NMSE Section 6.1 mentioned, A

Percentage Accurate: 72.7% → 98.9%

Time: 16.0s

Alternatives: 17

Speedup: 2.0×

• 37.4% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.7% 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

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \left(eps_m + -1\right)}\\ \mathbf{if}\;x \leq 2.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_0 + e^{x - x \cdot eps_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (+ eps_m -1.0)))))
   (if (<= x 2.6e-6) (/ (+ t_0 (exp (- x (* x eps_m)))) 2.0) (* t_0 0.5))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= 2.6e-6) {
		tmp = (t_0 + exp((x - (x * eps_m)))) / 2.0;
	} else {
		tmp = t_0 * 0.5;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((x * (eps_m + (-1.0d0))))
    if (x <= 2.6d-6) then
        tmp = (t_0 + exp((x - (x * eps_m)))) / 2.0d0
    else
        tmp = t_0 * 0.5d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= 2.6e-6) {
		tmp = (t_0 + Math.exp((x - (x * eps_m)))) / 2.0;
	} else {
		tmp = t_0 * 0.5;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp((x * (eps_m + -1.0)))
	tmp = 0
	if x <= 2.6e-6:
		tmp = (t_0 + math.exp((x - (x * eps_m)))) / 2.0
	else:
		tmp = t_0 * 0.5
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * Float64(eps_m + -1.0)))
	tmp = 0.0
	if (x <= 2.6e-6)
		tmp = Float64(Float64(t_0 + exp(Float64(x - Float64(x * eps_m)))) / 2.0);
	else
		tmp = Float64(t_0 * 0.5);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp((x * (eps_m + -1.0)));
	tmp = 0.0;
	if (x <= 2.6e-6)
		tmp = (t_0 + exp((x - (x * eps_m)))) / 2.0;
	else
		tmp = t_0 * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 2.6e-6], N[(N[(t$95$0 + N[Exp[N[(x - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(t$95$0 * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.60000000000000009e-6

   1. Initial program 57.9%

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

   3. Simplified 57.9%

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

   4. Taylor expanded in eps around inf 99.2%

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

   6. Simplified 99.2%

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

      1. neg-mul-1 99.2%

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

      2. +-commutative 99.2%

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

      3. associate-*r* 99.2%

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

      4. distribute-rgt-in 99.2%

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

      5. *-un-lft-identity 99.2%

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

      6. distribute-lft-in 99.2%

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

      7. neg-mul-1 99.2%

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

      8. add-sqr-sqrt 57.2%

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

      9. sqrt-unprod 93.6%

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

      10. sqr-neg 93.6%

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

      11. sqrt-unprod 42.2%

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

      12. add-sqr-sqrt 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. mul-1-neg 99.7%

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

      2. *-commutative 99.7%

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

      3. sub-neg 99.7%

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

   10. Simplified 99.7%

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

   if 2.60000000000000009e-6 < 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. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in x around 0 35.6%

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

   6. Taylor expanded in x around inf 79.4%

      \[\leadsto \color{blue}{0.5 \cdot e^{x \cdot \left(\varepsilon - 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{x - x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot \left(\varepsilon + -1\right)} \cdot 0.5\\ \end{array} \]

Alternative 2: 98.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 68.9%

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

3. Simplified 63.6%

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

4. Taylor expanded in eps around inf 99.4%

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

5. Final simplification 99.4%

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

Alternative 3: 98.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \left(eps_m + -1\right)}\\ \mathbf{if}\;x \leq 2.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_0 + e^{eps_m \cdot \left(-x\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (+ eps_m -1.0)))))
   (if (<= x 2.6e-6) (/ (+ t_0 (exp (* eps_m (- x)))) 2.0) (* t_0 0.5))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= 2.6e-6) {
		tmp = (t_0 + exp((eps_m * -x))) / 2.0;
	} else {
		tmp = t_0 * 0.5;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((x * (eps_m + (-1.0d0))))
    if (x <= 2.6d-6) then
        tmp = (t_0 + exp((eps_m * -x))) / 2.0d0
    else
        tmp = t_0 * 0.5d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= 2.6e-6) {
		tmp = (t_0 + Math.exp((eps_m * -x))) / 2.0;
	} else {
		tmp = t_0 * 0.5;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp((x * (eps_m + -1.0)))
	tmp = 0
	if x <= 2.6e-6:
		tmp = (t_0 + math.exp((eps_m * -x))) / 2.0
	else:
		tmp = t_0 * 0.5
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * Float64(eps_m + -1.0)))
	tmp = 0.0
	if (x <= 2.6e-6)
		tmp = Float64(Float64(t_0 + exp(Float64(eps_m * Float64(-x)))) / 2.0);
	else
		tmp = Float64(t_0 * 0.5);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = exp((x * (eps_m + -1.0)));
	tmp = 0.0;
	if (x <= 2.6e-6)
		tmp = (t_0 + exp((eps_m * -x))) / 2.0;
	else
		tmp = t_0 * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 2.6e-6], N[(N[(t$95$0 + N[Exp[N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(t$95$0 * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.60000000000000009e-6

   1. Initial program 57.9%

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

   3. Simplified 57.9%

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

   4. Taylor expanded in eps around inf 99.2%

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

   6. Simplified 99.2%

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

      1. neg-mul-1 99.2%

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

      2. +-commutative 99.2%

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

      3. associate-*r* 99.2%

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

      4. distribute-rgt-in 99.2%

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

      5. *-un-lft-identity 99.2%

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

      6. distribute-lft-in 99.2%

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

      7. neg-mul-1 99.2%

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

      8. add-sqr-sqrt 57.2%

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

      9. sqrt-unprod 93.6%

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

      10. sqr-neg 93.6%

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

      11. sqrt-unprod 42.2%

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

      12. add-sqr-sqrt 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. mul-1-neg 99.7%

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

      2. *-commutative 99.7%

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

      3. sub-neg 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in eps around inf 99.2%

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

       1. associate-*r* 99.2%

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

       2. neg-mul-1 99.2%

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

       3. *-commutative 99.2%

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

   13. Simplified 99.2%

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

   14. Taylor expanded in x around inf 99.2%

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

       1. associate-*r* 99.2%

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

       2. mul-1-neg 99.2%

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

       3. associate-*r* 99.2%

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

       4. neg-mul-1 99.2%

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

   16. Simplified 99.2%

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

   if 2.60000000000000009e-6 < 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. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in x around 0 35.6%

