NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.2% → 99.7%

Time: 14.2s

Alternatives: 16

Speedup: 2.0×

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

• could not determine a ground truth

  1. x = -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: 73.2% 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: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 0.00047:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-1 + \left(1 + x \cdot \left(-1 - eps\_m\right)\right)} + e^{eps\_m \cdot x}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 0.00047)
   (* (exp (- x)) (+ x 1.0))
   (/ (+ (exp (+ -1.0 (+ 1.0 (* x (- -1.0 eps_m))))) (exp (* eps_m x))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.00047) {
		tmp = exp(-x) * (x + 1.0);
	} else {
		tmp = (exp((-1.0 + (1.0 + (x * (-1.0 - eps_m))))) + exp((eps_m * x))) / 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 <= 0.00047d0) then
        tmp = exp(-x) * (x + 1.0d0)
    else
        tmp = (exp(((-1.0d0) + (1.0d0 + (x * ((-1.0d0) - eps_m))))) + exp((eps_m * x))) / 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 <= 0.00047) {
		tmp = Math.exp(-x) * (x + 1.0);
	} else {
		tmp = (Math.exp((-1.0 + (1.0 + (x * (-1.0 - eps_m))))) + Math.exp((eps_m * x))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 0.00047:
		tmp = math.exp(-x) * (x + 1.0)
	else:
		tmp = (math.exp((-1.0 + (1.0 + (x * (-1.0 - eps_m))))) + math.exp((eps_m * x))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 0.00047)
		tmp = Float64(exp(Float64(-x)) * Float64(x + 1.0));
	else
		tmp = Float64(Float64(exp(Float64(-1.0 + Float64(1.0 + Float64(x * Float64(-1.0 - eps_m))))) + exp(Float64(eps_m * x))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 0.00047)
		tmp = exp(-x) * (x + 1.0);
	else
		tmp = (exp((-1.0 + (1.0 + (x * (-1.0 - eps_m))))) + exp((eps_m * x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 0.00047], N[(N[Exp[(-x)], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[N[(-1.0 + N[(1.0 + N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(eps$95$m * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 4.69999999999999986e-4

   1. Initial program 56.1%

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

   3. Simplified 48.0%

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

   5. Taylor expanded in eps around 0 32.8%

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

      1. associate-+r+ 75.6%

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

      2. mul-1-neg 75.6%

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

      3. sub-neg 75.6%

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

      4. +-inverses 75.6%

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

      5. distribute-lft-out 75.6%

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

      6. distribute-rgt1-in 76.6%

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

      7. mul-1-neg 76.6%

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

   7. Simplified 76.6%

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

   8. Taylor expanded in eps around 0 76.6%

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

      1. +-commutative 76.6%

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

   10. Simplified 76.6%

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

   if 4.69999999999999986e-4 < 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 79.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

      3. *-commutative 100.0%

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

      4. expm1-log1p-u 67.6%

         \[\leadsto \frac{e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 + \varepsilon\right) \cdot \left(-x\right)\right)\right)}} + e^{x \cdot \left(\varepsilon - 1\right)}}{2} \]

      5. expm1-undefine 67.6%

         \[\leadsto \frac{e^{\color{blue}{e^{\mathsf{log1p}\left(\left(1 + \varepsilon\right) \cdot \left(-x\right)\right)} - 1}} + e^{x \cdot \left(\varepsilon - 1\right)}}{2} \]

   7. Applied egg-rr 67.6%

      \[\leadsto \frac{e^{\color{blue}{e^{\mathsf{log1p}\left(\left(1 + \varepsilon\right) \cdot \left(-x\right)\right)} - 1}} + e^{x \cdot \left(\varepsilon - 1\right)}}{2} \]
   8. Step-by-step derivation

      1. sub-neg 67.6%

         \[\leadsto \frac{e^{\color{blue}{e^{\mathsf{log1p}\left(\left(1 + \varepsilon\right) \cdot \left(-x\right)\right)} + \left(-1\right)}} + e^{x \cdot \left(\varepsilon - 1\right)}}{2} \]

      2. log1p-undefine 67.6%

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

      3. rem-exp-log 100.0%

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

      4. *-commutative 100.0%

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

      5. distribute-lft-neg-in 100.0%

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

      6. neg-mul-1 100.0%

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

      7. +-commutative 100.0%

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

      8. neg-mul-1 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. mul-1-neg 100.0%

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

      11. fma-define 100.0%

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

      12. distribute-lft-in 100.0%

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

      13. metadata-eval 100.0%

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

      14. mul-1-neg 100.0%

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

      15. unsub-neg 100.0%

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

      16. metadata-eval 100.0%

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

   9. Simplified 100.0%

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

       1. fma-undefine 100.0%

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

   11. Applied egg-rr 100.0%

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

   12. Taylor expanded in eps around inf 100.0%

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

       1. *-commutative 100.0%

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

   14. Simplified 100.0%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.00047:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-1 + \left(1 + x \cdot \left(-1 - \varepsilon\right)\right)} + e^{\varepsilon \cdot x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 5 \cdot 10^{-48}:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + e^{eps\_m \cdot \left(-x\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 5e-48)
   (* (exp (- x)) (+ x 1.0))
   (/ (+ (exp (* x (+ eps_m -1.0))) (exp (* eps_m (- x)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 5e-48) {
		tmp = exp(-x) * (x + 1.0);
	} else {
		tmp = (exp((x * (eps_m + -1.0))) + exp((eps_m * -x))) / 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 <= 5d-48) then
        tmp = exp(-x) * (x + 1.0d0)
    else
        tmp = (exp((x * (eps_m + (-1.0d0)))) + exp((eps_m * -x))) / 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 <= 5e-48) {
		tmp = Math.exp(-x) * (x + 1.0);
	} else {
		tmp = (Math.exp((x * (eps_m + -1.0))) + Math.exp((eps_m * -x))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 5e-48:
		tmp = math.exp(-x) * (x + 1.0)
	else:
		tmp = (math.exp((x * (eps_m + -1.0))) + math.exp((eps_m * -x))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 5e-48)
		tmp = Float64(exp(Float64(-x)) * Float64(x + 1.0));
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + exp(Float64(eps_m * Float64(-x)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 5e-48)
		tmp = exp(-x) * (x + 1.0);
	else
		tmp = (exp((x * (eps_m + -1.0))) + exp((eps_m * -x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 5e-48], N[(N[Exp[(-x)], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 4.9999999999999999e-48

