NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.0% → 99.4%

Time: 15.3s

Alternatives: 21

Speedup: 1.9×

• 37.6% 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: 73.0% 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: 99.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 0.00084:\\ \;\;\;\;\frac{2 \cdot e^{\mathsf{log1p}\left(x\right) - x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-eps\_m\right)} + e^{x \cdot eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 0.00084)
   (/ (* 2.0 (exp (- (log1p x) x))) 2.0)
   (/ (+ (exp (* x (- eps_m))) (exp (* x eps_m))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.00084) {
		tmp = (2.0 * exp((log1p(x) - x))) / 2.0;
	} else {
		tmp = (exp((x * -eps_m)) + exp((x * eps_m))) / 2.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 0.00084) {
		tmp = (2.0 * Math.exp((Math.log1p(x) - x))) / 2.0;
	} else {
		tmp = (Math.exp((x * -eps_m)) + Math.exp((x * eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 0.00084:
		tmp = (2.0 * math.exp((math.log1p(x) - x))) / 2.0
	else:
		tmp = (math.exp((x * -eps_m)) + math.exp((x * eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 0.00084)
		tmp = Float64(Float64(2.0 * exp(Float64(log1p(x) - x))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(-eps_m))) + exp(Float64(x * eps_m))) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 0.00084], N[(N[(2.0 * N[Exp[N[(N[Log[1 + x], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 8.4000000000000003e-4

   1. Initial program 61.4%

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

   3. Simplified 61.4%

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

   5. Taylor expanded in eps around 0 67.5%

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

      1. associate--l+ 67.5%

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

      2. neg-mul-1 67.5%

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

      3. neg-mul-1 67.5%

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

      4. distribute-lft-out 67.5%

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

      5. distribute-rgt1-in 67.5%

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

      6. neg-mul-1 67.5%

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

   7. Applied egg-rr 67.5%

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

      1. associate-+r- 67.5%

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

      2. distribute-rgt1-in 68.6%

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

      3. cancel-sign-sub-inv 68.6%

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

      4. metadata-eval 68.6%

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

      5. distribute-rgt1-in 68.6%

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

      6. metadata-eval 68.6%

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

      7. rem-exp-log 67.5%

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

      8. exp-sum 67.5%

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

      9. unsub-neg 67.5%

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

      10. +-commutative 67.5%

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

      11. log1p-define 67.5%

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

   9. Simplified 67.5%

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

   if 8.4000000000000003e-4 < 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 78.4%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in eps around inf 100.0%

