NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.5% → 99.9%

Time: 14.1s

Alternatives: 11

Speedup: 2.0×

• 38.4% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -1.99999999999999997e77
  2. eps = -2e-231

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 0.008:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot eps\_m} + e^{x \cdot \left(-eps\_m\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 0.008)
   (* (exp (- x)) (+ x 1.0))
   (/ (+ (exp (* x eps_m)) (exp (* x (- eps_m)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.008) {
		tmp = exp(-x) * (x + 1.0);
	} else {
		tmp = (exp((x * eps_m)) + exp((x * -eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 0.008d0) then
        tmp = exp(-x) * (x + 1.0d0)
    else
        tmp = (exp((x * eps_m)) + exp((x * -eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.008) {
		tmp = Math.exp(-x) * (x + 1.0);
	} else {
		tmp = (Math.exp((x * eps_m)) + Math.exp((x * -eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 0.008:
		tmp = math.exp(-x) * (x + 1.0)
	else:
		tmp = (math.exp((x * eps_m)) + math.exp((x * -eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 0.008)
		tmp = Float64(exp(Float64(-x)) * Float64(x + 1.0));
	else
		tmp = Float64(Float64(exp(Float64(x * eps_m)) + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 0.008)
		tmp = exp(-x) * (x + 1.0);
	else
		tmp = (exp((x * eps_m)) + exp((x * -eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 0.008], N[(N[Exp[(-x)], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 0.0080000000000000002

   1. Initial program 68.4%

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

   3. Simplified 61.4%

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

   5. Taylor expanded in eps around 0 34.9%

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

      1. associate-+r+ 67.4%

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

      2. mul-1-neg 67.4%

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

      3. sub-neg 67.4%

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

      4. +-inverses 67.4%

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

      5. distribute-lft-out 67.4%

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

      6. distribute-rgt1-in 67.4%

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

      7. mul-1-neg 67.4%

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

   7. Simplified 67.4%

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

   8. Taylor expanded in eps around 0 67.4%

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

   if 0.0080000000000000002 < eps

   1. Initial program 100.0%

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

   3. Simplified 89.7%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around -inf 100.0%

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

       1. *-commutative 100.0%

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

       2. fmm-def 100.0%

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

       3. mul-1-neg 100.0%

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

       4. remove-double-neg 100.0%

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

       5. *-rgt-identity 100.0%

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

       6. exp-neg 100.0%

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

       7. *-rgt-identity 100.0%

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

       8. remove-double-neg 100.0%

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

       9. mul-1-neg 100.0%

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

       10. fmm-def 100.0%

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

       11. *-commutative 100.0%

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

       12. distribute-rgt-out-- 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in eps around inf 100.0%

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

       1. associate-*r* 100.0%

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

       2. neg-mul-1 100.0%

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

   14. Simplified 100.0%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.008:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

3. Simplified 71.2%

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

5. Taylor expanded in eps around inf 99.5%

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

6. Final simplification 99.5%

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

Alternative 3: 84.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-220}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - eps\_m\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + \frac{1}{1 + x \cdot \left(eps\_m + 1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1e-220)
   (/ (+ 1.0 (exp (* x (- -1.0 eps_m)))) 2.0)
   (/ (+ (exp (* x (+ eps_m -1.0))) (/ 1.0 (+ 1.0 (* x (+ eps_m 1.0))))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1e-220) {
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	} else {
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 / (1.0 + (x * (eps_m + 1.0))))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1d-220)) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps_m)))) / 2.0d0
    else
        tmp = (exp((x * (eps_m + (-1.0d0)))) + (1.0d0 / (1.0d0 + (x * (eps_m + 1.0d0))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1e-220) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps_m)))) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps_m + -1.0))) + (1.0 / (1.0 + (x * (eps_m + 1.0))))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1e-220:
		tmp = (1.0 + math.exp((x * (-1.0 - eps_m)))) / 2.0
	else:
		tmp = (math.exp((x * (eps_m + -1.0))) + (1.0 / (1.0 + (x * (eps_m + 1.0))))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1e-220)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + Float64(1.0 / Float64(1.0 + Float64(x * Float64(eps_m + 1.0))))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1e-220)
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	else
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 / (1.0 + (x * (eps_m + 1.0))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1e-220], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(1.0 + N[(x * N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.99999999999999992e-221