   1. Initial program 55.3%

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

   3. Simplified 46.7%

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

   5. Taylor expanded in eps around 0 30.9%

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

      1. associate-+r+ 74.5%

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

      2. mul-1-neg 74.5%

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

      3. sub-neg 74.5%

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

      4. +-inverses 74.5%

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

      5. distribute-lft-out 74.5%

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

      6. distribute-rgt1-in 75.6%

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

      7. mul-1-neg 75.6%

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

   7. Simplified 75.6%

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

   8. Taylor expanded in eps around 0 75.6%

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

      1. +-commutative 75.6%

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

   10. Simplified 75.6%

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

   if 4.9999999999999999e-48 < eps

   1. Initial program 97.5%

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

   3. Simplified 78.7%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. mul-1-neg 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 5 \cdot 10^{-48}:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \left(eps\_m + -1\right)}\\ \mathbf{if}\;x \leq -4.7 \cdot 10^{-257}:\\ \;\;\;\;\frac{1 + e^{eps\_m \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 200000:\\ \;\;\;\;\frac{t\_0 + \left(1 + x \cdot \left(-1 - eps\_m\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+264}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t\_0}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (+ eps_m -1.0)))))
   (if (<= x -4.7e-257)
     (/ (+ 1.0 (exp (* eps_m (- x)))) 2.0)
     (if (<= x 200000.0)
       (/ (+ t_0 (+ 1.0 (* x (- -1.0 eps_m)))) 2.0)
       (if (<= x 1.05e+264) 0.0 (/ (+ 1.0 t_0) 2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= -4.7e-257) {
		tmp = (1.0 + exp((eps_m * -x))) / 2.0;
	} else if (x <= 200000.0) {
		tmp = (t_0 + (1.0 + (x * (-1.0 - eps_m)))) / 2.0;
	} else if (x <= 1.05e+264) {
		tmp = 0.0;
	} else {
		tmp = (1.0 + t_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) :: t_0
    real(8) :: tmp
    t_0 = exp((x * (eps_m + (-1.0d0))))
    if (x <= (-4.7d-257)) then
        tmp = (1.0d0 + exp((eps_m * -x))) / 2.0d0
    else if (x <= 200000.0d0) then
        tmp = (t_0 + (1.0d0 + (x * ((-1.0d0) - eps_m)))) / 2.0d0
    else if (x <= 1.05d+264) then
        tmp = 0.0d0
    else
        tmp = (1.0d0 + t_0) / 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 * (eps_m + -1.0)));
	double tmp;
	if (x <= -4.7e-257) {
		tmp = (1.0 + Math.exp((eps_m * -x))) / 2.0;
	} else if (x <= 200000.0) {
		tmp = (t_0 + (1.0 + (x * (-1.0 - eps_m)))) / 2.0;
	} else if (x <= 1.05e+264) {
		tmp = 0.0;
	} else {
		tmp = (1.0 + t_0) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp((x * (eps_m + -1.0)))
	tmp = 0
	if x <= -4.7e-257:
		tmp = (1.0 + math.exp((eps_m * -x))) / 2.0
	elif x <= 200000.0:
		tmp = (t_0 + (1.0 + (x * (-1.0 - eps_m)))) / 2.0
	elif x <= 1.05e+264:
		tmp = 0.0
	else:
		tmp = (1.0 + t_0) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * Float64(eps_m + -1.0)))
	tmp = 0.0
	if (x <= -4.7e-257)
		tmp = Float64(Float64(1.0 + exp(Float64(eps_m * Float64(-x)))) / 2.0);
	elseif (x <= 200000.0)
		tmp = Float64(Float64(t_0 + Float64(1.0 + Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	elseif (x <= 1.05e+264)
		tmp = 0.0;
	else
		tmp = Float64(Float64(1.0 + t_0) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp((x * (eps_m + -1.0)));
	tmp = 0.0;
	if (x <= -4.7e-257)
		tmp = (1.0 + exp((eps_m * -x))) / 2.0;
	elseif (x <= 200000.0)
		tmp = (t_0 + (1.0 + (x * (-1.0 - eps_m)))) / 2.0;
	elseif (x <= 1.05e+264)
		tmp = 0.0;
	else
		tmp = (1.0 + t_0) / 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[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -4.7e-257], N[(N[(1.0 + N[Exp[N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 200000.0], N[(N[(t$95$0 + N[(1.0 + N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.05e+264], 0.0, N[(N[(1.0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.6999999999999998e-257

   1. Initial program 66.5%

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

   3. Simplified 66.5%

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

   5. Taylor expanded in x around 0 45.5%

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

   6. Taylor expanded in eps around inf 73.8%

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

      1. mul-1-neg 73.8%

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

      2. neg-mul-1 73.8%

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

      3. distribute-lft-neg-in 73.8%

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

      4. +-commutative 73.8%

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

   8. Simplified 73.8%

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

   9. Taylor expanded in eps around inf 73.9%

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

   if -4.6999999999999998e-257 < x < 2e5

   1. Initial program 45.5%

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

   3. Simplified 21.3%

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

   5. Taylor expanded in eps around inf 98.6%

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

   6. Taylor expanded in x around 0 90.5%

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

      1. neg-mul-1 90.5%

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

      2. distribute-lft-neg-in 90.5%

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

      3. +-commutative 90.5%

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

   8. Simplified 90.5%

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

   if 2e5 < x < 1.05000000000000005e264

   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 62.3%

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

      1. div-sub 62.3%

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

      2. mul-1-neg 62.3%

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

      3. rec-exp 62.3%

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

      4. +-inverses 62.3%

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

      5. metadata-eval 62.3%

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

   7. Simplified 62.3%

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

   if 1.05000000000000005e264 < 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%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

      3. *-commutative 100.0%

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

      4. expm1-log1p-u 50.0%

         \[\leadsto \frac{e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 + \varepsilon\right) \cdot \left(-x\right)\right)\right)}} + e^{x \cdot \left(\varepsilon - 1\right)}}{2} \]

      5. expm1-undefine 50.0%

         \[\leadsto \frac{e^{\color{blue}{e^{\mathsf{log1p}\left(\left(1 + \varepsilon\right) \cdot \left(-x\right)\right)} - 1}} + e^{x \cdot \left(\varepsilon - 1\right)}}{2} \]

   7. Applied egg-rr 50.0%

      \[\leadsto \frac{e^{\color{blue}{e^{\mathsf{log1p}\left(\left(1 + \varepsilon\right) \cdot \left(-x\right)\right)} - 1}} + e^{x \cdot \left(\varepsilon - 1\right)}}{2} \]
   8. Step-by-step derivation

      1. sub-neg 50.0%

         \[\leadsto \frac{e^{\color{blue}{e^{\mathsf{log1p}\left(\left(1 + \varepsilon\right) \cdot \left(-x\right)\right)} + \left(-1\right)}} + e^{x \cdot \left(\varepsilon - 1\right)}}{2} \]