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

       1. *-commutative 100.0%

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

   11. Simplified 100.0%

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

Alternative 2: 98.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.7%

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

3. Simplified 60.0%

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

5. Taylor expanded in eps around inf 98.3%

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

6. Final simplification 98.3%

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

Alternative 3: 69.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := x \cdot \left(1 + eps\_m\right)\\ t_1 := \frac{t\_0}{eps\_m}\\ t_2 := x \cdot \left(-1 - eps\_m\right)\\ \mathbf{if}\;x \leq -1.05 \cdot 10^{+65}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 7.233796296296296 \cdot 10^{-5} - 0.16666666666666666 \cdot \frac{1}{eps\_m}\right) + \frac{1}{eps\_m} \cdot 0.5\right) + \frac{-1}{eps\_m}\right)}{2}\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{t\_0 \cdot t\_0 + t\_1 \cdot \frac{t\_2}{eps\_m}}{t\_2 - t\_1}}{2}\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+93}:\\ \;\;\;\;\frac{\left(1 + \frac{-1}{eps\_m}\right) - \left(-1 - \frac{1}{eps\_m}\right)}{2}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+166}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot 1.000072337962963\right)}^{-4}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* x (+ 1.0 eps_m)))
        (t_1 (/ t_0 eps_m))
        (t_2 (* x (- -1.0 eps_m))))
   (if (<= x -1.05e+65)
     (/
      (*
       x
       (+
        (*
         x
         (+
          (*
           x
           (-
            (* x 7.233796296296296e-5)
            (* 0.16666666666666666 (/ 1.0 eps_m))))
          (* (/ 1.0 eps_m) 0.5)))
        (/ -1.0 eps_m)))
      2.0)
     (if (<= x -4.2e-10)
       (/ (/ (+ (* t_0 t_0) (* t_1 (/ t_2 eps_m))) (- t_2 t_1)) 2.0)
       (if (<= x 1.42)
         (/ (- 2.0 (* x x)) 2.0)
         (if (<= x 1.5e+93)
           (/ (- (+ 1.0 (/ -1.0 eps_m)) (- -1.0 (/ 1.0 eps_m))) 2.0)
           (if (<= x 4e+166)
             (/
              (/
               (*
                x
                (+
                 -1.0
                 (*
                  x
                  (+
                   0.5
                   (* x (- (* x 0.041666666666666664) 0.16666666666666666))))))
               eps_m)
              2.0)
             (pow (* x 1.000072337962963) -4.0))))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = x * (1.0 + eps_m);
	double t_1 = t_0 / eps_m;
	double t_2 = x * (-1.0 - eps_m);
	double tmp;
	if (x <= -1.05e+65) {
		tmp = (x * ((x * ((x * ((x * 7.233796296296296e-5) - (0.16666666666666666 * (1.0 / eps_m)))) + ((1.0 / eps_m) * 0.5))) + (-1.0 / eps_m))) / 2.0;
	} else if (x <= -4.2e-10) {
		tmp = (((t_0 * t_0) + (t_1 * (t_2 / eps_m))) / (t_2 - t_1)) / 2.0;
	} else if (x <= 1.42) {
		tmp = (2.0 - (x * x)) / 2.0;
	} else if (x <= 1.5e+93) {
		tmp = ((1.0 + (-1.0 / eps_m)) - (-1.0 - (1.0 / eps_m))) / 2.0;
	} else if (x <= 4e+166) {
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0;
	} else {
		tmp = pow((x * 1.000072337962963), -4.0);
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x * (1.0d0 + eps_m)
    t_1 = t_0 / eps_m
    t_2 = x * ((-1.0d0) - eps_m)
    if (x <= (-1.05d+65)) then
        tmp = (x * ((x * ((x * ((x * 7.233796296296296d-5) - (0.16666666666666666d0 * (1.0d0 / eps_m)))) + ((1.0d0 / eps_m) * 0.5d0))) + ((-1.0d0) / eps_m))) / 2.0d0
    else if (x <= (-4.2d-10)) then
        tmp = (((t_0 * t_0) + (t_1 * (t_2 / eps_m))) / (t_2 - t_1)) / 2.0d0
    else if (x <= 1.42d0) then
        tmp = (2.0d0 - (x * x)) / 2.0d0
    else if (x <= 1.5d+93) then
        tmp = ((1.0d0 + ((-1.0d0) / eps_m)) - ((-1.0d0) - (1.0d0 / eps_m))) / 2.0d0
    else if (x <= 4d+166) then
        tmp = ((x * ((-1.0d0) + (x * (0.5d0 + (x * ((x * 0.041666666666666664d0) - 0.16666666666666666d0)))))) / eps_m) / 2.0d0
    else
        tmp = (x * 1.000072337962963d0) ** (-4.0d0)
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = x * (1.0 + eps_m);
	double t_1 = t_0 / eps_m;
	double t_2 = x * (-1.0 - eps_m);
	double tmp;
	if (x <= -1.05e+65) {
		tmp = (x * ((x * ((x * ((x * 7.233796296296296e-5) - (0.16666666666666666 * (1.0 / eps_m)))) + ((1.0 / eps_m) * 0.5))) + (-1.0 / eps_m))) / 2.0;
	} else if (x <= -4.2e-10) {
		tmp = (((t_0 * t_0) + (t_1 * (t_2 / eps_m))) / (t_2 - t_1)) / 2.0;
	} else if (x <= 1.42) {
		tmp = (2.0 - (x * x)) / 2.0;
	} else if (x <= 1.5e+93) {
		tmp = ((1.0 + (-1.0 / eps_m)) - (-1.0 - (1.0 / eps_m))) / 2.0;
	} else if (x <= 4e+166) {
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0;
	} else {
		tmp = Math.pow((x * 1.000072337962963), -4.0);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = x * (1.0 + eps_m)
	t_1 = t_0 / eps_m
	t_2 = x * (-1.0 - eps_m)
	tmp = 0
	if x <= -1.05e+65:
		tmp = (x * ((x * ((x * ((x * 7.233796296296296e-5) - (0.16666666666666666 * (1.0 / eps_m)))) + ((1.0 / eps_m) * 0.5))) + (-1.0 / eps_m))) / 2.0
	elif x <= -4.2e-10:
		tmp = (((t_0 * t_0) + (t_1 * (t_2 / eps_m))) / (t_2 - t_1)) / 2.0
	elif x <= 1.42:
		tmp = (2.0 - (x * x)) / 2.0
	elif x <= 1.5e+93:
		tmp = ((1.0 + (-1.0 / eps_m)) - (-1.0 - (1.0 / eps_m))) / 2.0
	elif x <= 4e+166:
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0
	else:
		tmp = math.pow((x * 1.000072337962963), -4.0)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(x * Float64(1.0 + eps_m))
	t_1 = Float64(t_0 / eps_m)
	t_2 = Float64(x * Float64(-1.0 - eps_m))
	tmp = 0.0
	if (x <= -1.05e+65)
		tmp = Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * 7.233796296296296e-5) - Float64(0.16666666666666666 * Float64(1.0 / eps_m)))) + Float64(Float64(1.0 / eps_m) * 0.5))) + Float64(-1.0 / eps_m))) / 2.0);
	elseif (x <= -4.2e-10)
		tmp = Float64(Float64(Float64(Float64(t_0 * t_0) + Float64(t_1 * Float64(t_2 / eps_m))) / Float64(t_2 - t_1)) / 2.0);
	elseif (x <= 1.42)
		tmp = Float64(Float64(2.0 - Float64(x * x)) / 2.0);
	elseif (x <= 1.5e+93)
		tmp = Float64(Float64(Float64(1.0 + Float64(-1.0 / eps_m)) - Float64(-1.0 - Float64(1.0 / eps_m))) / 2.0);
	elseif (x <= 4e+166)
		tmp = Float64(Float64(Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(Float64(x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0);
	else
		tmp = Float64(x * 1.000072337962963) ^ -4.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = x * (1.0 + eps_m);
	t_1 = t_0 / eps_m;
	t_2 = x * (-1.0 - eps_m);
	tmp = 0.0;
	if (x <= -1.05e+65)
		tmp = (x * ((x * ((x * ((x * 7.233796296296296e-5) - (0.16666666666666666 * (1.0 / eps_m)))) + ((1.0 / eps_m) * 0.5))) + (-1.0 / eps_m))) / 2.0;
	elseif (x <= -4.2e-10)
		tmp = (((t_0 * t_0) + (t_1 * (t_2 / eps_m))) / (t_2 - t_1)) / 2.0;
	elseif (x <= 1.42)
		tmp = (2.0 - (x * x)) / 2.0;
	elseif (x <= 1.5e+93)
		tmp = ((1.0 + (-1.0 / eps_m)) - (-1.0 - (1.0 / eps_m))) / 2.0;
	elseif (x <= 4e+166)
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0;
	else
		tmp = (x * 1.000072337962963) ^ -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(x * N[(1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / eps$95$m), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.05e+65], N[(N[(x * N[(N[(x * N[(N[(x * N[(N[(x * 7.233796296296296e-5), $MachinePrecision] - N[(0.16666666666666666 * N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / eps$95$m), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -4.2e-10], N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[(t$95$1 * N[(t$95$2 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 - t$95$1), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.42], N[(N[(2.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.5e+93], N[(N[(N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] - N[(-1.0 - N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 4e+166], N[(N[(N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(x * N[(N[(x * 0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], N[Power[N[(x * 1.000072337962963), $MachinePrecision], -4.0], $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < -1.04999999999999996e65