   1. Initial program 82.4%

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

   3. Simplified 71.1%

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

   5. Taylor expanded in eps around inf 99.3%

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

   6. Taylor expanded in x around 0 59.4%

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

   7. Taylor expanded in x around -inf 59.4%

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

      1. *-commutative 99.4%

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

      2. fmm-def 99.4%

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

      3. mul-1-neg 99.4%

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

      4. remove-double-neg 99.4%

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

      5. *-rgt-identity 99.4%

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

      6. exp-neg 99.4%

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

      7. *-rgt-identity 99.4%

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

      8. remove-double-neg 99.4%

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

      9. mul-1-neg 99.4%

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

      10. fmm-def 99.4%

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

      11. *-commutative 99.4%

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

      12. distribute-rgt-out-- 99.4%

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

   9. Simplified 59.4%

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

   if -9.99999999999999992e-221 < x

   1. Initial program 76.4%

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

   3. Simplified 71.2%

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

   5. Taylor expanded in eps around inf 99.6%

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

   6. Taylor expanded in x around 0 63.2%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-220}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \frac{1}{1 + x \cdot \left(\varepsilon + 1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;eps\_m \leq 0.008:\\ \;\;\;\;t\_0 \cdot \left(x + 1\right)\\ \mathbf{elif}\;eps\_m \leq 1.35 \cdot 10^{+173}:\\ \;\;\;\;\frac{1 + t\_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1 + eps\_m \cdot \left(eps\_m + 1\right)}{eps\_m} - eps\_m\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= eps_m 0.008)
     (* t_0 (+ x 1.0))
     (if (<= eps_m 1.35e+173)
       (/ (+ 1.0 t_0) 2.0)
       (/
        (+ 2.0 (* x (- (/ (+ 1.0 (* eps_m (+ eps_m 1.0))) eps_m) eps_m)))
        2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp(-x);
	double tmp;
	if (eps_m <= 0.008) {
		tmp = t_0 * (x + 1.0);
	} else if (eps_m <= 1.35e+173) {
		tmp = (1.0 + t_0) / 2.0;
	} else {
		tmp = (2.0 + (x * (((1.0 + (eps_m * (eps_m + 1.0))) / eps_m) - eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (eps_m <= 0.008d0) then
        tmp = t_0 * (x + 1.0d0)
    else if (eps_m <= 1.35d+173) then
        tmp = (1.0d0 + t_0) / 2.0d0
    else
        tmp = (2.0d0 + (x * (((1.0d0 + (eps_m * (eps_m + 1.0d0))) / eps_m) - eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp(-x);
	double tmp;
	if (eps_m <= 0.008) {
		tmp = t_0 * (x + 1.0);
	} else if (eps_m <= 1.35e+173) {
		tmp = (1.0 + t_0) / 2.0;
	} else {
		tmp = (2.0 + (x * (((1.0 + (eps_m * (eps_m + 1.0))) / eps_m) - eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp(-x)
	tmp = 0
	if eps_m <= 0.008:
		tmp = t_0 * (x + 1.0)
	elif eps_m <= 1.35e+173:
		tmp = (1.0 + t_0) / 2.0
	else:
		tmp = (2.0 + (x * (((1.0 + (eps_m * (eps_m + 1.0))) / eps_m) - eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (eps_m <= 0.008)
		tmp = Float64(t_0 * Float64(x + 1.0));
	elseif (eps_m <= 1.35e+173)
		tmp = Float64(Float64(1.0 + t_0) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(1.0 + Float64(eps_m * Float64(eps_m + 1.0))) / eps_m) - eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp(-x);
	tmp = 0.0;
	if (eps_m <= 0.008)
		tmp = t_0 * (x + 1.0);
	elseif (eps_m <= 1.35e+173)
		tmp = (1.0 + t_0) / 2.0;
	else
		tmp = (2.0 + (x * (((1.0 + (eps_m * (eps_m + 1.0))) / eps_m) - eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[eps$95$m, 0.008], N[(t$95$0 * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps$95$m, 1.35e+173], N[(N[(1.0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * N[(N[(N[(1.0 + N[(eps$95$m * N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < 0.0080000000000000002