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

   6. Taylor expanded in x around inf 79.4%

      \[\leadsto \color{blue}{0.5 \cdot e^{x \cdot \left(\varepsilon - 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot \left(\varepsilon + -1\right)} \cdot 0.5\\ \end{array} \]

Alternative 4: 98.7% 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 68.9%

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

3. Simplified 68.9%

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

4. Taylor expanded in eps around inf 99.4%

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

6. Simplified 99.4%

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

7. Final simplification 99.4%

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

Alternative 5: 98.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \left(eps_m + -1\right)}\\ \mathbf{if}\;x \leq -5 \cdot 10^{-273}:\\ \;\;\;\;\frac{1 + \frac{1}{e^{\mathsf{fma}\left(eps_m, x, x\right)}}}{2}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_0 + \frac{1}{1 + x \cdot \left(eps_m + 1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (+ eps_m -1.0)))))
   (if (<= x -5e-273)
     (/ (+ 1.0 (/ 1.0 (exp (fma eps_m x x)))) 2.0)
     (if (<= x 2.6e-6)
       (/ (+ t_0 (/ 1.0 (+ 1.0 (* x (+ eps_m 1.0))))) 2.0)
       (* t_0 0.5)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= -5e-273) {
		tmp = (1.0 + (1.0 / exp(fma(eps_m, x, x)))) / 2.0;
	} else if (x <= 2.6e-6) {
		tmp = (t_0 + (1.0 / (1.0 + (x * (eps_m + 1.0))))) / 2.0;
	} else {
		tmp = t_0 * 0.5;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * Float64(eps_m + -1.0)))
	tmp = 0.0
	if (x <= -5e-273)
		tmp = Float64(Float64(1.0 + Float64(1.0 / exp(fma(eps_m, x, x)))) / 2.0);
	elseif (x <= 2.6e-6)
		tmp = Float64(Float64(t_0 + Float64(1.0 / Float64(1.0 + Float64(x * Float64(eps_m + 1.0))))) / 2.0);
	else
		tmp = Float64(t_0 * 0.5);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -5e-273], N[(N[(1.0 + N[(1.0 / N[Exp[N[(eps$95$m * x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.6e-6], N[(N[(t$95$0 + N[(1.0 / N[(1.0 + N[(x * N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(t$95$0 * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.99999999999999965e-273

   1. Initial program 69.5%

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

   3. Simplified 64.2%

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

   4. Taylor expanded in eps around inf 99.0%

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

   5. Taylor expanded in x around 0 69.1%

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

   if -4.99999999999999965e-273 < x < 2.60000000000000009e-6

   1. Initial program 44.3%

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

   3. Simplified 34.9%

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

   4. Taylor expanded in eps around inf 99.3%

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

   5. Taylor expanded in x around 0 87.8%

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

   if 2.60000000000000009e-6 < 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. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in x around 0 35.6%

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

   6. Taylor expanded in x around inf 79.4%

      \[\leadsto \color{blue}{0.5 \cdot e^{x \cdot \left(\varepsilon - 1\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-273}:\\ \;\;\;\;\frac{1 + \frac{1}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}}{2}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \frac{1}{1 + x \cdot \left(\varepsilon + 1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot \left(\varepsilon + -1\right)} \cdot 0.5\\ \end{array} \]

Alternative 6: 98.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \left(eps_m + -1\right)}\\ \mathbf{if}\;x \leq -55000000:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{eps_m}}{2}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-267}:\\ \;\;\;\;\frac{\left(1 - x \cdot \left(1 - eps_m\right)\right) + e^{eps_m \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_0 + \frac{1}{1 + x \cdot \left(eps_m + 1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (+ eps_m -1.0)))))
   (if (<= x -55000000.0)
     (/ (/ (expm1 (- x)) eps_m) 2.0)
     (if (<= x -2e-267)
       (/ (+ (- 1.0 (* x (- 1.0 eps_m))) (exp (* eps_m (- x)))) 2.0)
       (if (<= x 2.6e-6)
         (/ (+ t_0 (/ 1.0 (+ 1.0 (* x (+ eps_m 1.0))))) 2.0)
         (* t_0 0.5))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= -55000000.0) {
		tmp = (expm1(-x) / eps_m) / 2.0;
	} else if (x <= -2e-267) {
		tmp = ((1.0 - (x * (1.0 - eps_m))) + exp((eps_m * -x))) / 2.0;
	} else if (x <= 2.6e-6) {
		tmp = (t_0 + (1.0 / (1.0 + (x * (eps_m + 1.0))))) / 2.0;
	} else {
		tmp = t_0 * 0.5;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= -55000000.0) {
		tmp = (Math.expm1(-x) / eps_m) / 2.0;
	} else if (x <= -2e-267) {
		tmp = ((1.0 - (x * (1.0 - eps_m))) + Math.exp((eps_m * -x))) / 2.0;
	} else if (x <= 2.6e-6) {
		tmp = (t_0 + (1.0 / (1.0 + (x * (eps_m + 1.0))))) / 2.0;
	} else {
		tmp = t_0 * 0.5;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp((x * (eps_m + -1.0)))
	tmp = 0
	if x <= -55000000.0:
		tmp = (math.expm1(-x) / eps_m) / 2.0
	elif x <= -2e-267:
		tmp = ((1.0 - (x * (1.0 - eps_m))) + math.exp((eps_m * -x))) / 2.0
	elif x <= 2.6e-6:
		tmp = (t_0 + (1.0 / (1.0 + (x * (eps_m + 1.0))))) / 2.0
	else:
		tmp = t_0 * 0.5
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * Float64(eps_m + -1.0)))
	tmp = 0.0
	if (x <= -55000000.0)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps_m) / 2.0);
	elseif (x <= -2e-267)
		tmp = Float64(Float64(Float64(1.0 - Float64(x * Float64(1.0 - eps_m))) + exp(Float64(eps_m * Float64(-x)))) / 2.0);
	elseif (x <= 2.6e-6)
		tmp = Float64(Float64(t_0 + Float64(1.0 / Float64(1.0 + Float64(x * Float64(eps_m + 1.0))))) / 2.0);
	else
		tmp = Float64(t_0 * 0.5);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -55000000.0], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -2e-267], N[(N[(N[(1.0 - N[(x * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Exp[N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.6e-6], N[(N[(t$95$0 + N[(1.0 / N[(1.0 + N[(x * N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(t$95$0 * 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.5e7

