NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.6% → 99.7%

Time: 14.0s

Alternatives: 15

Speedup: 2.0×

• 37.7% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 0.011:\\ \;\;\;\;\frac{\frac{eps\_m \cdot \left(e^{-x} \cdot \left(2 + x \cdot 2\right)\right)}{eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot eps\_m} + e^{x \cdot \left(-eps\_m\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 0.011)
   (/ (/ (* eps_m (* (exp (- x)) (+ 2.0 (* x 2.0)))) eps_m) 2.0)
   (/ (+ (exp (* x eps_m)) (exp (* x (- eps_m)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.011) {
		tmp = ((eps_m * (exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	} else {
		tmp = (exp((x * eps_m)) + exp((x * -eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 0.011d0) then
        tmp = ((eps_m * (exp(-x) * (2.0d0 + (x * 2.0d0)))) / eps_m) / 2.0d0
    else
        tmp = (exp((x * eps_m)) + exp((x * -eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.011) {
		tmp = ((eps_m * (Math.exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	} else {
		tmp = (Math.exp((x * eps_m)) + Math.exp((x * -eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 0.011:
		tmp = ((eps_m * (math.exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0
	else:
		tmp = (math.exp((x * eps_m)) + math.exp((x * -eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 0.011)
		tmp = Float64(Float64(Float64(eps_m * Float64(exp(Float64(-x)) * Float64(2.0 + Float64(x * 2.0)))) / eps_m) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * eps_m)) + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 0.011)
		tmp = ((eps_m * (exp(-x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	else
		tmp = (exp((x * eps_m)) + exp((x * -eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 0.011], N[(N[(N[(eps$95$m * N[(N[Exp[(-x)], $MachinePrecision] * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 0.010999999999999999

   1. Initial program 64.1%

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

   3. Simplified 57.8%

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

   5. Taylor expanded in eps around 0 31.5%

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

      1. associate-+r+ 67.5%

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

      2. distribute-rgt1-in 67.5%

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

      3. metadata-eval 67.5%

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

      4. mul0-lft 67.5%

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

      5. associate-*r* 67.5%

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

      6. distribute-rgt-out 68.1%

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

      7. mul-1-neg 68.1%

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

   7. Simplified 68.1%

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

   if 0.010999999999999999 < eps

   1. Initial program 100.0%

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

   3. Simplified 91.2%

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

   5. Taylor expanded in eps around inf 99.9%

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

   6. Taylor expanded in eps around inf 99.9%

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

      1. *-commutative 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around -inf 99.9%

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

       1. rec-exp 99.9%

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

       2. neg-mul-1 99.9%

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

       3. cancel-sign-sub-inv 99.9%

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

       4. metadata-eval 99.9%

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

       5. distribute-rgt-in 99.9%

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

       6. +-commutative 99.9%

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

       7. mul-1-neg 99.9%

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

       8. +-commutative 99.9%

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

       9. distribute-rgt-in 99.9%

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

       10. metadata-eval 99.9%

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

       11. cancel-sign-sub-inv 99.9%

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

       12. distribute-rgt-out-- 99.9%

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

   11. Simplified 99.9%

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

   12. Taylor expanded in eps around inf 100.0%

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

       1. mul-1-neg 100.0%

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

       2. distribute-lft-neg-out 100.0%

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

       3. *-commutative 100.0%

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

   14. Simplified 100.0%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.011:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(e^{-x} \cdot \left(2 + x \cdot 2\right)\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return (math.exp((x * (eps_m + -1.0))) + (1.0 / math.exp((x + (x * 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))) + Float64(1.0 / exp(Float64(x + Float64(x * eps_m))))) / 2.0)
end

MATLAB

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

Derivation

1. Initial program 75.0%

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

3. Simplified 67.9%

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

5. Taylor expanded in eps around inf 98.8%

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

6. Final simplification 98.8%

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

Alternative 3: 84.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2000:\\ \;\;\;\;0.5 + \frac{0.5}{e^{x}}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\frac{\left(1 + x \cdot eps\_m\right) + e^{x \cdot \left(-1 - eps\_m\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + \frac{1}{1 + x \cdot \left(eps\_m + 1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2000.0)
   (+ 0.5 (/ 0.5 (exp x)))
   (if (<= x -5e-308)
     (/ (+ (+ 1.0 (* x eps_m)) (exp (* x (- -1.0 eps_m)))) 2.0)
     (/
      (+ (exp (* x (+ eps_m -1.0))) (/ 1.0 (+ 1.0 (* x (+ eps_m 1.0)))))
      2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -2000.0) {
		tmp = 0.5 + (0.5 / exp(x));
	} else if (x <= -5e-308) {
		tmp = ((1.0 + (x * eps_m)) + exp((x * (-1.0 - eps_m)))) / 2.0;
	} else {
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 / (1.0 + (x * (eps_m + 1.0))))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-2000.0d0)) then
        tmp = 0.5d0 + (0.5d0 / exp(x))
    else if (x <= (-5d-308)) then
        tmp = ((1.0d0 + (x * eps_m)) + exp((x * ((-1.0d0) - eps_m)))) / 2.0d0
    else
        tmp = (exp((x * (eps_m + (-1.0d0)))) + (1.0d0 / (1.0d0 + (x * (eps_m + 1.0d0))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -2000.0) {
		tmp = 0.5 + (0.5 / Math.exp(x));
	} else if (x <= -5e-308) {
		tmp = ((1.0 + (x * eps_m)) + Math.exp((x * (-1.0 - eps_m)))) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps_m + -1.0))) + (1.0 / (1.0 + (x * (eps_m + 1.0))))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -2000.0:
		tmp = 0.5 + (0.5 / math.exp(x))
	elif x <= -5e-308:
		tmp = ((1.0 + (x * eps_m)) + math.exp((x * (-1.0 - eps_m)))) / 2.0
	else:
		tmp = (math.exp((x * (eps_m + -1.0))) + (1.0 / (1.0 + (x * (eps_m + 1.0))))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -2000.0)
		tmp = Float64(0.5 + Float64(0.5 / exp(x)));
	elseif (x <= -5e-308)
		tmp = Float64(Float64(Float64(1.0 + Float64(x * eps_m)) + exp(Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + Float64(1.0 / Float64(1.0 + Float64(x * Float64(eps_m + 1.0))))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -2000.0)
		tmp = 0.5 + (0.5 / exp(x));
	elseif (x <= -5e-308)
		tmp = ((1.0 + (x * eps_m)) + exp((x * (-1.0 - eps_m)))) / 2.0;
	else
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 / (1.0 + (x * (eps_m + 1.0))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -2000.0], N[(0.5 + N[(0.5 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5e-308], N[(N[(N[(1.0 + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(1.0 + N[(x * N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2e3