   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%

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

   5. Taylor expanded in x around 0 51.6%

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

   6. Taylor expanded in eps around 0 50.0%

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

      1. expm1-define 50.0%

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

      2. neg-mul-1 50.0%

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

   8. Simplified 50.0%

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

   9. Taylor expanded in x around 0 30.2%

      \[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(0.041666666666666664 \cdot \frac{x}{\varepsilon} - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}}{2} \]

   11. Applied egg-rr 88.7%

       \[\leadsto \frac{x \cdot \left(x \cdot \left(x \cdot \left(\color{blue}{\left(\left(-4 + x \cdot 7.233796296296296 \cdot 10^{-5}\right) - -4\right)} - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}{2} \]
   12. Step-by-step derivation

       1. +-commutative 88.7%

          \[\leadsto \frac{x \cdot \left(x \cdot \left(x \cdot \left(\left(\color{blue}{\left(x \cdot 7.233796296296296 \cdot 10^{-5} + -4\right)} - -4\right) - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}{2} \]

       2. associate--l+ 88.7%

          \[\leadsto \frac{x \cdot \left(x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot 7.233796296296296 \cdot 10^{-5} + \left(-4 - -4\right)\right)} - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}{2} \]

       3. metadata-eval 88.7%

          \[\leadsto \frac{x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot 7.233796296296296 \cdot 10^{-5} + \color{blue}{0}\right) - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}{2} \]

       4. +-rgt-identity 88.7%

          \[\leadsto \frac{x \cdot \left(x \cdot \left(x \cdot \left(\color{blue}{x \cdot 7.233796296296296 \cdot 10^{-5}} - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}{2} \]

   13. Simplified 88.7%

       \[\leadsto \frac{x \cdot \left(x \cdot \left(x \cdot \left(\color{blue}{x \cdot 7.233796296296296 \cdot 10^{-5}} - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}{2} \]

   if -1.04999999999999996e65 < x < -4.2e-10

   1. Initial program 82.5%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in x around 0 37.4%

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

   6. Taylor expanded in x around inf 3.3%

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

      1. mul-1-neg 3.3%

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

      2. distribute-rgt-neg-in 3.3%

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

      3. *-commutative 3.3%

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

      4. distribute-rgt-neg-in 3.3%

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

      5. mul-1-neg 3.3%

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

      6. distribute-lft-in 3.3%

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

      7. metadata-eval 3.3%

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

      8. neg-mul-1 3.3%

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

      9. distribute-neg-frac 3.3%

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

      10. metadata-eval 3.3%

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

   8. Simplified 3.3%

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

   10. Applied egg-rr 12.9%

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

   if -4.2e-10 < x < 1.4199999999999999

   1. Initial program 54.2%

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

   3. Simplified 54.2%

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

   5. Taylor expanded in eps around 0 74.2%

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

   6. Taylor expanded in x around 0 73.8%

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

      1. mul-1-neg 73.8%

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

      2. unsub-neg 73.8%

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

   8. Simplified 73.8%

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

   10. Applied egg-rr 73.8%

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

   if 1.4199999999999999 < x < 1.49999999999999989e93

   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%

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

   5. Taylor expanded in x around 0 43.4%

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

   6. Taylor expanded in x around 0 54.1%

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

   if 1.49999999999999989e93 < x < 3.99999999999999976e166

   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%

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

   5. Taylor expanded in x around 0 39.2%

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

   6. Taylor expanded in eps around 0 1.8%

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

      1. expm1-define 1.8%

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

      2. neg-mul-1 1.8%

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

   8. Simplified 1.8%

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

   9. Taylor expanded in x around 0 38.0%

      \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(x \cdot \left(0.5 + x \cdot \left(0.041666666666666664 \cdot x - 0.16666666666666666\right)\right) - 1\right)}}{\varepsilon}}{2} \]

   if 3.99999999999999976e166 < 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%

