NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.0% → 99.6%

Time: 13.3s

Alternatives: 14

Speedup: 0.7×

• 37.0% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -1.5197224285127093e+88
  2. eps = -1.6984431525599665e-20

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: 73.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\varepsilon} + 1\\ \mathbf{if}\;e^{\left(\varepsilon - 1\right) \cdot x} \cdot t\_0 - e^{\left(-1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right) \leq 2.2:\\ \;\;\;\;0.5 \cdot \left(\frac{x + 1}{e^{x}} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} \cdot t\_0 - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 eps) 1.0)))
   (if (<=
        (-
         (* (exp (* (- eps 1.0) x)) t_0)
         (* (exp (* (- -1.0 eps) x)) (- (/ 1.0 eps) 1.0)))
        2.2)
     (* 0.5 (* (/ (+ x 1.0) (exp x)) 2.0))
     (/ (- (* (exp (* x eps)) t_0) (- (exp (- (fma x eps x))))) 2.0))))

C

double code(double x, double eps) {
	double t_0 = (1.0 / eps) + 1.0;
	double tmp;
	if (((exp(((eps - 1.0) * x)) * t_0) - (exp(((-1.0 - eps) * x)) * ((1.0 / eps) - 1.0))) <= 2.2) {
		tmp = 0.5 * (((x + 1.0) / exp(x)) * 2.0);
	} else {
		tmp = ((exp((x * eps)) * t_0) - -exp(-fma(x, eps, x))) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(1.0 / eps) + 1.0)
	tmp = 0.0
	if (Float64(Float64(exp(Float64(Float64(eps - 1.0) * x)) * t_0) - Float64(exp(Float64(Float64(-1.0 - eps) * x)) * Float64(Float64(1.0 / eps) - 1.0))) <= 2.2)
		tmp = Float64(0.5 * Float64(Float64(Float64(x + 1.0) / exp(x)) * 2.0));
	else
		tmp = Float64(Float64(Float64(exp(Float64(x * eps)) * t_0) - Float64(-exp(Float64(-fma(x, eps, x))))) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(1.0 / eps), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[(N[(N[Exp[N[(N[(eps - 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] - N[(N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.2], N[(0.5 * N[(N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] * t$95$0), $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))))) < 2.2000000000000002

   1. Initial program 52.6%

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

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

   if 2.2000000000000002 < (-.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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\left(\varepsilon - 1\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right) - e^{\left(-1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right) \leq 2.2:\\ \;\;\;\;0.5 \cdot \left(\frac{x + 1}{e^{x}} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} \cdot \left(\frac{1}{\varepsilon} + 1\right) - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(1.0 / eps), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[(N[(N[Exp[N[(N[(eps - 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] - N[(N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4.0], N[(0.5 * N[(N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - (-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))))) < 4

   1. Initial program 52.9%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 99.5%

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

   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. 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%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-+.f64 N/A

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

       3. lower-/.f64 53.2

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

   13. Applied rewrites 53.2%

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

4. Final simplification 78.5%

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

Alternative 3: 69.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\varepsilon} + 1\\ t_1 := -e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\\ \mathbf{if}\;\varepsilon \leq 2300:\\ \;\;\;\;0.5 \cdot \left(\frac{x + 1}{e^{x}} \cdot 2\right)\\ \mathbf{elif}\;\varepsilon \leq 2.65 \cdot 10^{+247}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\varepsilon - 1, x, 1\right) \cdot t\_0 - t\_1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - t\_1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 eps) 1.0)) (t_1 (- (exp (- (fma x eps x))))))
   (if (<= eps 2300.0)
     (* 0.5 (* (/ (+ x 1.0) (exp x)) 2.0))
     (if (<= eps 2.65e+247)
       (/ (- (* (fma (- eps 1.0) x 1.0) t_0) t_1) 2.0)
       (/ (- t_0 t_1) 2.0)))))

C

double code(double x, double eps) {
	double t_0 = (1.0 / eps) + 1.0;
	double t_1 = -exp(-fma(x, eps, x));
	double tmp;
	if (eps <= 2300.0) {
		tmp = 0.5 * (((x + 1.0) / exp(x)) * 2.0);
	} else if (eps <= 2.65e+247) {
		tmp = ((fma((eps - 1.0), x, 1.0) * t_0) - t_1) / 2.0;
	} else {
		tmp = (t_0 - t_1) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(1.0 / eps) + 1.0)
	t_1 = Float64(-exp(Float64(-fma(x, eps, x))))
	tmp = 0.0
	if (eps <= 2300.0)
		tmp = Float64(0.5 * Float64(Float64(Float64(x + 1.0) / exp(x)) * 2.0));
	elseif (eps <= 2.65e+247)
		tmp = Float64(Float64(Float64(fma(Float64(eps - 1.0), x, 1.0) * t_0) - t_1) / 2.0);
	else
		tmp = Float64(Float64(t_0 - t_1) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(1.0 / eps), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = (-N[Exp[(-N[(x * eps + x), $MachinePrecision])], $MachinePrecision])}, If[LessEqual[eps, 2300.0], N[(0.5 * N[(N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 2.65e+247], N[(N[(N[(N[(N[(eps - 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision] - t$95$1), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 - t$95$1), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < 2300

