NMSE Section 6.1 mentioned, A

Percentage Accurate: 72.9% → 99.8%

Time: 6.9s

Alternatives: 13

Speedup: 0.7×

• 37.7% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -4.0410252533460064e+227
  2. eps = 1.624024092081062e-202

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: 72.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 eps) 1.0)))
   (if (<=
        (-
         (* (exp (* (+ -1.0 eps) x)) t_0)
         (* (exp (* (- -1.0 eps) x)) (- (/ 1.0 eps) 1.0)))
        2.0)
     (* (exp (- x)) (+ x 1.0))
     (/ (- (* (exp (* x eps)) t_0) (/ -1.0 (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(((-1.0 + eps) * x)) * t_0) - (exp(((-1.0 - eps) * x)) * ((1.0 / eps) - 1.0))) <= 2.0) {
		tmp = exp(-x) * (x + 1.0);
	} else {
		tmp = ((exp((x * eps)) * t_0) - (-1.0 / 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(-1.0 + eps) * x)) * t_0) - Float64(exp(Float64(Float64(-1.0 - eps) * x)) * Float64(Float64(1.0 / eps) - 1.0))) <= 2.0)
		tmp = Float64(exp(Float64(-x)) * Float64(x + 1.0));
	else
		tmp = Float64(Float64(Float64(exp(Float64(x * eps)) * t_0) - Float64(-1.0 / exp(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[(-1.0 + eps), $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], 2.0], N[(N[Exp[(-x)], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] - N[(-1.0 / N[Exp[N[(x * eps + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.1%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.8% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 eps) 1.0)))
   (if (<=
        (-
         (* (exp (* (+ -1.0 eps) x)) t_0)
         (* (exp (* (- -1.0 eps) x)) (- (/ 1.0 eps) 1.0)))
        2.0)
     (* (exp (- x)) (+ x 1.0))
     (/ (- (* (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(((-1.0 + eps) * x)) * t_0) - (exp(((-1.0 - eps) * x)) * ((1.0 / eps) - 1.0))) <= 2.0) {
		tmp = exp(-x) * (x + 1.0);
	} 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(-1.0 + eps) * x)) * t_0) - Float64(exp(Float64(Float64(-1.0 - eps) * x)) * Float64(Float64(1.0 / eps) - 1.0))) <= 2.0)
		tmp = Float64(exp(Float64(-x)) * Float64(x + 1.0));
	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[(-1.0 + eps), $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], 2.0], N[(N[Exp[(-x)], $MachinePrecision] * N[(x + 1.0), $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

   1. Initial program 55.1%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in eps around inf

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

      1. lower-*.f64 100.0

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

   10. Applied rewrites 100.0%

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

4. Final simplification 100.0%

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

Alternative 3: 79.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, eps)
	t_0 = Float64(Float64(1.0 / eps) - 1.0)
	t_1 = Float64(exp(Float64(Float64(-1.0 + eps) * x)) * Float64(Float64(1.0 / eps) + 1.0))
	tmp = 0.0
	if (Float64(t_1 - Float64(exp(Float64(Float64(-1.0 - eps) * x)) * t_0)) <= 0.0)
		tmp = Float64(exp(Float64(-x)) * Float64(x + 1.0));
	else
		tmp = Float64(Float64(t_1 - Float64(fma(Float64(-1.0 - eps), x, 1.0) * t_0)) / 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[(N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 - N[(N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[Exp[(-x)], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 - N[(N[(N[(-1.0 - eps), $MachinePrecision] * x + 1.0), $MachinePrecision] * t$95$0), $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))))) < 0.0

   1. Initial program 41.9%

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

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

   7. Applied rewrites 100.0%

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

   if 0.0 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. mul-1-neg N/A

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

      3. associate-*r* N/A

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

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

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

      5. distribute-lft1-in N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 61.3

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

   5. Applied rewrites 61.3%

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

4. Final simplification 78.6%

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

Alternative 4: 79.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 55.1%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 48.9%