      2. log1p-undefine 50.0%

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

      3. rem-exp-log 100.0%

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

      4. *-commutative 100.0%

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

      5. distribute-lft-neg-in 100.0%

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

      6. neg-mul-1 100.0%

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

      7. +-commutative 100.0%

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

      8. neg-mul-1 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. mul-1-neg 100.0%

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

      11. fma-define 100.0%

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

      12. distribute-lft-in 100.0%

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

      13. metadata-eval 100.0%

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

      14. mul-1-neg 100.0%

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

      15. unsub-neg 100.0%

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

      16. metadata-eval 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in x around 0 32.2%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{-257}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 200000:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \left(1 + x \cdot \left(-1 - \varepsilon\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+264}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-256}:\\ \;\;\;\;\frac{1 + e^{eps\_m \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 190000 \lor \neg \left(x \leq 1.45 \cdot 10^{+264}\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \left(eps\_m + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -4e-256)
   (/ (+ 1.0 (exp (* eps_m (- x)))) 2.0)
   (if (or (<= x 190000.0) (not (<= x 1.45e+264)))
     (/ (+ 1.0 (exp (* x (+ eps_m -1.0)))) 2.0)
     0.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -4e-256) {
		tmp = (1.0 + exp((eps_m * -x))) / 2.0;
	} else if ((x <= 190000.0) || !(x <= 1.45e+264)) {
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 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 <= (-4d-256)) then
        tmp = (1.0d0 + exp((eps_m * -x))) / 2.0d0
    else if ((x <= 190000.0d0) .or. (.not. (x <= 1.45d+264))) then
        tmp = (1.0d0 + exp((x * (eps_m + (-1.0d0))))) / 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 <= -4e-256) {
		tmp = (1.0 + Math.exp((eps_m * -x))) / 2.0;
	} else if ((x <= 190000.0) || !(x <= 1.45e+264)) {
		tmp = (1.0 + Math.exp((x * (eps_m + -1.0)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -4e-256:
		tmp = (1.0 + math.exp((eps_m * -x))) / 2.0
	elif (x <= 190000.0) or not (x <= 1.45e+264):
		tmp = (1.0 + math.exp((x * (eps_m + -1.0)))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -4e-256)
		tmp = Float64(Float64(1.0 + exp(Float64(eps_m * Float64(-x)))) / 2.0);
	elseif ((x <= 190000.0) || !(x <= 1.45e+264))
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps_m + -1.0)))) / 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 <= -4e-256)
		tmp = (1.0 + exp((eps_m * -x))) / 2.0;
	elseif ((x <= 190000.0) || ~((x <= 1.45e+264)))
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 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, -4e-256], N[(N[(1.0 + N[Exp[N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 190000.0], N[Not[LessEqual[x, 1.45e+264]], $MachinePrecision]], N[(N[(1.0 + N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -3.99999999999999991e-256

   1. Initial program 66.5%

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

   3. Simplified 66.5%

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

   5. Taylor expanded in x around 0 45.5%

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

   6. Taylor expanded in eps around inf 73.8%

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

      1. mul-1-neg 73.8%

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

      2. neg-mul-1 73.8%

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

      3. distribute-lft-neg-in 73.8%

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

      4. +-commutative 73.8%

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

   8. Simplified 73.8%

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

   9. Taylor expanded in eps around inf 73.9%

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

   if -3.99999999999999991e-256 < x < 1.9e5 or 1.4499999999999999e264 < x

   1. Initial program 50.6%

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

   3. Simplified 28.7%

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

   5. Taylor expanded in eps around inf 98.8%

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

      1. associate-*r* 98.8%

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

      2. neg-mul-1 98.8%

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

      3. *-commutative 98.8%

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

      4. expm1-log1p-u 86.4%

         \[\leadsto \frac{e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 + \varepsilon\right) \cdot \left(-x\right)\right)\right)}} + e^{x \cdot \left(\varepsilon - 1\right)}}{2} \]

      5. expm1-undefine 86.4%

         \[\leadsto \frac{e^{\color{blue}{e^{\mathsf{log1p}\left(\left(1 + \varepsilon\right) \cdot \left(-x\right)\right)} - 1}} + e^{x \cdot \left(\varepsilon - 1\right)}}{2} \]

   7. Applied egg-rr 86.4%

      \[\leadsto \frac{e^{\color{blue}{e^{\mathsf{log1p}\left(\left(1 + \varepsilon\right) \cdot \left(-x\right)\right)} - 1}} + e^{x \cdot \left(\varepsilon - 1\right)}}{2} \]
   8. Step-by-step derivation

      1. sub-neg 86.4%

         \[\leadsto \frac{e^{\color{blue}{e^{\mathsf{log1p}\left(\left(1 + \varepsilon\right) \cdot \left(-x\right)\right)} + \left(-1\right)}} + e^{x \cdot \left(\varepsilon - 1\right)}}{2} \]