   1. Initial program 62.8%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 68.8%

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

   if 2300 < eps < 2.6500000000000001e247

   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 x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 69.7

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

   10. Applied rewrites 69.7%

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

   if 2.6500000000000001e247 < 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 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%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-+.f64 N/A

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

       3. lower-/.f64 57.6

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

   13. Applied rewrites 57.6%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 2300:\\ \;\;\;\;0.5 \cdot \left(\frac{x + 1}{e^{x}} \cdot 2\right)\\ \mathbf{elif}\;\varepsilon \leq 2.65 \cdot 10^{+247}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\varepsilon - 1, x, 1\right) \cdot \left(\frac{1}{\varepsilon} + 1\right) - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\varepsilon} + 1\\ \mathbf{if}\;x \leq -4 \cdot 10^{-13}:\\ \;\;\;\;\frac{t\_0 - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)}{2}\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-211}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, t\_0, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-15}:\\ \;\;\;\;\left(\frac{x + 1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)} \cdot 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+245}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} \cdot t\_0 - -1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 eps) 1.0)))
   (if (<= x -4e-13)
     (/ (- t_0 (- (exp (- (fma x eps x))))) 2.0)
     (if (<= x -1.1e-211)
       (fma (* 0.5 x) (fma (- eps 1.0) t_0 (/ (- 1.0 (* eps eps)) eps)) 1.0)
       (if (<= x 1.75e-15)
         (* (* (/ (+ x 1.0) (fma (fma 0.5 x 1.0) x 1.0)) 2.0) 0.5)
         (if (<= x 1.65e+245)
           (/ (- (* (exp (* x eps)) t_0) -1.0) 2.0)
           (/ (- t_0 (- (/ 1.0 eps) 1.0)) 2.0)))))))

C

double code(double x, double eps) {
	double t_0 = (1.0 / eps) + 1.0;
	double tmp;
	if (x <= -4e-13) {
		tmp = (t_0 - -exp(-fma(x, eps, x))) / 2.0;
	} else if (x <= -1.1e-211) {
		tmp = fma((0.5 * x), fma((eps - 1.0), t_0, ((1.0 - (eps * eps)) / eps)), 1.0);
	} else if (x <= 1.75e-15) {
		tmp = (((x + 1.0) / fma(fma(0.5, x, 1.0), x, 1.0)) * 2.0) * 0.5;
	} else if (x <= 1.65e+245) {
		tmp = ((exp((x * eps)) * t_0) - -1.0) / 2.0;
	} else {
		tmp = (t_0 - ((1.0 / eps) - 1.0)) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(1.0 / eps) + 1.0)
	tmp = 0.0
	if (x <= -4e-13)
		tmp = Float64(Float64(t_0 - Float64(-exp(Float64(-fma(x, eps, x))))) / 2.0);
	elseif (x <= -1.1e-211)
		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 <= 1.75e-15)
		tmp = Float64(Float64(Float64(Float64(x + 1.0) / fma(fma(0.5, x, 1.0), x, 1.0)) * 2.0) * 0.5);
	elseif (x <= 1.65e+245)
		tmp = Float64(Float64(Float64(exp(Float64(x * eps)) * t_0) - -1.0) / 2.0);
	else
		tmp = Float64(Float64(t_0 - Float64(Float64(1.0 / eps) - 1.0)) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(1.0 / eps), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -4e-13], N[(N[(t$95$0 - (-N[Exp[(-N[(x * eps + x), $MachinePrecision])], $MachinePrecision])), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -1.1e-211], 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, 1.75e-15], N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[(N[(0.5 * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 1.65e+245], N[(N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 - N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -4.0000000000000001e-13

   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%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-+.f64 N/A

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

       3. lower-/.f64 58.2

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

   13. Applied rewrites 58.2%

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

   if -4.0000000000000001e-13 < x < -1.09999999999999999e-211

   1. Initial program 64.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]
   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 58.0%

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

   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 74.0%

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

   if -1.09999999999999999e-211 < x < 1.75e-15

   1. Initial program 45.4%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 89.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 89.7%

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

   if 1.75e-15 < x < 1.65000000000000005e245

   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 40.5%

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

   9. Taylor expanded in eps around inf

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

       1. lower-*.f64 40.2

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

   11. Applied rewrites 40.2%

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

   if 1.65000000000000005e245 < 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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 31.7