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 59.8%

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

   5. Taylor expanded in eps around 0 41.5%

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

      1. expm1-def 41.5%

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

      2. mul-1-neg 41.5%

         \[\leadsto \frac{\frac{\mathsf{expm1}\left(\color{blue}{-x}\right)}{\varepsilon}}{2} \]

   7. Simplified 41.5%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}}{2} \]

   if -5.5e7 < x < -2e-267

   1. Initial program 49.0%

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

   3. Simplified 49.0%

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

   4. Taylor expanded in eps around inf 98.4%

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

   6. Simplified 98.4%

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

      1. neg-mul-1 98.4%

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

      2. +-commutative 98.4%

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

      3. associate-*r* 98.4%

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

      4. distribute-rgt-in 98.4%

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

      5. *-un-lft-identity 98.4%

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

      6. distribute-lft-in 98.4%

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

      7. neg-mul-1 98.4%

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

      8. add-sqr-sqrt 98.4%

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

      9. sqrt-unprod 98.4%

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

      10. sqr-neg 98.4%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 99.4%

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

   8. Applied egg-rr 99.4%

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

      1. mul-1-neg 99.4%

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

      2. *-commutative 99.4%

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

      3. sub-neg 99.4%

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

   10. Simplified 99.4%

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

   11. Taylor expanded in eps around inf 98.5%

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

       1. associate-*r* 98.5%

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

       2. neg-mul-1 98.5%

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

       3. *-commutative 98.5%

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

   13. Simplified 98.5%

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

   14. Taylor expanded in x around 0 86.7%

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

       1. mul-1-neg 86.7%

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

       2. distribute-lft-neg-out 86.7%

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

       3. *-commutative 86.7%

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

   16. Simplified 86.7%

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

   if -2e-267 < x < 2.60000000000000009e-6

   1. Initial program 44.3%

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

   3. Simplified 34.9%

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

   4. Taylor expanded in eps around inf 99.3%

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

   5. Taylor expanded in x around 0 87.8%

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

   if 2.60000000000000009e-6 < 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. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in x around 0 35.6%

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

   6. Taylor expanded in x around inf 79.4%

      \[\leadsto \color{blue}{0.5 \cdot e^{x \cdot \left(\varepsilon - 1\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -55000000:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-267}:\\ \;\;\;\;\frac{\left(1 - x \cdot \left(1 - \varepsilon\right)\right) + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \frac{1}{1 + x \cdot \left(\varepsilon + 1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot \left(\varepsilon + -1\right)} \cdot 0.5\\ \end{array} \]

Alternative 7: 98.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \left(eps_m + -1\right)}\\ \mathbf{if}\;x \leq -4 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{eps_m}}{2}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-271}:\\ \;\;\;\;\frac{e^{x - x \cdot eps_m} + \left(1 - x \cdot \left(1 - eps_m\right)\right)}{2}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{t_0 + \frac{1}{1 + x \cdot \left(eps_m + 1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (+ eps_m -1.0)))))
   (if (<= x -4e+16)
     (/ (/ (expm1 (- x)) eps_m) 2.0)
     (if (<= x -5e-271)
       (/ (+ (exp (- x (* x eps_m))) (- 1.0 (* x (- 1.0 eps_m)))) 2.0)
       (if (<= x 2.6e-6)
         (/ (+ t_0 (/ 1.0 (+ 1.0 (* x (+ eps_m 1.0))))) 2.0)
         (* t_0 0.5))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= -4e+16) {
		tmp = (expm1(-x) / eps_m) / 2.0;
	} else if (x <= -5e-271) {
		tmp = (exp((x - (x * eps_m))) + (1.0 - (x * (1.0 - eps_m)))) / 2.0;
	} else if (x <= 2.6e-6) {
		tmp = (t_0 + (1.0 / (1.0 + (x * (eps_m + 1.0))))) / 2.0;
	} else {
		tmp = t_0 * 0.5;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= -4e+16) {
		tmp = (Math.expm1(-x) / eps_m) / 2.0;
	} else if (x <= -5e-271) {
		tmp = (Math.exp((x - (x * eps_m))) + (1.0 - (x * (1.0 - eps_m)))) / 2.0;
	} else if (x <= 2.6e-6) {
		tmp = (t_0 + (1.0 / (1.0 + (x * (eps_m + 1.0))))) / 2.0;
	} else {
		tmp = t_0 * 0.5;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp((x * (eps_m + -1.0)))
	tmp = 0
	if x <= -4e+16:
		tmp = (math.expm1(-x) / eps_m) / 2.0
	elif x <= -5e-271:
		tmp = (math.exp((x - (x * eps_m))) + (1.0 - (x * (1.0 - eps_m)))) / 2.0
	elif x <= 2.6e-6:
		tmp = (t_0 + (1.0 / (1.0 + (x * (eps_m + 1.0))))) / 2.0
	else:
		tmp = t_0 * 0.5
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * Float64(eps_m + -1.0)))
	tmp = 0.0
	if (x <= -4e+16)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps_m) / 2.0);
	elseif (x <= -5e-271)
		tmp = Float64(Float64(exp(Float64(x - Float64(x * eps_m))) + Float64(1.0 - Float64(x * Float64(1.0 - eps_m)))) / 2.0);
	elseif (x <= 2.6e-6)
		tmp = Float64(Float64(t_0 + Float64(1.0 / Float64(1.0 + Float64(x * Float64(eps_m + 1.0))))) / 2.0);
	else
		tmp = Float64(t_0 * 0.5);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -4e+16], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -5e-271], N[(N[(N[Exp[N[(x - N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 - N[(x * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.6e-6], N[(N[(t$95$0 + N[(1.0 / N[(1.0 + N[(x * N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(t$95$0 * 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4e16

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 59.8%

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

   5. Taylor expanded in eps around 0 41.5%

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

      1. expm1-def 41.5%

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

      2. mul-1-neg 41.5%

         \[\leadsto \frac{\frac{\mathsf{expm1}\left(\color{blue}{-x}\right)}{\varepsilon}}{2} \]