      2. log1p-undefine 86.4%

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

      3. rem-exp-log 98.8%

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

      4. *-commutative 98.8%

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

      5. distribute-lft-neg-in 98.8%

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

      6. neg-mul-1 98.8%

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

      7. +-commutative 98.8%

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

      8. neg-mul-1 98.8%

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

      9. distribute-rgt-neg-in 98.8%

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

      10. mul-1-neg 98.8%

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

      11. fma-define 98.8%

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

      12. distribute-lft-in 98.8%

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

      13. metadata-eval 98.8%

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

      14. mul-1-neg 98.8%

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

      15. unsub-neg 98.8%

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

      16. metadata-eval 98.8%

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

   9. Simplified 98.8%

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

   10. Taylor expanded in x around 0 83.9%

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

   if 1.9e5 < x < 1.4499999999999999e264

   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 62.3%

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

      1. div-sub 62.3%

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

      2. mul-1-neg 62.3%

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

      3. rec-exp 62.3%

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

      4. +-inverses 62.3%

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

      5. metadata-eval 62.3%

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

   7. Simplified 62.3%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-256}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 190000 \lor \neg \left(x \leq 1.45 \cdot 10^{+264}\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 405:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{eps\_m \cdot \left(\frac{1}{eps\_m} + \left(\frac{e^{x \cdot \left(eps\_m + -1\right)}}{eps\_m} - x\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 405.0)
   (* (exp (- x)) (+ x 1.0))
   (/
    (* eps_m (+ (/ 1.0 eps_m) (- (/ (exp (* x (+ eps_m -1.0))) eps_m) x)))
    2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 405.0) {
		tmp = exp(-x) * (x + 1.0);
	} else {
		tmp = (eps_m * ((1.0 / eps_m) + ((exp((x * (eps_m + -1.0))) / eps_m) - x))) / 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 <= 405.0d0) then
        tmp = exp(-x) * (x + 1.0d0)
    else
        tmp = (eps_m * ((1.0d0 / eps_m) + ((exp((x * (eps_m + (-1.0d0)))) / eps_m) - x))) / 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 <= 405.0) {
		tmp = Math.exp(-x) * (x + 1.0);
	} else {
		tmp = (eps_m * ((1.0 / eps_m) + ((Math.exp((x * (eps_m + -1.0))) / eps_m) - x))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 405.0:
		tmp = math.exp(-x) * (x + 1.0)
	else:
		tmp = (eps_m * ((1.0 / eps_m) + ((math.exp((x * (eps_m + -1.0))) / eps_m) - x))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 405.0)
		tmp = Float64(exp(Float64(-x)) * Float64(x + 1.0));
	else
		tmp = Float64(Float64(eps_m * Float64(Float64(1.0 / eps_m) + Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) / eps_m) - x))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 405.0)
		tmp = exp(-x) * (x + 1.0);
	else
		tmp = (eps_m * ((1.0 / eps_m) + ((exp((x * (eps_m + -1.0))) / eps_m) - x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 405.0], N[(N[Exp[(-x)], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(eps$95$m * N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / eps$95$m), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 405

   1. Initial program 56.4%

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

   3. Simplified 48.3%

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

   5. Taylor expanded in eps around 0 33.1%

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

      1. associate-+r+ 75.7%

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

      2. mul-1-neg 75.7%

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

      3. sub-neg 75.7%

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

      4. +-inverses 75.7%

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

      5. distribute-lft-out 75.7%

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

      6. distribute-rgt1-in 76.8%

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

      7. mul-1-neg 76.8%

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

   7. Simplified 76.8%

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

   8. Taylor expanded in eps around 0 76.8%

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

      1. +-commutative 76.8%

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

   10. Simplified 76.8%

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

   if 405 < 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 100.0%

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

   5. Taylor expanded in x around 0 65.5%

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

   6. Taylor expanded in eps around inf 77.4%

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

      1. associate--l+ 77.4%

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

      2. mul-1-neg 77.4%

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

   8. Simplified 77.4%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 405:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(\frac{1}{\varepsilon} + \left(\frac{e^{x \cdot \left(\varepsilon + -1\right)}}{\varepsilon} - x\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 0.95:\\ \;\;\;\;1 + x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+264}:\\ \;\;\;\;x \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{eps\_m \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 0.95)
   (+ 1.0 (* x (+ -1.0 (* x (+ 0.5 (* x -0.16666666666666666))))))
   (if (<= x 6e+264) (* x (exp (- x))) (/ (* eps_m x) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 0.95) {
		tmp = 1.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))));
	} else if (x <= 6e+264) {
		tmp = x * exp(-x);
	} else {
		tmp = (eps_m * x) / 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 <= 0.95d0) then
        tmp = 1.0d0 + (x * ((-1.0d0) + (x * (0.5d0 + (x * (-0.16666666666666666d0))))))
    else if (x <= 6d+264) then
        tmp = x * exp(-x)
    else
        tmp = (eps_m * x) / 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 <= 0.95) {
		tmp = 1.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))));
	} else if (x <= 6e+264) {
		tmp = x * Math.exp(-x);
	} else {
		tmp = (eps_m * x) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 0.95:
		tmp = 1.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))
	elif x <= 6e+264:
		tmp = x * math.exp(-x)
	else:
		tmp = (eps_m * x) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 0.95)
		tmp = Float64(1.0 + Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * -0.16666666666666666))))));
	elseif (x <= 6e+264)
		tmp = Float64(x * exp(Float64(-x)));
	else
		tmp = Float64(Float64(eps_m * x) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 0.95)
		tmp = 1.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))));
	elseif (x <= 6e+264)
		tmp = x * exp(-x);
	else
		tmp = (eps_m * x) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 0.95], N[(1.0 + N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6e+264], N[(x * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], N[(N[(eps$95$m * x), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.94999999999999996

   1. Initial program 55.7%

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

   3. Simplified 39.6%

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

   5. Taylor expanded in eps around inf 97.8%

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

   6. Taylor expanded in eps around 0 82.8%

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

      1. neg-mul-1 82.8%

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

   8. Simplified 82.8%

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

   9. Taylor expanded in x around 0 75.1%

      \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right) - 1\right)} \]

   if 0.94999999999999996 < x < 6e264

   1. Initial program 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in eps around 0 60.3%

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

      1. associate-+r+ 61.9%

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

      2. mul-1-neg 61.9%

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

      3. sub-neg 61.9%

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

      4. +-inverses 61.9%

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

      5. distribute-lft-out 61.9%

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

      6. distribute-rgt1-in 61.9%

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

      7. mul-1-neg 61.9%

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

   7. Simplified 61.9%

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

   8. Taylor expanded in x around inf 60.8%

      \[\leadsto \color{blue}{x \cdot e^{-x}} \]

   if 6e264 < 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%

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

   5. Taylor expanded in x around 0 30.3%

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

      1. +-commutative 30.3%

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

      2. mul-1-neg 30.3%

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

      3. unsub-neg 30.3%

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

      4. *-commutative 30.3%

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

   7. Simplified 30.3%

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

   8. Taylor expanded in eps around inf 30.4%

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot x}}{2} \]
   9. Step-by-step derivation

      1. *-commutative 30.4%

         \[\leadsto \frac{\color{blue}{x \cdot \varepsilon}}{2} \]

   10. Simplified 30.4%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.95:\\ \;\;\;\;1 + x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+264}:\\ \;\;\;\;x \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot x}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 0.00047:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(eps\_m + -1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 0.00047)
   (* (exp (- x)) (+ x 1.0))
   (/ (+ 1.0 (exp (* x (+ eps_m -1.0)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.00047) {
		tmp = exp(-x) * (x + 1.0);
	} else {
		tmp = (1.0 + exp((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 (eps_m <= 0.00047d0) then
        tmp = exp(-x) * (x + 1.0d0)
    else
        tmp = (1.0d0 + exp((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 (eps_m <= 0.00047) {
		tmp = Math.exp(-x) * (x + 1.0);
	} else {
		tmp = (1.0 + Math.exp((x * (eps_m + -1.0)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 0.00047:
		tmp = math.exp(-x) * (x + 1.0)
	else:
		tmp = (1.0 + math.exp((x * (eps_m + -1.0)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 0.00047)
		tmp = Float64(exp(Float64(-x)) * Float64(x + 1.0));
	else
		tmp = Float64(Float64(1.0 + exp(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 (eps_m <= 0.00047)
		tmp = exp(-x) * (x + 1.0);
	else
		tmp = (1.0 + exp((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[eps$95$m, 0.00047], N[(N[Exp[(-x)], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 4.69999999999999986e-4