   1. Initial program 68.4%

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

   3. Simplified 61.4%

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

   5. Taylor expanded in eps around 0 34.9%

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

      1. associate-+r+ 67.4%

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

      2. mul-1-neg 67.4%

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

      3. sub-neg 67.4%

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

      4. +-inverses 67.4%

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

      5. distribute-lft-out 67.4%

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

      6. distribute-rgt1-in 67.4%

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

      7. mul-1-neg 67.4%

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

   7. Simplified 67.4%

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

   8. Taylor expanded in eps around 0 67.4%

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

   if 0.0080000000000000002 < eps < 1.3500000000000001e173

   1. Initial program 100.0%

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

   3. Simplified 88.8%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 64.6%

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

   7. Taylor expanded in eps around 0 62.7%

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

      1. rec-exp 62.7%

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

   9. Simplified 62.7%

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

   if 1.3500000000000001e173 < eps

   1. Initial program 100.0%

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

   3. Simplified 91.3%

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

   5. Taylor expanded in x around 0 8.6%

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

   6. Taylor expanded in eps around inf 8.6%

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

   7. Taylor expanded in eps around 0 44.9%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.008:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{elif}\;\varepsilon \leq 1.35 \cdot 10^{+173}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{1 + \varepsilon \cdot \left(\varepsilon + 1\right)}{\varepsilon} - \varepsilon\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-275}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - eps\_m\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -8e-275)
   (/ (+ 1.0 (exp (* x (- -1.0 eps_m)))) 2.0)
   (/ (+ 1.0 (exp (* x eps_m))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -8e-275) {
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	} else {
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-8d-275)) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps_m)))) / 2.0d0
    else
        tmp = (1.0d0 + exp((x * eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -8e-275) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps_m)))) / 2.0;
	} else {
		tmp = (1.0 + Math.exp((x * eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -8e-275:
		tmp = (1.0 + math.exp((x * (-1.0 - eps_m)))) / 2.0
	else:
		tmp = (1.0 + math.exp((x * eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -8e-275)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(Float64(x * eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -8e-275)
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	else
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -8e-275], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.99999999999999947e-275

   1. Initial program 80.8%

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

   3. Simplified 69.6%

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

   5. Taylor expanded in eps around inf 99.4%

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

   6. Taylor expanded in x around 0 61.9%

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

   7. Taylor expanded in x around -inf 61.9%

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

      1. *-commutative 99.4%

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

      2. fmm-def 99.4%

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

      3. mul-1-neg 99.4%

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

      4. remove-double-neg 99.4%

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

      5. *-rgt-identity 99.4%

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

      6. exp-neg 99.4%

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

      7. *-rgt-identity 99.4%

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

      8. remove-double-neg 99.4%

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

      9. mul-1-neg 99.4%

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

      10. fmm-def 99.4%

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

      11. *-commutative 99.4%

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

      12. distribute-rgt-out-- 99.4%

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

   9. Simplified 61.9%

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

   if -7.99999999999999947e-275 < x

   1. Initial program 77.0%

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

   3. Simplified 72.1%

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

   5. Taylor expanded in eps around inf 99.5%

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

   6. Taylor expanded in eps around inf 78.3%

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

      1. *-commutative 78.3%

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

   8. Simplified 78.3%

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

   9. Taylor expanded in x around 0 61.4%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-275}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 0.008:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 0.008)
   (* (exp (- x)) (+ x 1.0))
   (/ (+ 1.0 (exp (* x eps_m))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.008) {
		tmp = exp(-x) * (x + 1.0);
	} else {
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 0.008d0) then
        tmp = exp(-x) * (x + 1.0d0)
    else
        tmp = (1.0d0 + exp((x * eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.008) {
		tmp = Math.exp(-x) * (x + 1.0);
	} else {
		tmp = (1.0 + Math.exp((x * eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 0.008:
		tmp = math.exp(-x) * (x + 1.0)
	else:
		tmp = (1.0 + math.exp((x * eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 0.008)
		tmp = Float64(exp(Float64(-x)) * Float64(x + 1.0));
	else
		tmp = Float64(Float64(1.0 + exp(Float64(x * eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 0.008)
		tmp = exp(-x) * (x + 1.0);
	else
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 0.008], N[(N[Exp[(-x)], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 0.0080000000000000002