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

   5. Taylor expanded in x around 0 21.2%

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

   6. Taylor expanded in eps around 0 2.0%

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

      1. expm1-define 2.0%

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

      2. neg-mul-1 2.0%

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

   8. Simplified 2.0%

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

   9. Taylor expanded in x around 0 20.2%

      \[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(0.041666666666666664 \cdot \frac{x}{\varepsilon} - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}}{2} \]

   11. Applied egg-rr 62.1%

       \[\leadsto \color{blue}{{\left(x \cdot 7.233796296296296 \cdot 10^{-5} + x\right)}^{-4}} \]
   12. Step-by-step derivation

       1. *-rgt-identity 62.1%

          \[\leadsto {\left(x \cdot 7.233796296296296 \cdot 10^{-5} + \color{blue}{x \cdot 1}\right)}^{-4} \]

       2. distribute-lft-out 62.1%

          \[\leadsto {\color{blue}{\left(x \cdot \left(7.233796296296296 \cdot 10^{-5} + 1\right)\right)}}^{-4} \]

       3. metadata-eval 62.1%

          \[\leadsto {\left(x \cdot \color{blue}{1.000072337962963}\right)}^{-4} \]

   13. Simplified 62.1%

       \[\leadsto \color{blue}{{\left(x \cdot 1.000072337962963\right)}^{-4}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+65}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 7.233796296296296 \cdot 10^{-5} - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + \frac{1}{\varepsilon} \cdot 0.5\right) + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{\left(x \cdot \left(1 + \varepsilon\right)\right) \cdot \left(x \cdot \left(1 + \varepsilon\right)\right) + \frac{x \cdot \left(1 + \varepsilon\right)}{\varepsilon} \cdot \frac{x \cdot \left(-1 - \varepsilon\right)}{\varepsilon}}{x \cdot \left(-1 - \varepsilon\right) - \frac{x \cdot \left(1 + \varepsilon\right)}{\varepsilon}}}{2}\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+93}:\\ \;\;\;\;\frac{\left(1 + \frac{-1}{\varepsilon}\right) - \left(-1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+166}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot 1.000072337962963\right)}^{-4}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := x \cdot \left(1 + eps\_m\right)\\ t_1 := \frac{t\_0}{eps\_m}\\ t_2 := x \cdot \left(-1 - eps\_m\right)\\ \mathbf{if}\;x \leq -1.05 \cdot 10^{+65}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 7.233796296296296 \cdot 10^{-5} - 0.16666666666666666 \cdot \frac{1}{eps\_m}\right) + \frac{1}{eps\_m} \cdot 0.5\right) + \frac{-1}{eps\_m}\right)}{2}\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{t\_0 \cdot t\_0 + t\_1 \cdot \frac{t\_2}{eps\_m}}{t\_2 - t\_1}}{2}\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+170}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot 1.000072337962963\right)}^{-4}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* x (+ 1.0 eps_m)))
        (t_1 (/ t_0 eps_m))
        (t_2 (* x (- -1.0 eps_m))))
   (if (<= x -1.05e+65)
     (/
      (*
       x
       (+
        (*
         x
         (+
          (*
           x
           (-
            (* x 7.233796296296296e-5)
            (* 0.16666666666666666 (/ 1.0 eps_m))))
          (* (/ 1.0 eps_m) 0.5)))
        (/ -1.0 eps_m)))
      2.0)
     (if (<= x -4.2e-10)
       (/ (/ (+ (* t_0 t_0) (* t_1 (/ t_2 eps_m))) (- t_2 t_1)) 2.0)
       (if (<= x 1.42)
         (/ (- 2.0 (* x x)) 2.0)
         (if (<= x 3e+170)
           (/ (/ (expm1 x) eps_m) 2.0)
           (pow (* x 1.000072337962963) -4.0)))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = x * (1.0 + eps_m);
	double t_1 = t_0 / eps_m;
	double t_2 = x * (-1.0 - eps_m);
	double tmp;
	if (x <= -1.05e+65) {
		tmp = (x * ((x * ((x * ((x * 7.233796296296296e-5) - (0.16666666666666666 * (1.0 / eps_m)))) + ((1.0 / eps_m) * 0.5))) + (-1.0 / eps_m))) / 2.0;
	} else if (x <= -4.2e-10) {
		tmp = (((t_0 * t_0) + (t_1 * (t_2 / eps_m))) / (t_2 - t_1)) / 2.0;
	} else if (x <= 1.42) {
		tmp = (2.0 - (x * x)) / 2.0;
	} else if (x <= 3e+170) {
		tmp = (expm1(x) / eps_m) / 2.0;
	} else {
		tmp = pow((x * 1.000072337962963), -4.0);
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = x * (1.0 + eps_m);
	double t_1 = t_0 / eps_m;
	double t_2 = x * (-1.0 - eps_m);
	double tmp;
	if (x <= -1.05e+65) {
		tmp = (x * ((x * ((x * ((x * 7.233796296296296e-5) - (0.16666666666666666 * (1.0 / eps_m)))) + ((1.0 / eps_m) * 0.5))) + (-1.0 / eps_m))) / 2.0;
	} else if (x <= -4.2e-10) {
		tmp = (((t_0 * t_0) + (t_1 * (t_2 / eps_m))) / (t_2 - t_1)) / 2.0;
	} else if (x <= 1.42) {
		tmp = (2.0 - (x * x)) / 2.0;
	} else if (x <= 3e+170) {
		tmp = (Math.expm1(x) / eps_m) / 2.0;
	} else {
		tmp = Math.pow((x * 1.000072337962963), -4.0);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = x * (1.0 + eps_m)
	t_1 = t_0 / eps_m
	t_2 = x * (-1.0 - eps_m)
	tmp = 0
	if x <= -1.05e+65:
		tmp = (x * ((x * ((x * ((x * 7.233796296296296e-5) - (0.16666666666666666 * (1.0 / eps_m)))) + ((1.0 / eps_m) * 0.5))) + (-1.0 / eps_m))) / 2.0
	elif x <= -4.2e-10:
		tmp = (((t_0 * t_0) + (t_1 * (t_2 / eps_m))) / (t_2 - t_1)) / 2.0
	elif x <= 1.42:
		tmp = (2.0 - (x * x)) / 2.0
	elif x <= 3e+170:
		tmp = (math.expm1(x) / eps_m) / 2.0
	else:
		tmp = math.pow((x * 1.000072337962963), -4.0)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(x * Float64(1.0 + eps_m))
	t_1 = Float64(t_0 / eps_m)
	t_2 = Float64(x * Float64(-1.0 - eps_m))
	tmp = 0.0
	if (x <= -1.05e+65)
		tmp = Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * 7.233796296296296e-5) - Float64(0.16666666666666666 * Float64(1.0 / eps_m)))) + Float64(Float64(1.0 / eps_m) * 0.5))) + Float64(-1.0 / eps_m))) / 2.0);
	elseif (x <= -4.2e-10)
		tmp = Float64(Float64(Float64(Float64(t_0 * t_0) + Float64(t_1 * Float64(t_2 / eps_m))) / Float64(t_2 - t_1)) / 2.0);
	elseif (x <= 1.42)
		tmp = Float64(Float64(2.0 - Float64(x * x)) / 2.0);
	elseif (x <= 3e+170)
		tmp = Float64(Float64(expm1(x) / eps_m) / 2.0);
	else
		tmp = Float64(x * 1.000072337962963) ^ -4.0;
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(x * N[(1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / eps$95$m), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.05e+65], N[(N[(x * N[(N[(x * N[(N[(x * N[(N[(x * 7.233796296296296e-5), $MachinePrecision] - N[(0.16666666666666666 * N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / eps$95$m), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -4.2e-10], N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[(t$95$1 * N[(t$95$2 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 - t$95$1), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.42], N[(N[(2.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3e+170], N[(N[(N[(Exp[x] - 1), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], N[Power[N[(x * 1.000072337962963), $MachinePrecision], -4.0], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1.04999999999999996e65