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

4. Final simplification 78.6%

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

Alternative 5: 78.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\left(-1 + \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right) - e^{\left(-1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right) \leq 40:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\varepsilon} - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (-
       (* (exp (* (+ -1.0 eps) x)) (+ (/ 1.0 eps) 1.0))
       (* (exp (* (- -1.0 eps) x)) (- (/ 1.0 eps) 1.0)))
      40.0)
   (* (exp (- x)) (+ x 1.0))
   (/ (- (/ 1.0 eps) (/ -1.0 (exp (fma x eps x)))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (((exp(((-1.0 + eps) * x)) * ((1.0 / eps) + 1.0)) - (exp(((-1.0 - eps) * x)) * ((1.0 / eps) - 1.0))) <= 40.0) {
		tmp = exp(-x) * (x + 1.0);
	} else {
		tmp = ((1.0 / eps) - (-1.0 / exp(fma(x, eps, x)))) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(Float64(exp(Float64(Float64(-1.0 + eps) * x)) * Float64(Float64(1.0 / eps) + 1.0)) - Float64(exp(Float64(Float64(-1.0 - eps) * x)) * Float64(Float64(1.0 / eps) - 1.0))) <= 40.0)
		tmp = Float64(exp(Float64(-x)) * Float64(x + 1.0));
	else
		tmp = Float64(Float64(Float64(1.0 / eps) - Float64(-1.0 / exp(fma(x, eps, x)))) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[(N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 40.0], N[(N[Exp[(-x)], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / eps), $MachinePrecision] - N[(-1.0 / N[Exp[N[(x * eps + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 40

   1. Initial program 56.3%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 98.8%

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

   7. Applied rewrites 98.8%

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

   if 40 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 60.5

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

   8. Applied rewrites 60.5%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 59.8%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\left(-1 + \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right) - e^{\left(-1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right) \leq 40:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\varepsilon} - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	t_0 = (1.0 / eps) + 1.0
	tmp = 0
	if ((math.exp(((-1.0 + eps) * x)) * t_0) - (math.exp(((-1.0 - eps) * x)) * ((1.0 / eps) - 1.0))) <= 4.0:
		tmp = math.exp(-x) * (x + 1.0)
	else:
		tmp = (t_0 - (-1.0 / math.exp(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(-1.0 + eps) * x)) * t_0) - Float64(exp(Float64(Float64(-1.0 - eps) * x)) * Float64(Float64(1.0 / eps) - 1.0))) <= 4.0)
		tmp = Float64(exp(Float64(-x)) * Float64(x + 1.0));
	else
		tmp = Float64(Float64(t_0 - Float64(-1.0 / exp(x))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (1.0 / eps) + 1.0;
	tmp = 0.0;
	if (((exp(((-1.0 + eps) * x)) * t_0) - (exp(((-1.0 - eps) * x)) * ((1.0 / eps) - 1.0))) <= 4.0)
		tmp = exp(-x) * (x + 1.0);
	else
		tmp = (t_0 - (-1.0 / exp(x))) / 2.0;
	end
	tmp_2 = 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[(-1.0 + eps), $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[(N[Exp[(-x)], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 - N[(-1.0 / N[Exp[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 56.0%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 99.4%

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

   7. Applied rewrites 99.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 60.1

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

   8. Applied rewrites 60.1%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 43.3%

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

4. Final simplification 76.6%

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

Alternative 7: 71.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\left(-1 + \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right) - e^{\left(-1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right) \leq 40:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\varepsilon} - \frac{-1}{e^{x}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<=
      (-
       (* (exp (* (+ -1.0 eps) x)) (+ (/ 1.0 eps) 1.0))
       (* (exp (* (- -1.0 eps) x)) (- (/ 1.0 eps) 1.0)))
      40.0)
   (* (exp (- x)) (+ x 1.0))
   (/ (- (/ 1.0 eps) (/ -1.0 (exp x))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (((exp(((-1.0 + eps) * x)) * ((1.0 / eps) + 1.0)) - (exp(((-1.0 - eps) * x)) * ((1.0 / eps) - 1.0))) <= 40.0) {
		tmp = exp(-x) * (x + 1.0);
	} else {
		tmp = ((1.0 / eps) - (-1.0 / exp(x))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (((exp((((-1.0d0) + eps) * x)) * ((1.0d0 / eps) + 1.0d0)) - (exp((((-1.0d0) - eps) * x)) * ((1.0d0 / eps) - 1.0d0))) <= 40.0d0) then
        tmp = exp(-x) * (x + 1.0d0)
    else
        tmp = ((1.0d0 / eps) - ((-1.0d0) / exp(x))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (((Math.exp(((-1.0 + eps) * x)) * ((1.0 / eps) + 1.0)) - (Math.exp(((-1.0 - eps) * x)) * ((1.0 / eps) - 1.0))) <= 40.0) {
		tmp = Math.exp(-x) * (x + 1.0);
	} else {
		tmp = ((1.0 / eps) - (-1.0 / Math.exp(x))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if ((math.exp(((-1.0 + eps) * x)) * ((1.0 / eps) + 1.0)) - (math.exp(((-1.0 - eps) * x)) * ((1.0 / eps) - 1.0))) <= 40.0:
		tmp = math.exp(-x) * (x + 1.0)
	else:
		tmp = ((1.0 / eps) - (-1.0 / math.exp(x))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(Float64(exp(Float64(Float64(-1.0 + eps) * x)) * Float64(Float64(1.0 / eps) + 1.0)) - Float64(exp(Float64(Float64(-1.0 - eps) * x)) * Float64(Float64(1.0 / eps) - 1.0))) <= 40.0)
		tmp = Float64(exp(Float64(-x)) * Float64(x + 1.0));
	else
		tmp = Float64(Float64(Float64(1.0 / eps) - Float64(-1.0 / exp(x))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (((exp(((-1.0 + eps) * x)) * ((1.0 / eps) + 1.0)) - (exp(((-1.0 - eps) * x)) * ((1.0 / eps) - 1.0))) <= 40.0)
		tmp = exp(-x) * (x + 1.0);
	else
		tmp = ((1.0 / eps) - (-1.0 / exp(x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[(N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Exp[N[(N[(-1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 40.0], N[(N[Exp[(-x)], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / eps), $MachinePrecision] - N[(-1.0 / N[Exp[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))))) < 40

