NMSE Section 6.1 mentioned, A

Percentage Accurate: 74.2% → 99.9%

Time: 15.5s

Alternatives: 14

Speedup: 2.0×

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

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.2% 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: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps_m \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 2, 2\right)}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps_m + -1\right)} + e^{x \cdot \left(-eps_m\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 5e-7)
   (/ (/ (fma x 2.0 2.0) (exp x)) 2.0)
   (/ (+ (exp (* x (+ eps_m -1.0))) (exp (* x (- eps_m)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 5e-7) {
		tmp = (fma(x, 2.0, 2.0) / exp(x)) / 2.0;
	} else {
		tmp = (exp((x * (eps_m + -1.0))) + exp((x * -eps_m))) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 5e-7)
		tmp = Float64(Float64(fma(x, 2.0, 2.0) / exp(x)) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + exp(Float64(x * Float64(-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, 5e-7], N[(N[(N[(x * 2.0 + 2.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $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 < 4.99999999999999977e-7

   1. Initial program 58.5%

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

   3. Simplified 58.5%

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

   5. Taylor expanded in eps around 0 72.5%

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

      1. associate--r+ 72.5%

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

      2. associate-*r* 72.5%

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

      3. mul-1-neg 72.5%

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

      4. cancel-sign-sub 72.5%

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

      5. distribute-rgt1-in 72.5%

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

      6. distribute-rgt-out-- 72.5%

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

      7. mul-1-neg 72.5%

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

      8. mul-1-neg 72.5%

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

   7. Simplified 72.5%

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

   8. Taylor expanded in x around inf 72.5%

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

      1. distribute-lft-out 72.5%

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

      2. neg-mul-1 72.5%

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

      3. neg-mul-1 72.5%

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

      4. distribute-lft-out 72.5%

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

      5. associate-*r* 72.5%

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

      6. distribute-rgt-out 72.5%

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

      7. neg-mul-1 72.5%

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

   10. Simplified 72.5%

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

       1. *-commutative 72.5%

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

       2. exp-neg 72.5%

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

       3. un-div-inv 72.5%

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

       4. +-commutative 72.5%

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

       5. *-commutative 72.5%

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

       6. fma-def 72.5%

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

   12. Applied egg-rr 72.5%

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

   if 4.99999999999999977e-7 < eps

   1. Initial program 100.0%

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

   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 eps around inf 100.0%

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

   6. Taylor expanded in eps around -inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

      3. mul-1-neg 100.0%

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

      4. sub-neg 100.0%

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

      5. mul-1-neg 100.0%

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

      6. associate-*r* 100.0%

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

      7. neg-mul-1 100.0%

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

      8. mul-1-neg 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in eps around inf 100.0%

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

       1. associate-*r* 100.0%

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

       2. neg-mul-1 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x around inf 100.0%

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

       1. mul-1-neg 100.0%

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

       2. distribute-rgt-neg-in 100.0%

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

       3. sub-neg 100.0%

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

       4. neg-mul-1 100.0%

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

       5. distribute-neg-in 100.0%

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

       6. metadata-eval 100.0%

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

       7. neg-mul-1 100.0%

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

       8. remove-double-neg 100.0%

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

   14. Simplified 100.0%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 2, 2\right)}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-300}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-eps_m\right)}}{2}\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;\frac{1 + e^{x + x \cdot eps_m}}{2}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+216}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 2, 2\right)}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} \cdot \left(x + \left(x + 2\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1e-300)
   (/ (+ 1.0 (exp (* x (- eps_m)))) 2.0)
   (if (<= x 0.95)
     (/ (+ 1.0 (exp (+ x (* x eps_m)))) 2.0)
     (if (<= x 8e+216)
       (/ (/ (fma x 2.0 2.0) (exp x)) 2.0)
       (/ (* (exp x) (+ x (+ x 2.0))) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1e-300) {
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	} else if (x <= 0.95) {
		tmp = (1.0 + exp((x + (x * eps_m)))) / 2.0;
	} else if (x <= 8e+216) {
		tmp = (fma(x, 2.0, 2.0) / exp(x)) / 2.0;
	} else {
		tmp = (exp(x) * (x + (x + 2.0))) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1e-300)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	elseif (x <= 0.95)
		tmp = Float64(Float64(1.0 + exp(Float64(x + Float64(x * eps_m)))) / 2.0);
	elseif (x <= 8e+216)
		tmp = Float64(Float64(fma(x, 2.0, 2.0) / exp(x)) / 2.0);
	else
		tmp = Float64(Float64(exp(x) * Float64(x + Float64(x + 2.0))) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1e-300], N[(N[(1.0 + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 0.95], N[(N[(1.0 + N[Exp[N[(x + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 8e+216], N[(N[(N[(x * 2.0 + 2.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[x], $MachinePrecision] * N[(x + N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.00000000000000003e-300