   1. Initial program 68.4%

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

   3. Simplified 61.4%

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

   5. Taylor expanded in eps around 0 34.9%

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

      1. associate-+r+ 67.4%

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

      2. mul-1-neg 67.4%

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

      3. sub-neg 67.4%

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

      4. +-inverses 67.4%

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

      5. distribute-lft-out 67.4%

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

      6. distribute-rgt1-in 67.4%

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

      7. mul-1-neg 67.4%

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

   7. Simplified 67.4%

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

   8. Taylor expanded in eps around 0 67.4%

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

   if 0.0080000000000000002 < eps

   1. Initial program 100.0%

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

   3. Simplified 89.7%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 68.2%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.008:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 0.008:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{elif}\;eps\_m \leq 2.6 \cdot 10^{+264}:\\ \;\;\;\;\frac{\frac{eps\_m \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot \left(1 + x \cdot \left(-1 + x \cdot 0.5\right)\right)\right)\right)}{eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(eps\_m + \frac{-1}{eps\_m}\right) + \left(\frac{1 + 2 \cdot \frac{eps\_m}{x}}{eps\_m} - eps\_m\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 0.008)
   (* (exp (- x)) (+ x 1.0))
   (if (<= eps_m 2.6e+264)
     (/
      (/
       (* eps_m (* 2.0 (* (+ x 1.0) (+ 1.0 (* x (+ -1.0 (* x 0.5)))))))
       eps_m)
      2.0)
     (/
      (*
       x
       (+
        (+ eps_m (/ -1.0 eps_m))
        (- (/ (+ 1.0 (* 2.0 (/ eps_m x))) eps_m) eps_m)))
      2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.008) {
		tmp = exp(-x) * (x + 1.0);
	} else if (eps_m <= 2.6e+264) {
		tmp = ((eps_m * (2.0 * ((x + 1.0) * (1.0 + (x * (-1.0 + (x * 0.5))))))) / eps_m) / 2.0;
	} else {
		tmp = (x * ((eps_m + (-1.0 / eps_m)) + (((1.0 + (2.0 * (eps_m / x))) / eps_m) - eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 0.008d0) then
        tmp = exp(-x) * (x + 1.0d0)
    else if (eps_m <= 2.6d+264) then
        tmp = ((eps_m * (2.0d0 * ((x + 1.0d0) * (1.0d0 + (x * ((-1.0d0) + (x * 0.5d0))))))) / eps_m) / 2.0d0
    else
        tmp = (x * ((eps_m + ((-1.0d0) / eps_m)) + (((1.0d0 + (2.0d0 * (eps_m / x))) / eps_m) - eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.008) {
		tmp = Math.exp(-x) * (x + 1.0);
	} else if (eps_m <= 2.6e+264) {
		tmp = ((eps_m * (2.0 * ((x + 1.0) * (1.0 + (x * (-1.0 + (x * 0.5))))))) / eps_m) / 2.0;
	} else {
		tmp = (x * ((eps_m + (-1.0 / eps_m)) + (((1.0 + (2.0 * (eps_m / x))) / eps_m) - eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 0.008:
		tmp = math.exp(-x) * (x + 1.0)
	elif eps_m <= 2.6e+264:
		tmp = ((eps_m * (2.0 * ((x + 1.0) * (1.0 + (x * (-1.0 + (x * 0.5))))))) / eps_m) / 2.0
	else:
		tmp = (x * ((eps_m + (-1.0 / eps_m)) + (((1.0 + (2.0 * (eps_m / x))) / eps_m) - eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 0.008)
		tmp = Float64(exp(Float64(-x)) * Float64(x + 1.0));
	elseif (eps_m <= 2.6e+264)
		tmp = Float64(Float64(Float64(eps_m * Float64(2.0 * Float64(Float64(x + 1.0) * Float64(1.0 + Float64(x * Float64(-1.0 + Float64(x * 0.5))))))) / eps_m) / 2.0);
	else
		tmp = Float64(Float64(x * Float64(Float64(eps_m + Float64(-1.0 / eps_m)) + Float64(Float64(Float64(1.0 + Float64(2.0 * Float64(eps_m / x))) / eps_m) - eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 0.008)
		tmp = exp(-x) * (x + 1.0);
	elseif (eps_m <= 2.6e+264)
		tmp = ((eps_m * (2.0 * ((x + 1.0) * (1.0 + (x * (-1.0 + (x * 0.5))))))) / eps_m) / 2.0;
	else
		tmp = (x * ((eps_m + (-1.0 / eps_m)) + (((1.0 + (2.0 * (eps_m / x))) / eps_m) - eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 0.008], N[(N[Exp[(-x)], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps$95$m, 2.6e+264], N[(N[(N[(eps$95$m * N[(2.0 * N[(N[(x + 1.0), $MachinePrecision] * N[(1.0 + N[(x * N[(-1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * N[(N[(eps$95$m + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 + N[(2.0 * N[(eps$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 0.0080000000000000002