   1. Initial program 56.1%

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

   3. Simplified 48.0%

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

   5. Taylor expanded in eps around 0 32.8%

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

      1. associate-+r+ 75.6%

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

      2. mul-1-neg 75.6%

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

      3. sub-neg 75.6%

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

      4. +-inverses 75.6%

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

      5. distribute-lft-out 75.6%

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

      6. distribute-rgt1-in 76.6%

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

      7. mul-1-neg 76.6%

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

   7. Simplified 76.6%

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

   8. Taylor expanded in eps around 0 76.6%

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

      1. +-commutative 76.6%

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

   10. Simplified 76.6%

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

   if 4.69999999999999986e-4 < 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 79.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

      3. *-commutative 100.0%

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

      4. expm1-log1p-u 67.6%

         \[\leadsto \frac{e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 + \varepsilon\right) \cdot \left(-x\right)\right)\right)}} + e^{x \cdot \left(\varepsilon - 1\right)}}{2} \]

      5. expm1-undefine 67.6%

         \[\leadsto \frac{e^{\color{blue}{e^{\mathsf{log1p}\left(\left(1 + \varepsilon\right) \cdot \left(-x\right)\right)} - 1}} + e^{x \cdot \left(\varepsilon - 1\right)}}{2} \]

   7. Applied egg-rr 67.6%

      \[\leadsto \frac{e^{\color{blue}{e^{\mathsf{log1p}\left(\left(1 + \varepsilon\right) \cdot \left(-x\right)\right)} - 1}} + e^{x \cdot \left(\varepsilon - 1\right)}}{2} \]
   8. Step-by-step derivation

      1. sub-neg 67.6%

         \[\leadsto \frac{e^{\color{blue}{e^{\mathsf{log1p}\left(\left(1 + \varepsilon\right) \cdot \left(-x\right)\right)} + \left(-1\right)}} + e^{x \cdot \left(\varepsilon - 1\right)}}{2} \]

      2. log1p-undefine 67.6%

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

      3. rem-exp-log 100.0%

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

      4. *-commutative 100.0%

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

      5. distribute-lft-neg-in 100.0%

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

      6. neg-mul-1 100.0%

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

      7. +-commutative 100.0%

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

      8. neg-mul-1 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. mul-1-neg 100.0%

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

      11. fma-define 100.0%

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

      12. distribute-lft-in 100.0%

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

      13. metadata-eval 100.0%

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

      14. mul-1-neg 100.0%

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

      15. unsub-neg 100.0%

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

      16. metadata-eval 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in x around 0 56.8%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.00047:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 0.00047:\\ \;\;\;\;e^{-x} \cdot \left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(x - 2\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 0.00047)
   (* (exp (- x)) (+ x 1.0))
   (/ (+ 2.0 (* x (- x 2.0))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.00047) {
		tmp = exp(-x) * (x + 1.0);
	} else {
		tmp = (2.0 + (x * (x - 2.0))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 0.00047d0) then
        tmp = exp(-x) * (x + 1.0d0)
    else
        tmp = (2.0d0 + (x * (x - 2.0d0))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.00047) {
		tmp = Math.exp(-x) * (x + 1.0);
	} else {
		tmp = (2.0 + (x * (x - 2.0))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 0.00047:
		tmp = math.exp(-x) * (x + 1.0)
	else:
		tmp = (2.0 + (x * (x - 2.0))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 0.00047)
		tmp = Float64(exp(Float64(-x)) * Float64(x + 1.0));
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(x - 2.0))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 0.00047)
		tmp = exp(-x) * (x + 1.0);
	else
		tmp = (2.0 + (x * (x - 2.0))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 0.00047], N[(N[Exp[(-x)], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(x * N[(x - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 4.69999999999999986e-4

   1. Initial program 56.1%

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

   3. Simplified 48.0%

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

   5. Taylor expanded in eps around 0 32.8%

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

      1. associate-+r+ 75.6%

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

      2. mul-1-neg 75.6%

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

      3. sub-neg 75.6%

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

      4. +-inverses 75.6%

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

      5. distribute-lft-out 75.6%

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

      6. distribute-rgt1-in 76.6%

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

      7. mul-1-neg 76.6%

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

   7. Simplified 76.6%

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

   8. Taylor expanded in eps around 0 76.6%

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

      1. +-commutative 76.6%

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

   10. Simplified 76.6%

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

   if 4.69999999999999986e-4 < 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 79.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around 0 54.2%

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

      1. neg-mul-1 54.2%

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

   8. Simplified 54.2%

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

   9. Taylor expanded in x around 0 56.4%

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

Alternative 9: 70.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 6.4 \cdot 10^{+264}:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{eps\_m \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 6.4e+264) (/ (/ 2.0 (exp x)) 2.0) (/ (* eps_m x) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 6.4e+264) {
		tmp = (2.0 / exp(x)) / 2.0;
	} else {
		tmp = (eps_m * x) / 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 <= 6.4d+264) then
        tmp = (2.0d0 / exp(x)) / 2.0d0
    else
        tmp = (eps_m * x) / 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 <= 6.4e+264) {
		tmp = (2.0 / Math.exp(x)) / 2.0;
	} else {
		tmp = (eps_m * x) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 6.4e+264:
		tmp = (2.0 / math.exp(x)) / 2.0
	else:
		tmp = (eps_m * x) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 6.4e+264)
		tmp = Float64(Float64(2.0 / exp(x)) / 2.0);
	else
		tmp = Float64(Float64(eps_m * x) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 6.4e+264)
		tmp = (2.0 / exp(x)) / 2.0;
	else
		tmp = (eps_m * x) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 6.4e+264], N[(N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(eps$95$m * x), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.4000000000000001e264

   1. Initial program 66.5%

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

   3. Simplified 54.5%

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

   5. Taylor expanded in eps around inf 98.0%

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

      1. associate-*r* 98.0%

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

      2. neg-mul-1 98.0%

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

      3. *-commutative 98.0%

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

      4. expm1-log1p-u 67.0%

         \[\leadsto \frac{e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 + \varepsilon\right) \cdot \left(-x\right)\right)\right)}} + e^{x \cdot \left(\varepsilon - 1\right)}}{2} \]