   1. Initial program 56.3%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   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 98.8%

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

   7. Applied rewrites 98.8%

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

   if 40 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x)))))

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 60.5

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

   8. Applied rewrites 60.5%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 43.6%

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

   12. Taylor expanded in eps around 0

       \[\leadsto \frac{\frac{1}{\color{blue}{\varepsilon}} - \frac{-1}{e^{x}}}{2} \]
   13. Step-by-step derivation

   14. Applied rewrites 43.1%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\left(-1 + \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right) - e^{\left(-1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right) \leq 40:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\varepsilon} - \frac{-1}{e^{x}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\varepsilon} + 1\\ \mathbf{if}\;\varepsilon \leq 7.4 \cdot 10^{+19}:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{elif}\;\varepsilon \leq 1.55 \cdot 10^{+249}:\\ \;\;\;\;\frac{t\_0 - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(-1 + \varepsilon\right) \cdot x} \cdot t\_0 - -1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 eps) 1.0)))
   (if (<= eps 7.4e+19)
     (* (exp (- x)) (+ x 1.0))
     (if (<= eps 1.55e+249)
       (/ (- t_0 (/ -1.0 (exp (fma x eps x)))) 2.0)
       (/ (- (* (exp (* (+ -1.0 eps) 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 (eps <= 7.4e+19) {
		tmp = exp(-x) * (x + 1.0);
	} else if (eps <= 1.55e+249) {
		tmp = (t_0 - (-1.0 / exp(fma(x, eps, x)))) / 2.0;
	} else {
		tmp = ((exp(((-1.0 + eps) * 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 (eps <= 7.4e+19)
		tmp = Float64(exp(Float64(-x)) * Float64(x + 1.0));
	elseif (eps <= 1.55e+249)
		tmp = Float64(Float64(t_0 - Float64(-1.0 / exp(fma(x, eps, x)))) / 2.0);
	else
		tmp = Float64(Float64(Float64(exp(Float64(Float64(-1.0 + eps) * 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[eps, 7.4e+19], N[(N[Exp[(-x)], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 1.55e+249], N[(N[(t$95$0 - N[(-1.0 / N[Exp[N[(x * eps + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < 7.4e19