   1. Initial program 68.4%

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

   3. Simplified 61.4%

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

   5. Taylor expanded in eps around 0 34.9%

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

      1. associate-+r+ 67.4%

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

      2. mul-1-neg 67.4%

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

      3. sub-neg 67.4%

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

      4. +-inverses 67.4%

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

      5. distribute-lft-out 67.4%

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

      6. distribute-rgt1-in 67.4%

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

      7. mul-1-neg 67.4%

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

   7. Simplified 67.4%

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

   8. Taylor expanded in eps around 0 67.4%

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

   if 0.0080000000000000002 < eps < 2.6e264

   1. Initial program 100.0%

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

   3. Simplified 70.7%

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

   5. Taylor expanded in eps around 0 34.2%

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

      1. associate-+r+ 34.2%

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

      2. mul-1-neg 34.2%

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

      3. sub-neg 34.2%

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

      4. +-inverses 34.2%

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

      5. distribute-lft-out 34.2%

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

      6. distribute-rgt1-in 34.2%

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

      7. mul-1-neg 34.2%

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

   7. Simplified 34.2%

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

   8. Taylor expanded in x around 0 57.6%

      \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot \color{blue}{\left(1 + x \cdot \left(0.5 \cdot x - 1\right)\right)}\right)\right)}{\varepsilon}}{2} \]

   if 2.6e264 < eps

   1. Initial program 100.0%

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

   3. Simplified 85.1%

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

   5. Taylor expanded in x around 0 3.1%

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

   6. Taylor expanded in x around inf 1.6%

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

      1. associate--l+ 1.6%

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

      2. sub-neg 1.6%

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

      3. metadata-eval 1.6%

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

      4. *-commutative 1.6%

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

      5. distribute-lft-in 1.6%

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

      6. *-rgt-identity 1.6%

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

      7. *-commutative 1.6%

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

      8. distribute-rgt-in 1.6%

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

      9. rgt-mult-inverse 1.6%

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

      10. neg-mul-1 1.6%

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

      11. sub-neg 1.6%

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

      12. associate-+r+ 1.6%

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

      13. sub-neg 1.6%

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

      14. associate-+r+ 1.6%

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

      15. metadata-eval 1.6%

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

      16. distribute-neg-frac 1.6%

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

      17. metadata-eval 1.6%

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

   8. Simplified 1.6%

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

   9. Taylor expanded in eps around 0 47.0%

      \[\leadsto \frac{x \cdot \left(\left(\varepsilon + \left(0 + \frac{-1}{\varepsilon}\right)\right) + \left(\color{blue}{\frac{1 + 2 \cdot \frac{\varepsilon}{x}}{\varepsilon}} - \varepsilon\right)\right)}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.008:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{elif}\;\varepsilon \leq 2.6 \cdot 10^{+264}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot \left(1 + x \cdot \left(-1 + x \cdot 0.5\right)\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(\left(\varepsilon + \frac{-1}{\varepsilon}\right) + \left(\frac{1 + 2 \cdot \frac{\varepsilon}{x}}{\varepsilon} - \varepsilon\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.8% accurate, 18.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 21.5:\\ \;\;\;\;\frac{2 - x \cdot eps\_m}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 21.5) (/ (- 2.0 (* x eps_m)) 2.0) 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 21.5) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 21.5d0) then
        tmp = (2.0d0 - (x * eps_m)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 21.5) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 21.5:
		tmp = (2.0 - (x * eps_m)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 21.5)
		tmp = Float64(Float64(2.0 - Float64(x * eps_m)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 21.5)
		tmp = (2.0 - (x * eps_m)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 21.5], N[(N[(2.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 21.5