   7. Simplified 41.5%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}}{2} \]

   if -4e16 < x < -5.0000000000000002e-271

   1. Initial program 49.0%

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

   3. Simplified 49.0%

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

   4. Taylor expanded in eps around inf 98.4%

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

   6. Simplified 98.4%

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

      1. neg-mul-1 98.4%

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

      2. +-commutative 98.4%

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

      3. associate-*r* 98.4%

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

      4. distribute-rgt-in 98.4%

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

      5. *-un-lft-identity 98.4%

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

      6. distribute-lft-in 98.4%

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

      7. neg-mul-1 98.4%

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

      8. add-sqr-sqrt 98.4%

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

      9. sqrt-unprod 98.4%

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

      10. sqr-neg 98.4%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 99.4%

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

   8. Applied egg-rr 99.4%

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

      1. mul-1-neg 99.4%

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

      2. *-commutative 99.4%

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

      3. sub-neg 99.4%

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

   10. Simplified 99.4%

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

   11. Taylor expanded in x around 0 87.6%

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

       1. mul-1-neg 86.7%

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

       2. distribute-lft-neg-out 86.7%

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

       3. *-commutative 86.7%

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

   13. Simplified 87.6%

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

   if -5.0000000000000002e-271 < x < 2.60000000000000009e-6

   1. Initial program 44.3%

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

   3. Simplified 34.9%

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

   4. Taylor expanded in eps around inf 99.3%

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

   5. Taylor expanded in x around 0 87.8%

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

   if 2.60000000000000009e-6 < 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. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in x around 0 35.6%

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

   6. Taylor expanded in x around inf 79.4%

      \[\leadsto \color{blue}{0.5 \cdot e^{x \cdot \left(\varepsilon - 1\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-271}:\\ \;\;\;\;\frac{e^{x - x \cdot \varepsilon} + \left(1 - x \cdot \left(1 - \varepsilon\right)\right)}{2}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \frac{1}{1 + x \cdot \left(\varepsilon + 1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot \left(\varepsilon + -1\right)} \cdot 0.5\\ \end{array} \]

Alternative 8: 97.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \left(eps_m + -1\right)}\\ \mathbf{if}\;x \leq -4 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{eps_m}}{2}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-271}:\\ \;\;\;\;\frac{\left(1 - x \cdot \left(1 - eps_m\right)\right) + e^{eps_m \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{1 + t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (+ eps_m -1.0)))))
   (if (<= x -4e+16)
     (/ (/ (expm1 (- x)) eps_m) 2.0)
     (if (<= x -5e-271)
       (/ (+ (- 1.0 (* x (- 1.0 eps_m))) (exp (* eps_m (- x)))) 2.0)
       (if (<= x 2.6e-6) (/ (+ 1.0 t_0) 2.0) (* t_0 0.5))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= -4e+16) {
		tmp = (expm1(-x) / eps_m) / 2.0;
	} else if (x <= -5e-271) {
		tmp = ((1.0 - (x * (1.0 - eps_m))) + exp((eps_m * -x))) / 2.0;
	} else if (x <= 2.6e-6) {
		tmp = (1.0 + t_0) / 2.0;
	} else {
		tmp = t_0 * 0.5;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= -4e+16) {
		tmp = (Math.expm1(-x) / eps_m) / 2.0;
	} else if (x <= -5e-271) {
		tmp = ((1.0 - (x * (1.0 - eps_m))) + Math.exp((eps_m * -x))) / 2.0;
	} else if (x <= 2.6e-6) {
		tmp = (1.0 + t_0) / 2.0;
	} else {
		tmp = t_0 * 0.5;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp((x * (eps_m + -1.0)))
	tmp = 0
	if x <= -4e+16:
		tmp = (math.expm1(-x) / eps_m) / 2.0
	elif x <= -5e-271:
		tmp = ((1.0 - (x * (1.0 - eps_m))) + math.exp((eps_m * -x))) / 2.0
	elif x <= 2.6e-6:
		tmp = (1.0 + t_0) / 2.0
	else:
		tmp = t_0 * 0.5
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * Float64(eps_m + -1.0)))
	tmp = 0.0
	if (x <= -4e+16)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps_m) / 2.0);
	elseif (x <= -5e-271)
		tmp = Float64(Float64(Float64(1.0 - Float64(x * Float64(1.0 - eps_m))) + exp(Float64(eps_m * Float64(-x)))) / 2.0);
	elseif (x <= 2.6e-6)
		tmp = Float64(Float64(1.0 + t_0) / 2.0);
	else
		tmp = Float64(t_0 * 0.5);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -4e+16], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -5e-271], N[(N[(N[(1.0 - N[(x * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Exp[N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.6e-6], N[(N[(1.0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(t$95$0 * 0.5), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4e16

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 59.8%

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

   5. Taylor expanded in eps around 0 41.5%

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

      1. expm1-def 41.5%

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

      2. mul-1-neg 41.5%

         \[\leadsto \frac{\frac{\mathsf{expm1}\left(\color{blue}{-x}\right)}{\varepsilon}}{2} \]

