NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.9% → 98.9%

Time: 19.2s

Alternatives: 11

Speedup: 2.0×

• 38.0% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps_m \leq 0.55:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot eps_m} + e^{x \cdot \left(-eps_m\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 0.55)
   (/ (* 2.0 (exp (- x))) 2.0)
   (/ (+ (exp (* x eps_m)) (exp (* x (- eps_m)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.55) {
		tmp = (2.0 * exp(-x)) / 2.0;
	} else {
		tmp = (exp((x * eps_m)) + exp((x * -eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 0.55d0) then
        tmp = (2.0d0 * exp(-x)) / 2.0d0
    else
        tmp = (exp((x * eps_m)) + exp((x * -eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.55) {
		tmp = (2.0 * Math.exp(-x)) / 2.0;
	} else {
		tmp = (Math.exp((x * eps_m)) + Math.exp((x * -eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 0.55:
		tmp = (2.0 * math.exp(-x)) / 2.0
	else:
		tmp = (math.exp((x * eps_m)) + math.exp((x * -eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 0.55)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * eps_m)) + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 0.55)
		tmp = (2.0 * exp(-x)) / 2.0;
	else
		tmp = (exp((x * eps_m)) + exp((x * -eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 0.55], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 0.55000000000000004

   1. Initial program 62.2%

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

      1. fma-neg 62.3%

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

      2. /-rgt-identity 62.3%

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

      3. fma-neg 62.2%

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

      4. /-rgt-identity 62.2%

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

      5. distribute-rgt-neg-in 62.2%

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

      6. sub-neg 62.2%

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

      7. metadata-eval 62.2%

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

      8. distribute-rgt-neg-in 62.2%

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

   3. Simplified 62.2%

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

   5. Taylor expanded in eps around inf 98.3%

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

   6. Taylor expanded in eps around 0 82.0%

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

      1. cancel-sign-sub-inv 82.0%

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

      2. metadata-eval 82.0%

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

      3. distribute-rgt1-in 82.0%

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

      4. metadata-eval 82.0%

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

      5. mul-1-neg 82.0%

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

   8. Simplified 82.0%

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

   if 0.55000000000000004 < 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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 99.9%

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

   6. Taylor expanded in eps around -inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. neg-mul-1 99.9%

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

      3. sub-neg 99.9%

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

      4. distribute-rgt-neg-in 99.9%

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

      5. mul-1-neg 99.9%

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

      6. associate-*r* 99.9%

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

      7. mul-1-neg 99.9%

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

      8. neg-mul-1 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in eps around inf 99.9%

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

       1. mul-1-neg 99.9%

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

       2. distribute-rgt-neg-in 99.9%

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

   11. Simplified 99.9%

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

   12. Taylor expanded in eps around inf 100.0%

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

       1. *-commutative 100.0%

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

   14. Simplified 100.0%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.55:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.9%

