NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.4% → 99.7%

Time: 30.7s

Alternatives: 12

Speedup: 2.0×

• 39.1% 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: 73.4% 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

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 2, 2\right)}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{e}^{\left(x \cdot \left(-1 - eps\_m\right)\right)} + e^{x \cdot eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 8e-6)
   (/ (/ (fma x 2.0 2.0) (exp x)) 2.0)
   (/ (+ (pow E (* x (- -1.0 eps_m))) (exp (* x eps_m))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 8e-6) {
		tmp = (fma(x, 2.0, 2.0) / exp(x)) / 2.0;
	} else {
		tmp = (pow(((double) M_E), (x * (-1.0 - eps_m))) + exp((x * eps_m))) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 8e-6)
		tmp = Float64(Float64(fma(x, 2.0, 2.0) / exp(x)) / 2.0);
	else
		tmp = Float64(Float64((exp(1) ^ Float64(x * Float64(-1.0 - eps_m))) + exp(Float64(x * eps_m))) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 8e-6], N[(N[(N[(x * 2.0 + 2.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Power[E, N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 7.99999999999999964e-6

   1. Initial program 61.3%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in eps around 0 27.7%

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

   7. Simplified 67.5%

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

   8. Taylor expanded in eps around 0 67.5%

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

      1. *-commutative 67.5%

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

      2. +-commutative 67.5%

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

      3. *-commutative 67.5%

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

      4. fma-undefine 67.5%

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

      5. exp-neg 67.5%

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

      6. associate-*r/ 67.5%

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

      7. *-rgt-identity 67.5%

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

   10. Simplified 67.5%

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

   if 7.99999999999999964e-6 < eps

   1. Initial program 99.8%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in eps around inf 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. exp-prod 100.0%

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

      3. +-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. fma-define 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. pow-flip 100.0%

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

      2. exp-1-e 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. fma-undefine 100.0%

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

       2. distribute-neg-in 100.0%

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

       3. distribute-rgt-neg-out 100.0%

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

       4. mul-1-neg 100.0%

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

       5. *-commutative 100.0%

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

       6. +-commutative 100.0%

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

       7. distribute-rgt-out 100.0%

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

       8. sub-neg 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in eps around inf 100.0%

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

       1. *-commutative 100.0%

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

   14. Simplified 100.0%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 2, 2\right)}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{e}^{\left(x \cdot \left(-1 - \varepsilon\right)\right)} + e^{x \cdot \varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 2, 2\right)}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - eps\_m\right)} + e^{x \cdot eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 8e-6)
   (/ (/ (fma x 2.0 2.0) (exp x)) 2.0)
   (/ (+ (exp (* x (- -1.0 eps_m))) (exp (* x eps_m))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 8e-6) {
		tmp = (fma(x, 2.0, 2.0) / exp(x)) / 2.0;
	} else {
		tmp = (exp((x * (-1.0 - eps_m))) + exp((x * eps_m))) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 8e-6)
		tmp = Float64(Float64(fma(x, 2.0, 2.0) / exp(x)) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 - eps_m))) + exp(Float64(x * eps_m))) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 8e-6], N[(N[(N[(x * 2.0 + 2.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 7.99999999999999964e-6

   1. Initial program 61.3%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in eps around 0 27.7%

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

   7. Simplified 67.5%

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

   8. Taylor expanded in eps around 0 67.5%

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

      1. *-commutative 67.5%

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

      2. +-commutative 67.5%

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

      3. *-commutative 67.5%

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

      4. fma-undefine 67.5%

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

      5. exp-neg 67.5%

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

      6. associate-*r/ 67.5%

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

      7. *-rgt-identity 67.5%

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

   10. Simplified 67.5%

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

   if 7.99999999999999964e-6 < eps

   1. Initial program 99.8%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around -inf 100.0%

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

      1. *-commutative 100.0%

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

      2. fma-neg 100.0%

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

      3. mul-1-neg 100.0%

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

      4. remove-double-neg 100.0%

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

      5. fma-undefine 100.0%

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

      6. *-commutative 100.0%

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

      7. exp-sum 100.0%

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

      8. *-commutative 100.0%

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

      9. exp-prod 85.9%

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

      10. pow-plus 85.9%

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

      11. +-commutative 85.9%

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

      12. exp-prod 100.0%

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

      13. exp-neg 100.0%

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

      14. distribute-rgt-neg-in 100.0%

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

      15. distribute-neg-in 100.0%

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

      16. metadata-eval 100.0%

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

      17. unsub-neg 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in eps around inf 100.0%