   1. Initial program 66.8%

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

   3. Simplified 66.8%

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

   5. Taylor expanded in eps around inf 98.8%

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

   6. Taylor expanded in eps around -inf 98.8%

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

      1. associate-*r* 98.8%

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

      2. neg-mul-1 98.8%

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

      3. mul-1-neg 98.8%

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

      4. sub-neg 98.8%

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

      5. mul-1-neg 98.8%

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

      6. associate-*r* 98.8%

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

      7. neg-mul-1 98.8%

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

      8. mul-1-neg 98.8%

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

   8. Simplified 98.8%

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

   9. Taylor expanded in eps around inf 98.8%

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

       1. associate-*r* 98.8%

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

       2. neg-mul-1 98.8%

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

   11. Simplified 98.8%

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

   12. Taylor expanded in x around 0 79.6%

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

   if -1.00000000000000003e-300 < x < 0.94999999999999996

   1. Initial program 52.7%

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

   3. Simplified 52.7%

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

   5. Taylor expanded in x around 0 43.4%

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

      1. *-commutative 43.4%

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

      2. +-commutative 43.4%

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

      3. distribute-lft-in 43.4%

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

      4. add-sqr-sqrt 1.2%

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

      5. sqrt-unprod 33.9%

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

      6. sqr-neg 33.9%

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

      7. sqrt-unprod 34.3%

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

      8. add-sqr-sqrt 35.5%

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

      9. *-commutative 35.5%

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

      10. *-un-lft-identity 35.5%

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

      11. add-sqr-sqrt 1.2%

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

      12. sqrt-unprod 35.5%

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

      13. sqr-neg 35.5%

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

      14. sqrt-unprod 34.3%

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

      15. add-sqr-sqrt 35.5%

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

   7. Applied egg-rr 35.5%

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

   8. Taylor expanded in eps around inf 82.8%

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

   if 0.94999999999999996 < x < 8.0000000000000002e216

   1. Initial program 95.9%

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

   3. Simplified 95.9%

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

   5. Taylor expanded in eps around 0 60.1%

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

      1. associate--r+ 60.2%

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

      2. associate-*r* 60.2%

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

      3. mul-1-neg 60.2%

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

      4. cancel-sign-sub 60.2%

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

      5. distribute-rgt1-in 60.2%

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

      6. distribute-rgt-out-- 60.2%

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

      7. mul-1-neg 60.2%

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

      8. mul-1-neg 60.2%

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

   7. Simplified 60.2%

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

   8. Taylor expanded in x around inf 60.1%

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

      1. distribute-lft-out 60.1%

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

      2. neg-mul-1 60.1%

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

      3. neg-mul-1 60.1%

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

      4. distribute-lft-out 60.1%

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

      5. associate-*r* 60.1%

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

      6. distribute-rgt-out 60.2%

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

      7. neg-mul-1 60.2%

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

   10. Simplified 60.2%

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

       1. *-commutative 60.2%

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

       2. exp-neg 60.2%

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

       3. un-div-inv 60.2%

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

       4. +-commutative 60.2%

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

       5. *-commutative 60.2%

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

       6. fma-def 60.2%

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

   12. Applied egg-rr 60.2%

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

   if 8.0000000000000002e216 < x

   1. Initial program 100.0%

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

   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 eps around 0 36.0%

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

      1. associate--r+ 36.0%

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

      2. associate-*r* 36.0%

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

      3. mul-1-neg 36.0%

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

      4. cancel-sign-sub 36.0%

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

      5. distribute-rgt1-in 36.0%

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

      6. distribute-rgt-out-- 36.0%

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

      7. mul-1-neg 36.0%

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

      8. mul-1-neg 36.0%

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

   7. Simplified 36.0%

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

      1. *-un-lft-identity 36.0%

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

      2. fma-def 36.0%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 65.5%

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

      5. sqr-neg 65.5%

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

      6. sqrt-unprod 65.5%

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

      7. add-sqr-sqrt 65.5%

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

      8. associate--l+ 65.5%

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

      9. metadata-eval 65.5%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 65.5%

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

      12. sqr-neg 65.5%

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

      13. sqrt-unprod 65.5%

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

      14. add-sqr-sqrt 65.5%

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

   9. Applied egg-rr 65.5%

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

       1. fma-udef 65.5%

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

       2. *-lft-identity 65.5%

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

       3. *-commutative 65.5%

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

       4. distribute-lft-out 65.5%

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

   11. Simplified 65.5%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-300}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;\frac{1 + e^{x + x \cdot \varepsilon}}{2}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+216}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, 2, 2\right)}{e^{x}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} \cdot \left(x + \left(x + 2\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% 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 70.0%

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

3. Simplified 70.0%

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

5. Taylor expanded in eps around inf 98.8%

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

6. Taylor expanded in eps around -inf 98.8%

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

   1. associate-*r* 98.8%

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

   2. neg-mul-1 98.8%

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

   3. mul-1-neg 98.8%

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

   4. sub-neg 98.8%

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

   5. mul-1-neg 98.8%

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

   6. associate-*r* 98.8%

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

   7. neg-mul-1 98.8%

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

   8. mul-1-neg 98.8%

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

8. Simplified 98.8%

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

9. Final simplification 98.8%

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

Alternative 4: 83.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-301}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-eps_m\right)}}{2}\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;\frac{1 + e^{x + x \cdot eps_m}}{2}\\ \mathbf{elif}\;x \leq 10^{+217}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} \cdot \left(x + \left(x + 2\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -7e-301)
   (/ (+ 1.0 (exp (* x (- eps_m)))) 2.0)
   (if (<= x 0.95)
     (/ (+ 1.0 (exp (+ x (* x eps_m)))) 2.0)
     (if (<= x 1e+217)
       (/ (* (exp (- x)) (+ 2.0 (* x 2.0))) 2.0)
       (/ (* (exp x) (+ x (+ x 2.0))) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -7e-301) {
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	} else if (x <= 0.95) {
		tmp = (1.0 + exp((x + (x * eps_m)))) / 2.0;
	} else if (x <= 1e+217) {
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	} else {
		tmp = (exp(x) * (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 (x <= (-7d-301)) then
        tmp = (1.0d0 + exp((x * -eps_m))) / 2.0d0
    else if (x <= 0.95d0) then
        tmp = (1.0d0 + exp((x + (x * eps_m)))) / 2.0d0
    else if (x <= 1d+217) then
        tmp = (exp(-x) * (2.0d0 + (x * 2.0d0))) / 2.0d0
    else
        tmp = (exp(x) * (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 (x <= -7e-301) {
		tmp = (1.0 + Math.exp((x * -eps_m))) / 2.0;
	} else if (x <= 0.95) {
		tmp = (1.0 + Math.exp((x + (x * eps_m)))) / 2.0;
	} else if (x <= 1e+217) {
		tmp = (Math.exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	} else {
		tmp = (Math.exp(x) * (x + (x + 2.0))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -7e-301:
		tmp = (1.0 + math.exp((x * -eps_m))) / 2.0
	elif x <= 0.95:
		tmp = (1.0 + math.exp((x + (x * eps_m)))) / 2.0
	elif x <= 1e+217:
		tmp = (math.exp(-x) * (2.0 + (x * 2.0))) / 2.0
	else:
		tmp = (math.exp(x) * (x + (x + 2.0))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -7e-301)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	elseif (x <= 0.95)
		tmp = Float64(Float64(1.0 + exp(Float64(x + Float64(x * eps_m)))) / 2.0);
	elseif (x <= 1e+217)
		tmp = Float64(Float64(exp(Float64(-x)) * Float64(2.0 + Float64(x * 2.0))) / 2.0);
	else
		tmp = Float64(Float64(exp(x) * 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 (x <= -7e-301)
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	elseif (x <= 0.95)
		tmp = (1.0 + exp((x + (x * eps_m)))) / 2.0;
	elseif (x <= 1e+217)
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	else
		tmp = (exp(x) * (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[x, -7e-301], N[(N[(1.0 + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 0.95], N[(N[(1.0 + N[Exp[N[(x + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1e+217], N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[x], $MachinePrecision] * N[(x + N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6.99999999999999984e-301