   7. Simplified 41.5%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}}{2} \]

   if -4e16 < x < -5.0000000000000002e-271

   1. Initial program 49.0%

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

   3. Simplified 49.0%

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

   4. Taylor expanded in eps around inf 98.4%

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

   6. Simplified 98.4%

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

      1. neg-mul-1 98.4%

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

      2. +-commutative 98.4%

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

      3. associate-*r* 98.4%

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

      4. distribute-rgt-in 98.4%

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

      5. *-un-lft-identity 98.4%

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

      6. distribute-lft-in 98.4%

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

      7. neg-mul-1 98.4%

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

      8. add-sqr-sqrt 98.4%

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

      9. sqrt-unprod 98.4%

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

      10. sqr-neg 98.4%

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

      11. sqrt-unprod 0.0%

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

      12. add-sqr-sqrt 99.4%

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

   8. Applied egg-rr 99.4%

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

      1. mul-1-neg 99.4%

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

      2. *-commutative 99.4%

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

      3. sub-neg 99.4%

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

   10. Simplified 99.4%

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

   11. Taylor expanded in eps around inf 98.5%

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

       1. associate-*r* 98.5%

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

       2. neg-mul-1 98.5%

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

       3. *-commutative 98.5%

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

   13. Simplified 98.5%

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

   14. Taylor expanded in x around 0 86.7%

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

       1. mul-1-neg 86.7%

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

       2. distribute-lft-neg-out 86.7%

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

       3. *-commutative 86.7%

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

   16. Simplified 86.7%

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

   if -5.0000000000000002e-271 < x < 2.60000000000000009e-6

   1. Initial program 44.3%

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

   3. Simplified 44.3%

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

   4. Taylor expanded in x around 0 32.8%

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

   5. Taylor expanded in eps around inf 87.7%

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

      1. sub-neg 87.7%

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

      2. neg-mul-1 87.7%

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

      3. associate-*r* 87.7%

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

      4. mul-1-neg 87.7%

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

      5. neg-mul-1 87.7%

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

      6. sub-neg 87.7%

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

   7. Simplified 87.7%

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

   if 2.60000000000000009e-6 < 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. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in x around 0 35.6%

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

   6. Taylor expanded in x around inf 79.4%

      \[\leadsto \color{blue}{0.5 \cdot e^{x \cdot \left(\varepsilon - 1\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-271}:\\ \;\;\;\;\frac{\left(1 - x \cdot \left(1 - \varepsilon\right)\right) + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot \left(\varepsilon + -1\right)} \cdot 0.5\\ \end{array} \]

Alternative 9: 84.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -320:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{eps_m}}{2}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-97}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-38} \lor \neg \left(x \leq 6.6 \cdot 10^{-15}\right):\\ \;\;\;\;e^{x \cdot \left(eps_m + -1\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -320.0)
   (/ (/ (expm1 (- x)) eps_m) 2.0)
   (if (<= x 1.35e-97)
     1.0
     (if (or (<= x 1.7e-38) (not (<= x 6.6e-15)))
       (* (exp (* x (+ eps_m -1.0))) 0.5)
       1.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -320.0) {
		tmp = (expm1(-x) / eps_m) / 2.0;
	} else if (x <= 1.35e-97) {
		tmp = 1.0;
	} else if ((x <= 1.7e-38) || !(x <= 6.6e-15)) {
		tmp = exp((x * (eps_m + -1.0))) * 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -320.0) {
		tmp = (Math.expm1(-x) / eps_m) / 2.0;
	} else if (x <= 1.35e-97) {
		tmp = 1.0;
	} else if ((x <= 1.7e-38) || !(x <= 6.6e-15)) {
		tmp = Math.exp((x * (eps_m + -1.0))) * 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -320.0:
		tmp = (math.expm1(-x) / eps_m) / 2.0
	elif x <= 1.35e-97:
		tmp = 1.0
	elif (x <= 1.7e-38) or not (x <= 6.6e-15):
		tmp = math.exp((x * (eps_m + -1.0))) * 0.5
	else:
		tmp = 1.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -320.0)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps_m) / 2.0);
	elseif (x <= 1.35e-97)
		tmp = 1.0;
	elseif ((x <= 1.7e-38) || !(x <= 6.6e-15))
		tmp = Float64(exp(Float64(x * Float64(eps_m + -1.0))) * 0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -320.0], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.35e-97], 1.0, If[Or[LessEqual[x, 1.7e-38], N[Not[LessEqual[x, 6.6e-15]], $MachinePrecision]], N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if x < -320

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 59.8%

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

   5. Taylor expanded in eps around 0 41.5%

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

      1. expm1-def 41.5%

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

      2. mul-1-neg 41.5%

         \[\leadsto \frac{\frac{\mathsf{expm1}\left(\color{blue}{-x}\right)}{\varepsilon}}{2} \]

   7. Simplified 41.5%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}}{2} \]

   if -320 < x < 1.34999999999999993e-97 or 1.7000000000000001e-38 < x < 6.6e-15

   1. Initial program 43.0%

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

   3. Simplified 43.0%

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

   4. Taylor expanded in x around 0 85.2%

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

   if 1.34999999999999993e-97 < x < 1.7000000000000001e-38 or 6.6e-15 < x

   1. Initial program 91.3%

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

   3. Simplified 84.8%

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

   4. Taylor expanded in eps around inf 99.4%

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

   5. Taylor expanded in x around 0 43.2%

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

   6. Taylor expanded in x around inf 69.5%

      \[\leadsto \color{blue}{0.5 \cdot e^{x \cdot \left(\varepsilon - 1\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -320:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-97}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-38} \lor \neg \left(x \leq 6.6 \cdot 10^{-15}\right):\\ \;\;\;\;e^{x \cdot \left(\varepsilon + -1\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10: 90.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \left(eps_m + -1\right)}\\ \mathbf{if}\;x \leq -380:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{eps_m}}{2}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{1 + t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot 0.5\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (+ eps_m -1.0)))))
   (if (<= x -380.0)
     (/ (/ (expm1 (- x)) eps_m) 2.0)
     (if (<= x 2.6e-6) (/ (+ 1.0 t_0) 2.0) (* t_0 0.5)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= -380.0) {
		tmp = (expm1(-x) / eps_m) / 2.0;
	} else if (x <= 2.6e-6) {
		tmp = (1.0 + t_0) / 2.0;
	} else {
		tmp = t_0 * 0.5;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp((x * (eps_m + -1.0)));
	double tmp;
	if (x <= -380.0) {
		tmp = (Math.expm1(-x) / eps_m) / 2.0;
	} else if (x <= 2.6e-6) {
		tmp = (1.0 + t_0) / 2.0;
	} else {
		tmp = t_0 * 0.5;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp((x * (eps_m + -1.0)))
	tmp = 0
	if x <= -380.0:
		tmp = (math.expm1(-x) / eps_m) / 2.0
	elif x <= 2.6e-6:
		tmp = (1.0 + t_0) / 2.0
	else:
		tmp = t_0 * 0.5
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * Float64(eps_m + -1.0)))
	tmp = 0.0
	if (x <= -380.0)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps_m) / 2.0);
	elseif (x <= 2.6e-6)
		tmp = Float64(Float64(1.0 + t_0) / 2.0);
	else
		tmp = Float64(t_0 * 0.5);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -380.0], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.6e-6], N[(N[(1.0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(t$95$0 * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -380