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

   5. Applied rewrites 31.7%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 61.2

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

   8. Applied rewrites 61.2%

      \[\leadsto \frac{\left(\frac{1}{\varepsilon} + 1\right) - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-13}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)}{2}\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-211}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, \frac{1}{\varepsilon} + 1, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-15}:\\ \;\;\;\;\left(\frac{x + 1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)} \cdot 2\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+245}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} \cdot \left(\frac{1}{\varepsilon} + 1\right) - -1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\varepsilon} + 1\\ \mathbf{if}\;x \leq -4 \cdot 10^{-13}:\\ \;\;\;\;\frac{t\_0 - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)}{2}\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-211}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, t\_0, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-15}:\\ \;\;\;\;\left(\frac{x + 1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)} \cdot 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(\varepsilon - 1\right) \cdot x} \cdot t\_0 - -1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 eps) 1.0)))
   (if (<= x -4e-13)
     (/ (- t_0 (- (exp (- (fma x eps x))))) 2.0)
     (if (<= x -1.1e-211)
       (fma (* 0.5 x) (fma (- eps 1.0) t_0 (/ (- 1.0 (* eps eps)) eps)) 1.0)
       (if (<= x 1.75e-15)
         (* (* (/ (+ x 1.0) (fma (fma 0.5 x 1.0) x 1.0)) 2.0) 0.5)
         (/ (- (* (exp (* (- eps 1.0) x)) t_0) -1.0) 2.0))))))

C

double code(double x, double eps) {
	double t_0 = (1.0 / eps) + 1.0;
	double tmp;
	if (x <= -4e-13) {
		tmp = (t_0 - -exp(-fma(x, eps, x))) / 2.0;
	} else if (x <= -1.1e-211) {
		tmp = fma((0.5 * x), fma((eps - 1.0), t_0, ((1.0 - (eps * eps)) / eps)), 1.0);
	} else if (x <= 1.75e-15) {
		tmp = (((x + 1.0) / fma(fma(0.5, x, 1.0), x, 1.0)) * 2.0) * 0.5;
	} else {
		tmp = ((exp(((eps - 1.0) * x)) * t_0) - -1.0) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(1.0 / eps) + 1.0)
	tmp = 0.0
	if (x <= -4e-13)
		tmp = Float64(Float64(t_0 - Float64(-exp(Float64(-fma(x, eps, x))))) / 2.0);
	elseif (x <= -1.1e-211)
		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 <= 1.75e-15)
		tmp = Float64(Float64(Float64(Float64(x + 1.0) / fma(fma(0.5, x, 1.0), x, 1.0)) * 2.0) * 0.5);
	else
		tmp = Float64(Float64(Float64(exp(Float64(Float64(eps - 1.0) * x)) * t_0) - -1.0) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(1.0 / eps), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -4e-13], N[(N[(t$95$0 - (-N[Exp[(-N[(x * eps + x), $MachinePrecision])], $MachinePrecision])), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -1.1e-211], 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, 1.75e-15], N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[(N[(0.5 * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[Exp[N[(N[(eps - 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.0000000000000001e-13

   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%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-+.f64 N/A

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

       3. lower-/.f64 58.2

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

   13. Applied rewrites 58.2%

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

   if -4.0000000000000001e-13 < x < -1.09999999999999999e-211

   1. Initial program 64.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]
   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 58.0%

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

   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 74.0%

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

   if -1.09999999999999999e-211 < x < 1.75e-15

   1. Initial program 45.4%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 89.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 89.7%

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

   if 1.75e-15 < 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 33.3%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-13}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) - \left(-e^{-\mathsf{fma}\left(x, \varepsilon, x\right)}\right)}{2}\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-211}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, \frac{1}{\varepsilon} + 1, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-15}:\\ \;\;\;\;\left(\frac{x + 1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, 1\right), x, 1\right)} \cdot 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(\varepsilon - 1\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right) - -1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\varepsilon} + 1\\ \mathbf{if}\;x \leq -8.2 \cdot 10^{+57}:\\ \;\;\;\;\frac{e^{-x} \cdot t\_0 - -1}{2}\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-211}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, t\_0, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{elif}\;x \leq 10^{+109}:\\ \;\;\;\;0.5 \cdot \left(\frac{x + 1}{e^{x}} \cdot 2\right)\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+245}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 eps) 1.0)))
   (if (<= x -8.2e+57)
     (/ (- (* (exp (- x)) t_0) -1.0) 2.0)
     (if (<= x -1.1e-211)
       (fma (* 0.5 x) (fma (- eps 1.0) t_0 (/ (- 1.0 (* eps eps)) eps)) 1.0)
       (if (<= x 1e+109)
         (* 0.5 (* (/ (+ x 1.0) (exp x)) 2.0))
         (if (<= x 1.65e+245)
           (fma (* 0.3333333333333333 x) (* x x) 1.0)
           (/ (- t_0 (- (/ 1.0 eps) 1.0)) 2.0)))))))