   1. Initial program 66.8%

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

   3. Simplified 66.8%

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

   5. Taylor expanded in eps around inf 98.8%

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

   6. Taylor expanded in eps around -inf 98.8%

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

      1. associate-*r* 98.8%

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

      2. neg-mul-1 98.8%

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

      3. mul-1-neg 98.8%

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

      4. sub-neg 98.8%

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

      5. mul-1-neg 98.8%

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

      6. associate-*r* 98.8%

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

      7. neg-mul-1 98.8%

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

      8. mul-1-neg 98.8%

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

   8. Simplified 98.8%

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

   9. Taylor expanded in eps around inf 98.8%

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

       1. associate-*r* 98.8%

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

       2. neg-mul-1 98.8%

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

   11. Simplified 98.8%

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

   12. Taylor expanded in x around 0 79.6%

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

   if -6.99999999999999984e-301 < x < 0.94999999999999996

   1. Initial program 52.7%

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

   3. Simplified 52.7%

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

   5. Taylor expanded in x around 0 43.4%

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

      1. *-commutative 43.4%

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

      2. +-commutative 43.4%

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

      3. distribute-lft-in 43.4%

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

      4. add-sqr-sqrt 1.2%

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

      5. sqrt-unprod 33.9%

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

      6. sqr-neg 33.9%

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

      7. sqrt-unprod 34.3%

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

      8. add-sqr-sqrt 35.5%

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

      9. *-commutative 35.5%

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

      10. *-un-lft-identity 35.5%

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

      11. add-sqr-sqrt 1.2%

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

      12. sqrt-unprod 35.5%

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

      13. sqr-neg 35.5%

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

      14. sqrt-unprod 34.3%

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

      15. add-sqr-sqrt 35.5%

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

   7. Applied egg-rr 35.5%

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

   8. Taylor expanded in eps around inf 82.8%

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

   if 0.94999999999999996 < x < 9.9999999999999996e216

   1. Initial program 95.9%

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

   3. Simplified 95.9%

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

   5. Taylor expanded in eps around 0 60.1%

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

      1. associate--r+ 60.2%

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

      2. associate-*r* 60.2%

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

      3. mul-1-neg 60.2%

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

      4. cancel-sign-sub 60.2%

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

      5. distribute-rgt1-in 60.2%

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

      6. distribute-rgt-out-- 60.2%

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

      7. mul-1-neg 60.2%

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

      8. mul-1-neg 60.2%

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

   7. Simplified 60.2%

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

   8. Taylor expanded in x around inf 60.1%

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

      1. distribute-lft-out 60.1%

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

      2. neg-mul-1 60.1%

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

      3. neg-mul-1 60.1%

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

      4. distribute-lft-out 60.1%

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

      5. associate-*r* 60.1%

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

      6. distribute-rgt-out 60.2%

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

      7. neg-mul-1 60.2%

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

   10. Simplified 60.2%

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

   if 9.9999999999999996e216 < x

   1. Initial program 100.0%

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

   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 eps around 0 36.0%

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

      1. associate--r+ 36.0%

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

      2. associate-*r* 36.0%

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

      3. mul-1-neg 36.0%

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

      4. cancel-sign-sub 36.0%

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

      5. distribute-rgt1-in 36.0%

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

      6. distribute-rgt-out-- 36.0%

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

      7. mul-1-neg 36.0%

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

      8. mul-1-neg 36.0%

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

   7. Simplified 36.0%

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

      1. *-un-lft-identity 36.0%

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

      2. fma-def 36.0%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 65.5%

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

      5. sqr-neg 65.5%

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

      6. sqrt-unprod 65.5%

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

      7. add-sqr-sqrt 65.5%

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

      8. associate--l+ 65.5%

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

      9. metadata-eval 65.5%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 65.5%

