (* (fma 0.3333333333333333 x -0.5) x) x)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 2.9e+32) {
		tmp = 1.0;
	} else {
		tmp = (fma(0.3333333333333333, x, -0.5) * x) * x;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 2.9e+32)
		tmp = 1.0;
	else
		tmp = Float64(Float64(fma(0.3333333333333333, x, -0.5) * x) * x);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 2.9e+32], 1.0, N[(N[(N[(0.3333333333333333 * x + -0.5), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.90000000000000003e32

   1. Initial program 65.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 58.3%

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

   if 2.90000000000000003e32 < 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 48.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 39.6%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 39.6%

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

Alternative 16: 53.3% accurate, 15.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.5%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
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 58.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 52.5%

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

Alternative 17: 53.2% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.5%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
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 58.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 52.5%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 52.5%

    \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right) \]
12. Add Preprocessing

Alternative 18: 43.8% accurate, 273.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 1.0

Julia

function code(x, eps)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 1.0

Derivation

1. Initial program 75.5%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 42.3%

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

Reproduce

herbie shell --seed 2024309 
(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))