   1. Initial program 68.8%

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

   3. Simplified 58.3%

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

   5. Taylor expanded in x around 0 56.7%

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

   6. Taylor expanded in eps around 0 61.3%

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

   7. Taylor expanded in x around 0 61.3%

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

      1. associate-*r* 61.3%

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

      2. neg-mul-1 61.3%

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

   9. Simplified 61.3%

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

   if 21.5 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 52.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{e^{-1 \cdot x} - \frac{1}{e^{x}}}{\varepsilon}} \]
   6. Step-by-step derivation

      1. div-sub 52.7%

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

      2. mul-1-neg 52.7%

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

      3. rec-exp 52.7%

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

      4. +-inverses 52.7%

         \[\leadsto 0.5 \cdot \color{blue}{0} \]

      5. metadata-eval 52.7%

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

   7. Simplified 52.7%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 21.5:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.3% accurate, 22.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{2 - x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 2.0) (/ (- 2.0 x) 2.0) 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.0) {
		tmp = (2.0 - x) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

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

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.0) {
		tmp = (2.0 - x) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 2.0:
		tmp = (2.0 - x) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 2.0)
		tmp = Float64(Float64(2.0 - x) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = (2.0 - x) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 2.0], N[(N[(2.0 - x), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 68.8%

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

   3. Simplified 58.3%

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

   5. Taylor expanded in eps around inf 99.2%

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

   6. Taylor expanded in eps around inf 99.3%

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

      1. *-commutative 99.3%

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

   8. Simplified 99.3%

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

   9. Taylor expanded in x around 0 56.9%

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

       1. mul-1-neg 56.9%

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

       2. unsub-neg 56.9%

          \[\leadsto \frac{\color{blue}{2 - x}}{2} \]

   11. Simplified 56.9%

       \[\leadsto \frac{\color{blue}{2 - x}}{2} \]

   if 2 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 52.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{e^{-1 \cdot x} - \frac{1}{e^{x}}}{\varepsilon}} \]
   6. Step-by-step derivation

      1. div-sub 52.7%

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

      2. mul-1-neg 52.7%

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

      3. rec-exp 52.7%

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

      4. +-inverses 52.7%

         \[\leadsto 0.5 \cdot \color{blue}{0} \]

      5. metadata-eval 52.7%

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

   7. Simplified 52.7%

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

Alternative 10: 57.2% accurate, 37.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 550:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 550.0) 1.0 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 550.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 550.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 550.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 550.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 550.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 550.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 550.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 550

   1. Initial program 69.0%

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

   3. Simplified 69.0%

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

   5. Taylor expanded in x around 0 56.4%

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

   if 550 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 53.3%

      \[\leadsto \color{blue}{0.5 \cdot \frac{e^{-1 \cdot x} - \frac{1}{e^{x}}}{\varepsilon}} \]
   6. Step-by-step derivation

      1. div-sub 53.3%

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

      2. mul-1-neg 53.3%

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

      3. rec-exp 53.3%

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

      4. +-inverses 53.3%

         \[\leadsto 0.5 \cdot \color{blue}{0} \]

      5. metadata-eval 53.3%

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

   7. Simplified 53.3%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 550:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 16.2% accurate, 227.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ 0 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 0.0)

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return 0.0;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    code = 0.0d0
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return 0.0;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return 0.0

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return 0.0
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = 0.0;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := 0.0

Derivation

1. Initial program 78.4%

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

3. Simplified 71.2%

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

5. Taylor expanded in eps around 0 17.7%

   \[\leadsto \color{blue}{0.5 \cdot \frac{e^{-1 \cdot x} - \frac{1}{e^{x}}}{\varepsilon}} \]
6. Step-by-step derivation

   1. div-sub 17.7%

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

   2. mul-1-neg 17.7%

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

   3. rec-exp 17.7%

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

   4. +-inverses 17.9%

      \[\leadsto 0.5 \cdot \color{blue}{0} \]

   5. metadata-eval 17.9%

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

7. Simplified 17.9%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024157 
(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))