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

       1. *-commutative 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 2, 2\right)}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)} + e^{x \cdot \varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + (exp(1) ^ Float64(x * Float64(-1.0 - eps_m)))) / 2.0)
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = (exp((x * (eps_m + -1.0))) + (2.71828182845904523536 ^ (x * (-1.0 - eps_m)))) / 2.0;
end

Wolfram

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

Derivation

1. Initial program 71.6%

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

3. Simplified 64.5%

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

5. Taylor expanded in eps around inf 98.7%

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

   1. *-un-lft-identity 98.7%

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

   2. exp-prod 98.7%

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

   3. +-commutative 98.7%

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

   4. *-commutative 98.7%

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

   5. fma-define 98.7%

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

7. Applied egg-rr 98.7%

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

   1. pow-flip 98.7%

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

   2. exp-1-e 98.7%

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

9. Applied egg-rr 98.7%

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

    1. fma-undefine 98.7%

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

    2. distribute-neg-in 98.7%

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

    3. distribute-rgt-neg-out 98.7%

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

    4. mul-1-neg 98.7%

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

    5. *-commutative 98.7%

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

    6. +-commutative 98.7%

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

    7. distribute-rgt-out 98.7%

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

    8. sub-neg 98.7%

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

11. Simplified 98.7%

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

12. Final simplification 98.7%

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

Alternative 4: 98.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.6%

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

3. Simplified 64.5%

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

5. Taylor expanded in eps around inf 98.7%

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

6. Taylor expanded in x around -inf 98.7%

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

   1. *-commutative 98.7%

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

   2. fma-neg 98.7%

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

   3. mul-1-neg 98.7%

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

   4. remove-double-neg 98.7%

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

   5. fma-undefine 98.7%

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

   6. *-commutative 98.7%

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

   7. exp-sum 80.0%

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

   8. *-commutative 80.0%

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

   9. exp-prod 71.7%

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

   10. pow-plus 93.2%

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

   11. +-commutative 93.2%

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

   12. exp-prod 98.7%

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

   13. exp-neg 98.7%

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

   14. distribute-rgt-neg-in 98.7%

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

   15. distribute-neg-in 98.7%

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

   16. metadata-eval 98.7%

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

   17. unsub-neg 98.7%

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

8. Simplified 98.7%

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

9. Final simplification 98.7%

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

Alternative 5: 85.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 2, 2\right)}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{eps\_m \cdot \left(x + \frac{1 + e^{x \cdot \left(-1 - eps\_m\right)}}{eps\_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 8e-6)
   (/ (/ (fma x 2.0 2.0) (exp x)) 2.0)
   (/ (* eps_m (+ x (/ (+ 1.0 (exp (* x (- -1.0 eps_m)))) eps_m))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 8e-6) {
		tmp = (fma(x, 2.0, 2.0) / exp(x)) / 2.0;
	} else {
		tmp = (eps_m * (x + ((1.0 + exp((x * (-1.0 - eps_m)))) / eps_m))) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 8e-6)
		tmp = Float64(Float64(fma(x, 2.0, 2.0) / exp(x)) / 2.0);
	else
		tmp = Float64(Float64(eps_m * Float64(x + Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps_m)))) / eps_m))) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 8e-6], N[(N[(N[(x * 2.0 + 2.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(eps$95$m * N[(x + N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 7.99999999999999964e-6