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

   1. fma-neg 73.9%

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

   2. /-rgt-identity 73.9%

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

   3. fma-neg 73.9%

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

   4. /-rgt-identity 73.9%

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

   5. distribute-rgt-neg-in 73.9%

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

   6. sub-neg 73.9%

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

   7. metadata-eval 73.9%

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

   8. distribute-rgt-neg-in 73.9%

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

3. Simplified 73.9%

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

5. Taylor expanded in eps around inf 98.8%

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

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. mul-1-neg 98.8%

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

   2. neg-mul-1 98.8%

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

   3. sub-neg 98.8%

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

   4. distribute-rgt-neg-in 98.8%

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

   5. mul-1-neg 98.8%

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

   6. associate-*r* 98.8%

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

   7. mul-1-neg 98.8%

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

   8. neg-mul-1 98.8%

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

8. Simplified 98.8%

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

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 3: 76.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 26000:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - eps_m\right)}}{2}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+185}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps_m}\right) + e^{x + x \cdot eps_m} \cdot \left(\frac{-1}{eps_m} - -1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 26000.0)
   (/ (+ 1.0 (exp (* x (- -1.0 eps_m)))) 2.0)
   (if (<= x 9e+185)
     (/
      (+
       (+ 1.0 (/ 1.0 eps_m))
       (* (exp (+ x (* x eps_m))) (- (/ -1.0 eps_m) -1.0)))
      2.0)
     (/ (/ 2.0 (exp x)) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 26000.0) {
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	} else if (x <= 9e+185) {
		tmp = ((1.0 + (1.0 / eps_m)) + (exp((x + (x * eps_m))) * ((-1.0 / eps_m) - -1.0))) / 2.0;
	} else {
		tmp = (2.0 / exp(x)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 26000.0d0) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps_m)))) / 2.0d0
    else if (x <= 9d+185) then
        tmp = ((1.0d0 + (1.0d0 / eps_m)) + (exp((x + (x * eps_m))) * (((-1.0d0) / eps_m) - (-1.0d0)))) / 2.0d0
    else
        tmp = (2.0d0 / exp(x)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 26000.0) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps_m)))) / 2.0;
	} else if (x <= 9e+185) {
		tmp = ((1.0 + (1.0 / eps_m)) + (Math.exp((x + (x * eps_m))) * ((-1.0 / eps_m) - -1.0))) / 2.0;
	} else {
		tmp = (2.0 / Math.exp(x)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 26000.0:
		tmp = (1.0 + math.exp((x * (-1.0 - eps_m)))) / 2.0
	elif x <= 9e+185:
		tmp = ((1.0 + (1.0 / eps_m)) + (math.exp((x + (x * eps_m))) * ((-1.0 / eps_m) - -1.0))) / 2.0
	else:
		tmp = (2.0 / math.exp(x)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 26000.0)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	elseif (x <= 9e+185)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(exp(Float64(x + Float64(x * eps_m))) * Float64(Float64(-1.0 / eps_m) - -1.0))) / 2.0);
	else
		tmp = Float64(Float64(2.0 / exp(x)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 26000.0)
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	elseif (x <= 9e+185)
		tmp = ((1.0 + (1.0 / eps_m)) + (exp((x + (x * eps_m))) * ((-1.0 / eps_m) - -1.0))) / 2.0;
	else
		tmp = (2.0 / exp(x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 26000.0], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 9e+185], N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(-1.0 / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 26000

   1. Initial program 64.2%

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

      1. fma-neg 64.3%

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

      2. /-rgt-identity 64.3%

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

      3. fma-neg 64.2%

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

      4. /-rgt-identity 64.2%

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

      5. distribute-rgt-neg-in 64.2%

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

      6. sub-neg 64.2%

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

      7. metadata-eval 64.2%

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

      8. distribute-rgt-neg-in 64.2%

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

   3. Simplified 64.2%

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

   5. Taylor expanded in x around 0 45.1%

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

   6. Taylor expanded in eps around inf 79.2%

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

      1. mul-1-neg 79.2%

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

      2. neg-mul-1 79.2%

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

      3. distribute-rgt-neg-in 79.2%

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

      4. +-commutative 79.2%

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

   8. Simplified 79.2%

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

   9. Taylor expanded in x around inf 79.2%

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

       1. associate-*r* 79.2%

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

       2. mul-1-neg 79.2%

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

       3. +-commutative 79.2%

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

   11. Simplified 79.2%

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

   if 26000 < x < 9.0000000000000004e185

   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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 30.1%

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

      1. add-sqr-sqrt 0.0%

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

      2. +-commutative 0.0%

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

      3. sqrt-unprod 38.4%

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

      4. sqr-neg 38.4%

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

      5. sqrt-unprod 38.4%

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

      6. add-sqr-sqrt 38.4%

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

      7. distribute-lft1-in 38.4%

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

   7. Applied egg-rr 38.4%

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

   if 9.0000000000000004e185 < 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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in 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. mul-1-neg 100.0%

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

      2. neg-mul-1 100.0%

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

      3. sub-neg 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. mul-1-neg 100.0%

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

      6. associate-*r* 100.0%

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

      7. mul-1-neg 100.0%

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

      8. neg-mul-1 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in eps around 0 61.5%