C

double code(double x, double eps) {
	double t_0 = (1.0 / eps) + 1.0;
	double tmp;
	if (x <= -8.2e+57) {
		tmp = ((exp(-x) * t_0) - -1.0) / 2.0;
	} else if (x <= -1.1e-211) {
		tmp = fma((0.5 * x), fma((eps - 1.0), t_0, ((1.0 - (eps * eps)) / eps)), 1.0);
	} else if (x <= 1e+109) {
		tmp = 0.5 * (((x + 1.0) / exp(x)) * 2.0);
	} else if (x <= 1.65e+245) {
		tmp = fma((0.3333333333333333 * x), (x * x), 1.0);
	} else {
		tmp = (t_0 - ((1.0 / eps) - 1.0)) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(1.0 / eps) + 1.0)
	tmp = 0.0
	if (x <= -8.2e+57)
		tmp = Float64(Float64(Float64(exp(Float64(-x)) * t_0) - -1.0) / 2.0);
	elseif (x <= -1.1e-211)
		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 <= 1e+109)
		tmp = Float64(0.5 * Float64(Float64(Float64(x + 1.0) / exp(x)) * 2.0));
	elseif (x <= 1.65e+245)
		tmp = fma(Float64(0.3333333333333333 * x), Float64(x * x), 1.0);
	else
		tmp = Float64(Float64(t_0 - Float64(Float64(1.0 / eps) - 1.0)) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(1.0 / eps), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -8.2e+57], N[(N[(N[(N[Exp[(-x)], $MachinePrecision] * t$95$0), $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -1.1e-211], 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, 1e+109], N[(0.5 * N[(N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.65e+245], N[(N[(0.3333333333333333 * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(t$95$0 - N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -8.2e57

   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 43.9%

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

   9. Taylor expanded in eps around 0

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

       1. neg-mul-1 N/A

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

       2. lower-neg.f64 100.0

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

   11. Applied rewrites 100.0%

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

   if -8.2e57 < x < -1.09999999999999999e-211

   1. Initial program 69.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]
   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 51.0%

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

   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 68.8%

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

   if -1.09999999999999999e-211 < x < 9.99999999999999982e108

   1. Initial program 58.3%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 79.3%

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

   if 9.99999999999999982e108 < x < 1.65000000000000005e245

   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 26.2%

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

   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 75.4%

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

   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 75.4%

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

   if 1.65000000000000005e245 < 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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 31.7

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

   5. Applied rewrites 31.7%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 61.2

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

   8. Applied rewrites 61.2%

      \[\leadsto \frac{\left(\frac{1}{\varepsilon} + 1\right) - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{+57}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(\frac{1}{\varepsilon} + 1\right) - -1}{2}\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-211}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, \frac{1}{\varepsilon} + 1, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{elif}\;x \leq 10^{+109}:\\ \;\;\;\;0.5 \cdot \left(\frac{x + 1}{e^{x}} \cdot 2\right)\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+245}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\varepsilon} + 1\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{-13}:\\ \;\;\;\;\frac{t\_0 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon - 1, x, x - 1\right), \varepsilon, 1 - x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-211}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, t\_0, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{elif}\;x \leq 10^{+109}:\\ \;\;\;\;0.5 \cdot \left(\frac{x + 1}{e^{x}} \cdot 2\right)\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+245}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 eps) 1.0)))
   (if (<= x -7.5e-13)
     (/ (- t_0 (/ (fma (fma (- eps 1.0) x (- x 1.0)) eps (- 1.0 x)) eps)) 2.0)
     (if (<= x -1.1e-211)
       (fma (* 0.5 x) (fma (- eps 1.0) t_0 (/ (- 1.0 (* eps eps)) eps)) 1.0)
       (if (<= x 1e+109)
         (* 0.5 (* (/ (+ x 1.0) (exp x)) 2.0))
         (if (<= x 1.65e+245)
           (fma (* 0.3333333333333333 x) (* x x) 1.0)
           (/ (- t_0 (- (/ 1.0 eps) 1.0)) 2.0)))))))

C

double code(double x, double eps) {
	double t_0 = (1.0 / eps) + 1.0;
	double tmp;
	if (x <= -7.5e-13) {
		tmp = (t_0 - (fma(fma((eps - 1.0), x, (x - 1.0)), eps, (1.0 - x)) / eps)) / 2.0;
	} else if (x <= -1.1e-211) {
		tmp = fma((0.5 * x), fma((eps - 1.0), t_0, ((1.0 - (eps * eps)) / eps)), 1.0);
	} else if (x <= 1e+109) {
		tmp = 0.5 * (((x + 1.0) / exp(x)) * 2.0);
	} else if (x <= 1.65e+245) {
		tmp = fma((0.3333333333333333 * x), (x * x), 1.0);
	} else {
		tmp = (t_0 - ((1.0 / eps) - 1.0)) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(1.0 / eps) + 1.0)
	tmp = 0.0
	if (x <= -7.5e-13)
		tmp = Float64(Float64(t_0 - Float64(fma(fma(Float64(eps - 1.0), x, Float64(x - 1.0)), eps, Float64(1.0 - x)) / eps)) / 2.0);
	elseif (x <= -1.1e-211)
		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 <= 1e+109)
		tmp = Float64(0.5 * Float64(Float64(Float64(x + 1.0) / exp(x)) * 2.0));
	elseif (x <= 1.65e+245)
		tmp = fma(Float64(0.3333333333333333 * x), Float64(x * x), 1.0);
	else
		tmp = Float64(Float64(t_0 - Float64(Float64(1.0 / eps) - 1.0)) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(1.0 / eps), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -7.5e-13], N[(N[(t$95$0 - N[(N[(N[(N[(eps - 1.0), $MachinePrecision] * x + N[(x - 1.0), $MachinePrecision]), $MachinePrecision] * eps + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -1.1e-211], 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, 1e+109], N[(0.5 * N[(N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.65e+245], N[(N[(0.3333333333333333 * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(t$95$0 - N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -7.5000000000000004e-13