   1. Initial program 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in eps around inf 97.3%

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

   6. Taylor expanded in x around 0 55.4%

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

   7. Taylor expanded in eps around 0 97.3%

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

      1. neg-mul-1 97.3%

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

   9. Simplified 97.3%

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

   10. Taylor expanded in x around inf 97.3%

       \[\leadsto \color{blue}{0.5 \cdot \left(1 + e^{-x}\right)} \]
   11. Step-by-step derivation

       1. distribute-lft-in 97.3%

          \[\leadsto \color{blue}{0.5 \cdot 1 + 0.5 \cdot e^{-x}} \]

       2. metadata-eval 97.3%

          \[\leadsto \color{blue}{0.5} + 0.5 \cdot e^{-x} \]

       3. rec-exp 97.3%

          \[\leadsto 0.5 + 0.5 \cdot \color{blue}{\frac{1}{e^{x}}} \]

       4. associate-*r/ 97.3%

          \[\leadsto 0.5 + \color{blue}{\frac{0.5 \cdot 1}{e^{x}}} \]

       5. metadata-eval 97.3%

          \[\leadsto 0.5 + \frac{\color{blue}{0.5}}{e^{x}} \]

   12. Simplified 97.3%

       \[\leadsto \color{blue}{0.5 + \frac{0.5}{e^{x}}} \]

   if -2e3 < x < -4.99999999999999955e-308

   1. Initial program 54.1%

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

   3. Simplified 38.3%

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

   5. Taylor expanded in eps around inf 98.7%

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

   6. Taylor expanded in eps around inf 98.7%

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

      1. *-commutative 98.7%

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

   8. Simplified 98.7%

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

   9. Taylor expanded in x around -inf 98.7%

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

       1. rec-exp 98.7%

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

       2. neg-mul-1 98.7%

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

       3. cancel-sign-sub-inv 98.7%

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

       4. metadata-eval 98.7%

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

       5. distribute-rgt-in 98.7%

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

       6. +-commutative 98.7%

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

       7. mul-1-neg 98.7%

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

       8. +-commutative 98.7%

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

       9. distribute-rgt-in 98.7%

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

       10. metadata-eval 98.7%

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

       11. cancel-sign-sub-inv 98.7%

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

       12. distribute-rgt-out-- 98.7%

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

   11. Simplified 98.7%

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

   12. Taylor expanded in x around 0 83.9%

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

   if -4.99999999999999955e-308 < x

   1. Initial program 80.4%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in eps around inf 99.3%

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

   6. Taylor expanded in x around 0 54.8%

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

      1. +-commutative 54.8%

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

   8. Simplified 54.8%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2000:\\ \;\;\;\;0.5 + \frac{0.5}{e^{x}}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-308}:\\ \;\;\;\;\frac{\left(1 + x \cdot \varepsilon\right) + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \frac{1}{1 + x \cdot \left(\varepsilon + 1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2000:\\ \;\;\;\;0.5 + \frac{0.5}{e^{x}}\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-169}:\\ \;\;\;\;\frac{\left(1 + x \cdot eps\_m\right) + e^{x \cdot \left(-1 - eps\_m\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(eps\_m + -1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2000.0)
   (+ 0.5 (/ 0.5 (exp x)))
   (if (<= x 8.8e-169)
     (/ (+ (+ 1.0 (* x eps_m)) (exp (* x (- -1.0 eps_m)))) 2.0)
     (/ (+ 1.0 (exp (* x (+ eps_m -1.0)))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -2000.0) {
		tmp = 0.5 + (0.5 / exp(x));
	} else if (x <= 8.8e-169) {
		tmp = ((1.0 + (x * eps_m)) + exp((x * (-1.0 - eps_m)))) / 2.0;
	} else {
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-2000.0d0)) then
        tmp = 0.5d0 + (0.5d0 / exp(x))
    else if (x <= 8.8d-169) then
        tmp = ((1.0d0 + (x * eps_m)) + exp((x * ((-1.0d0) - eps_m)))) / 2.0d0
    else
        tmp = (1.0d0 + exp((x * (eps_m + (-1.0d0))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -2000.0) {
		tmp = 0.5 + (0.5 / Math.exp(x));
	} else if (x <= 8.8e-169) {
		tmp = ((1.0 + (x * eps_m)) + Math.exp((x * (-1.0 - eps_m)))) / 2.0;
	} else {
		tmp = (1.0 + Math.exp((x * (eps_m + -1.0)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -2000.0:
		tmp = 0.5 + (0.5 / math.exp(x))
	elif x <= 8.8e-169:
		tmp = ((1.0 + (x * eps_m)) + math.exp((x * (-1.0 - eps_m)))) / 2.0
	else:
		tmp = (1.0 + math.exp((x * (eps_m + -1.0)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -2000.0)
		tmp = Float64(0.5 + Float64(0.5 / exp(x)));
	elseif (x <= 8.8e-169)
		tmp = Float64(Float64(Float64(1.0 + Float64(x * eps_m)) + exp(Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps_m + -1.0)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -2000.0)
		tmp = 0.5 + (0.5 / exp(x));
	elseif (x <= 8.8e-169)
		tmp = ((1.0 + (x * eps_m)) + exp((x * (-1.0 - eps_m)))) / 2.0;
	else
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -2000.0], N[(0.5 + N[(0.5 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.8e-169], N[(N[(N[(1.0 + N[(x * eps$95$m), $MachinePrecision]), $MachinePrecision] + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2e3

   1. Initial program 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in eps around inf 97.3%

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

   6. Taylor expanded in x around 0 55.4%

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

   7. Taylor expanded in eps around 0 97.3%

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

      1. neg-mul-1 97.3%

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

   9. Simplified 97.3%

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

   10. Taylor expanded in x around inf 97.3%

       \[\leadsto \color{blue}{0.5 \cdot \left(1 + e^{-x}\right)} \]
   11. Step-by-step derivation

       1. distribute-lft-in 97.3%

          \[\leadsto \color{blue}{0.5 \cdot 1 + 0.5 \cdot e^{-x}} \]

       2. metadata-eval 97.3%

          \[\leadsto \color{blue}{0.5} + 0.5 \cdot e^{-x} \]

       3. rec-exp 97.3%

          \[\leadsto 0.5 + 0.5 \cdot \color{blue}{\frac{1}{e^{x}}} \]