   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%

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

   5. Taylor expanded in x around 0 51.6%

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

   6. Taylor expanded in eps around 0 50.0%

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

      1. expm1-define 50.0%

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

      2. neg-mul-1 50.0%

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

   8. Simplified 50.0%

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

   9. Taylor expanded in x around 0 30.2%

      \[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(0.041666666666666664 \cdot \frac{x}{\varepsilon} - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}}{2} \]

   11. Applied egg-rr 88.7%

       \[\leadsto \frac{x \cdot \left(x \cdot \left(x \cdot \left(\color{blue}{\left(\left(-4 + x \cdot 7.233796296296296 \cdot 10^{-5}\right) - -4\right)} - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}{2} \]
   12. Step-by-step derivation

       1. +-commutative 88.7%

          \[\leadsto \frac{x \cdot \left(x \cdot \left(x \cdot \left(\left(\color{blue}{\left(x \cdot 7.233796296296296 \cdot 10^{-5} + -4\right)} - -4\right) - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}{2} \]

       2. associate--l+ 88.7%

          \[\leadsto \frac{x \cdot \left(x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot 7.233796296296296 \cdot 10^{-5} + \left(-4 - -4\right)\right)} - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}{2} \]

       3. metadata-eval 88.7%

          \[\leadsto \frac{x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot 7.233796296296296 \cdot 10^{-5} + \color{blue}{0}\right) - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}{2} \]

       4. +-rgt-identity 88.7%

          \[\leadsto \frac{x \cdot \left(x \cdot \left(x \cdot \left(\color{blue}{x \cdot 7.233796296296296 \cdot 10^{-5}} - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}{2} \]

   13. Simplified 88.7%

       \[\leadsto \frac{x \cdot \left(x \cdot \left(x \cdot \left(\color{blue}{x \cdot 7.233796296296296 \cdot 10^{-5}} - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}{2} \]