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

      12. sqr-neg 65.5%

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

      13. sqrt-unprod 65.5%

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

      14. add-sqr-sqrt 65.5%

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

   9. Applied egg-rr 65.5%

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

       1. fma-udef 65.5%

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

       2. *-lft-identity 65.5%

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

       3. *-commutative 65.5%

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

       4. distribute-lft-out 65.5%

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

   11. Simplified 65.5%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-301}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;\frac{1 + e^{x + x \cdot \varepsilon}}{2}\\ \mathbf{elif}\;x \leq 10^{+217}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} \cdot \left(x + \left(x + 2\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -5.3 \cdot 10^{-226}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-eps_m\right)}}{2}\\ \mathbf{elif}\;x \leq 10^{+217}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} \cdot \left(x + \left(x + 2\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -5.3e-226)
   (/ (+ 1.0 (exp (* x (- eps_m)))) 2.0)
   (if (<= x 1e+217)
     (/ (* (exp (- x)) (+ 2.0 (* x 2.0))) 2.0)
     (/ (* (exp x) (+ x (+ x 2.0))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -5.3e-226) {
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	} else if (x <= 1e+217) {
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	} else {
		tmp = (exp(x) * (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 (x <= (-5.3d-226)) then
        tmp = (1.0d0 + exp((x * -eps_m))) / 2.0d0
    else if (x <= 1d+217) then
        tmp = (exp(-x) * (2.0d0 + (x * 2.0d0))) / 2.0d0
    else
        tmp = (exp(x) * (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 (x <= -5.3e-226) {
		tmp = (1.0 + Math.exp((x * -eps_m))) / 2.0;
	} else if (x <= 1e+217) {
		tmp = (Math.exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	} else {
		tmp = (Math.exp(x) * (x + (x + 2.0))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -5.3e-226:
		tmp = (1.0 + math.exp((x * -eps_m))) / 2.0
	elif x <= 1e+217:
		tmp = (math.exp(-x) * (2.0 + (x * 2.0))) / 2.0
	else:
		tmp = (math.exp(x) * (x + (x + 2.0))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -5.3e-226)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	elseif (x <= 1e+217)
		tmp = Float64(Float64(exp(Float64(-x)) * Float64(2.0 + Float64(x * 2.0))) / 2.0);
	else
		tmp = Float64(Float64(exp(x) * 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 (x <= -5.3e-226)
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	elseif (x <= 1e+217)
		tmp = (exp(-x) * (2.0 + (x * 2.0))) / 2.0;
	else
		tmp = (exp(x) * (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[x, -5.3e-226], N[(N[(1.0 + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1e+217], N[(N[(N[Exp[(-x)], $MachinePrecision] * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[x], $MachinePrecision] * N[(x + N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.3000000000000004e-226

   1. Initial program 70.0%

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

   3. Simplified 70.0%

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

   5. Taylor expanded in eps around inf 98.5%

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

   6. Taylor expanded in eps around -inf 98.5%

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

      1. associate-*r* 98.5%

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

      2. neg-mul-1 98.5%

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

      3. mul-1-neg 98.5%

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

      4. sub-neg 98.5%

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

      5. mul-1-neg 98.5%

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

      6. associate-*r* 98.5%

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

      7. neg-mul-1 98.5%

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

      8. mul-1-neg 98.5%

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

   8. Simplified 98.5%

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

   9. Taylor expanded in eps around inf 98.6%

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

       1. associate-*r* 98.6%

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

       2. neg-mul-1 98.6%

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

   11. Simplified 98.6%

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

   12. Taylor expanded in x around 0 76.1%

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

   if -5.3000000000000004e-226 < x < 9.9999999999999996e216

   1. Initial program 66.0%

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

   3. Simplified 66.0%

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

   5. Taylor expanded in eps around 0 73.2%

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

      1. associate--r+ 73.2%

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

      2. associate-*r* 73.2%

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

      3. mul-1-neg 73.2%

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

      4. cancel-sign-sub 73.2%

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

      5. distribute-rgt1-in 73.2%

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

      6. distribute-rgt-out-- 73.2%

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

      7. mul-1-neg 73.2%

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

      8. mul-1-neg 73.2%

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

   7. Simplified 73.2%

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

   8. Taylor expanded in x around inf 73.2%

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

      1. distribute-lft-out 73.2%

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

      2. neg-mul-1 73.2%

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

      3. neg-mul-1 73.2%

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

      4. distribute-lft-out 73.2%

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

      5. associate-*r* 73.2%

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

      6. distribute-rgt-out 73.2%

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

      7. neg-mul-1 73.2%

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

   10. Simplified 73.2%

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

   if 9.9999999999999996e216 < x

   1. Initial program 100.0%

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

   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 eps around 0 36.0%

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

      1. associate--r+ 36.0%

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

      2. associate-*r* 36.0%

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

      3. mul-1-neg 36.0%

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

      4. cancel-sign-sub 36.0%

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

      5. distribute-rgt1-in 36.0%

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

      6. distribute-rgt-out-- 36.0%

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

      7. mul-1-neg 36.0%

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

      8. mul-1-neg 36.0%

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

   7. Simplified 36.0%

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

      1. *-un-lft-identity 36.0%

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

      2. fma-def 36.0%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 65.5%

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

      5. sqr-neg 65.5%

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

      6. sqrt-unprod 65.5%

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

      7. add-sqr-sqrt 65.5%

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

      8. associate--l+ 65.5%

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

      9. metadata-eval 65.5%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 65.5%

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

      12. sqr-neg 65.5%

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

      13. sqrt-unprod 65.5%

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

      14. add-sqr-sqrt 65.5%

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

   9. Applied egg-rr 65.5%

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

       1. fma-udef 65.5%

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

       2. *-lft-identity 65.5%

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

       3. *-commutative 65.5%

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

       4. distribute-lft-out 65.5%

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

   11. Simplified 65.5%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.3 \cdot 10^{-226}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 10^{+217}:\\ \;\;\;\;\frac{e^{-x} \cdot \left(2 + x \cdot 2\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} \cdot \left(x + \left(x + 2\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-226}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-eps_m\right)}}{2}\\ \mathbf{elif}\;x \leq 10^{+217}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} \cdot \left(x + \left(x + 2\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -4.8e-226)
   (/ (+ 1.0 (exp (* x (- eps_m)))) 2.0)
   (if (<= x 1e+217)
     (/ (* 2.0 (exp (- x))) 2.0)
     (/ (* (exp x) (+ x (+ x 2.0))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -4.8e-226) {
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	} else if (x <= 1e+217) {
		tmp = (2.0 * exp(-x)) / 2.0;
	} else {
		tmp = (exp(x) * (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 (x <= (-4.8d-226)) then
        tmp = (1.0d0 + exp((x * -eps_m))) / 2.0d0
    else if (x <= 1d+217) then
        tmp = (2.0d0 * exp(-x)) / 2.0d0
    else
        tmp = (exp(x) * (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 (x <= -4.8e-226) {
		tmp = (1.0 + Math.exp((x * -eps_m))) / 2.0;
	} else if (x <= 1e+217) {
		tmp = (2.0 * Math.exp(-x)) / 2.0;
	} else {
		tmp = (Math.exp(x) * (x + (x + 2.0))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -4.8e-226:
		tmp = (1.0 + math.exp((x * -eps_m))) / 2.0
	elif x <= 1e+217:
		tmp = (2.0 * math.exp(-x)) / 2.0
	else:
		tmp = (math.exp(x) * (x + (x + 2.0))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -4.8e-226)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	elseif (x <= 1e+217)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(Float64(exp(x) * 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 (x <= -4.8e-226)
		tmp = (1.0 + exp((x * -eps_m))) / 2.0;
	elseif (x <= 1e+217)
		tmp = (2.0 * exp(-x)) / 2.0;
	else
		tmp = (exp(x) * (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[x, -4.8e-226], N[(N[(1.0 + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1e+217], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[x], $MachinePrecision] * N[(x + N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.7999999999999999e-226