   1. Initial program 61.3%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in eps around 0 27.7%

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

   7. Simplified 67.5%

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

   8. Taylor expanded in eps around 0 67.5%

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

      1. *-commutative 67.5%

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

      2. +-commutative 67.5%

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

      3. *-commutative 67.5%

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

      4. fma-undefine 67.5%

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

      5. exp-neg 67.5%

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

      6. associate-*r/ 67.5%

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

      7. *-rgt-identity 67.5%

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

   10. Simplified 67.5%

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

   if 7.99999999999999964e-6 < eps

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 70.3%

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

      1. +-commutative 70.3%

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

      2. associate-+r+ 70.3%

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

      3. mul-1-neg 70.3%

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

      4. distribute-rgt-neg-in 70.3%

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

      5. *-commutative 70.3%

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

      6. distribute-rgt-neg-in 70.3%

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

      7. distribute-neg-in 70.3%

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

      8. metadata-eval 70.3%

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

      9. distribute-neg-frac 70.3%

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

      10. metadata-eval 70.3%

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

   7. Simplified 70.3%

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

   8. Taylor expanded in eps around inf 82.1%

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

   10. Simplified 82.1%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 2, 2\right)}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 8 \cdot 10^{-6}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{eps\_m \cdot \left(x + \frac{1 + e^{x \cdot \left(-1 - eps\_m\right)}}{eps\_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 8e-6)
   (/ (* (exp (- x)) (+ 2.0 (* x 2.0))) 2.0)
   (/ (* eps_m (+ x (/ (+ 1.0 (exp (* x (- -1.0 eps_m)))) eps_m))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 8e-6) {
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	} else {
		tmp = (eps_m * (x + ((1.0 + exp((x * (-1.0 - eps_m)))) / eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 8d-6) then
        tmp = (exp(-x) * (2.0d0 + (x * 2.0d0))) / 2.0d0
    else
        tmp = (eps_m * (x + ((1.0d0 + exp((x * ((-1.0d0) - eps_m)))) / eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 8e-6) {
		tmp = (Math.exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	} else {
		tmp = (eps_m * (x + ((1.0 + Math.exp((x * (-1.0 - eps_m)))) / eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 8e-6:
		tmp = (math.exp(-x) * (2.0 + (x * 2.0))) / 2.0
	else:
		tmp = (eps_m * (x + ((1.0 + math.exp((x * (-1.0 - eps_m)))) / eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 8e-6)
		tmp = Float64(Float64(exp(Float64(-x)) * Float64(2.0 + Float64(x * 2.0))) / 2.0);
	else
		tmp = Float64(Float64(eps_m * Float64(x + Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps_m)))) / eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 8e-6)
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	else
		tmp = (eps_m * (x + ((1.0 + exp((x * (-1.0 - eps_m)))) / eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 8e-6], N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(eps$95$m * N[(x + N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 7.99999999999999964e-6

   1. Initial program 61.3%

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

   3. Simplified 52.8%

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

   5. Taylor expanded in eps around 0 27.7%

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

   7. Simplified 67.5%

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

   8. Taylor expanded in eps around 0 67.5%

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

   if 7.99999999999999964e-6 < eps

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 70.3%

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

      1. +-commutative 70.3%

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

      2. associate-+r+ 70.3%

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

      3. mul-1-neg 70.3%

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

      4. distribute-rgt-neg-in 70.3%

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

      5. *-commutative 70.3%

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

      6. distribute-rgt-neg-in 70.3%

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

      7. distribute-neg-in 70.3%

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

      8. metadata-eval 70.3%

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

      9. distribute-neg-frac 70.3%

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

      10. metadata-eval 70.3%

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

   7. Simplified 70.3%

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

   8. Taylor expanded in eps around inf 82.1%

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

   10. Simplified 82.1%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 8 \cdot 10^{-6}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-249}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-eps\_m\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(eps\_m + -1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2.8e-249)
   (/ (+ 1.0 (exp (* x (- eps_m)))) 2.0)
   (/ (+ 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 (x <= -2.8e-249) {
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	} 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 (x <= (-2.8d-249)) then
        tmp = (1.0d0 + exp((x * -eps_m))) / 2.0d0
    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 (x <= -2.8e-249) {
		tmp = (1.0 + Math.exp((x * -eps_m))) / 2.0;
	} 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 x <= -2.8e-249:
		tmp = (1.0 + math.exp((x * -eps_m))) / 2.0
	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 (x <= -2.8e-249)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	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 (x <= -2.8e-249)
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	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[x, -2.8e-249], N[(N[(1.0 + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $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 x < -2.7999999999999999e-249