   if -1.04999999999999996e65 < x < -4.2e-10

   1. Initial program 82.5%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in x around 0 37.4%

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

   6. Taylor expanded in x around inf 3.3%

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

      1. mul-1-neg 3.3%

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

      2. distribute-rgt-neg-in 3.3%

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

      3. *-commutative 3.3%

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

      4. distribute-rgt-neg-in 3.3%

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

      5. mul-1-neg 3.3%

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

      6. distribute-lft-in 3.3%

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

      7. metadata-eval 3.3%

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

      8. neg-mul-1 3.3%

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

      9. distribute-neg-frac 3.3%

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

      10. metadata-eval 3.3%

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

   8. Simplified 3.3%

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

   10. Applied egg-rr 12.9%

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

   if -4.2e-10 < x < 1.4199999999999999

   1. Initial program 54.2%

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

   3. Simplified 54.2%

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

   5. Taylor expanded in eps around 0 74.2%

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

   6. Taylor expanded in x around 0 73.8%

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

      1. mul-1-neg 73.8%

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

      2. unsub-neg 73.8%

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

   8. Simplified 73.8%

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

   10. Applied egg-rr 73.8%

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

   if 1.4199999999999999 < x < 2.99999999999999997e170

   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%

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

   5. Taylor expanded in x around 0 41.1%

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

   6. Taylor expanded in eps around 0 1.8%

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

      1. expm1-define 1.8%

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

      2. neg-mul-1 1.8%

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

   8. Simplified 1.8%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}}{2} \]
   9. Step-by-step derivation

      1. expm1-undefine 1.8%

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

      2. div-sub 1.8%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 40.2%

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

      5. sqr-neg 40.2%

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

      6. sqrt-unprod 40.2%

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

      7. add-sqr-sqrt 40.2%

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

   10. Applied egg-rr 40.2%

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

       1. div-sub 40.2%

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

       2. expm1-define 40.2%

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

   12. Simplified 40.2%

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

   if 2.99999999999999997e170 < 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%

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

   5. Taylor expanded in x around 0 21.2%

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

   6. Taylor expanded in eps around 0 2.0%

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

      1. expm1-define 2.0%

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

      2. neg-mul-1 2.0%

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

   8. Simplified 2.0%

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

   9. Taylor expanded in x around 0 20.2%

      \[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(0.041666666666666664 \cdot \frac{x}{\varepsilon} - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}}{2} \]

   11. Applied egg-rr 62.1%

       \[\leadsto \color{blue}{{\left(x \cdot 7.233796296296296 \cdot 10^{-5} + x\right)}^{-4}} \]
   12. Step-by-step derivation

       1. *-rgt-identity 62.1%

          \[\leadsto {\left(x \cdot 7.233796296296296 \cdot 10^{-5} + \color{blue}{x \cdot 1}\right)}^{-4} \]

       2. distribute-lft-out 62.1%

          \[\leadsto {\color{blue}{\left(x \cdot \left(7.233796296296296 \cdot 10^{-5} + 1\right)\right)}}^{-4} \]

       3. metadata-eval 62.1%

          \[\leadsto {\left(x \cdot \color{blue}{1.000072337962963}\right)}^{-4} \]

   13. Simplified 62.1%

       \[\leadsto \color{blue}{{\left(x \cdot 1.000072337962963\right)}^{-4}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+65}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 7.233796296296296 \cdot 10^{-5} - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + \frac{1}{\varepsilon} \cdot 0.5\right) + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{\left(x \cdot \left(1 + \varepsilon\right)\right) \cdot \left(x \cdot \left(1 + \varepsilon\right)\right) + \frac{x \cdot \left(1 + \varepsilon\right)}{\varepsilon} \cdot \frac{x \cdot \left(-1 - \varepsilon\right)}{\varepsilon}}{x \cdot \left(-1 - \varepsilon\right) - \frac{x \cdot \left(1 + \varepsilon\right)}{\varepsilon}}}{2}\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+170}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot 1.000072337962963\right)}^{-4}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-295}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(1 - eps\_m\right)}}{2}\\ \mathbf{elif}\;x \leq 6.7 \cdot 10^{+170}:\\ \;\;\;\;\frac{1 + e^{x \cdot eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot 1.000072337962963\right)}^{-4}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -4e-295)
   (/ (+ 1.0 (exp (* x (- 1.0 eps_m)))) 2.0)
   (if (<= x 6.7e+170)
     (/ (+ 1.0 (exp (* x eps_m))) 2.0)
     (pow (* x 1.000072337962963) -4.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -4e-295) {
		tmp = (1.0 + exp((x * (1.0 - eps_m)))) / 2.0;
	} else if (x <= 6.7e+170) {
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	} else {
		tmp = pow((x * 1.000072337962963), -4.0);
	}
	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 <= (-4d-295)) then
        tmp = (1.0d0 + exp((x * (1.0d0 - eps_m)))) / 2.0d0
    else if (x <= 6.7d+170) then
        tmp = (1.0d0 + exp((x * eps_m))) / 2.0d0
    else
        tmp = (x * 1.000072337962963d0) ** (-4.0d0)
    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 <= -4e-295) {
		tmp = (1.0 + Math.exp((x * (1.0 - eps_m)))) / 2.0;
	} else if (x <= 6.7e+170) {
		tmp = (1.0 + Math.exp((x * eps_m))) / 2.0;
	} else {
		tmp = Math.pow((x * 1.000072337962963), -4.0);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -4e-295:
		tmp = (1.0 + math.exp((x * (1.0 - eps_m)))) / 2.0
	elif x <= 6.7e+170:
		tmp = (1.0 + math.exp((x * eps_m))) / 2.0
	else:
		tmp = math.pow((x * 1.000072337962963), -4.0)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -4e-295)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(1.0 - eps_m)))) / 2.0);
	elseif (x <= 6.7e+170)
		tmp = Float64(Float64(1.0 + exp(Float64(x * eps_m))) / 2.0);
	else
		tmp = Float64(x * 1.000072337962963) ^ -4.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -4e-295)
		tmp = (1.0 + exp((x * (1.0 - eps_m)))) / 2.0;
	elseif (x <= 6.7e+170)
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	else
		tmp = (x * 1.000072337962963) ^ -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -4e-295], N[(N[(1.0 + N[Exp[N[(x * N[(1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 6.7e+170], N[(N[(1.0 + N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[Power[N[(x * 1.000072337962963), $MachinePrecision], -4.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.00000000000000024e-295