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

       1. neg-mul-1 61.5%

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

       2. exp-neg 61.5%

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

       3. associate-*r/ 61.5%

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

       4. metadata-eval 61.5%

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

   11. Simplified 61.5%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 26000:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+185}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + e^{x + x \cdot \varepsilon} \cdot \left(\frac{-1}{\varepsilon} - -1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{e^{x}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \left(-1 - eps_m\right)}\\ \mathbf{if}\;eps_m \leq 1.1 \cdot 10^{+164}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{elif}\;eps_m \leq 1.05 \cdot 10^{+271}:\\ \;\;\;\;\frac{1 + \left(t_0 + x \cdot eps_m\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + t_0}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (- -1.0 eps_m)))))
   (if (<= eps_m 1.1e+164)
     (/ (* 2.0 (exp (- x))) 2.0)
     (if (<= eps_m 1.05e+271)
       (/ (+ 1.0 (+ t_0 (* x eps_m))) 2.0)
       (/ (+ 1.0 t_0) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * (-1.0 - eps_m)));
	double tmp;
	if (eps_m <= 1.1e+164) {
		tmp = (2.0 * exp(-x)) / 2.0;
	} else if (eps_m <= 1.05e+271) {
		tmp = (1.0 + (t_0 + (x * eps_m))) / 2.0;
	} else {
		tmp = (1.0 + t_0) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((x * ((-1.0d0) - eps_m)))
    if (eps_m <= 1.1d+164) then
        tmp = (2.0d0 * exp(-x)) / 2.0d0
    else if (eps_m <= 1.05d+271) then
        tmp = (1.0d0 + (t_0 + (x * eps_m))) / 2.0d0
    else
        tmp = (1.0d0 + t_0) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp((x * (-1.0 - eps_m)));
	double tmp;
	if (eps_m <= 1.1e+164) {
		tmp = (2.0 * Math.exp(-x)) / 2.0;
	} else if (eps_m <= 1.05e+271) {
		tmp = (1.0 + (t_0 + (x * eps_m))) / 2.0;
	} else {
		tmp = (1.0 + t_0) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp((x * (-1.0 - eps_m)))
	tmp = 0
	if eps_m <= 1.1e+164:
		tmp = (2.0 * math.exp(-x)) / 2.0
	elif eps_m <= 1.05e+271:
		tmp = (1.0 + (t_0 + (x * eps_m))) / 2.0
	else:
		tmp = (1.0 + t_0) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * Float64(-1.0 - eps_m)))
	tmp = 0.0
	if (eps_m <= 1.1e+164)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0);
	elseif (eps_m <= 1.05e+271)
		tmp = Float64(Float64(1.0 + Float64(t_0 + Float64(x * eps_m))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + t_0) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp((x * (-1.0 - eps_m)));
	tmp = 0.0;
	if (eps_m <= 1.1e+164)
		tmp = (2.0 * exp(-x)) / 2.0;
	elseif (eps_m <= 1.05e+271)
		tmp = (1.0 + (t_0 + (x * eps_m))) / 2.0;
	else
		tmp = (1.0 + t_0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eps$95$m, 1.1e+164], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps$95$m, 1.05e+271], N[(N[(1.0 + N[(t$95$0 + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eps < 1.10000000000000003e164

   1. Initial program 68.6%

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

      1. fma-neg 68.6%

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

      2. /-rgt-identity 68.6%

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

      3. fma-neg 68.6%

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

      4. /-rgt-identity 68.6%

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

      5. distribute-rgt-neg-in 68.6%

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

      6. sub-neg 68.6%

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

      7. metadata-eval 68.6%

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

      8. distribute-rgt-neg-in 68.6%

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

   3. Simplified 68.6%

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

   5. Taylor expanded in eps around inf 98.6%

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

   6. Taylor expanded in eps around 0 79.5%

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

      1. cancel-sign-sub-inv 79.5%

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

      2. metadata-eval 79.5%

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

      3. distribute-rgt1-in 79.5%

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

      4. metadata-eval 79.5%

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

      5. mul-1-neg 79.5%

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

   8. Simplified 79.5%

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

   if 1.10000000000000003e164 < eps < 1.05e271

   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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 69.9%