   1. Initial program 71.9%

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

   3. Simplified 71.9%

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

   5. Taylor expanded in x around 0 40.2%

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

   6. Taylor expanded in eps around inf 67.9%

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

      1. cancel-sign-sub-inv 67.9%

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

      2. metadata-eval 67.9%

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

      3. *-lft-identity 67.9%

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

      4. mul-1-neg 67.9%

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

      5. distribute-rgt-neg-in 67.9%

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

      6. distribute-neg-in 67.9%

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

      7. metadata-eval 67.9%

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

      8. unsub-neg 67.9%

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

   8. Simplified 67.9%

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

   9. Taylor expanded in eps around inf 68.8%

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

       1. mul-1-neg 68.8%

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

       2. distribute-lft-neg-out 68.8%

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

       3. *-commutative 68.8%

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

   11. Simplified 68.8%

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

   if -2.7999999999999999e-249 < x

   1. Initial program 71.3%

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

   3. Simplified 66.5%

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

   5. Taylor expanded in eps around inf 99.2%

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

      1. *-un-lft-identity 99.2%

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

      2. exp-prod 99.2%

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

      3. +-commutative 99.2%

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

      4. *-commutative 99.2%

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

      5. fma-define 99.2%

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

   7. Applied egg-rr 99.2%

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

      1. pow-flip 99.2%

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

      2. exp-1-e 99.2%

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

   9. Applied egg-rr 99.2%

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

       1. fma-undefine 99.2%

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

       2. distribute-neg-in 99.2%

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

       3. distribute-rgt-neg-out 99.2%

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

       4. mul-1-neg 99.2%

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

       5. *-commutative 99.2%

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

       6. +-commutative 99.2%

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

       7. distribute-rgt-out 99.2%

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

       8. sub-neg 99.2%

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

   11. Simplified 99.2%

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

   12. Taylor expanded in x around 0 60.4%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-249}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 480:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-eps\_m\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 480.0) (/ (+ 1.0 (exp (* x (- eps_m)))) 2.0) 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 480.0) {
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 480.0d0) then
        tmp = (1.0d0 + exp((x * -eps_m))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 480.0) {
		tmp = (1.0 + Math.exp((x * -eps_m))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 480.0:
		tmp = (1.0 + math.exp((x * -eps_m))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 480.0)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 480.0)
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 480.0], N[(N[(1.0 + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 480

   1. Initial program 61.5%

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

   3. Simplified 61.5%

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

   5. Taylor expanded in x around 0 38.8%

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

   6. Taylor expanded in eps around inf 76.5%

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

      1. cancel-sign-sub-inv 76.5%

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

      2. metadata-eval 76.5%

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

      3. *-lft-identity 76.5%

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

      4. mul-1-neg 76.5%

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

      5. distribute-rgt-neg-in 76.5%

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

      6. distribute-neg-in 76.5%

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

      7. metadata-eval 76.5%

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

      8. unsub-neg 76.5%

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

   8. Simplified 76.5%

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

   9. Taylor expanded in eps around inf 77.4%

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

       1. mul-1-neg 77.4%

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

       2. distribute-lft-neg-out 77.4%

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

       3. *-commutative 77.4%

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

   11. Simplified 77.4%

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

   if 480 < 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 0 45.6%

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

      1. mul-1-neg 45.6%

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

      2. mul-1-neg 45.6%

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

      3. rec-exp 45.6%

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

      4. sub-neg 45.6%

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

      5. div-sub 45.6%

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

      6. mul-1-neg 45.6%

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

      7. rec-exp 45.6%

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

      8. +-inverses 45.6%

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

   7. Simplified 45.6%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 480:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 580:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 580.0) (/ (+ 1.0 (exp (- x))) 2.0) 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 580.0) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 580.0d0) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 580.0) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 580.0:
		tmp = (1.0 + math.exp(-x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 580.0)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 580.0)
		tmp = (1.0 + exp(-x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 580.0], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 580