   1. Initial program 63.6%

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

   3. Simplified 63.6%

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

   5. Taylor expanded in x around 0 44.1%

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

   6. Taylor expanded in eps around inf 76.8%

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

      1. associate-*r* 76.8%

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

      2. neg-mul-1 76.8%

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

      3. *-commutative 76.8%

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

   8. Simplified 76.8%

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

      1. neg-sub0 76.8%

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

      2. sub-neg 76.8%

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

      3. add-sqr-sqrt 76.8%

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

      4. sqrt-unprod 76.1%

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

      5. sqr-neg 76.1%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 73.4%

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

   10. Applied egg-rr 73.4%

       \[\leadsto \frac{1 + e^{\left(1 - \varepsilon\right) \cdot \color{blue}{\left(0 + x\right)}}}{2} \]
   11. Step-by-step derivation

       1. +-lft-identity 73.4%

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

   12. Simplified 73.4%

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

   if -4.00000000000000024e-295 < x < 6.69999999999999984e170

   1. Initial program 75.3%

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

   3. Simplified 75.3%

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

   5. Taylor expanded in x around 0 39.3%

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

   6. Taylor expanded in eps around inf 62.9%

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

      1. associate-*r* 62.9%

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

      2. neg-mul-1 62.9%

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

      3. *-commutative 62.9%

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

   8. Simplified 62.9%

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

   9. Taylor expanded in eps around inf 63.2%

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

   if 6.69999999999999984e170 < 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%

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

   5. Taylor expanded in x around 0 21.2%

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

   6. Taylor expanded in eps around 0 2.0%

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

      1. expm1-define 2.0%

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

      2. neg-mul-1 2.0%

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

   8. Simplified 2.0%

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

   9. Taylor expanded in x around 0 20.2%

      \[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(0.041666666666666664 \cdot \frac{x}{\varepsilon} - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}}{2} \]

   11. Applied egg-rr 62.1%

       \[\leadsto \color{blue}{{\left(x \cdot 7.233796296296296 \cdot 10^{-5} + x\right)}^{-4}} \]
   12. Step-by-step derivation

       1. *-rgt-identity 62.1%

          \[\leadsto {\left(x \cdot 7.233796296296296 \cdot 10^{-5} + \color{blue}{x \cdot 1}\right)}^{-4} \]

       2. distribute-lft-out 62.1%

          \[\leadsto {\color{blue}{\left(x \cdot \left(7.233796296296296 \cdot 10^{-5} + 1\right)\right)}}^{-4} \]

       3. metadata-eval 62.1%

          \[\leadsto {\left(x \cdot \color{blue}{1.000072337962963}\right)}^{-4} \]

   13. Simplified 62.1%

       \[\leadsto \color{blue}{{\left(x \cdot 1.000072337962963\right)}^{-4}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-295}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 6.7 \cdot 10^{+170}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot 1.000072337962963\right)}^{-4}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-295}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+171}:\\ \;\;\;\;\frac{1 + e^{x \cdot eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot 1.000072337962963\right)}^{-4}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -4e-295)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (<= x 1.9e+171)
     (/ (+ 1.0 (exp (* x eps_m))) 2.0)
     (pow (* x 1.000072337962963) -4.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -4e-295) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 1.9e+171) {
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	} else {
		tmp = pow((x * 1.000072337962963), -4.0);
	}
	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 <= (-4d-295)) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if (x <= 1.9d+171) then
        tmp = (1.0d0 + exp((x * eps_m))) / 2.0d0
    else
        tmp = (x * 1.000072337962963d0) ** (-4.0d0)
    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 <= -4e-295) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 1.9e+171) {
		tmp = (1.0 + Math.exp((x * eps_m))) / 2.0;
	} else {
		tmp = Math.pow((x * 1.000072337962963), -4.0);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -4e-295:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 1.9e+171:
		tmp = (1.0 + math.exp((x * eps_m))) / 2.0
	else:
		tmp = math.pow((x * 1.000072337962963), -4.0)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -4e-295)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 1.9e+171)
		tmp = Float64(Float64(1.0 + exp(Float64(x * eps_m))) / 2.0);
	else
		tmp = Float64(x * 1.000072337962963) ^ -4.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -4e-295)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif (x <= 1.9e+171)
		tmp = (1.0 + exp((x * eps_m))) / 2.0;
	else
		tmp = (x * 1.000072337962963) ^ -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -4e-295], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.9e+171], N[(N[(1.0 + N[Exp[N[(x * eps$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[Power[N[(x * 1.000072337962963), $MachinePrecision], -4.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.00000000000000024e-295