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 59.8%

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

   5. Taylor expanded in eps around 0 41.5%

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

      1. expm1-def 41.5%

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

      2. mul-1-neg 41.5%

         \[\leadsto \frac{\frac{\mathsf{expm1}\left(\color{blue}{-x}\right)}{\varepsilon}}{2} \]

   7. Simplified 41.5%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}}{2} \]

   if -380 < x < 2.60000000000000009e-6

   1. Initial program 46.2%

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

   3. Simplified 46.2%

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

   4. Taylor expanded in x around 0 34.9%

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

   5. Taylor expanded in eps around inf 87.6%

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

      1. sub-neg 87.6%

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

      2. neg-mul-1 87.6%

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

      3. associate-*r* 87.6%

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

      4. mul-1-neg 87.6%

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

      5. neg-mul-1 87.6%

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

      6. sub-neg 87.6%

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

   7. Simplified 87.6%

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

   if 2.60000000000000009e-6 < 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. Taylor expanded in eps around inf 100.0%

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

   5. Taylor expanded in x around 0 35.6%

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

   6. Taylor expanded in x around inf 79.4%

      \[\leadsto \color{blue}{0.5 \cdot e^{x \cdot \left(\varepsilon - 1\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -380:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;e^{x \cdot \left(\varepsilon + -1\right)} \cdot 0.5\\ \end{array} \]

Alternative 11: 69.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -550:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{eps_m}}{2}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-6}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - x \cdot \left(1 - eps_m\right)\right) \cdot \left(1 + \frac{1}{eps_m}\right) + \frac{x + -1}{eps_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -550.0)
   (/ (/ (expm1 (- x)) eps_m) 2.0)
   (if (<= x 2.6e-6)
     1.0
     (/
      (+
       (* (- 1.0 (* x (- 1.0 eps_m))) (+ 1.0 (/ 1.0 eps_m)))
       (/ (+ x -1.0) eps_m))
      2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -550.0) {
		tmp = (expm1(-x) / eps_m) / 2.0;
	} else if (x <= 2.6e-6) {
		tmp = 1.0;
	} else {
		tmp = (((1.0 - (x * (1.0 - eps_m))) * (1.0 + (1.0 / eps_m))) + ((x + -1.0) / eps_m)) / 2.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -550.0) {
		tmp = (Math.expm1(-x) / eps_m) / 2.0;
	} else if (x <= 2.6e-6) {
		tmp = 1.0;
	} else {
		tmp = (((1.0 - (x * (1.0 - eps_m))) * (1.0 + (1.0 / eps_m))) + ((x + -1.0) / eps_m)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -550.0:
		tmp = (math.expm1(-x) / eps_m) / 2.0
	elif x <= 2.6e-6:
		tmp = 1.0
	else:
		tmp = (((1.0 - (x * (1.0 - eps_m))) * (1.0 + (1.0 / eps_m))) + ((x + -1.0) / eps_m)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -550.0)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps_m) / 2.0);
	elseif (x <= 2.6e-6)
		tmp = 1.0;
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - Float64(x * Float64(1.0 - eps_m))) * Float64(1.0 + Float64(1.0 / eps_m))) + Float64(Float64(x + -1.0) / eps_m)) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -550.0], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.6e-6], 1.0, N[(N[(N[(N[(1.0 - N[(x * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x + -1.0), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -550

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 59.8%

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

   5. Taylor expanded in eps around 0 41.5%

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

      1. expm1-def 41.5%

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

      2. mul-1-neg 41.5%

         \[\leadsto \frac{\frac{\mathsf{expm1}\left(\color{blue}{-x}\right)}{\varepsilon}}{2} \]

   7. Simplified 41.5%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}}{2} \]

   if -550 < x < 2.60000000000000009e-6

   1. Initial program 46.2%

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

   3. Simplified 46.2%

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

   4. Taylor expanded in x around 0 78.1%

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

   if 2.60000000000000009e-6 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 19.4%

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

   5. Taylor expanded in x around 0 25.7%

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

      1. mul-1-neg 25.7%

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

      2. unsub-neg 25.7%

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

   7. Simplified 25.7%

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

   8. Taylor expanded in eps around 0 39.5%

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

      1. mul-1-neg 39.5%

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

      2. unsub-neg 39.5%

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

   10. Simplified 39.5%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -550:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-6}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) + \frac{x + -1}{\varepsilon}}{2}\\ \end{array} \]

Alternative 12: 62.6% accurate, 9.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0016:\\ \;\;\;\;\left(x \cdot eps_m\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-6}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - x \cdot \left(1 - eps_m\right)\right) \cdot \left(1 + \frac{1}{eps_m}\right) + \frac{x + -1}{eps_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -0.0016)
   (* (* x eps_m) -0.5)
   (if (<= x 2.6e-6)
     1.0
     (/
      (+
       (* (- 1.0 (* x (- 1.0 eps_m))) (+ 1.0 (/ 1.0 eps_m)))
       (/ (+ x -1.0) eps_m))
      2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -0.0016) {
		tmp = (x * eps_m) * -0.5;
	} else if (x <= 2.6e-6) {
		tmp = 1.0;
	} else {
		tmp = (((1.0 - (x * (1.0 - eps_m))) * (1.0 + (1.0 / eps_m))) + ((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 (x <= (-0.0016d0)) then
        tmp = (x * eps_m) * (-0.5d0)
    else if (x <= 2.6d-6) then
        tmp = 1.0d0
    else
        tmp = (((1.0d0 - (x * (1.0d0 - eps_m))) * (1.0d0 + (1.0d0 / eps_m))) + ((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 (x <= -0.0016) {
		tmp = (x * eps_m) * -0.5;
	} else if (x <= 2.6e-6) {
		tmp = 1.0;
	} else {
		tmp = (((1.0 - (x * (1.0 - eps_m))) * (1.0 + (1.0 / eps_m))) + ((x + -1.0) / eps_m)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -0.0016:
		tmp = (x * eps_m) * -0.5
	elif x <= 2.6e-6:
		tmp = 1.0
	else:
		tmp = (((1.0 - (x * (1.0 - eps_m))) * (1.0 + (1.0 / eps_m))) + ((x + -1.0) / eps_m)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -0.0016)
		tmp = Float64(Float64(x * eps_m) * -0.5);
	elseif (x <= 2.6e-6)
		tmp = 1.0;
	else
		tmp = Float64(Float64(Float64(Float64(1.0 - Float64(x * Float64(1.0 - eps_m))) * Float64(1.0 + Float64(1.0 / eps_m))) + Float64(Float64(x + -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 (x <= -0.0016)
		tmp = (x * eps_m) * -0.5;
	elseif (x <= 2.6e-6)
		tmp = 1.0;
	else
		tmp = (((1.0 - (x * (1.0 - eps_m))) * (1.0 + (1.0 / eps_m))) + ((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[x, -0.0016], N[(N[(x * eps$95$m), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[x, 2.6e-6], 1.0, N[(N[(N[(N[(1.0 - N[(x * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x + -1.0), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00160000000000000008