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

   6. Taylor expanded in eps around inf 69.9%

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

      1. associate-*r* 69.9%

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

      2. remove-double-neg 69.9%

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

      3. neg-mul-1 69.9%

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

      4. sub-neg 69.9%

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

      5. associate-*r* 69.9%

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

      6. associate--l+ 69.9%

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

      7. *-commutative 69.9%

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

      8. mul-1-neg 69.9%

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

      9. mul-1-neg 69.9%

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

      10. distribute-rgt-neg-in 69.9%

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

      11. sub-neg 69.9%

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

      12. neg-mul-1 69.9%

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

      13. remove-double-neg 69.9%

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

      14. +-commutative 69.9%

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

   8. Simplified 69.9%

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

   9. Taylor expanded in x around inf 69.9%

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

       1. neg-mul-1 69.9%

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

       2. distribute-rgt-neg-in 69.9%

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

       3. *-commutative 69.9%

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

   11. Simplified 69.9%

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

   if 1.05e271 < 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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 73.6%

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

   6. Taylor expanded in eps around inf 73.6%

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

      1. mul-1-neg 73.6%

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

      2. neg-mul-1 73.6%

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

      3. distribute-rgt-neg-in 73.6%

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

      4. +-commutative 73.6%

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

   8. Simplified 73.6%

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

   9. Taylor expanded in x around inf 73.6%

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

       1. associate-*r* 73.6%

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

       2. mul-1-neg 73.6%

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

       3. +-commutative 73.6%

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

   11. Simplified 73.6%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.1 \cdot 10^{+164}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.05 \cdot 10^{+271}:\\ \;\;\;\;\frac{1 + \left(e^{x \cdot \left(-1 - \varepsilon\right)} + x \cdot \varepsilon\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps_m \leq 0.55:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - eps_m\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 0.55)
   (/ (* 2.0 (exp (- x))) 2.0)
   (/ (+ 1.0 (exp (* x (- -1.0 eps_m)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.55) {
		tmp = (2.0 * exp(-x)) / 2.0;
	} else {
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 0.55d0) then
        tmp = (2.0d0 * exp(-x)) / 2.0d0
    else
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps_m)))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.55) {
		tmp = (2.0 * Math.exp(-x)) / 2.0;
	} else {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps_m)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 0.55:
		tmp = (2.0 * math.exp(-x)) / 2.0
	else:
		tmp = (1.0 + math.exp((x * (-1.0 - eps_m)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 0.55)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 0.55)
		tmp = (2.0 * exp(-x)) / 2.0;
	else
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 0.55], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 0.55000000000000004

   1. Initial program 62.2%

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

      1. fma-neg 62.3%

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

      2. /-rgt-identity 62.3%

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

      3. fma-neg 62.2%

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

      4. /-rgt-identity 62.2%

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

      5. distribute-rgt-neg-in 62.2%

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

      6. sub-neg 62.2%

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

      7. metadata-eval 62.2%

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

      8. distribute-rgt-neg-in 62.2%

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

   3. Simplified 62.2%

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

   5. Taylor expanded in eps around inf 98.3%

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

   6. Taylor expanded in eps around 0 82.0%

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

      1. cancel-sign-sub-inv 82.0%

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

      2. metadata-eval 82.0%

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

      3. distribute-rgt1-in 82.0%

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

      4. metadata-eval 82.0%

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

      5. mul-1-neg 82.0%

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

   8. Simplified 82.0%

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

   if 0.55000000000000004 < 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

      1. fma-neg 100.0%

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

      2. /-rgt-identity 100.0%

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

      3. fma-neg 100.0%

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

      4. /-rgt-identity 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-neg-in 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 63.1%

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

   6. Taylor expanded in eps around inf 63.1%

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

      1. mul-1-neg 63.1%

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

      2. neg-mul-1 63.1%

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

      3. distribute-rgt-neg-in 63.1%

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

      4. +-commutative 63.1%

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

   8. Simplified 63.1%

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

   9. Taylor expanded in x around inf 63.1%

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

       1. associate-*r* 63.1%

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

       2. mul-1-neg 63.1%

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

       3. +-commutative 63.1%

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

   11. Simplified 63.1%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.55:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \frac{2 \cdot e^{-x}}{2} \end{array} \]