   1. Initial program 65.5%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 69.4%

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

   7. Applied rewrites 69.4%

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

   if 7.4e19 < eps < 1.55000000000000008e249

   1. Initial program 99.9%

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

   3. Taylor expanded in eps around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 76.5

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

   8. Applied rewrites 76.5%

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

   if 1.55000000000000008e249 < 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. 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 56.8%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 7.4 \cdot 10^{+19}:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{elif}\;\varepsilon \leq 1.55 \cdot 10^{+249}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) - \frac{-1}{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(-1 + \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right) - -1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\varepsilon} + 1\\ \mathbf{if}\;\varepsilon \leq 55000000000000:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{elif}\;\varepsilon \leq 8 \cdot 10^{+243}:\\ \;\;\;\;\frac{t\_0 - \frac{-1}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(-1 + \varepsilon\right) \cdot x} \cdot t\_0 - -1}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 eps) 1.0)))
   (if (<= eps 55000000000000.0)
     (* (exp (- x)) (+ x 1.0))
     (if (<= eps 8e+243)
       (/ (- t_0 (/ -1.0 (exp x))) 2.0)
       (/ (- (* (exp (* (+ -1.0 eps) 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 (eps <= 55000000000000.0) {
		tmp = exp(-x) * (x + 1.0);
	} else if (eps <= 8e+243) {
		tmp = (t_0 - (-1.0 / exp(x))) / 2.0;
	} else {
		tmp = ((exp(((-1.0 + eps) * x)) * t_0) - -1.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / eps) + 1.0d0
    if (eps <= 55000000000000.0d0) then
        tmp = exp(-x) * (x + 1.0d0)
    else if (eps <= 8d+243) then
        tmp = (t_0 - ((-1.0d0) / exp(x))) / 2.0d0
    else
        tmp = ((exp((((-1.0d0) + eps) * x)) * t_0) - (-1.0d0)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = (1.0 / eps) + 1.0;
	double tmp;
	if (eps <= 55000000000000.0) {
		tmp = Math.exp(-x) * (x + 1.0);
	} else if (eps <= 8e+243) {
		tmp = (t_0 - (-1.0 / Math.exp(x))) / 2.0;
	} else {
		tmp = ((Math.exp(((-1.0 + eps) * x)) * t_0) - -1.0) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (1.0 / eps) + 1.0
	tmp = 0
	if eps <= 55000000000000.0:
		tmp = math.exp(-x) * (x + 1.0)
	elif eps <= 8e+243:
		tmp = (t_0 - (-1.0 / math.exp(x))) / 2.0
	else:
		tmp = ((math.exp(((-1.0 + eps) * 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 (eps <= 55000000000000.0)
		tmp = Float64(exp(Float64(-x)) * Float64(x + 1.0));
	elseif (eps <= 8e+243)
		tmp = Float64(Float64(t_0 - Float64(-1.0 / exp(x))) / 2.0);
	else
		tmp = Float64(Float64(Float64(exp(Float64(Float64(-1.0 + eps) * x)) * t_0) - -1.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (1.0 / eps) + 1.0;
	tmp = 0.0;
	if (eps <= 55000000000000.0)
		tmp = exp(-x) * (x + 1.0);
	elseif (eps <= 8e+243)
		tmp = (t_0 - (-1.0 / exp(x))) / 2.0;
	else
		tmp = ((exp(((-1.0 + eps) * x)) * t_0) - -1.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(1.0 / eps), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[eps, 55000000000000.0], N[(N[Exp[(-x)], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 8e+243], N[(N[(t$95$0 - N[(-1.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[Exp[N[(N[(-1.0 + eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] - -1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < 5.5e13

   1. Initial program 65.2%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 69.6%

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

   7. Applied rewrites 69.6%

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

   if 5.5e13 < eps < 8.0000000000000006e243

   1. Initial program 99.9%

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

   3. Taylor expanded in eps around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 74.3

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

   8. Applied rewrites 74.3%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 73.8%

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

   if 8.0000000000000006e243 < 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. 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 53.6%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 55000000000000:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{elif}\;\varepsilon \leq 8 \cdot 10^{+243}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) - \frac{-1}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(-1 + \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} + 1\right) - -1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 3.05 \cdot 10^{-6}:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{elif}\;\varepsilon \leq 7.3 \cdot 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right) \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) - \mathsf{fma}\left(\varepsilon - -1, x, -1\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 3.05e-6)
   (* (exp (- x)) (+ x 1.0))
   (if (<= eps 7.3e+106)
     (* (fma (fma 0.5 x -1.0) x 1.0) (+ x 1.0))
     (/ (- (+ (/ 1.0 eps) 1.0) (fma (- eps -1.0) x -1.0)) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 3.05e-6) {
		tmp = exp(-x) * (x + 1.0);
	} else if (eps <= 7.3e+106) {
		tmp = fma(fma(0.5, x, -1.0), x, 1.0) * (x + 1.0);
	} else {
		tmp = (((1.0 / eps) + 1.0) - fma((eps - -1.0), x, -1.0)) / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 3.05e-6)
		tmp = Float64(exp(Float64(-x)) * Float64(x + 1.0));
	elseif (eps <= 7.3e+106)
		tmp = Float64(fma(fma(0.5, x, -1.0), x, 1.0) * Float64(x + 1.0));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / eps) + 1.0) - fma(Float64(eps - -1.0), x, -1.0)) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 3.05e-6], N[(N[Exp[(-x)], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 7.3e+106], N[(N[(N[(0.5 * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 / eps), $MachinePrecision] + 1.0), $MachinePrecision] - N[(N[(eps - -1.0), $MachinePrecision] * x + -1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 3.05000000000000002e-6