       4. associate-*r/ 97.3%

          \[\leadsto 0.5 + \color{blue}{\frac{0.5 \cdot 1}{e^{x}}} \]

       5. metadata-eval 97.3%

          \[\leadsto 0.5 + \frac{\color{blue}{0.5}}{e^{x}} \]

   12. Simplified 97.3%

       \[\leadsto \color{blue}{0.5 + \frac{0.5}{e^{x}}} \]

   if -2e3 < x < 8.80000000000000029e-169

   1. Initial program 54.4%

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

   3. Simplified 41.5%

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

   5. Taylor expanded in eps around inf 99.1%

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

   6. Taylor expanded in eps around inf 99.1%

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

      1. *-commutative 99.1%

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

   8. Simplified 99.1%

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

   9. Taylor expanded in x around -inf 99.1%

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

       1. rec-exp 99.1%

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

       2. neg-mul-1 99.1%

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

       3. cancel-sign-sub-inv 99.1%

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

       4. metadata-eval 99.1%

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

       5. distribute-rgt-in 99.1%

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

       6. +-commutative 99.1%

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

       7. mul-1-neg 99.1%

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

       8. +-commutative 99.1%

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

       9. distribute-rgt-in 99.1%

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

       10. metadata-eval 99.1%

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

       11. cancel-sign-sub-inv 99.1%

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

       12. distribute-rgt-out-- 99.1%

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

   11. Simplified 99.1%

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

   12. Taylor expanded in x around 0 87.2%

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

   if 8.80000000000000029e-169 < x

   1. Initial program 86.5%

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

   3. Simplified 82.3%

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

   5. Taylor expanded in eps around inf 99.1%

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

   6. Taylor expanded in x around 0 44.4%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2000:\\ \;\;\;\;0.5 + \frac{0.5}{e^{x}}\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-169}:\\ \;\;\;\;\frac{\left(1 + x \cdot \varepsilon\right) + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 8.8 \cdot 10^{-169}:\\ \;\;\;\;0.5 + \frac{0.5}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(eps\_m + -1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 8.8e-169)
   (+ 0.5 (/ 0.5 (exp x)))
   (/ (+ 1.0 (exp (* x (+ eps_m -1.0)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 8.8e-169) {
		tmp = 0.5 + (0.5 / exp(x));
	} else {
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 8.8d-169) then
        tmp = 0.5d0 + (0.5d0 / exp(x))
    else
        tmp = (1.0d0 + exp((x * (eps_m + (-1.0d0))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 8.8e-169) {
		tmp = 0.5 + (0.5 / Math.exp(x));
	} else {
		tmp = (1.0 + Math.exp((x * (eps_m + -1.0)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 8.8e-169:
		tmp = 0.5 + (0.5 / math.exp(x))
	else:
		tmp = (1.0 + math.exp((x * (eps_m + -1.0)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 8.8e-169)
		tmp = Float64(0.5 + Float64(0.5 / exp(x)));
	else
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps_m + -1.0)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 8.8e-169)
		tmp = 0.5 + (0.5 / exp(x));
	else
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 8.8e-169], N[(0.5 + N[(0.5 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.80000000000000029e-169

   1. Initial program 65.6%

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

   3. Simplified 56.1%

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

   5. Taylor expanded in eps around inf 98.6%

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

   6. Taylor expanded in x around 0 78.8%

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

   7. Taylor expanded in eps around 0 82.8%

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

      1. neg-mul-1 82.8%

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

   9. Simplified 82.8%

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

   10. Taylor expanded in x around inf 82.8%

       \[\leadsto \color{blue}{0.5 \cdot \left(1 + e^{-x}\right)} \]
   11. Step-by-step derivation

       1. distribute-lft-in 82.8%

          \[\leadsto \color{blue}{0.5 \cdot 1 + 0.5 \cdot e^{-x}} \]

       2. metadata-eval 82.8%

          \[\leadsto \color{blue}{0.5} + 0.5 \cdot e^{-x} \]

       3. rec-exp 82.8%

          \[\leadsto 0.5 + 0.5 \cdot \color{blue}{\frac{1}{e^{x}}} \]

       4. associate-*r/ 82.8%

          \[\leadsto 0.5 + \color{blue}{\frac{0.5 \cdot 1}{e^{x}}} \]

       5. metadata-eval 82.8%

          \[\leadsto 0.5 + \frac{\color{blue}{0.5}}{e^{x}} \]

   12. Simplified 82.8%

       \[\leadsto \color{blue}{0.5 + \frac{0.5}{e^{x}}} \]

   if 8.80000000000000029e-169 < x

   1. Initial program 86.5%

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

   3. Simplified 82.3%

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

   5. Taylor expanded in eps around inf 99.1%

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

   6. Taylor expanded in x around 0 44.4%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.8 \cdot 10^{-169}:\\ \;\;\;\;0.5 + \frac{0.5}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 8.8 \cdot 10^{-169}:\\ \;\;\;\;0.5 + \frac{0.5}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 8.8e-169)
   (+ 0.5 (/ 0.5 (exp x)))
   (/ (+ 1.0 (exp (* x eps_m))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 8.8e-169) {
		tmp = 0.5 + (0.5 / exp(x));
	} else {
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 8.8d-169) then
        tmp = 0.5d0 + (0.5d0 / exp(x))
    else
        tmp = (1.0d0 + exp((x * eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 8.8e-169) {
		tmp = 0.5 + (0.5 / Math.exp(x));
	} else {
		tmp = (1.0 + Math.exp((x * eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 8.8e-169:
		tmp = 0.5 + (0.5 / math.exp(x))
	else:
		tmp = (1.0 + math.exp((x * eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 8.8e-169)
		tmp = Float64(0.5 + Float64(0.5 / exp(x)));
	else
		tmp = Float64(Float64(1.0 + exp(Float64(x * eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 8.8e-169)
		tmp = 0.5 + (0.5 / exp(x));
	else
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 8.8e-169], N[(0.5 + N[(0.5 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.80000000000000029e-169

   1. Initial program 65.6%

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

   3. Simplified 56.1%

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

   5. Taylor expanded in eps around inf 98.6%

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

   6. Taylor expanded in x around 0 78.8%

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

   7. Taylor expanded in eps around 0 82.8%

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

      1. neg-mul-1 82.8%

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

   9. Simplified 82.8%

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

   10. Taylor expanded in x around inf 82.8%

       \[\leadsto \color{blue}{0.5 \cdot \left(1 + e^{-x}\right)} \]
   11. Step-by-step derivation