      5. expm1-undefine 67.0%

         \[\leadsto \frac{e^{\color{blue}{e^{\mathsf{log1p}\left(\left(1 + \varepsilon\right) \cdot \left(-x\right)\right)} - 1}} + e^{x \cdot \left(\varepsilon - 1\right)}}{2} \]

   7. Applied egg-rr 67.0%

      \[\leadsto \frac{e^{\color{blue}{e^{\mathsf{log1p}\left(\left(1 + \varepsilon\right) \cdot \left(-x\right)\right)} - 1}} + e^{x \cdot \left(\varepsilon - 1\right)}}{2} \]
   8. Step-by-step derivation

      1. sub-neg 67.0%

         \[\leadsto \frac{e^{\color{blue}{e^{\mathsf{log1p}\left(\left(1 + \varepsilon\right) \cdot \left(-x\right)\right)} + \left(-1\right)}} + e^{x \cdot \left(\varepsilon - 1\right)}}{2} \]

      2. log1p-undefine 67.0%

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

      3. rem-exp-log 98.0%

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

      4. *-commutative 98.0%

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

      5. distribute-lft-neg-in 98.0%

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

      6. neg-mul-1 98.0%

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

      7. +-commutative 98.0%

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

      8. neg-mul-1 98.0%

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

      9. distribute-rgt-neg-in 98.0%

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

      10. mul-1-neg 98.0%

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

      11. fma-define 98.0%

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

      12. distribute-lft-in 98.0%

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

      13. metadata-eval 98.0%

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

      14. mul-1-neg 98.0%

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

      15. unsub-neg 98.0%

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

      16. metadata-eval 98.0%

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

   9. Simplified 98.0%

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

       1. fma-undefine 98.0%

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

   11. Applied egg-rr 98.0%

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

   12. Taylor expanded in eps around 0 77.2%

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

       1. neg-mul-1 77.2%

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

       2. exp-neg 77.2%

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

       3. associate-*r/ 77.2%

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

       4. metadata-eval 77.2%

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

   14. Simplified 77.2%

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

   if 6.4000000000000001e264 < 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%

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

   5. Taylor expanded in x around 0 30.3%

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

      1. +-commutative 30.3%

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

      2. mul-1-neg 30.3%

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

      3. unsub-neg 30.3%

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

      4. *-commutative 30.3%

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

   7. Simplified 30.3%

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

   8. Taylor expanded in eps around inf 30.4%

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot x}}{2} \]
   9. Step-by-step derivation

      1. *-commutative 30.4%

         \[\leadsto \frac{\color{blue}{x \cdot \varepsilon}}{2} \]

   10. Simplified 30.4%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.4 \cdot 10^{+264}:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot x}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.1% accurate, 11.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0064:\\ \;\;\;\;eps\_m \cdot \left(x \cdot -0.5\right)\\ \mathbf{elif}\;x \leq 150000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+264}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{eps\_m \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -0.0064)
   (* eps_m (* x -0.5))
   (if (<= x 150000.0) 1.0 (if (<= x 4.8e+264) 0.0 (/ (* eps_m x) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -0.0064) {
		tmp = eps_m * (x * -0.5);
	} else if (x <= 150000.0) {
		tmp = 1.0;
	} else if (x <= 4.8e+264) {
		tmp = 0.0;
	} else {
		tmp = (eps_m * x) / 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 <= (-0.0064d0)) then
        tmp = eps_m * (x * (-0.5d0))
    else if (x <= 150000.0d0) then
        tmp = 1.0d0
    else if (x <= 4.8d+264) then
        tmp = 0.0d0
    else
        tmp = (eps_m * x) / 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 <= -0.0064) {
		tmp = eps_m * (x * -0.5);
	} else if (x <= 150000.0) {
		tmp = 1.0;
	} else if (x <= 4.8e+264) {
		tmp = 0.0;
	} else {
		tmp = (eps_m * x) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -0.0064:
		tmp = eps_m * (x * -0.5)
	elif x <= 150000.0:
		tmp = 1.0
	elif x <= 4.8e+264:
		tmp = 0.0
	else:
		tmp = (eps_m * x) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -0.0064)
		tmp = Float64(eps_m * Float64(x * -0.5));
	elseif (x <= 150000.0)
		tmp = 1.0;
	elseif (x <= 4.8e+264)
		tmp = 0.0;
	else
		tmp = Float64(Float64(eps_m * x) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -0.0064)
		tmp = eps_m * (x * -0.5);
	elseif (x <= 150000.0)
		tmp = 1.0;
	elseif (x <= 4.8e+264)
		tmp = 0.0;
	else
		tmp = (eps_m * x) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -0.0064], N[(eps$95$m * N[(x * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 150000.0], 1.0, If[LessEqual[x, 4.8e+264], 0.0, N[(N[(eps$95$m * x), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -0.00640000000000000031

   1. Initial program 92.7%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in x around 0 72.0%

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

   6. Taylor expanded in x around inf 35.5%

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

      1. +-commutative 35.5%

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

      2. sub-neg 35.5%

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

      3. metadata-eval 35.5%

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

      4. +-commutative 35.5%

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

   8. Simplified 35.5%

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

   9. Taylor expanded in eps around inf 35.5%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\varepsilon \cdot x\right)} \]
   10. Step-by-step derivation

       1. *-commutative 35.5%

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

       2. *-commutative 35.5%

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

       3. *-commutative 35.5%

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

       4. associate-*r* 35.5%

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

   11. Simplified 35.5%

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

   if -0.00640000000000000031 < x < 1.5e5

   1. Initial program 45.2%

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

   3. Simplified 45.2%

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

   5. Taylor expanded in x around 0 80.7%

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

   if 1.5e5 < x < 4.79999999999999985e264

   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 62.3%

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

      1. div-sub 62.3%

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

      2. mul-1-neg 62.3%

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

      3. rec-exp 62.3%

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

      4. +-inverses 62.3%

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

      5. metadata-eval 62.3%

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

   7. Simplified 62.3%

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

   if 4.79999999999999985e264 < 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%

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

   5. Taylor expanded in x around 0 30.3%

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

      1. +-commutative 30.3%

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

      2. mul-1-neg 30.3%

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

      3. unsub-neg 30.3%

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

      4. *-commutative 30.3%

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

   7. Simplified 30.3%

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

   8. Taylor expanded in eps around inf 30.4%

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot x}}{2} \]
   9. Step-by-step derivation

      1. *-commutative 30.4%

         \[\leadsto \frac{\color{blue}{x \cdot \varepsilon}}{2} \]