   1. Initial program 70.0%

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

   3. Simplified 70.0%

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

   5. Taylor expanded in eps around inf 98.5%

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

   6. Taylor expanded in eps around -inf 98.5%

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

      1. associate-*r* 98.5%

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

      2. neg-mul-1 98.5%

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

      3. mul-1-neg 98.5%

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

      4. sub-neg 98.5%

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

      5. mul-1-neg 98.5%

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

      6. associate-*r* 98.5%

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

      7. neg-mul-1 98.5%

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

      8. mul-1-neg 98.5%

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

   8. Simplified 98.5%

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

   9. Taylor expanded in eps around inf 98.6%

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

       1. associate-*r* 98.6%

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

       2. neg-mul-1 98.6%

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

   11. Simplified 98.6%

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

   12. Taylor expanded in x around 0 76.1%

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

   if -4.7999999999999999e-226 < x < 9.9999999999999996e216

   1. Initial program 66.0%

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

   3. Simplified 66.0%

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

   5. Taylor expanded in eps around inf 98.9%

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

   6. Taylor expanded in eps around 0 72.0%

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

      1. cancel-sign-sub-inv 72.0%

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

      2. metadata-eval 72.0%

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

      3. distribute-rgt1-in 72.0%

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

      4. metadata-eval 72.0%

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

      5. neg-mul-1 72.0%

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

   8. Simplified 72.0%

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

   if 9.9999999999999996e216 < x

   1. Initial program 100.0%

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

   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 eps around 0 36.0%

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

      1. associate--r+ 36.0%

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

      2. associate-*r* 36.0%

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

      3. mul-1-neg 36.0%

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

      4. cancel-sign-sub 36.0%

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

      5. distribute-rgt1-in 36.0%

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

      6. distribute-rgt-out-- 36.0%

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

      7. mul-1-neg 36.0%

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

      8. mul-1-neg 36.0%

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

   7. Simplified 36.0%

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

      1. *-un-lft-identity 36.0%

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

      2. fma-def 36.0%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 65.5%

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

      5. sqr-neg 65.5%

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

      6. sqrt-unprod 65.5%

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

      7. add-sqr-sqrt 65.5%

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

      8. associate--l+ 65.5%

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

      9. metadata-eval 65.5%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 65.5%

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

      12. sqr-neg 65.5%

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

      13. sqrt-unprod 65.5%

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

      14. add-sqr-sqrt 65.5%

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

   9. Applied egg-rr 65.5%

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

       1. fma-udef 65.5%

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

       2. *-lft-identity 65.5%

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

       3. *-commutative 65.5%

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

       4. distribute-lft-out 65.5%

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

   11. Simplified 65.5%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-226}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 10^{+217}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} \cdot \left(x + \left(x + 2\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps_m \leq 1.1 \cdot 10^{+219}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} \cdot \left(x + \left(x + 2\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 1.1e+219)
   (/ (* 2.0 (exp (- x))) 2.0)
   (/ (* (exp x) (+ x (+ x 2.0))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.1e+219) {
		tmp = (2.0 * exp(-x)) / 2.0;
	} else {
		tmp = (exp(x) * (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 <= 1.1d+219) then
        tmp = (2.0d0 * exp(-x)) / 2.0d0
    else
        tmp = (exp(x) * (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 <= 1.1e+219) {
		tmp = (2.0 * Math.exp(-x)) / 2.0;
	} else {
		tmp = (Math.exp(x) * (x + (x + 2.0))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 1.1e+219:
		tmp = (2.0 * math.exp(-x)) / 2.0
	else:
		tmp = (math.exp(x) * (x + (x + 2.0))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 1.1e+219)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(Float64(exp(x) * 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 <= 1.1e+219)
		tmp = (2.0 * exp(-x)) / 2.0;
	else
		tmp = (exp(x) * (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, 1.1e+219], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[x], $MachinePrecision] * N[(x + N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.1000000000000001e219