       1. distribute-lft-in 82.8%

          \[\leadsto \color{blue}{0.5 \cdot 1 + 0.5 \cdot e^{-x}} \]

       2. metadata-eval 82.8%

          \[\leadsto \color{blue}{0.5} + 0.5 \cdot e^{-x} \]

       3. rec-exp 82.8%

          \[\leadsto 0.5 + 0.5 \cdot \color{blue}{\frac{1}{e^{x}}} \]

       4. associate-*r/ 82.8%

          \[\leadsto 0.5 + \color{blue}{\frac{0.5 \cdot 1}{e^{x}}} \]

       5. metadata-eval 82.8%

          \[\leadsto 0.5 + \frac{\color{blue}{0.5}}{e^{x}} \]

   12. Simplified 82.8%

       \[\leadsto \color{blue}{0.5 + \frac{0.5}{e^{x}}} \]

   if 8.80000000000000029e-169 < x

   1. Initial program 86.5%

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

   3. Simplified 82.3%

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

   5. Taylor expanded in eps around inf 99.1%

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

   6. Taylor expanded in eps around inf 74.6%

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

      1. *-commutative 74.6%

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

   8. Simplified 74.6%

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

   9. Taylor expanded in x around 0 44.3%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.8 \cdot 10^{-169}:\\ \;\;\;\;0.5 + \frac{0.5}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 315:\\ \;\;\;\;0.5 + \frac{0.5}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 315.0) (+ 0.5 (/ 0.5 (exp x))) (/ (+ 1.0 (exp x)) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 315.0) {
		tmp = 0.5 + (0.5 / exp(x));
	} else {
		tmp = (1.0 + exp(x)) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 315.0d0) then
        tmp = 0.5d0 + (0.5d0 / exp(x))
    else
        tmp = (1.0d0 + exp(x)) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 315.0) {
		tmp = 0.5 + (0.5 / Math.exp(x));
	} else {
		tmp = (1.0 + Math.exp(x)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 315.0:
		tmp = 0.5 + (0.5 / math.exp(x))
	else:
		tmp = (1.0 + math.exp(x)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 315.0)
		tmp = Float64(0.5 + Float64(0.5 / exp(x)));
	else
		tmp = Float64(Float64(1.0 + exp(x)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 315.0)
		tmp = 0.5 + (0.5 / exp(x));
	else
		tmp = (1.0 + exp(x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 315.0], N[(0.5 + N[(0.5 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 315

   1. Initial program 63.0%

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

   3. Simplified 52.5%

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

   5. Taylor expanded in eps around inf 98.3%

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

   6. Taylor expanded in x around 0 79.3%

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

   7. Taylor expanded in eps around 0 79.3%

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

      1. neg-mul-1 79.3%

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

   9. Simplified 79.3%

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

   10. Taylor expanded in x around inf 79.3%

       \[\leadsto \color{blue}{0.5 \cdot \left(1 + e^{-x}\right)} \]
   11. Step-by-step derivation

       1. distribute-lft-in 79.3%

          \[\leadsto \color{blue}{0.5 \cdot 1 + 0.5 \cdot e^{-x}} \]

       2. metadata-eval 79.3%

          \[\leadsto \color{blue}{0.5} + 0.5 \cdot e^{-x} \]

       3. rec-exp 79.3%

          \[\leadsto 0.5 + 0.5 \cdot \color{blue}{\frac{1}{e^{x}}} \]

       4. associate-*r/ 79.3%

          \[\leadsto 0.5 + \color{blue}{\frac{0.5 \cdot 1}{e^{x}}} \]

       5. metadata-eval 79.3%

          \[\leadsto 0.5 + \frac{\color{blue}{0.5}}{e^{x}} \]

   12. Simplified 79.3%

       \[\leadsto \color{blue}{0.5 + \frac{0.5}{e^{x}}} \]