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 58.2

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

   5. Applied rewrites 58.2%

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

   6. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. mul-1-neg N/A

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

      3. associate-*r* N/A

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

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

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

      5. distribute-lft1-in N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. distribute-lft-in N/A

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

      12. metadata-eval N/A

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

      13. mul-1-neg N/A

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

      14. unsub-neg N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-/.f64 25.2

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

   8. Applied rewrites 25.2%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 48.2%

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

   if -7.5000000000000004e-13 < x < -1.09999999999999999e-211

   1. Initial program 64.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]
   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 58.0%

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

   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 74.0%

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

   if -1.09999999999999999e-211 < x < 9.99999999999999982e108

   1. Initial program 58.3%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 79.3%

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

   if 9.99999999999999982e108 < x < 1.65000000000000005e245

   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 26.2%

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

   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 75.4%

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

   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 75.4%

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

   if 1.65000000000000005e245 < 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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 31.7

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

   5. Applied rewrites 31.7%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 61.2

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

   8. Applied rewrites 61.2%

      \[\leadsto \frac{\left(\frac{1}{\varepsilon} + 1\right) - \color{blue}{\left(\frac{1}{\varepsilon} - 1\right)}}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) - \frac{\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon - 1, x, x - 1\right), \varepsilon, 1 - x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-211}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, \frac{1}{\varepsilon} + 1, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{elif}\;x \leq 10^{+109}:\\ \;\;\;\;0.5 \cdot \left(\frac{x + 1}{e^{x}} \cdot 2\right)\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+245}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right)\\ t_1 := \frac{1}{\varepsilon} + 1\\ t_2 := \frac{t\_1 - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{-13}:\\ \;\;\;\;\frac{t\_1 - \frac{\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon - 1, x, x - 1\right), \varepsilon, 1 - x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-211}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, t\_1, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+110}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+245}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (fma (* 0.3333333333333333 x) (* x x) 1.0))
        (t_1 (+ (/ 1.0 eps) 1.0))
        (t_2 (/ (- t_1 (- (/ 1.0 eps) 1.0)) 2.0)))
   (if (<= x -7.5e-13)
     (/ (- t_1 (/ (fma (fma (- eps 1.0) x (- x 1.0)) eps (- 1.0 x)) eps)) 2.0)
     (if (<= x -1.1e-211)
       (fma (* 0.5 x) (fma (- eps 1.0) t_1 (/ (- 1.0 (* eps eps)) eps)) 1.0)
       (if (<= x 5e+17)
         t_0
         (if (<= x 3.2e+110) t_2 (if (<= x 1.65e+245) t_0 t_2)))))))

C

double code(double x, double eps) {
	double t_0 = fma((0.3333333333333333 * x), (x * x), 1.0);
	double t_1 = (1.0 / eps) + 1.0;
	double t_2 = (t_1 - ((1.0 / eps) - 1.0)) / 2.0;
	double tmp;
	if (x <= -7.5e-13) {
		tmp = (t_1 - (fma(fma((eps - 1.0), x, (x - 1.0)), eps, (1.0 - x)) / eps)) / 2.0;
	} else if (x <= -1.1e-211) {
		tmp = fma((0.5 * x), fma((eps - 1.0), t_1, ((1.0 - (eps * eps)) / eps)), 1.0);
	} else if (x <= 5e+17) {
		tmp = t_0;
	} else if (x <= 3.2e+110) {
		tmp = t_2;
	} else if (x <= 1.65e+245) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = fma(Float64(0.3333333333333333 * x), Float64(x * x), 1.0)
	t_1 = Float64(Float64(1.0 / eps) + 1.0)
	t_2 = Float64(Float64(t_1 - Float64(Float64(1.0 / eps) - 1.0)) / 2.0)
	tmp = 0.0
	if (x <= -7.5e-13)
		tmp = Float64(Float64(t_1 - Float64(fma(fma(Float64(eps - 1.0), x, Float64(x - 1.0)), eps, Float64(1.0 - x)) / eps)) / 2.0);
	elseif (x <= -1.1e-211)
		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 <= 5e+17)
		tmp = t_0;
	elseif (x <= 3.2e+110)
		tmp = t_2;
	elseif (x <= 1.65e+245)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(0.3333333333333333 * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 / eps), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 - N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -7.5e-13], N[(N[(t$95$1 - N[(N[(N[(N[(eps - 1.0), $MachinePrecision] * x + N[(x - 1.0), $MachinePrecision]), $MachinePrecision] * eps + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -1.1e-211], 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, 5e+17], t$95$0, If[LessEqual[x, 3.2e+110], t$95$2, If[LessEqual[x, 1.65e+245], t$95$0, t$95$2]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.5000000000000004e-13