   1. Initial program 61.5%

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

   3. Simplified 61.5%

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

   5. Taylor expanded in x around 0 41.4%

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

      1. +-commutative 41.4%

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

      2. associate-+r+ 41.4%

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

      3. mul-1-neg 41.4%

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

      4. distribute-rgt-neg-in 41.4%

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

      5. *-commutative 41.4%

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

      6. distribute-rgt-neg-in 41.4%

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

      7. distribute-neg-in 41.4%

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

      8. metadata-eval 41.4%

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

      9. distribute-neg-frac 41.4%

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

      10. metadata-eval 41.4%

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

   7. Simplified 41.4%

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

   8. Taylor expanded in eps around inf 82.3%

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

   10. Simplified 82.3%

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

   11. Taylor expanded in eps around 0 79.5%

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

       1. mul-1-neg 79.5%

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

   13. Simplified 79.5%

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

   if 580 < 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 0 45.6%

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

      1. mul-1-neg 45.6%

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

      2. mul-1-neg 45.6%

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

      3. rec-exp 45.6%

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

      4. sub-neg 45.6%

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

      5. div-sub 45.6%

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

      6. mul-1-neg 45.6%

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

      7. rec-exp 45.6%

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

      8. +-inverses 45.6%

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

   7. Simplified 45.6%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 580:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.2% accurate, 18.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 72:\\ \;\;\;\;\frac{2 - x \cdot eps\_m}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 72.0) (/ (- 2.0 (* x eps_m)) 2.0) 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 72.0) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 72.0d0) then
        tmp = (2.0d0 - (x * eps_m)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 72.0) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 72.0:
		tmp = (2.0 - (x * eps_m)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 72.0)
		tmp = Float64(Float64(2.0 - Float64(x * eps_m)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 72.0)
		tmp = (2.0 - (x * eps_m)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 72.0], N[(N[(2.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 72

   1. Initial program 61.3%

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

   3. Simplified 61.3%

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

   5. Taylor expanded in x around 0 38.4%

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

   6. Taylor expanded in x around 0 40.8%

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

   7. Taylor expanded in eps around inf 61.6%

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

      1. associate-*r* 61.6%

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

      2. neg-mul-1 61.6%

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

   9. Simplified 61.6%

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

   if 72 < 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 0 45.0%

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

      1. mul-1-neg 45.0%

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

      2. mul-1-neg 45.0%

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

      3. rec-exp 45.0%

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

      4. sub-neg 45.0%

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

      5. div-sub 45.0%

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

      6. mul-1-neg 45.0%

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

      7. rec-exp 45.0%

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

      8. +-inverses 45.0%

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

   7. Simplified 45.0%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 72:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.4% accurate, 37.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 470:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 470.0) 1.0 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 470.0) {
		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 <= 470.0d0) 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 <= 470.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 470.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 470.0)
		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 <= 470.0)
		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, 470.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 470

   1. Initial program 61.5%

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

   3. Simplified 61.5%

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

   5. Taylor expanded in x around 0 57.4%

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

   if 470 < 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 0 45.6%

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

      1. mul-1-neg 45.6%

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

      2. mul-1-neg 45.6%

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

      3. rec-exp 45.6%

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

      4. sub-neg 45.6%

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

      5. div-sub 45.6%

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

      6. mul-1-neg 45.6%

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

      7. rec-exp 45.6%

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

      8. +-inverses 45.6%

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

   7. Simplified 45.6%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 470:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 15.8% 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 71.6%

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

3. Simplified 61.5%

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

5. Taylor expanded in eps around 0 13.5%

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

   1. mul-1-neg 13.5%

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

   2. mul-1-neg 13.5%

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

   3. rec-exp 13.5%

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

   4. sub-neg 13.5%

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

   5. div-sub 13.5%

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

   6. mul-1-neg 13.5%

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

   7. rec-exp 13.5%

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

   8. +-inverses 13.8%

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

7. Simplified 13.8%

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

8. Final simplification 13.8%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024076 
(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))