   if 315 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 30.0%

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

   7. Taylor expanded in eps around 0 3.1%

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

      1. neg-mul-1 3.1%

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

   9. Simplified 3.1%

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

       1. expm1-log1p-u 3.1%

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

       2. log1p-define 3.1%

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

       3. +-commutative 3.1%

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

       4. expm1-undefine 3.1%

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

       5. add-exp-log 3.1%

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

       6. +-commutative 3.1%

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

       7. add-sqr-sqrt 0.0%

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

       8. sqrt-unprod 52.6%

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

       9. sqr-neg 52.6%

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

       10. sqrt-unprod 52.6%

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

       11. add-sqr-sqrt 52.6%

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

   11. Applied egg-rr 52.6%

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

       1. associate--l+ 52.6%

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

       2. expm1-undefine 52.6%

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

       3. rem-exp-log 52.6%

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

       4. log1p-define 52.6%

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

       5. log1p-expm1 52.6%

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

   13. Simplified 52.6%

       \[\leadsto \frac{\color{blue}{e^{x}} + 1}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 315:\\ \;\;\;\;0.5 + \frac{0.5}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.22 \cdot 10^{-32}:\\ \;\;\;\;0.5 + \frac{0.5}{e^{x}}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+35}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{2 + eps\_m \cdot \left(eps\_m - 2\right)}{eps\_m} - eps\_m\right)}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+166}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot 0.25 - 0.5\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1.22e-32)
   (+ 0.5 (/ 0.5 (exp x)))
   (if (<= x 1.8e+35)
     (/ (+ 2.0 (* x (- (/ (+ 2.0 (* eps_m (- eps_m 2.0))) eps_m) eps_m))) 2.0)
     (if (<= x 2e+166) 0.0 (+ 1.0 (* x (- (* x 0.25) 0.5)))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.22e-32) {
		tmp = 0.5 + (0.5 / exp(x));
	} else if (x <= 1.8e+35) {
		tmp = (2.0 + (x * (((2.0 + (eps_m * (eps_m - 2.0))) / eps_m) - eps_m))) / 2.0;
	} else if (x <= 2e+166) {
		tmp = 0.0;
	} else {
		tmp = 1.0 + (x * ((x * 0.25) - 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) :: tmp
    if (x <= 1.22d-32) then
        tmp = 0.5d0 + (0.5d0 / exp(x))
    else if (x <= 1.8d+35) then
        tmp = (2.0d0 + (x * (((2.0d0 + (eps_m * (eps_m - 2.0d0))) / eps_m) - eps_m))) / 2.0d0
    else if (x <= 2d+166) then
        tmp = 0.0d0
    else
        tmp = 1.0d0 + (x * ((x * 0.25d0) - 0.5d0))
    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 <= 1.22e-32) {
		tmp = 0.5 + (0.5 / Math.exp(x));
	} else if (x <= 1.8e+35) {
		tmp = (2.0 + (x * (((2.0 + (eps_m * (eps_m - 2.0))) / eps_m) - eps_m))) / 2.0;
	} else if (x <= 2e+166) {
		tmp = 0.0;
	} else {
		tmp = 1.0 + (x * ((x * 0.25) - 0.5));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 1.22e-32:
		tmp = 0.5 + (0.5 / math.exp(x))
	elif x <= 1.8e+35:
		tmp = (2.0 + (x * (((2.0 + (eps_m * (eps_m - 2.0))) / eps_m) - eps_m))) / 2.0
	elif x <= 2e+166:
		tmp = 0.0
	else:
		tmp = 1.0 + (x * ((x * 0.25) - 0.5))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 1.22e-32)
		tmp = Float64(0.5 + Float64(0.5 / exp(x)));
	elseif (x <= 1.8e+35)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(2.0 + Float64(eps_m * Float64(eps_m - 2.0))) / eps_m) - eps_m))) / 2.0);
	elseif (x <= 2e+166)
		tmp = 0.0;
	else
		tmp = Float64(1.0 + Float64(x * Float64(Float64(x * 0.25) - 0.5)));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 1.22e-32)
		tmp = 0.5 + (0.5 / exp(x));
	elseif (x <= 1.8e+35)
		tmp = (2.0 + (x * (((2.0 + (eps_m * (eps_m - 2.0))) / eps_m) - eps_m))) / 2.0;
	elseif (x <= 2e+166)
		tmp = 0.0;
	else
		tmp = 1.0 + (x * ((x * 0.25) - 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 1.22e-32], N[(0.5 + N[(0.5 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.8e+35], N[(N[(2.0 + N[(x * N[(N[(N[(2.0 + N[(eps$95$m * N[(eps$95$m - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2e+166], 0.0, N[(1.0 + N[(x * N[(N[(x * 0.25), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.22e-32

   1. Initial program 62.2%

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

   3. Simplified 52.2%

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

   5. Taylor expanded in eps around inf 98.8%

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

   6. Taylor expanded in x around 0 80.6%

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

   7. Taylor expanded in eps around 0 82.3%

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

      1. neg-mul-1 82.3%

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

   9. Simplified 82.3%

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

   10. Taylor expanded in x around inf 82.3%

       \[\leadsto \color{blue}{0.5 \cdot \left(1 + e^{-x}\right)} \]
   11. Step-by-step derivation

       1. distribute-lft-in 82.3%

          \[\leadsto \color{blue}{0.5 \cdot 1 + 0.5 \cdot e^{-x}} \]

       2. metadata-eval 82.3%

          \[\leadsto \color{blue}{0.5} + 0.5 \cdot e^{-x} \]

       3. rec-exp 82.3%

          \[\leadsto 0.5 + 0.5 \cdot \color{blue}{\frac{1}{e^{x}}} \]

       4. associate-*r/ 82.3%

          \[\leadsto 0.5 + \color{blue}{\frac{0.5 \cdot 1}{e^{x}}} \]

       5. metadata-eval 82.3%

          \[\leadsto 0.5 + \frac{\color{blue}{0.5}}{e^{x}} \]

   12. Simplified 82.3%

       \[\leadsto \color{blue}{0.5 + \frac{0.5}{e^{x}}} \]

   if 1.22e-32 < x < 1.8e35

   1. Initial program 90.8%

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

   3. Simplified 81.4%

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

   5. Taylor expanded in x around 0 13.2%

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

      1. add-sqr-sqrt 5.9%

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

      2. sqrt-unprod 7.3%

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

      3. frac-times 7.3%

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

      4. metadata-eval 7.3%

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

      5. metadata-eval 7.3%

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

      6. frac-times 7.3%

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

      7. sqrt-unprod 1.4%

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

      8. add-sqr-sqrt 7.1%

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

      9. div-inv 7.1%

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

   7. Applied egg-rr 7.1%

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

   8. Taylor expanded in eps around 0 20.0%

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

   if 1.8e35 < x < 1.99999999999999988e166

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 61.9%

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

      1. div-sub 61.9%

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

      2. mul-1-neg 61.9%

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

      3. rec-exp 61.9%

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

      4. +-inverses 61.9%

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

      5. metadata-eval 61.9%

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

   7. Simplified 61.9%

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

   if 1.99999999999999988e166 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 37.0%