   1. Initial program 65.0%

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

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

   7. Applied rewrites 69.5%

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

   if 3.05000000000000002e-6 < eps < 7.30000000000000003e106

   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 56.4%

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

   7. Applied rewrites 56.4%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 75.3%

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

   if 7.30000000000000003e106 < eps

   1. Initial program 99.9%

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

   3. Taylor expanded in eps around inf

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}}{2} \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 71.9

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

   8. Applied rewrites 71.9%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 59.4%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 45.1%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 3.05 \cdot 10^{-6}:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{elif}\;\varepsilon \leq 7.3 \cdot 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right) \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) - \mathsf{fma}\left(\varepsilon - -1, x, -1\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.2% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -195:\\ \;\;\;\;\frac{\left(\frac{1}{\varepsilon} + 1\right) - \mathsf{fma}\left(\varepsilon - -1, x, -1\right)}{2}\\ \mathbf{elif}\;x \leq 1.85:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.3333333333333333\right), x, -0.5\right) \cdot x, x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -195.0)
   (/ (- (+ (/ 1.0 eps) 1.0) (fma (- eps -1.0) x -1.0)) 2.0)
   (if (<= x 1.85)
     (fma (* (fma (fma -0.125 x 0.3333333333333333) x -0.5) x) x 1.0)
     (/ 0.0 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -195.0) {
		tmp = (((1.0 / eps) + 1.0) - fma((eps - -1.0), x, -1.0)) / 2.0;
	} else if (x <= 1.85) {
		tmp = fma((fma(fma(-0.125, x, 0.3333333333333333), x, -0.5) * x), x, 1.0);
	} else {
		tmp = 0.0 / 2.0;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -195.0)
		tmp = Float64(Float64(Float64(Float64(1.0 / eps) + 1.0) - fma(Float64(eps - -1.0), x, -1.0)) / 2.0);
	elseif (x <= 1.85)
		tmp = fma(Float64(fma(fma(-0.125, x, 0.3333333333333333), x, -0.5) * x), x, 1.0);
	else
		tmp = Float64(0.0 / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -195.0], N[(N[(N[(N[(1.0 / eps), $MachinePrecision] + 1.0), $MachinePrecision] - N[(N[(eps - -1.0), $MachinePrecision] * x + -1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.85], N[(N[(N[(N[(-0.125 * x + 0.3333333333333333), $MachinePrecision] * x + -0.5), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision], N[(0.0 / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -195

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around inf

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

      1. exp-neg N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 50.4

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

   8. Applied rewrites 50.4%

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

   9. Taylor expanded in eps around 0

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

   11. Applied rewrites 100.0%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 31.6%

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

   if -195 < x < 1.8500000000000001

   1. Initial program 50.5%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 78.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 77.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.3333333333333333\right), x, -0.5\right), \color{blue}{x \cdot x}, 1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 77.5%

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

   if 1.8500000000000001 < 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. Step-by-step derivation

      1. +-lft-identity N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 79.5%

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

   5. Taylor expanded in eps around 0

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

      1. div-sub N/A

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

      2. neg-mul-1 N/A

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

      3. +-inverses 59.6

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   7. Applied rewrites 59.6%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 57.4% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1650000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 1650000000000.0) 1.0 (/ 0.0 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 1650000000000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0 / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 1650000000000.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0 / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 1650000000000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0 / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 1650000000000.0:
		tmp = 1.0
	else:
		tmp = 0.0 / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 1650000000000.0)
		tmp = 1.0;
	else
		tmp = Float64(0.0 / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 1650000000000.0)
		tmp = 1.0;
	else
		tmp = 0.0 / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 1650000000000.0], 1.0, N[(0.0 / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.65e12

   1. Initial program 63.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 57.5%

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

   if 1.65e12 < 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. Step-by-step derivation

      1. +-lft-identity N/A

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

      2. flip-+ N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 78.1%

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

   5. Taylor expanded in eps around 0

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

      1. div-sub N/A

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

      2. neg-mul-1 N/A

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

      3. +-inverses 63.6

         \[\leadsto \frac{\color{blue}{0}}{2} \]

   7. Applied rewrites 63.6%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 44.3% 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 73.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 42.0%

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

Reproduce

herbie shell --seed 2024332 
(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))