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 58.2

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

   5. Applied rewrites 58.2%

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

   6. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. mul-1-neg N/A

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

      3. associate-*r* N/A

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

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

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

      5. distribute-lft1-in N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. distribute-lft-in N/A

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

      12. metadata-eval N/A

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

      13. mul-1-neg N/A

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

      14. unsub-neg N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-/.f64 25.2

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

   8. Applied rewrites 25.2%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 48.2%

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

   if -7.5000000000000004e-13 < x < -1.09999999999999999e-211

   1. Initial program 64.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{1}{2} \cdot \left(x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\varepsilon - 1\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)\right)} \]
   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 58.0%

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

   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 74.0%

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

   if -1.09999999999999999e-211 < x < 5e17 or 3.19999999999999994e110 < x < 1.65000000000000005e245

   1. Initial program 59.6%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 72.2%

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

   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 83.1%

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

   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 83.1%

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

   if 5e17 < x < 3.19999999999999994e110 or 1.65000000000000005e245 < 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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 26.9

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

   5. Applied rewrites 26.9%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 58.2

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

   8. Applied rewrites 58.2%

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

Alternative 9: 62.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right)\\ t_1 := \frac{1}{\varepsilon} + 1\\ t_2 := \frac{1}{\varepsilon} - 1\\ t_3 := \frac{t\_1 - t\_2}{2}\\ \mathbf{if}\;x \leq -1.05 \cdot 10^{+124}:\\ \;\;\;\;\frac{\frac{1}{\varepsilon} - \mathsf{fma}\left(-1 - \varepsilon, x, 1\right) \cdot t\_2}{2}\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-211}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, t\_1, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+110}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+245}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (fma (* 0.3333333333333333 x) (* x x) 1.0))
        (t_1 (+ (/ 1.0 eps) 1.0))
        (t_2 (- (/ 1.0 eps) 1.0))
        (t_3 (/ (- t_1 t_2) 2.0)))
   (if (<= x -1.05e+124)
     (/ (- (/ 1.0 eps) (* (fma (- -1.0 eps) x 1.0) t_2)) 2.0)
     (if (<= x -1.1e-211)
       (fma (* 0.5 x) (fma (- eps 1.0) t_1 (/ (- 1.0 (* eps eps)) eps)) 1.0)
       (if (<= x 5e+17)
         t_0
         (if (<= x 3.2e+110) t_3 (if (<= x 1.65e+245) t_0 t_3)))))))

C

double code(double x, double eps) {
	double t_0 = fma((0.3333333333333333 * x), (x * x), 1.0);
	double t_1 = (1.0 / eps) + 1.0;
	double t_2 = (1.0 / eps) - 1.0;
	double t_3 = (t_1 - t_2) / 2.0;
	double tmp;
	if (x <= -1.05e+124) {
		tmp = ((1.0 / eps) - (fma((-1.0 - eps), x, 1.0) * t_2)) / 2.0;
	} else if (x <= -1.1e-211) {
		tmp = fma((0.5 * x), fma((eps - 1.0), t_1, ((1.0 - (eps * eps)) / eps)), 1.0);
	} else if (x <= 5e+17) {
		tmp = t_0;
	} else if (x <= 3.2e+110) {
		tmp = t_3;
	} else if (x <= 1.65e+245) {
		tmp = t_0;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = fma(Float64(0.3333333333333333 * x), Float64(x * x), 1.0)
	t_1 = Float64(Float64(1.0 / eps) + 1.0)
	t_2 = Float64(Float64(1.0 / eps) - 1.0)
	t_3 = Float64(Float64(t_1 - t_2) / 2.0)
	tmp = 0.0
	if (x <= -1.05e+124)
		tmp = Float64(Float64(Float64(1.0 / eps) - Float64(fma(Float64(-1.0 - eps), x, 1.0) * t_2)) / 2.0);
	elseif (x <= -1.1e-211)
		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 <= 5e+17)
		tmp = t_0;
	elseif (x <= 3.2e+110)
		tmp = t_3;
	elseif (x <= 1.65e+245)
		tmp = t_0;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(0.3333333333333333 * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 / eps), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 - t$95$2), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -1.05e+124], N[(N[(N[(1.0 / eps), $MachinePrecision] - N[(N[(N[(-1.0 - eps), $MachinePrecision] * x + 1.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -1.1e-211], 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, 5e+17], t$95$0, If[LessEqual[x, 3.2e+110], t$95$3, If[LessEqual[x, 1.65e+245], t$95$0, t$95$3]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.05000000000000006e124