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

   7. Taylor expanded in eps around 0 3.1%

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

      1. neg-mul-1 3.1%

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

   9. Simplified 3.1%

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

   10. Taylor expanded in x around 0 58.2%

       \[\leadsto \color{blue}{1 + x \cdot \left(0.25 \cdot x - 0.5\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.22 \cdot 10^{-32}:\\ \;\;\;\;0.5 + \frac{0.5}{e^{x}}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+35}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{2 + \varepsilon \cdot \left(\varepsilon - 2\right)}{\varepsilon} - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+166}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot 0.25 - 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.3% accurate, 8.4× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{-29}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.25 + x \cdot -0.08333333333333333\right) - 0.5\right)\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+35}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{2 + eps\_m \cdot \left(eps\_m - 2\right)}{eps\_m} - eps\_m\right)}{2}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+166}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot 0.25 - 0.5\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 9e-29)
   (+ 1.0 (* x (- (* x (+ 0.25 (* x -0.08333333333333333))) 0.5)))
   (if (<= x 3.1e+35)
     (/ (+ 2.0 (* x (- (/ (+ 2.0 (* eps_m (- eps_m 2.0))) eps_m) eps_m))) 2.0)
     (if (<= x 4.2e+166) 0.0 (+ 1.0 (* x (- (* x 0.25) 0.5)))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 9e-29) {
		tmp = 1.0 + (x * ((x * (0.25 + (x * -0.08333333333333333))) - 0.5));
	} else if (x <= 3.1e+35) {
		tmp = (2.0 + (x * (((2.0 + (eps_m * (eps_m - 2.0))) / eps_m) - eps_m))) / 2.0;
	} else if (x <= 4.2e+166) {
		tmp = 0.0;
	} else {
		tmp = 1.0 + (x * ((x * 0.25) - 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) :: tmp
    if (x <= 9d-29) then
        tmp = 1.0d0 + (x * ((x * (0.25d0 + (x * (-0.08333333333333333d0)))) - 0.5d0))
    else if (x <= 3.1d+35) then
        tmp = (2.0d0 + (x * (((2.0d0 + (eps_m * (eps_m - 2.0d0))) / eps_m) - eps_m))) / 2.0d0
    else if (x <= 4.2d+166) then
        tmp = 0.0d0
    else
        tmp = 1.0d0 + (x * ((x * 0.25d0) - 0.5d0))
    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 <= 9e-29) {
		tmp = 1.0 + (x * ((x * (0.25 + (x * -0.08333333333333333))) - 0.5));
	} else if (x <= 3.1e+35) {
		tmp = (2.0 + (x * (((2.0 + (eps_m * (eps_m - 2.0))) / eps_m) - eps_m))) / 2.0;
	} else if (x <= 4.2e+166) {
		tmp = 0.0;
	} else {
		tmp = 1.0 + (x * ((x * 0.25) - 0.5));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 9e-29:
		tmp = 1.0 + (x * ((x * (0.25 + (x * -0.08333333333333333))) - 0.5))
	elif x <= 3.1e+35:
		tmp = (2.0 + (x * (((2.0 + (eps_m * (eps_m - 2.0))) / eps_m) - eps_m))) / 2.0
	elif x <= 4.2e+166:
		tmp = 0.0
	else:
		tmp = 1.0 + (x * ((x * 0.25) - 0.5))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 9e-29)
		tmp = Float64(1.0 + Float64(x * Float64(Float64(x * Float64(0.25 + Float64(x * -0.08333333333333333))) - 0.5)));
	elseif (x <= 3.1e+35)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(2.0 + Float64(eps_m * Float64(eps_m - 2.0))) / eps_m) - eps_m))) / 2.0);
	elseif (x <= 4.2e+166)
		tmp = 0.0;
	else
		tmp = Float64(1.0 + Float64(x * Float64(Float64(x * 0.25) - 0.5)));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 9e-29)
		tmp = 1.0 + (x * ((x * (0.25 + (x * -0.08333333333333333))) - 0.5));
	elseif (x <= 3.1e+35)
		tmp = (2.0 + (x * (((2.0 + (eps_m * (eps_m - 2.0))) / eps_m) - eps_m))) / 2.0;
	elseif (x <= 4.2e+166)
		tmp = 0.0;
	else
		tmp = 1.0 + (x * ((x * 0.25) - 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 9e-29], N[(1.0 + N[(x * N[(N[(x * N[(0.25 + N[(x * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.1e+35], N[(N[(2.0 + N[(x * N[(N[(N[(2.0 + N[(eps$95$m * N[(eps$95$m - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 4.2e+166], 0.0, N[(1.0 + N[(x * N[(N[(x * 0.25), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 8.9999999999999996e-29

   1. Initial program 62.2%

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

   3. Simplified 52.2%

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

   5. Taylor expanded in eps around inf 98.8%

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

   6. Taylor expanded in x around 0 80.6%

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

   7. Taylor expanded in eps around 0 82.3%

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

      1. neg-mul-1 82.3%

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

   9. Simplified 82.3%

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

   10. Taylor expanded in x around 0 75.4%

       \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(0.25 + -0.08333333333333333 \cdot x\right) - 0.5\right)} \]

   if 8.9999999999999996e-29 < x < 3.09999999999999987e35

   1. Initial program 90.8%

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

   3. Simplified 81.4%

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

   5. Taylor expanded in x around 0 13.2%

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

      1. add-sqr-sqrt 5.9%

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

      2. sqrt-unprod 7.3%

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

      3. frac-times 7.3%

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

      4. metadata-eval 7.3%

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

      5. metadata-eval 7.3%

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

      6. frac-times 7.3%

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

      7. sqrt-unprod 1.4%

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

      8. add-sqr-sqrt 7.1%

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

      9. div-inv 7.1%

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

   7. Applied egg-rr 7.1%

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

   8. Taylor expanded in eps around 0 20.0%

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

   if 3.09999999999999987e35 < x < 4.2000000000000001e166

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 61.9%

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

      1. div-sub 61.9%

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

      2. mul-1-neg 61.9%

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

      3. rec-exp 61.9%

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

      4. +-inverses 61.9%

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

      5. metadata-eval 61.9%

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

   7. Simplified 61.9%

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

   if 4.2000000000000001e166 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 37.0%

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

   7. Taylor expanded in eps around 0 3.1%

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

      1. neg-mul-1 3.1%

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

   9. Simplified 3.1%

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

   10. Taylor expanded in x around 0 58.2%

       \[\leadsto \color{blue}{1 + x \cdot \left(0.25 \cdot x - 0.5\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{-29}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.25 + x \cdot -0.08333333333333333\right) - 0.5\right)\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+35}:\\ \;\;\;\;\frac{2 + x \cdot \left(\frac{2 + \varepsilon \cdot \left(\varepsilon - 2\right)}{\varepsilon} - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+166}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot 0.25 - 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.2% accurate, 11.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 560 \lor \neg \left(x \leq 10^{+170}\right):\\ \;\;\;\;1 + x \cdot \left(x \cdot 0.25 - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (or (<= x 560.0) (not (<= x 1e+170)))
   (+ 1.0 (* x (- (* x 0.25) 0.5)))
   0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if ((x <= 560.0) || !(x <= 1e+170)) {
		tmp = 1.0 + (x * ((x * 0.25) - 0.5));
	} 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 <= 560.0d0) .or. (.not. (x <= 1d+170))) then
        tmp = 1.0d0 + (x * ((x * 0.25d0) - 0.5d0))
    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 <= 560.0) || !(x <= 1e+170)) {
		tmp = 1.0 + (x * ((x * 0.25) - 0.5));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if (x <= 560.0) or not (x <= 1e+170):
		tmp = 1.0 + (x * ((x * 0.25) - 0.5))
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if ((x <= 560.0) || !(x <= 1e+170))
		tmp = Float64(1.0 + Float64(x * Float64(Float64(x * 0.25) - 0.5)));
	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 <= 560.0) || ~((x <= 1e+170)))
		tmp = 1.0 + (x * ((x * 0.25) - 0.5));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[Or[LessEqual[x, 560.0], N[Not[LessEqual[x, 1e+170]], $MachinePrecision]], N[(1.0 + N[(x * N[(N[(x * 0.25), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 560 or 1.00000000000000003e170 < x