   1. Initial program 66.9%

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

   3. Simplified 66.9%

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

   5. Taylor expanded in eps around inf 98.7%

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

   6. Taylor expanded in eps around 0 77.1%

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

      1. cancel-sign-sub-inv 77.1%

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

      2. metadata-eval 77.1%

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

      3. distribute-rgt1-in 77.1%

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

      4. metadata-eval 77.1%

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

      5. neg-mul-1 77.1%

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

   8. Simplified 77.1%

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

   if 1.1000000000000001e219 < eps

   1. Initial program 100.0%

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

   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 eps around 0 14.2%

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

      1. associate--r+ 14.2%

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

      2. associate-*r* 14.2%

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

      3. mul-1-neg 14.2%

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

      4. cancel-sign-sub 14.2%

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

      5. distribute-rgt1-in 14.2%

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

      6. distribute-rgt-out-- 14.2%

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

      7. mul-1-neg 14.2%

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

      8. mul-1-neg 14.2%

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

   7. Simplified 14.2%

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

      1. *-un-lft-identity 14.2%

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

      2. fma-def 14.2%

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

      3. add-sqr-sqrt 8.9%

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

      4. sqrt-unprod 47.0%

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

      5. sqr-neg 47.0%

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

      6. sqrt-unprod 38.2%

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

      7. add-sqr-sqrt 47.0%

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

      8. associate--l+ 47.0%

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

      9. metadata-eval 47.0%

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

      10. add-sqr-sqrt 8.9%

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

      11. sqrt-unprod 47.0%

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

      12. sqr-neg 47.0%

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

      13. sqrt-unprod 38.2%

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

      14. add-sqr-sqrt 47.3%

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

   9. Applied egg-rr 47.3%

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

       1. fma-udef 47.3%

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

       2. *-lft-identity 47.3%

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

       3. *-commutative 47.3%

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

       4. distribute-lft-out 47.3%

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

   11. Simplified 47.3%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.1 \cdot 10^{+219}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} \cdot \left(x + \left(x + 2\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps_m \leq 4.4 \cdot 10^{+219}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot eps_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 4.4e+219)
   (/ (* 2.0 (exp (- x))) 2.0)
   (/ (+ 2.0 (* x eps_m)) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 4.4e+219) {
		tmp = (2.0 * exp(-x)) / 2.0;
	} else {
		tmp = (2.0 + (x * eps_m)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 4.4d+219) then
        tmp = (2.0d0 * exp(-x)) / 2.0d0
    else
        tmp = (2.0d0 + (x * eps_m)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 4.4e+219) {
		tmp = (2.0 * Math.exp(-x)) / 2.0;
	} else {
		tmp = (2.0 + (x * eps_m)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 4.4e+219:
		tmp = (2.0 * math.exp(-x)) / 2.0
	else:
		tmp = (2.0 + (x * eps_m)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 4.4e+219)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * eps_m)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 4.4e+219)
		tmp = (2.0 * exp(-x)) / 2.0;
	else
		tmp = (2.0 + (x * eps_m)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 4.4e+219], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 4.4000000000000003e219

   1. Initial program 66.9%

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

   3. Simplified 66.9%

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

   5. Taylor expanded in eps around inf 98.7%

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

   6. Taylor expanded in eps around 0 77.1%

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

      1. cancel-sign-sub-inv 77.1%

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

      2. metadata-eval 77.1%

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

      3. distribute-rgt1-in 77.1%

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

      4. metadata-eval 77.1%

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

      5. neg-mul-1 77.1%

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

   8. Simplified 77.1%

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

   if 4.4000000000000003e219 < eps

   1. Initial program 100.0%

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

   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 59.7%

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

      1. *-commutative 59.7%

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

      2. neg-mul-1 59.7%

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

      3. unsub-neg 59.7%

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

      4. *-commutative 59.7%

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

   7. Simplified 59.7%

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

   8. Taylor expanded in x around 0 42.0%

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

   9. Taylor expanded in eps around inf 42.0%

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

       1. +-commutative 42.0%

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

       2. *-commutative 42.0%

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

   11. Simplified 42.0%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 4.4 \cdot 10^{+219}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.9% accurate, 11.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -0.96:\\ \;\;\;\;\left(x \cdot eps_m\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 360:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+241}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps_m}\right) - \left(-1 + \frac{1}{eps_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -0.96)
   (* (* x eps_m) -0.5)
   (if (<= x 360.0)
     1.0
     (if (<= x 8e+241)
       (/ (- (+ 1.0 (/ 1.0 eps_m)) (+ -1.0 (/ 1.0 eps_m))) 2.0)
       (/ (* x eps_m) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -0.96) {
		tmp = (x * eps_m) * -0.5;
	} else if (x <= 360.0) {
		tmp = 1.0;
	} else if (x <= 8e+241) {
		tmp = ((1.0 + (1.0 / eps_m)) - (-1.0 + (1.0 / eps_m))) / 2.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-0.96d0)) then
        tmp = (x * eps_m) * (-0.5d0)
    else if (x <= 360.0d0) then
        tmp = 1.0d0
    else if (x <= 8d+241) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) - ((-1.0d0) + (1.0d0 / eps_m))) / 2.0d0
    else
        tmp = (x * eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -0.96) {
		tmp = (x * eps_m) * -0.5;
	} else if (x <= 360.0) {
		tmp = 1.0;
	} else if (x <= 8e+241) {
		tmp = ((1.0 + (1.0 / eps_m)) - (-1.0 + (1.0 / eps_m))) / 2.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -0.96:
		tmp = (x * eps_m) * -0.5
	elif x <= 360.0:
		tmp = 1.0
	elif x <= 8e+241:
		tmp = ((1.0 + (1.0 / eps_m)) - (-1.0 + (1.0 / eps_m))) / 2.0
	else:
		tmp = (x * eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -0.96)
		tmp = Float64(Float64(x * eps_m) * -0.5);
	elseif (x <= 360.0)
		tmp = 1.0;
	elseif (x <= 8e+241)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) - Float64(-1.0 + Float64(1.0 / eps_m))) / 2.0);
	else
		tmp = Float64(Float64(x * eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -0.96)
		tmp = (x * eps_m) * -0.5;
	elseif (x <= 360.0)
		tmp = 1.0;
	elseif (x <= 8e+241)
		tmp = ((1.0 + (1.0 / eps_m)) - (-1.0 + (1.0 / eps_m))) / 2.0;
	else
		tmp = (x * eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -0.96], N[(N[(x * eps$95$m), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[x, 360.0], 1.0, If[LessEqual[x, 8e+241], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] - N[(-1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -0.95999999999999996