FPCore

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

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return (2.0 * exp(-x)) / 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 = (2.0d0 * exp(-x)) / 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.9%

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

   1. fma-neg 73.9%

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

   2. /-rgt-identity 73.9%

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

   3. fma-neg 73.9%

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

   4. /-rgt-identity 73.9%

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

   5. distribute-rgt-neg-in 73.9%

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

   6. sub-neg 73.9%

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

   7. metadata-eval 73.9%

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

   8. distribute-rgt-neg-in 73.9%

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

3. Simplified 73.9%

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

5. Taylor expanded in eps around inf 98.8%

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

6. Taylor expanded in eps around 0 71.9%

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

   1. cancel-sign-sub-inv 71.9%

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

   2. metadata-eval 71.9%

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

   3. distribute-rgt1-in 71.9%

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

   4. metadata-eval 71.9%

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

   5. mul-1-neg 71.9%

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

8. Simplified 71.9%

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

9. Final simplification 71.9%

   \[\leadsto \frac{2 \cdot e^{-x}}{2} \]
10. Add Preprocessing

Alternative 7: 70.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \frac{\frac{2}{e^{x}}}{2} \end{array} \]

FPCore

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

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return (2.0 / exp(x)) / 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 = (2.0d0 / exp(x)) / 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.9%

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

   1. fma-neg 73.9%

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

   2. /-rgt-identity 73.9%

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

   3. fma-neg 73.9%

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

   4. /-rgt-identity 73.9%

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

   5. distribute-rgt-neg-in 73.9%

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

   6. sub-neg 73.9%

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

   7. metadata-eval 73.9%

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

   8. distribute-rgt-neg-in 73.9%

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

3. Simplified 73.9%

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

5. Taylor expanded in eps around inf 98.8%

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

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. mul-1-neg 98.8%

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

   2. neg-mul-1 98.8%

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

   3. sub-neg 98.8%

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

   4. distribute-rgt-neg-in 98.8%

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

   5. mul-1-neg 98.8%

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

   6. associate-*r* 98.8%

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

   7. mul-1-neg 98.8%

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

   8. neg-mul-1 98.8%

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

8. Simplified 98.8%

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

9. Taylor expanded in eps around 0 71.9%

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

    1. neg-mul-1 71.9%

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

    2. exp-neg 71.9%

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

    3. associate-*r/ 71.9%

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

    4. metadata-eval 71.9%

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

11. Simplified 71.9%

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

12. Final simplification 71.9%

    \[\leadsto \frac{\frac{2}{e^{x}}}{2} \]
13. Add Preprocessing

Alternative 8: 64.4% accurate, 18.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 330:\\ \;\;\;\;\frac{2 - x \cdot eps_m}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{eps_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 330.0) (/ (- 2.0 (* x eps_m)) 2.0) (/ (/ 0.0 eps_m) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 330.0) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else {
		tmp = (0.0 / 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 <= 330.0d0) then
        tmp = (2.0d0 - (x * eps_m)) / 2.0d0
    else
        tmp = (0.0d0 / 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 <= 330.0) {
		tmp = (2.0 - (x * eps_m)) / 2.0;
	} else {
		tmp = (0.0 / eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 330.0:
		tmp = (2.0 - (x * eps_m)) / 2.0
	else:
		tmp = (0.0 / eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 330.0)
		tmp = Float64(Float64(2.0 - Float64(x * eps_m)) / 2.0);
	else
		tmp = Float64(Float64(0.0 / 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 <= 330.0)
		tmp = (2.0 - (x * eps_m)) / 2.0;
	else
		tmp = (0.0 / eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 330.0], N[(N[(2.0 - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(0.0 / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 330