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 52.3%

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

   5. Taylor expanded in eps around inf 22.1%

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

      1. mul-1-neg 22.1%

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

      2. *-commutative 22.1%

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

      3. distribute-rgt-neg-in 22.1%

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

   7. Simplified 22.1%

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

   8. Taylor expanded in x around 0 22.1%

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

      1. *-commutative 22.1%

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

   10. Simplified 22.1%

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

   if -0.00160000000000000008 < x < 2.60000000000000009e-6

   1. Initial program 45.5%

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

   3. Simplified 45.5%

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

   4. Taylor expanded in x around 0 79.2%

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

   if 2.60000000000000009e-6 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 19.4%

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

   5. Taylor expanded in x around 0 25.7%

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

      1. mul-1-neg 25.7%

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

      2. unsub-neg 25.7%

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

   7. Simplified 25.7%

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

   8. Taylor expanded in eps around 0 39.5%

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

      1. mul-1-neg 39.5%

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

      2. unsub-neg 39.5%

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

   10. Simplified 39.5%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0016:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-6}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - x \cdot \left(1 - \varepsilon\right)\right) \cdot \left(1 + \frac{1}{\varepsilon}\right) + \frac{x + -1}{\varepsilon}}{2}\\ \end{array} \]

Alternative 13: 64.5% accurate, 13.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0016:\\ \;\;\;\;\left(x \cdot eps_m\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 480:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+253}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps_m + \left(1 + \frac{1}{eps_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -0.0016)
   (* (* x eps_m) -0.5)
   (if (<= x 480.0)
     1.0
     (if (<= x 2.2e+253) 0.0 (/ (+ (* x eps_m) (+ 1.0 (/ 1.0 eps_m))) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -0.0016) {
		tmp = (x * eps_m) * -0.5;
	} else if (x <= 480.0) {
		tmp = 1.0;
	} else if (x <= 2.2e+253) {
		tmp = 0.0;
	} else {
		tmp = ((x * eps_m) + (1.0 + (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 (x <= (-0.0016d0)) then
        tmp = (x * eps_m) * (-0.5d0)
    else if (x <= 480.0d0) then
        tmp = 1.0d0
    else if (x <= 2.2d+253) then
        tmp = 0.0d0
    else
        tmp = ((x * eps_m) + (1.0d0 + (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 (x <= -0.0016) {
		tmp = (x * eps_m) * -0.5;
	} else if (x <= 480.0) {
		tmp = 1.0;
	} else if (x <= 2.2e+253) {
		tmp = 0.0;
	} else {
		tmp = ((x * eps_m) + (1.0 + (1.0 / eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -0.0016:
		tmp = (x * eps_m) * -0.5
	elif x <= 480.0:
		tmp = 1.0
	elif x <= 2.2e+253:
		tmp = 0.0
	else:
		tmp = ((x * eps_m) + (1.0 + (1.0 / eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -0.0016)
		tmp = Float64(Float64(x * eps_m) * -0.5);
	elseif (x <= 480.0)
		tmp = 1.0;
	elseif (x <= 2.2e+253)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(x * eps_m) + Float64(1.0 + 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 (x <= -0.0016)
		tmp = (x * eps_m) * -0.5;
	elseif (x <= 480.0)
		tmp = 1.0;
	elseif (x <= 2.2e+253)
		tmp = 0.0;
	else
		tmp = ((x * eps_m) + (1.0 + (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[x, -0.0016], N[(N[(x * eps$95$m), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[x, 480.0], 1.0, If[LessEqual[x, 2.2e+253], 0.0, N[(N[(N[(x * eps$95$m), $MachinePrecision] + N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -0.00160000000000000008

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 52.3%

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

   5. Taylor expanded in eps around inf 22.1%

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

      1. mul-1-neg 22.1%

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

      2. *-commutative 22.1%

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

      3. distribute-rgt-neg-in 22.1%

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

   7. Simplified 22.1%

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

   8. Taylor expanded in x around 0 22.1%

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

      1. *-commutative 22.1%

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

   10. Simplified 22.1%

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

   if -0.00160000000000000008 < x < 480

   1. Initial program 45.9%

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

   3. Simplified 45.9%

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

   4. Taylor expanded in x around 0 78.7%

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

   if 480 < x < 2.20000000000000006e253

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 17.7%

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

   5. Taylor expanded in x around 0 34.4%

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

      1. mul-1-neg 34.4%

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

      2. unsub-neg 34.4%

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

   7. Simplified 34.4%

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

   8. Taylor expanded in eps around 0 52.8%

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

      1. associate-*r/ 52.8%

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

      2. distribute-rgt1-in 52.8%

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

      3. metadata-eval 52.8%

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

      4. mul0-lft 52.8%

         \[\leadsto \frac{\frac{-1 \cdot \color{blue}{0}}{\varepsilon}}{2} \]

      5. metadata-eval 52.8%

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

      6. +-inverses 52.8%

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

      7. div-sub 34.8%

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

      8. +-inverses 52.8%

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

   10. Simplified 52.8%

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

   if 2.20000000000000006e253 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 19.7%