   1. Initial program 100.0%

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

   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 54.8%

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

      1. *-commutative 54.8%

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

      2. neg-mul-1 54.8%

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

      3. unsub-neg 54.8%

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

      4. *-commutative 54.8%

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

   7. Simplified 54.8%

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

   8. Taylor expanded in x around inf 19.6%

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

   9. Taylor expanded in eps around inf 19.6%

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

       1. *-commutative 19.6%

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

   11. Simplified 19.6%

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

       1. frac-2neg 19.6%

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

       2. *-commutative 19.6%

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

       3. distribute-lft-neg-out 19.6%

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

       4. div-inv 19.6%

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

       5. add-sqr-sqrt 19.5%

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

       6. sqrt-unprod 58.1%

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

       7. sqr-neg 58.1%

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

       8. sqrt-unprod 22.5%

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

       9. add-sqr-sqrt 22.6%

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

       10. *-commutative 22.6%

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

       11. metadata-eval 22.6%

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

       12. metadata-eval 22.6%

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

   13. Applied egg-rr 22.6%

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

   if -0.95999999999999996 < x < 360

   1. Initial program 51.4%

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

   3. Simplified 51.4%

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

   5. Taylor expanded in x around 0 76.8%

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

   if 360 < x < 8.0000000000000004e241

   1. Initial program 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in x around 0 22.8%

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

   6. Taylor expanded in x around 0 57.7%

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

   if 8.0000000000000004e241 < x

   1. Initial program 100.0%

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

   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 23.1%

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

      1. *-commutative 23.1%

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

      2. neg-mul-1 23.1%

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

      3. unsub-neg 23.1%

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

      4. *-commutative 23.1%

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

   7. Simplified 23.1%

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

   8. Taylor expanded in x around inf 23.1%

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

   9. Taylor expanded in eps around inf 23.5%

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

       1. *-commutative 23.5%

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

   11. Simplified 23.5%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.96:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 360:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+241}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) - \left(-1 + \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.9% accurate, 13.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\left(x \cdot eps_m\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 360:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+233}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps_m}\right) + \frac{-1}{eps_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.0)
   (* (* x eps_m) -0.5)
   (if (<= x 360.0)
     1.0
     (if (<= x 2.4e+233)
       (/ (+ (+ 1.0 (/ 1.0 eps_m)) (/ -1.0 eps_m)) 2.0)
       (/ (* x eps_m) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = (x * eps_m) * -0.5;
	} else if (x <= 360.0) {
		tmp = 1.0;
	} else if (x <= 2.4e+233) {
		tmp = ((1.0 + (1.0 / eps_m)) + (-1.0 / eps_m)) / 2.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (x * eps_m) * (-0.5d0)
    else if (x <= 360.0d0) then
        tmp = 1.0d0
    else if (x <= 2.4d+233) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + ((-1.0d0) / eps_m)) / 2.0d0
    else
        tmp = (x * eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = (x * eps_m) * -0.5;
	} else if (x <= 360.0) {
		tmp = 1.0;
	} else if (x <= 2.4e+233) {
		tmp = ((1.0 + (1.0 / eps_m)) + (-1.0 / eps_m)) / 2.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.0:
		tmp = (x * eps_m) * -0.5
	elif x <= 360.0:
		tmp = 1.0
	elif x <= 2.4e+233:
		tmp = ((1.0 + (1.0 / eps_m)) + (-1.0 / eps_m)) / 2.0
	else:
		tmp = (x * eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(x * eps_m) * -0.5);
	elseif (x <= 360.0)
		tmp = 1.0;
	elseif (x <= 2.4e+233)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(-1.0 / eps_m)) / 2.0);
	else
		tmp = Float64(Float64(x * eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (x * eps_m) * -0.5;
	elseif (x <= 360.0)
		tmp = 1.0;
	elseif (x <= 2.4e+233)
		tmp = ((1.0 + (1.0 / eps_m)) + (-1.0 / eps_m)) / 2.0;
	else
		tmp = (x * eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.0], N[(N[(x * eps$95$m), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[x, 360.0], 1.0, If[LessEqual[x, 2.4e+233], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1

   1. Initial program 100.0%

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

   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 54.8%

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

      1. *-commutative 54.8%

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

      2. neg-mul-1 54.8%

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

      3. unsub-neg 54.8%

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

      4. *-commutative 54.8%

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

   7. Simplified 54.8%

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

   8. Taylor expanded in x around inf 19.6%

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

   9. Taylor expanded in eps around inf 19.6%

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

       1. *-commutative 19.6%

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

   11. Simplified 19.6%

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

       1. frac-2neg 19.6%

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

       2. *-commutative 19.6%

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

       3. distribute-lft-neg-out 19.6%

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

       4. div-inv 19.6%

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

       5. add-sqr-sqrt 19.5%

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

       6. sqrt-unprod 58.1%

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

       7. sqr-neg 58.1%

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

       8. sqrt-unprod 22.5%

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

       9. add-sqr-sqrt 22.6%

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

       10. *-commutative 22.6%

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

       11. metadata-eval 22.6%

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

       12. metadata-eval 22.6%

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

   13. Applied egg-rr 22.6%

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

   if -1 < x < 360

   1. Initial program 51.4%

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

   3. Simplified 51.4%

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

   5. Taylor expanded in x around 0 76.8%

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

   if 360 < x < 2.40000000000000003e233

   1. Initial program 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in x around 0 22.8%

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

   6. Taylor expanded in x around 0 57.7%

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

   7. Taylor expanded in eps around 0 57.7%

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

   if 2.40000000000000003e233 < x

   1. Initial program 100.0%

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

   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 23.1%

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

      1. *-commutative 23.1%

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

      2. neg-mul-1 23.1%

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

      3. unsub-neg 23.1%

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

      4. *-commutative 23.1%

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

   7. Simplified 23.1%

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

   8. Taylor expanded in x around inf 23.1%

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

   9. Taylor expanded in eps around inf 23.5%

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

       1. *-commutative 23.5%

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

   11. Simplified 23.5%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 360:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+233}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \frac{-1}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.1% accurate, 24.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\left(x \cdot eps_m\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 102:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.0)
   (* (* x eps_m) -0.5)
   (if (<= x 102.0) 1.0 (/ (* x eps_m) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = (x * eps_m) * -0.5;
	} else if (x <= 102.0) {
		tmp = 1.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (x * eps_m) * (-0.5d0)
    else if (x <= 102.0d0) then
        tmp = 1.0d0
    else
        tmp = (x * eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = (x * eps_m) * -0.5;
	} else if (x <= 102.0) {
		tmp = 1.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.0:
		tmp = (x * eps_m) * -0.5
	elif x <= 102.0:
		tmp = 1.0
	else:
		tmp = (x * eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(x * eps_m) * -0.5);
	elseif (x <= 102.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(x * eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (x * eps_m) * -0.5;
	elseif (x <= 102.0)
		tmp = 1.0;
	else
		tmp = (x * eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.0], N[(N[(x * eps$95$m), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[x, 102.0], 1.0, N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 100.0%