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 59.4

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

   5. Applied rewrites 59.4%

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

   6. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. mul-1-neg N/A

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

      3. associate-*r* N/A

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

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

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

      5. distribute-lft1-in N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. distribute-lft-in N/A

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

      12. metadata-eval N/A

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

      13. mul-1-neg N/A

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

      14. unsub-neg N/A

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

      15. lower--.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lower-/.f64 31.2

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

   8. Applied rewrites 31.2%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 31.2%

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

   if -1.05000000000000006e124 < x < -1.09999999999999999e-211

   1. Initial program 73.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 44.7%

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

   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 65.6%

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

   if -1.09999999999999999e-211 < x < 5e17 or 3.19999999999999994e110 < x < 1.65000000000000005e245

   1. Initial program 59.6%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 72.2%

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

   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 83.1%

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

   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 83.1%

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

   if 5e17 < x < 3.19999999999999994e110 or 1.65000000000000005e245 < 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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 26.9

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

   5. Applied rewrites 26.9%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 58.2

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

   8. Applied rewrites 58.2%

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

Alternative 10: 62.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\varepsilon} + 1\\ t_1 := \frac{t\_0 - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ t_2 := \mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right)\\ \mathbf{if}\;x \leq -1.05 \cdot 10^{+124}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, \frac{1}{\varepsilon}, -\varepsilon\right), 1\right)\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-211}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, t\_0, \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon}\right), 1\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+17}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+110}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+245}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 eps) 1.0))
        (t_1 (/ (- t_0 (- (/ 1.0 eps) 1.0)) 2.0))
        (t_2 (fma (* 0.3333333333333333 x) (* x x) 1.0)))
   (if (<= x -1.05e+124)
     (fma (* 0.5 x) (fma (- eps 1.0) (/ 1.0 eps) (- eps)) 1.0)
     (if (<= x -1.1e-211)
       (fma (* 0.5 x) (fma (- eps 1.0) t_0 (/ (- 1.0 (* eps eps)) eps)) 1.0)
       (if (<= x 5e+17)
         t_2
         (if (<= x 3.2e+110) t_1 (if (<= x 1.65e+245) t_2 t_1)))))))

C

double code(double x, double eps) {
	double t_0 = (1.0 / eps) + 1.0;
	double t_1 = (t_0 - ((1.0 / eps) - 1.0)) / 2.0;
	double t_2 = fma((0.3333333333333333 * x), (x * x), 1.0);
	double tmp;
	if (x <= -1.05e+124) {
		tmp = fma((0.5 * x), fma((eps - 1.0), (1.0 / eps), -eps), 1.0);
	} else if (x <= -1.1e-211) {
		tmp = fma((0.5 * x), fma((eps - 1.0), t_0, ((1.0 - (eps * eps)) / eps)), 1.0);
	} else if (x <= 5e+17) {
		tmp = t_2;
	} else if (x <= 3.2e+110) {
		tmp = t_1;
	} else if (x <= 1.65e+245) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(1.0 / eps) + 1.0)
	t_1 = Float64(Float64(t_0 - Float64(Float64(1.0 / eps) - 1.0)) / 2.0)
	t_2 = fma(Float64(0.3333333333333333 * x), Float64(x * x), 1.0)
	tmp = 0.0
	if (x <= -1.05e+124)
		tmp = fma(Float64(0.5 * x), fma(Float64(eps - 1.0), Float64(1.0 / eps), Float64(-eps)), 1.0);
	elseif (x <= -1.1e-211)
		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 <= 5e+17)
		tmp = t_2;
	elseif (x <= 3.2e+110)
		tmp = t_1;
	elseif (x <= 1.65e+245)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(1.0 / eps), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.3333333333333333 * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -1.05e+124], N[(N[(0.5 * x), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * N[(1.0 / eps), $MachinePrecision] + (-eps)), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, -1.1e-211], 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, 5e+17], t$95$2, If[LessEqual[x, 3.2e+110], t$95$1, If[LessEqual[x, 1.65e+245], t$95$2, t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.05000000000000006e124

   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 3.4%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 3.4%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 31.2%

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

   if -1.05000000000000006e124 < x < -1.09999999999999999e-211

   1. Initial program 73.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 44.7%

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

   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 65.6%

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

   if -1.09999999999999999e-211 < x < 5e17 or 3.19999999999999994e110 < x < 1.65000000000000005e245