   10. Simplified 30.4%

       \[\leadsto \frac{\color{blue}{x \cdot \varepsilon}}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0064:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot -0.5\right)\\ \mathbf{elif}\;x \leq 150000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+264}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot x}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.2% accurate, 12.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;1 + x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+264}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{eps\_m \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1.6)
   (+ 1.0 (* x (+ -1.0 (* x (+ 0.5 (* x -0.16666666666666666))))))
   (if (<= x 2.6e+264) 0.0 (/ (* eps_m x) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.6) {
		tmp = 1.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))));
	} else if (x <= 2.6e+264) {
		tmp = 0.0;
	} else {
		tmp = (eps_m * x) / 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 <= 1.6d0) then
        tmp = 1.0d0 + (x * ((-1.0d0) + (x * (0.5d0 + (x * (-0.16666666666666666d0))))))
    else if (x <= 2.6d+264) then
        tmp = 0.0d0
    else
        tmp = (eps_m * x) / 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 <= 1.6) {
		tmp = 1.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))));
	} else if (x <= 2.6e+264) {
		tmp = 0.0;
	} else {
		tmp = (eps_m * x) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 1.6:
		tmp = 1.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))
	elif x <= 2.6e+264:
		tmp = 0.0
	else:
		tmp = (eps_m * x) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 1.6)
		tmp = Float64(1.0 + Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * -0.16666666666666666))))));
	elseif (x <= 2.6e+264)
		tmp = 0.0;
	else
		tmp = Float64(Float64(eps_m * x) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 1.6)
		tmp = 1.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))));
	elseif (x <= 2.6e+264)
		tmp = 0.0;
	else
		tmp = (eps_m * x) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 1.6], N[(1.0 + N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.6e+264], 0.0, N[(N[(eps$95$m * x), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.6000000000000001

   1. Initial program 55.7%

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

   3. Simplified 39.6%

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

   5. Taylor expanded in eps around inf 97.8%

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

   6. Taylor expanded in eps around 0 82.8%

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

      1. neg-mul-1 82.8%

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

   8. Simplified 82.8%

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

   9. Taylor expanded in x around 0 75.1%

      \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right) - 1\right)} \]

   if 1.6000000000000001 < x < 2.6e264

   1. Initial program 98.4%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in eps around 0 60.3%

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

      1. div-sub 60.3%

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

      2. mul-1-neg 60.3%

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

      3. rec-exp 60.3%

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

      4. +-inverses 60.3%

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

      5. metadata-eval 60.3%

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

   7. Simplified 60.3%

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

   if 2.6e264 < 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%

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

   5. Taylor expanded in x around 0 30.3%

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

      1. +-commutative 30.3%

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

      2. mul-1-neg 30.3%

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

      3. unsub-neg 30.3%

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

      4. *-commutative 30.3%

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

   7. Simplified 30.3%

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

   8. Taylor expanded in eps around inf 30.4%

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot x}}{2} \]
   9. Step-by-step derivation

      1. *-commutative 30.4%

         \[\leadsto \frac{\color{blue}{x \cdot \varepsilon}}{2} \]

   10. Simplified 30.4%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;1 + x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+264}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot x}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.8% accurate, 15.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 150000:\\ \;\;\;\;\frac{2 + x \cdot \left(x - 2\right)}{2}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+264}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{eps\_m \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 150000.0)
   (/ (+ 2.0 (* x (- x 2.0))) 2.0)
   (if (<= x 3.9e+264) 0.0 (/ (* eps_m x) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 150000.0) {
		tmp = (2.0 + (x * (x - 2.0))) / 2.0;
	} else if (x <= 3.9e+264) {
		tmp = 0.0;
	} else {
		tmp = (eps_m * x) / 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 <= 150000.0d0) then
        tmp = (2.0d0 + (x * (x - 2.0d0))) / 2.0d0
    else if (x <= 3.9d+264) then
        tmp = 0.0d0
    else
        tmp = (eps_m * x) / 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 <= 150000.0) {
		tmp = (2.0 + (x * (x - 2.0))) / 2.0;
	} else if (x <= 3.9e+264) {
		tmp = 0.0;
	} else {
		tmp = (eps_m * x) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 150000.0:
		tmp = (2.0 + (x * (x - 2.0))) / 2.0
	elif x <= 3.9e+264:
		tmp = 0.0
	else:
		tmp = (eps_m * x) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 150000.0)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(x - 2.0))) / 2.0);
	elseif (x <= 3.9e+264)
		tmp = 0.0;
	else
		tmp = Float64(Float64(eps_m * x) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 150000.0)
		tmp = (2.0 + (x * (x - 2.0))) / 2.0;
	elseif (x <= 3.9e+264)
		tmp = 0.0;
	else
		tmp = (eps_m * x) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 150000.0], N[(N[(2.0 + N[(x * N[(x - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3.9e+264], 0.0, N[(N[(eps$95$m * x), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.5e5

   1. Initial program 55.7%

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

   3. Simplified 39.8%

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

   5. Taylor expanded in eps around inf 97.3%

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

   6. Taylor expanded in eps around 0 82.0%

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

      1. neg-mul-1 82.0%

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

   8. Simplified 82.0%

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

   9. Taylor expanded in x around 0 72.8%

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

   if 1.5e5 < x < 3.89999999999999994e264

   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 62.3%

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

      1. div-sub 62.3%

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

      2. mul-1-neg 62.3%

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

      3. rec-exp 62.3%

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

      4. +-inverses 62.3%

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

      5. metadata-eval 62.3%

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

   7. Simplified 62.3%

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

   if 3.89999999999999994e264 < 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%

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

   5. Taylor expanded in x around 0 30.3%

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

      1. +-commutative 30.3%

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

      2. mul-1-neg 30.3%

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

      3. unsub-neg 30.3%

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

      4. *-commutative 30.3%

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

   7. Simplified 30.3%

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

   8. Taylor expanded in eps around inf 30.4%

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot x}}{2} \]
   9. Step-by-step derivation

      1. *-commutative 30.4%

         \[\leadsto \frac{\color{blue}{x \cdot \varepsilon}}{2} \]