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

   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 54.8%

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

      1. *-commutative 54.8%

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

      2. neg-mul-1 54.8%

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

      3. unsub-neg 54.8%

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

      4. *-commutative 54.8%

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

   7. Simplified 54.8%

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

   8. Taylor expanded in x around inf 19.6%

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

   9. Taylor expanded in eps around inf 19.6%

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

       1. *-commutative 19.6%

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

   11. Simplified 19.6%

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

       1. frac-2neg 19.6%

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

       2. *-commutative 19.6%

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

       3. distribute-lft-neg-out 19.6%

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

       4. div-inv 19.6%

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

       5. add-sqr-sqrt 19.5%

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

       6. sqrt-unprod 58.1%

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

       7. sqr-neg 58.1%

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

       8. sqrt-unprod 22.5%

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

       9. add-sqr-sqrt 22.6%

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

       10. *-commutative 22.6%

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

       11. metadata-eval 22.6%

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

       12. metadata-eval 22.6%

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

   13. Applied egg-rr 22.6%

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

   if -1 < x < 102

   1. Initial program 51.4%

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

   3. Simplified 51.4%

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

   5. Taylor expanded in x around 0 76.8%

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

   if 102 < x

   1. Initial program 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in x around 0 27.2%

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

      1. *-commutative 27.2%

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

      2. neg-mul-1 27.2%

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

      3. unsub-neg 27.2%

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

      4. *-commutative 27.2%

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

   7. Simplified 27.2%

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

   8. Taylor expanded in x around inf 16.6%

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

   9. Taylor expanded in eps around inf 17.8%

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

       1. *-commutative 17.8%

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

   11. Simplified 17.8%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 102:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.5% accurate, 25.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\left(x \cdot eps_m\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot eps_m}{2}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 100.0%

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

   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 54.8%

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

      1. *-commutative 54.8%

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

      2. neg-mul-1 54.8%

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

      3. unsub-neg 54.8%

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

      4. *-commutative 54.8%

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

   7. Simplified 54.8%

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

   8. Taylor expanded in x around inf 19.6%

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

   9. Taylor expanded in eps around inf 19.6%

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

       1. *-commutative 19.6%

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

   11. Simplified 19.6%

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

       1. frac-2neg 19.6%

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

       2. *-commutative 19.6%

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

       3. distribute-lft-neg-out 19.6%

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

       4. div-inv 19.6%

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

       5. add-sqr-sqrt 19.5%

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

       6. sqrt-unprod 58.1%

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

       7. sqr-neg 58.1%

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

       8. sqrt-unprod 22.5%

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

       9. add-sqr-sqrt 22.6%

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

       10. *-commutative 22.6%

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

       11. metadata-eval 22.6%

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

       12. metadata-eval 22.6%

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

   13. Applied egg-rr 22.6%

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

   if -1 < x

   1. Initial program 65.4%

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

   3. Simplified 65.4%

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

   5. Taylor expanded in x around 0 38.1%

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

      1. *-commutative 38.1%

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

      2. neg-mul-1 38.1%

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

      3. unsub-neg 38.1%

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

      4. *-commutative 38.1%

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

   7. Simplified 38.1%

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

   8. Taylor expanded in x around 0 24.6%

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

   9. Taylor expanded in eps around inf 58.6%

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

       1. +-commutative 58.6%

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

       2. *-commutative 58.6%

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

   11. Simplified 58.6%

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

4. Final simplification 53.8%

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

Alternative 13: 50.7% accurate, 32.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(x * eps_m) * -0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (x * eps_m) * -0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.0], N[(N[(x * eps$95$m), $MachinePrecision] * -0.5), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 100.0%

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

   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 54.8%

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

      1. *-commutative 54.8%

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

      2. neg-mul-1 54.8%

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

      3. unsub-neg 54.8%

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

      4. *-commutative 54.8%

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

   7. Simplified 54.8%

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

   8. Taylor expanded in x around inf 19.6%

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

   9. Taylor expanded in eps around inf 19.6%

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

       1. *-commutative 19.6%

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

   11. Simplified 19.6%

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

       1. frac-2neg 19.6%

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

       2. *-commutative 19.6%

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

       3. distribute-lft-neg-out 19.6%

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

       4. div-inv 19.6%

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

       5. add-sqr-sqrt 19.5%

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

       6. sqrt-unprod 58.1%

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

       7. sqr-neg 58.1%

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

       8. sqrt-unprod 22.5%

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

       9. add-sqr-sqrt 22.6%

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

       10. *-commutative 22.6%

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

       11. metadata-eval 22.6%

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

       12. metadata-eval 22.6%

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

   13. Applied egg-rr 22.6%

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

   if -1 < x

   1. Initial program 65.4%

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

   3. Simplified 65.4%

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

   5. Taylor expanded in x around 0 54.9%

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

4. Final simplification 50.6%

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

Alternative 14: 43.9% accurate, 227.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ 1 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 1.0)

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return 1.0;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    code = 1.0d0
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return 1.0;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return 1.0

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return 1.0
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = 1.0;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := 1.0

Derivation

1. Initial program 70.0%

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

3. Simplified 70.0%

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

5. Taylor expanded in x around 0 48.0%

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

6. Final simplification 48.0%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024011 
(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))