   1. Initial program 70.1%

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

   3. Simplified 61.6%

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

   5. Taylor expanded in eps around inf 98.6%

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

   6. Taylor expanded in x around 0 71.5%

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

   7. Taylor expanded in eps around 0 64.7%

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

      1. neg-mul-1 64.7%

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

   9. Simplified 64.7%

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

   10. Taylor expanded in x around 0 67.0%

       \[\leadsto \color{blue}{1 + x \cdot \left(0.25 \cdot x - 0.5\right)} \]

   if 560 < x < 1.00000000000000003e170

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 53.1%

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

      1. div-sub 53.1%

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

      2. mul-1-neg 53.1%

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

      3. rec-exp 53.1%

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

      4. +-inverses 53.1%

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

      5. metadata-eval 53.1%

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

   7. Simplified 53.1%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 560 \lor \neg \left(x \leq 10^{+170}\right):\\ \;\;\;\;1 + x \cdot \left(x \cdot 0.25 - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.2% accurate, 11.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2.5:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.25 + x \cdot -0.08333333333333333\right) - 0.5\right)\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+166}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot 0.25 - 0.5\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 2.5)
   (+ 1.0 (* x (- (* x (+ 0.25 (* x -0.08333333333333333))) 0.5)))
   (if (<= x 1.5e+166) 0.0 (+ 1.0 (* x (- (* x 0.25) 0.5))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.5) {
		tmp = 1.0 + (x * ((x * (0.25 + (x * -0.08333333333333333))) - 0.5));
	} else if (x <= 1.5e+166) {
		tmp = 0.0;
	} else {
		tmp = 1.0 + (x * ((x * 0.25) - 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) :: tmp
    if (x <= 2.5d0) then
        tmp = 1.0d0 + (x * ((x * (0.25d0 + (x * (-0.08333333333333333d0)))) - 0.5d0))
    else if (x <= 1.5d+166) then
        tmp = 0.0d0
    else
        tmp = 1.0d0 + (x * ((x * 0.25d0) - 0.5d0))
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.5) {
		tmp = 1.0 + (x * ((x * (0.25 + (x * -0.08333333333333333))) - 0.5));
	} else if (x <= 1.5e+166) {
		tmp = 0.0;
	} else {
		tmp = 1.0 + (x * ((x * 0.25) - 0.5));
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 2.5:
		tmp = 1.0 + (x * ((x * (0.25 + (x * -0.08333333333333333))) - 0.5))
	elif x <= 1.5e+166:
		tmp = 0.0
	else:
		tmp = 1.0 + (x * ((x * 0.25) - 0.5))
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 2.5)
		tmp = Float64(1.0 + Float64(x * Float64(Float64(x * Float64(0.25 + Float64(x * -0.08333333333333333))) - 0.5)));
	elseif (x <= 1.5e+166)
		tmp = 0.0;
	else
		tmp = Float64(1.0 + Float64(x * Float64(Float64(x * 0.25) - 0.5)));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 2.5)
		tmp = 1.0 + (x * ((x * (0.25 + (x * -0.08333333333333333))) - 0.5));
	elseif (x <= 1.5e+166)
		tmp = 0.0;
	else
		tmp = 1.0 + (x * ((x * 0.25) - 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 2.5], N[(1.0 + N[(x * N[(N[(x * N[(0.25 + N[(x * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.5e+166], 0.0, N[(1.0 + N[(x * N[(N[(x * 0.25), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.5

   1. Initial program 63.0%

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

   3. Simplified 52.5%

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

   5. Taylor expanded in eps around inf 98.3%

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

   6. Taylor expanded in x around 0 79.3%

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

   7. Taylor expanded in eps around 0 79.3%

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

      1. neg-mul-1 79.3%

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

   9. Simplified 79.3%

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

   10. Taylor expanded in x around 0 72.7%

       \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot \left(0.25 + -0.08333333333333333 \cdot x\right) - 0.5\right)} \]

   if 2.5 < x < 1.49999999999999999e166

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 51.9%

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

      1. div-sub 51.9%

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

      2. mul-1-neg 51.9%

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

      3. rec-exp 51.9%

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

      4. +-inverses 51.9%

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

      5. metadata-eval 51.9%

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

   7. Simplified 51.9%

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

   if 1.49999999999999999e166 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 37.0%

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

   7. Taylor expanded in eps around 0 3.1%

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

      1. neg-mul-1 3.1%

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

   9. Simplified 3.1%

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

   10. Taylor expanded in x around 0 58.2%

       \[\leadsto \color{blue}{1 + x \cdot \left(0.25 \cdot x - 0.5\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.25 + x \cdot -0.08333333333333333\right) - 0.5\right)\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+166}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot 0.25 - 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.0% accurate, 18.9× speedup

Math

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

FPCore

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

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.0) {
		tmp = (x * (1.0 - eps_m)) / eps_m;
	} else if (x <= 480.0) {
		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 <= (-1.0d0)) then
        tmp = (x * (1.0d0 - eps_m)) / eps_m
    else if (x <= 480.0d0) 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 <= -1.0) {
		tmp = (x * (1.0 - eps_m)) / eps_m;
	} else if (x <= 480.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

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

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(x * Float64(1.0 - eps_m)) / eps_m);
	elseif (x <= 480.0)
		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 <= -1.0)
		tmp = (x * (1.0 - eps_m)) / eps_m;
	elseif (x <= 480.0)
		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, -1.0], N[(N[(x * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision], If[LessEqual[x, 480.0], 1.0, 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 94.9%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in x around 0 3.0%

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

      1. add-sqr-sqrt 1.4%

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

      2. sqrt-unprod 3.0%

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

      3. frac-times 3.0%

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

      4. metadata-eval 3.0%

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

      5. metadata-eval 3.0%

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

      6. frac-times 3.0%

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

      7. sqrt-unprod 1.6%

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

      8. add-sqr-sqrt 3.1%

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

      9. div-inv 3.1%

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

   7. Applied egg-rr 3.1%

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

   8. Taylor expanded in eps around 0 53.1%

      \[\leadsto \color{blue}{\frac{x + 0.5 \cdot \left(\varepsilon \cdot \left(2 + -2 \cdot x\right)\right)}{\varepsilon}} \]
   9. Step-by-step derivation

      1. *-commutative 53.1%

         \[\leadsto \frac{x + \color{blue}{\left(\varepsilon \cdot \left(2 + -2 \cdot x\right)\right) \cdot 0.5}}{\varepsilon} \]

      2. associate-*l* 53.1%

         \[\leadsto \frac{x + \color{blue}{\varepsilon \cdot \left(\left(2 + -2 \cdot x\right) \cdot 0.5\right)}}{\varepsilon} \]

      3. *-commutative 53.1%

         \[\leadsto \frac{x + \varepsilon \cdot \color{blue}{\left(0.5 \cdot \left(2 + -2 \cdot x\right)\right)}}{\varepsilon} \]