   1. Initial program 59.6%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 72.2%

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

   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 83.1%

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

   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 83.1%

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

   if 5e17 < x < 3.19999999999999994e110 or 1.65000000000000005e245 < 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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 26.9

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

   5. Applied rewrites 26.9%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 58.2

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

   8. Applied rewrites 58.2%

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

Alternative 11: 59.4% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right)\\ t_1 := \frac{\left(\frac{1}{\varepsilon} + 1\right) - \left(\frac{1}{\varepsilon} - 1\right)}{2}\\ \mathbf{if}\;x \leq -9.8 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, \frac{1}{\varepsilon}, -\varepsilon\right), 1\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+110}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+245}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (fma (* 0.3333333333333333 x) (* x x) 1.0))
        (t_1 (/ (- (+ (/ 1.0 eps) 1.0) (- (/ 1.0 eps) 1.0)) 2.0)))
   (if (<= x -9.8e-15)
     (fma (* 0.5 x) (fma (- eps 1.0) (/ 1.0 eps) (- eps)) 1.0)
     (if (<= x 5e+17)
       t_0
       (if (<= x 3.2e+110) t_1 (if (<= x 1.65e+245) t_0 t_1))))))

C

double code(double x, double eps) {
	double t_0 = fma((0.3333333333333333 * x), (x * x), 1.0);
	double t_1 = (((1.0 / eps) + 1.0) - ((1.0 / eps) - 1.0)) / 2.0;
	double tmp;
	if (x <= -9.8e-15) {
		tmp = fma((0.5 * x), fma((eps - 1.0), (1.0 / eps), -eps), 1.0);
	} else if (x <= 5e+17) {
		tmp = t_0;
	} else if (x <= 3.2e+110) {
		tmp = t_1;
	} else if (x <= 1.65e+245) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = fma(Float64(0.3333333333333333 * x), Float64(x * x), 1.0)
	t_1 = Float64(Float64(Float64(Float64(1.0 / eps) + 1.0) - Float64(Float64(1.0 / eps) - 1.0)) / 2.0)
	tmp = 0.0
	if (x <= -9.8e-15)
		tmp = fma(Float64(0.5 * x), fma(Float64(eps - 1.0), Float64(1.0 / eps), Float64(-eps)), 1.0);
	elseif (x <= 5e+17)
		tmp = t_0;
	elseif (x <= 3.2e+110)
		tmp = t_1;
	elseif (x <= 1.65e+245)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(0.3333333333333333 * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(1.0 / eps), $MachinePrecision] + 1.0), $MachinePrecision] - N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -9.8e-15], N[(N[(0.5 * x), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * N[(1.0 / eps), $MachinePrecision] + (-eps)), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[x, 5e+17], t$95$0, If[LessEqual[x, 3.2e+110], t$95$1, If[LessEqual[x, 1.65e+245], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.7999999999999999e-15

   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 3.2%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 3.2%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 24.8%

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

   if -9.7999999999999999e-15 < x < 5e17 or 3.19999999999999994e110 < x < 1.65000000000000005e245

   1. Initial program 60.6%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 69.2%

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

   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 77.4%

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

   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 77.4%

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

   if 5e17 < x < 3.19999999999999994e110 or 1.65000000000000005e245 < 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

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 26.9

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

   5. Applied rewrites 26.9%

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

   6. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 58.2

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

   8. Applied rewrites 58.2%

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

Alternative 12: 56.1% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot x, \mathsf{fma}\left(\varepsilon - 1, \frac{1}{\varepsilon}, -\varepsilon\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -9.8e-15)
   (fma (* 0.5 x) (fma (- eps 1.0) (/ 1.0 eps) (- eps)) 1.0)
   (fma (* 0.3333333333333333 x) (* x x) 1.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -9.8e-15) {
		tmp = fma((0.5 * x), fma((eps - 1.0), (1.0 / eps), -eps), 1.0);
	} else {
		tmp = fma((0.3333333333333333 * x), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -9.8e-15)
		tmp = fma(Float64(0.5 * x), fma(Float64(eps - 1.0), Float64(1.0 / eps), Float64(-eps)), 1.0);
	else
		tmp = fma(Float64(0.3333333333333333 * x), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -9.8e-15], N[(N[(0.5 * x), $MachinePrecision] * N[(N[(eps - 1.0), $MachinePrecision] * N[(1.0 / eps), $MachinePrecision] + (-eps)), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(0.3333333333333333 * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.7999999999999999e-15

   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 3.2%

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

   6. Taylor expanded in eps around inf

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

   8. Applied rewrites 3.2%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 24.8%

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

   if -9.7999999999999999e-15 < x

   1. Initial program 68.8%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 66.7%

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

   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 65.6%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 65.6%

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

Alternative 13: 53.3% 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 74.3%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 55.0%

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

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 54.1%

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

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 54.1%

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

Alternative 14: 44.5% 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 74.3%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 43.7%

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

Reproduce

herbie shell --seed 2024288 
(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))