   1. Initial program 64.1%

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

      1. fma-neg 64.1%

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

      2. /-rgt-identity 64.1%

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

      3. fma-neg 64.1%

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

      4. /-rgt-identity 64.1%

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

      5. distribute-rgt-neg-in 64.1%

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

      6. sub-neg 64.1%

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

      7. metadata-eval 64.1%

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

      8. distribute-rgt-neg-in 64.1%

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

   3. Simplified 64.1%

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

   5. Taylor expanded in x around 0 45.4%

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

   6. Taylor expanded in x around 0 33.5%

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

   7. Taylor expanded in eps around inf 68.0%

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

      1. mul-1-neg 68.0%

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

      2. *-commutative 68.0%

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

      3. unsub-neg 68.0%

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

   9. Simplified 68.0%

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

   if 330 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 45.2%

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

   6. Taylor expanded in x around 0 45.2%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 330:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.8% accurate, 22.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 54:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 54.0) 1.0 (/ (* x eps_m) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 54.0) {
		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 <= 54.0d0) 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 <= 54.0) {
		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 <= 54.0:
		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 <= 54.0)
		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 <= 54.0)
		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, 54.0], 1.0, N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 54

   1. Initial program 64.4%

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

      1. fma-neg 64.4%

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

      2. /-rgt-identity 64.4%

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

      3. fma-neg 64.4%

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

      4. /-rgt-identity 64.4%

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

      5. distribute-rgt-neg-in 64.4%

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

      6. sub-neg 64.4%

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

      7. metadata-eval 64.4%

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

      8. distribute-rgt-neg-in 64.4%

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

   3. Simplified 64.4%

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

   5. Taylor expanded in x around 0 62.6%

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

   if 54 < x

   1. Initial program 98.7%

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

      1. fma-neg 98.6%

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

      2. /-rgt-identity 98.6%

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

      3. fma-neg 98.7%

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

      4. /-rgt-identity 98.7%

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

      5. distribute-rgt-neg-in 98.7%

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

      6. sub-neg 98.7%

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

      7. metadata-eval 98.7%

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

      8. distribute-rgt-neg-in 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in x around 0 32.6%

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

   6. Taylor expanded in x around inf 18.6%

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

      1. mul-1-neg 18.6%

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

      2. associate-*r* 18.6%

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

      3. *-commutative 18.6%

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

      4. distribute-rgt-neg-in 18.6%

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

      5. distribute-rgt-neg-in 18.6%

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

      6. distribute-neg-in 18.6%

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

      7. metadata-eval 18.6%

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

      8. distribute-neg-frac 18.6%

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

      9. metadata-eval 18.6%

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

   8. Simplified 18.6%

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

   9. Taylor expanded in eps around inf 19.4%

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

       1. *-commutative 19.4%

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

   11. Simplified 19.4%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 54:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.6% accurate, 22.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 520:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{eps_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 520.0) 1.0 (/ (/ 0.0 eps_m) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 520.0) {
		tmp = 1.0;
	} else {
		tmp = (0.0 / 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 <= 520.0d0) then
        tmp = 1.0d0
    else
        tmp = (0.0d0 / 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 <= 520.0) {
		tmp = 1.0;
	} else {
		tmp = (0.0 / eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 520.0:
		tmp = 1.0
	else:
		tmp = (0.0 / eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 520.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(0.0 / 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 <= 520.0)
		tmp = 1.0;
	else
		tmp = (0.0 / eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 520.0], 1.0, N[(N[(0.0 / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 520

   1. Initial program 64.1%

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

      1. fma-neg 64.1%

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

      2. /-rgt-identity 64.1%

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

      3. fma-neg 64.1%

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

      4. /-rgt-identity 64.1%

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

      5. distribute-rgt-neg-in 64.1%

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

      6. sub-neg 64.1%

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

      7. metadata-eval 64.1%

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

      8. distribute-rgt-neg-in 64.1%

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

   3. Simplified 64.1%

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

   5. Taylor expanded in x around 0 62.3%

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

   if 520 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 45.2%

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

   6. Taylor expanded in x around 0 45.2%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 520:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 43.5% 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 73.9%

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

   1. fma-neg 73.9%

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

   2. /-rgt-identity 73.9%

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

   3. fma-neg 73.9%

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

   4. /-rgt-identity 73.9%

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

   5. distribute-rgt-neg-in 73.9%

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

   6. sub-neg 73.9%

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

   7. metadata-eval 73.9%

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

   8. distribute-rgt-neg-in 73.9%

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

3. Simplified 73.9%

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

5. Taylor expanded in x around 0 46.1%

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

6. Final simplification 46.1%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024017 
(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))