      4. distribute-lft-in 53.1%

         \[\leadsto \frac{x + \varepsilon \cdot \color{blue}{\left(0.5 \cdot 2 + 0.5 \cdot \left(-2 \cdot x\right)\right)}}{\varepsilon} \]

      5. metadata-eval 53.1%

         \[\leadsto \frac{x + \varepsilon \cdot \left(\color{blue}{1} + 0.5 \cdot \left(-2 \cdot x\right)\right)}{\varepsilon} \]

      6. associate-*r* 53.1%

         \[\leadsto \frac{x + \varepsilon \cdot \left(1 + \color{blue}{\left(0.5 \cdot -2\right) \cdot x}\right)}{\varepsilon} \]

      7. metadata-eval 53.1%

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

      8. mul-1-neg 53.1%

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

   10. Simplified 53.1%

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

   11. Taylor expanded in x around inf 53.1%

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

       1. neg-mul-1 53.1%

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

       2. sub-neg 53.1%

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

   13. Simplified 53.1%

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

   if -1 < x < 480

   1. Initial program 54.1%

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

   3. Simplified 40.6%

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

   5. Taylor expanded in x around 0 75.0%

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

      1. add-sqr-sqrt 38.7%

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

      2. sqrt-unprod 52.5%

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

      3. frac-times 51.2%

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

      4. metadata-eval 51.2%

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

      5. metadata-eval 51.2%

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

      6. frac-times 52.5%

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

      7. sqrt-unprod 22.6%

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

      8. add-sqr-sqrt 57.4%

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

      9. div-inv 57.4%

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

   7. Applied egg-rr 57.4%

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

   8. Taylor expanded in x around 0 75.0%

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

   if 480 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 48.4%

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

      1. div-sub 48.4%

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

      2. mul-1-neg 48.4%

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

      3. rec-exp 48.4%

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

      4. +-inverses 48.4%

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

      5. metadata-eval 48.4%

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

   7. Simplified 48.4%

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

Alternative 13: 58.1% accurate, 28.3× 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 63.0%

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

   3. Simplified 52.5%

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

   5. Taylor expanded in x around 0 59.2%

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

      1. add-sqr-sqrt 30.5%

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

      2. sqrt-unprod 41.6%

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

      3. frac-times 40.6%

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

      4. metadata-eval 40.6%

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

      5. metadata-eval 40.6%

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

      6. frac-times 41.6%

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

      7. sqrt-unprod 18.0%

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

      8. add-sqr-sqrt 45.5%

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

      9. div-inv 45.5%

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

   7. Applied egg-rr 45.5%

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

   8. Taylor expanded in eps around 0 56.8%

      \[\leadsto \color{blue}{\frac{x + 0.5 \cdot \left(\varepsilon \cdot \left(2 + -2 \cdot x\right)\right)}{\varepsilon}} \]
   9. Step-by-step derivation

      1. *-commutative 56.8%

         \[\leadsto \frac{x + \color{blue}{\left(\varepsilon \cdot \left(2 + -2 \cdot x\right)\right) \cdot 0.5}}{\varepsilon} \]

      2. associate-*l* 56.8%

         \[\leadsto \frac{x + \color{blue}{\varepsilon \cdot \left(\left(2 + -2 \cdot x\right) \cdot 0.5\right)}}{\varepsilon} \]

      3. *-commutative 56.8%

         \[\leadsto \frac{x + \varepsilon \cdot \color{blue}{\left(0.5 \cdot \left(2 + -2 \cdot x\right)\right)}}{\varepsilon} \]

      4. distribute-lft-in 56.8%

         \[\leadsto \frac{x + \varepsilon \cdot \color{blue}{\left(0.5 \cdot 2 + 0.5 \cdot \left(-2 \cdot x\right)\right)}}{\varepsilon} \]

      5. metadata-eval 56.8%

         \[\leadsto \frac{x + \varepsilon \cdot \left(\color{blue}{1} + 0.5 \cdot \left(-2 \cdot x\right)\right)}{\varepsilon} \]

      6. associate-*r* 56.8%

         \[\leadsto \frac{x + \varepsilon \cdot \left(1 + \color{blue}{\left(0.5 \cdot -2\right) \cdot x}\right)}{\varepsilon} \]

      7. metadata-eval 56.8%

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

      8. mul-1-neg 56.8%

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

   10. Simplified 56.8%

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

   11. Taylor expanded in eps around inf 59.6%

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

   if 1 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 47.8%

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

      1. div-sub 47.8%

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

      2. mul-1-neg 47.8%

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

      3. rec-exp 47.8%

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

      4. +-inverses 47.8%

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

      5. metadata-eval 47.8%

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

   7. Simplified 47.8%

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

Alternative 14: 58.1% accurate, 37.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 500:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 500.0) 1.0 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 500.0) {
		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 <= 500.0d0) 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 <= 500.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 500.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 500.0)
		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 <= 500.0)
		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, 500.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 500

   1. Initial program 63.2%

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

   3. Simplified 52.7%

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

   5. Taylor expanded in x around 0 58.9%

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

      1. add-sqr-sqrt 30.4%

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

      2. sqrt-unprod 41.4%

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

      3. frac-times 40.4%

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

      4. metadata-eval 40.4%

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

      5. metadata-eval 40.4%

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

      6. frac-times 41.4%

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

      7. sqrt-unprod 17.9%

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

      8. add-sqr-sqrt 45.3%

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

      9. div-inv 45.3%

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

   7. Applied egg-rr 45.3%

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

   8. Taylor expanded in x around 0 58.9%

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

   if 500 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 48.4%

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

      1. div-sub 48.4%

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

      2. mul-1-neg 48.4%

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

      3. rec-exp 48.4%

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

      4. +-inverses 48.4%

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

      5. metadata-eval 48.4%

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

   7. Simplified 48.4%

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

Alternative 15: 16.8% accurate, 227.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ 0 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 0.0)

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return 0.0;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    code = 0.0d0
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return 0.0;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return 0.0

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return 0.0
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = 0.0;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := 0.0

Derivation

1. Initial program 75.0%

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

3. Simplified 67.9%

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

5. Taylor expanded in eps around 0 17.0%

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

   1. div-sub 17.0%

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

   2. mul-1-neg 17.0%

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

   3. rec-exp 17.0%

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

   4. +-inverses 17.2%

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

   5. metadata-eval 17.2%

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

7. Simplified 17.2%

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

Reproduce

herbie shell --seed 2024136 
(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))