   10. Simplified 30.4%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 150000:\\ \;\;\;\;\frac{2 + x \cdot \left(x - 2\right)}{2}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+264}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot x}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.5% accurate, 15.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2.95 \cdot 10^{-8}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - eps\_m\right)}{2}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+264}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{eps\_m \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 2.95e-8)
   (/ (+ 2.0 (* x (- -1.0 eps_m))) 2.0)
   (if (<= x 2.9e+264) 0.0 (/ (* eps_m x) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.95e-8) {
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0;
	} else if (x <= 2.9e+264) {
		tmp = 0.0;
	} else {
		tmp = (eps_m * x) / 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 <= 2.95d-8) then
        tmp = (2.0d0 + (x * ((-1.0d0) - eps_m))) / 2.0d0
    else if (x <= 2.9d+264) then
        tmp = 0.0d0
    else
        tmp = (eps_m * x) / 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 <= 2.95e-8) {
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0;
	} else if (x <= 2.9e+264) {
		tmp = 0.0;
	} else {
		tmp = (eps_m * x) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 2.95e-8:
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0
	elif x <= 2.9e+264:
		tmp = 0.0
	else:
		tmp = (eps_m * x) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 2.95e-8)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 - eps_m))) / 2.0);
	elseif (x <= 2.9e+264)
		tmp = 0.0;
	else
		tmp = Float64(Float64(eps_m * x) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 2.95e-8)
		tmp = (2.0 + (x * (-1.0 - eps_m))) / 2.0;
	elseif (x <= 2.9e+264)
		tmp = 0.0;
	else
		tmp = (eps_m * x) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 2.95e-8], N[(N[(2.0 + N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.9e+264], 0.0, N[(N[(eps$95$m * x), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.9499999999999999e-8

   1. Initial program 55.2%

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

   3. Simplified 55.2%

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

   5. Taylor expanded in x around 0 40.4%

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

   6. Taylor expanded in eps around inf 82.4%

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

      1. mul-1-neg 82.4%

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

      2. neg-mul-1 82.4%

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

      3. distribute-lft-neg-in 82.4%

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

      4. +-commutative 82.4%

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

   8. Simplified 82.4%

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

   9. Taylor expanded in x around 0 71.0%

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

       1. associate-*r* 71.0%

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

       2. +-commutative 71.0%

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

       3. associate-*r* 71.0%

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

       4. *-commutative 71.0%

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

       5. distribute-rgt1-in 71.0%

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

       6. distribute-lft-out 71.0%

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

       7. associate-*r* 71.0%

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

       8. distribute-rgt-in 71.0%

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

       9. mul-1-neg 71.0%

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

       10. sub-neg 71.0%

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

   11. Simplified 71.0%

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

   if 2.9499999999999999e-8 < x < 2.8999999999999998e264

   1. Initial program 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in eps around 0 58.5%

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

      1. div-sub 58.5%

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

      2. mul-1-neg 58.5%

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

      3. rec-exp 58.5%

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

      4. +-inverses 58.5%

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

      5. metadata-eval 58.5%

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

   7. Simplified 58.5%

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

   if 2.8999999999999998e264 < 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%

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

   5. Taylor expanded in x around 0 30.3%

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

      1. +-commutative 30.3%

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

      2. mul-1-neg 30.3%

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

      3. unsub-neg 30.3%

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

      4. *-commutative 30.3%

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

   7. Simplified 30.3%

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

   8. Taylor expanded in eps around inf 30.4%

      \[\leadsto \frac{\color{blue}{\varepsilon \cdot x}}{2} \]
   9. Step-by-step derivation

      1. *-commutative 30.4%

         \[\leadsto \frac{\color{blue}{x \cdot \varepsilon}}{2} \]

   10. Simplified 30.4%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.95 \cdot 10^{-8}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+264}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot x}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.3% accurate, 20.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0064:\\ \;\;\;\;eps\_m \cdot \left(x \cdot -0.5\right)\\ \mathbf{elif}\;x \leq 150000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -0.0064) (* eps_m (* x -0.5)) (if (<= x 150000.0) 1.0 0.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -0.0064) {
		tmp = eps_m * (x * -0.5);
	} else if (x <= 150000.0) {
		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 <= (-0.0064d0)) then
        tmp = eps_m * (x * (-0.5d0))
    else if (x <= 150000.0d0) 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 <= -0.0064) {
		tmp = eps_m * (x * -0.5);
	} else if (x <= 150000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -0.0064:
		tmp = eps_m * (x * -0.5)
	elif x <= 150000.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -0.0064)
		tmp = Float64(eps_m * Float64(x * -0.5));
	elseif (x <= 150000.0)
		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 <= -0.0064)
		tmp = eps_m * (x * -0.5);
	elseif (x <= 150000.0)
		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, -0.0064], N[(eps$95$m * N[(x * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 150000.0], 1.0, 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00640000000000000031

   1. Initial program 92.7%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in x around 0 72.0%

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

   6. Taylor expanded in x around inf 35.5%

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

      1. +-commutative 35.5%

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

      2. sub-neg 35.5%

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

      3. metadata-eval 35.5%

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

      4. +-commutative 35.5%

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

   8. Simplified 35.5%

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

   9. Taylor expanded in eps around inf 35.5%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\varepsilon \cdot x\right)} \]
   10. Step-by-step derivation

       1. *-commutative 35.5%

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

       2. *-commutative 35.5%

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

       3. *-commutative 35.5%

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

       4. associate-*r* 35.5%

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

   11. Simplified 35.5%

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

   if -0.00640000000000000031 < x < 1.5e5

   1. Initial program 45.2%

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

   3. Simplified 45.2%

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

   5. Taylor expanded in x around 0 80.7%

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

   if 1.5e5 < 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 56.4%

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

      1. div-sub 56.4%

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

      2. mul-1-neg 56.4%

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

      3. rec-exp 56.4%

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

      4. +-inverses 56.4%

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

      5. metadata-eval 56.4%

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

   7. Simplified 56.4%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0064:\\ \;\;\;\;\varepsilon \cdot \left(x \cdot -0.5\right)\\ \mathbf{elif}\;x \leq 150000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 57.0% accurate, 37.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 150000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 150000.0) 1.0 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 150000.0) {
		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 <= 150000.0d0) 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 <= 150000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 150000.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 150000.0)
		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 <= 150000.0)
		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, 150000.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.5e5

   1. Initial program 55.7%

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

   3. Simplified 55.7%

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

   5. Taylor expanded in x around 0 63.5%

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

   if 1.5e5 < 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 56.4%

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

      1. div-sub 56.4%

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

      2. mul-1-neg 56.4%

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

      3. rec-exp 56.4%

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

      4. +-inverses 56.4%

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

      5. metadata-eval 56.4%

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

   7. Simplified 56.4%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 150000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 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 67.8%

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

3. Simplified 63.7%

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

5. Taylor expanded in eps around 0 17.0%

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

   1. div-sub 17.0%

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

   2. mul-1-neg 17.0%

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

   3. rec-exp 17.1%

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

   4. +-inverses 17.3%

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

   5. metadata-eval 17.3%

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

7. Simplified 17.3%

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

Reproduce

herbie shell --seed 2024144 
(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))