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

   5. Taylor expanded in x around 0 19.7%

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

   6. Taylor expanded in eps around inf 19.9%

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

      1. *-commutative 19.9%

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

   8. Simplified 19.9%

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

      1. add-log-exp 25.5%

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

      2. sub-neg 25.5%

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

      3. distribute-rgt-neg-out 25.5%

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

      4. exp-sum 25.0%

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

      5. exp-prod 25.0%

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

      6. add-sqr-sqrt 25.0%

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

      7. sqrt-unprod 69.2%

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

      8. sqr-neg 69.2%

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

      9. sqrt-unprod 44.1%

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

      10. add-sqr-sqrt 44.2%

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

      11. exp-prod 44.2%

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

      12. sum-log 44.2%

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

      13. add-log-exp 44.2%

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

      14. add-log-exp 44.4%

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

   10. Applied egg-rr 44.4%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0016:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 480:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+253}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon + \left(1 + \frac{1}{\varepsilon}\right)}{2}\\ \end{array} \]

Alternative 14: 64.9% accurate, 32.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0016:\\ \;\;\;\;\left(x \cdot eps_m\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 550:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -0.0016) (* (* x eps_m) -0.5) (if (<= x 550.0) 1.0 0.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -0.0016) {
		tmp = (x * eps_m) * -0.5;
	} else if (x <= 550.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-0.0016d0)) then
        tmp = (x * eps_m) * (-0.5d0)
    else if (x <= 550.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -0.0016) {
		tmp = (x * eps_m) * -0.5;
	} else if (x <= 550.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -0.0016:
		tmp = (x * eps_m) * -0.5
	elif x <= 550.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -0.0016)
		tmp = Float64(Float64(x * eps_m) * -0.5);
	elseif (x <= 550.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -0.0016)
		tmp = (x * eps_m) * -0.5;
	elseif (x <= 550.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -0.0016], N[(N[(x * eps$95$m), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[x, 550.0], 1.0, 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00160000000000000008

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 52.3%

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

   5. Taylor expanded in eps around inf 22.1%

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

      1. mul-1-neg 22.1%

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

      2. *-commutative 22.1%

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

      3. distribute-rgt-neg-in 22.1%

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

   7. Simplified 22.1%

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

   8. Taylor expanded in x around 0 22.1%

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

      1. *-commutative 22.1%

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

   10. Simplified 22.1%

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

   if -0.00160000000000000008 < x < 550

   1. Initial program 45.9%

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

   3. Simplified 45.9%

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

   4. Taylor expanded in x around 0 78.7%

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

   if 550 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 18.2%

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

   5. Taylor expanded in x around 0 26.1%

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

      1. mul-1-neg 26.1%

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

      2. unsub-neg 26.1%

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

   7. Simplified 26.1%

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

   8. Taylor expanded in eps around 0 47.8%

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

      1. associate-*r/ 47.8%

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

      2. distribute-rgt1-in 47.8%

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

      3. metadata-eval 47.8%

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

      4. mul0-lft 47.8%

         \[\leadsto \frac{\frac{-1 \cdot \color{blue}{0}}{\varepsilon}}{2} \]

      5. metadata-eval 47.8%

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

      6. +-inverses 47.8%

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

      7. div-sub 26.6%

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

      8. +-inverses 47.8%

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

   10. Simplified 47.8%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0016:\\ \;\;\;\;\left(x \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{elif}\;x \leq 550:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 15: 58.2% accurate, 44.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 58.1%

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

   3. Simplified 51.0%

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

   4. Taylor expanded in eps around inf 99.2%

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

   5. Taylor expanded in x around 0 81.4%

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

   6. Taylor expanded in x around 0 61.5%

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

      1. neg-mul-1 61.5%

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

      2. unsub-neg 61.5%

         \[\leadsto \color{blue}{1 - x} \]

   8. Simplified 61.5%

      \[\leadsto \color{blue}{1 - x} \]

   if 1 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 18.2%

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

   5. Taylor expanded in x around 0 26.1%

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

      1. mul-1-neg 26.1%

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

      2. unsub-neg 26.1%

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

   7. Simplified 26.1%

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

   8. Taylor expanded in eps around 0 47.8%

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

      1. associate-*r/ 47.8%

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

      2. distribute-rgt1-in 47.8%

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

      3. metadata-eval 47.8%

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

      4. mul0-lft 47.8%

         \[\leadsto \frac{\frac{-1 \cdot \color{blue}{0}}{\varepsilon}}{2} \]

      5. metadata-eval 47.8%

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

      6. +-inverses 47.8%

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

      7. div-sub 26.6%

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

      8. +-inverses 47.8%

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

   10. Simplified 47.8%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 16: 58.2% accurate, 74.1× speedup

Math

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

FPCore

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

C

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

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 470.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 470.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

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

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 470.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 470.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 470.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 470

   1. Initial program 58.1%

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

   3. Simplified 58.1%

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

   4. Taylor expanded in x around 0 61.6%

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

   if 470 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 18.2%

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

   5. Taylor expanded in x around 0 26.1%

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

      1. mul-1-neg 26.1%

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

      2. unsub-neg 26.1%

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

   7. Simplified 26.1%

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

   8. Taylor expanded in eps around 0 47.8%

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

      1. associate-*r/ 47.8%

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

      2. distribute-rgt1-in 47.8%

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

      3. metadata-eval 47.8%

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

      4. mul0-lft 47.8%

         \[\leadsto \frac{\frac{-1 \cdot \color{blue}{0}}{\varepsilon}}{2} \]

      5. metadata-eval 47.8%

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

      6. +-inverses 47.8%

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

      7. div-sub 26.6%

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

      8. +-inverses 47.8%

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

   10. Simplified 47.8%

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

4. Final simplification 58.0%

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

Alternative 17: 44.3% accurate, 75.7× speedup

Math

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

FPCore

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

C

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

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 - x
end function

Java

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

Python

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

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return Float64(1.0 - x)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.9%

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

3. Simplified 63.6%

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

4. Taylor expanded in eps around inf 99.4%

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

5. Taylor expanded in x around 0 69.4%

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

6. Taylor expanded in x around 0 45.9%

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

   1. neg-mul-1 45.9%

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

   2. unsub-neg 45.9%

      \[\leadsto \color{blue}{1 - x} \]

8. Simplified 45.9%

   \[\leadsto \color{blue}{1 - x} \]

9. Final simplification 45.9%

   \[\leadsto 1 - x \]

Reproduce

herbie shell --seed 2023334 
(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))