   1. Initial program 63.6%

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

   3. Simplified 47.2%

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

   5. Taylor expanded in eps around inf 97.3%

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

   6. Taylor expanded in eps around inf 97.3%

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

      1. mul-1-neg 97.3%

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

      2. *-commutative 97.3%

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

      3. distribute-rgt-neg-in 97.3%

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

   8. Simplified 97.3%

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

   9. Taylor expanded in eps around 0 86.0%

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

       1. neg-mul-1 86.0%

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

   11. Simplified 86.0%

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

   if -4.00000000000000024e-295 < x < 1.9000000000000001e171

   1. Initial program 75.3%

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

   3. Simplified 75.3%

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

   5. Taylor expanded in x around 0 39.3%

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

   6. Taylor expanded in eps around inf 62.9%

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

      1. associate-*r* 62.9%

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

      2. neg-mul-1 62.9%

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

      3. *-commutative 62.9%

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

   8. Simplified 62.9%

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

   9. Taylor expanded in eps around inf 63.2%

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

   if 1.9000000000000001e171 < 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%

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

   5. Taylor expanded in x around 0 21.2%

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

   6. Taylor expanded in eps around 0 2.0%

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

      1. expm1-define 2.0%

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

      2. neg-mul-1 2.0%

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

   8. Simplified 2.0%

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

   9. Taylor expanded in x around 0 20.2%

      \[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(0.041666666666666664 \cdot \frac{x}{\varepsilon} - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}}{2} \]

   11. Applied egg-rr 62.1%

       \[\leadsto \color{blue}{{\left(x \cdot 7.233796296296296 \cdot 10^{-5} + x\right)}^{-4}} \]
   12. Step-by-step derivation

       1. *-rgt-identity 62.1%

          \[\leadsto {\left(x \cdot 7.233796296296296 \cdot 10^{-5} + \color{blue}{x \cdot 1}\right)}^{-4} \]

       2. distribute-lft-out 62.1%

          \[\leadsto {\color{blue}{\left(x \cdot \left(7.233796296296296 \cdot 10^{-5} + 1\right)\right)}}^{-4} \]

       3. metadata-eval 62.1%

          \[\leadsto {\left(x \cdot \color{blue}{1.000072337962963}\right)}^{-4} \]

   13. Simplified 62.1%

       \[\leadsto \color{blue}{{\left(x \cdot 1.000072337962963\right)}^{-4}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-295}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+171}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot 1.000072337962963\right)}^{-4}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 600:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+167}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(x \cdot 1.000072337962963\right)}^{-4}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 600.0)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (<= x 7e+167)
     (/ (/ (expm1 x) eps_m) 2.0)
     (pow (* x 1.000072337962963) -4.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 600.0) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 7e+167) {
		tmp = (expm1(x) / eps_m) / 2.0;
	} else {
		tmp = pow((x * 1.000072337962963), -4.0);
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 600.0) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 7e+167) {
		tmp = (Math.expm1(x) / eps_m) / 2.0;
	} else {
		tmp = Math.pow((x * 1.000072337962963), -4.0);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 600.0:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 7e+167:
		tmp = (math.expm1(x) / eps_m) / 2.0
	else:
		tmp = math.pow((x * 1.000072337962963), -4.0)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 600.0)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 7e+167)
		tmp = Float64(Float64(expm1(x) / eps_m) / 2.0);
	else
		tmp = Float64(x * 1.000072337962963) ^ -4.0;
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 600.0], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 7e+167], N[(N[(N[(Exp[x] - 1), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], N[Power[N[(x * 1.000072337962963), $MachinePrecision], -4.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 600

   1. Initial program 62.6%

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

   3. Simplified 45.3%

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

   5. Taylor expanded in eps around inf 97.6%

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

   6. Taylor expanded in eps around inf 97.7%

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

      1. mul-1-neg 97.7%

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

      2. *-commutative 97.7%

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

      3. distribute-rgt-neg-in 97.7%

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

   8. Simplified 97.7%

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

   9. Taylor expanded in eps around 0 76.5%

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

       1. neg-mul-1 76.5%

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

   11. Simplified 76.5%

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

   if 600 < x < 6.99999999999999975e167

   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%

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

   5. Taylor expanded in x around 0 41.1%

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

   6. Taylor expanded in eps around 0 1.8%

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

      1. expm1-define 1.8%

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

      2. neg-mul-1 1.8%

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

   8. Simplified 1.8%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}}{2} \]
   9. Step-by-step derivation

      1. expm1-undefine 1.8%

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

      2. div-sub 1.8%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 40.2%

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

      5. sqr-neg 40.2%

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

      6. sqrt-unprod 40.2%

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

      7. add-sqr-sqrt 40.2%

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

   10. Applied egg-rr 40.2%

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

       1. div-sub 40.2%

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

       2. expm1-define 40.2%

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

   12. Simplified 40.2%

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

   if 6.99999999999999975e167 < 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%

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

   5. Taylor expanded in x around 0 21.2%

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

   6. Taylor expanded in eps around 0 2.0%

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

      1. expm1-define 2.0%

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

      2. neg-mul-1 2.0%

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

   8. Simplified 2.0%

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

   9. Taylor expanded in x around 0 20.2%

      \[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(0.041666666666666664 \cdot \frac{x}{\varepsilon} - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}}{2} \]

   11. Applied egg-rr 62.1%

       \[\leadsto \color{blue}{{\left(x \cdot 7.233796296296296 \cdot 10^{-5} + x\right)}^{-4}} \]
   12. Step-by-step derivation

       1. *-rgt-identity 62.1%

          \[\leadsto {\left(x \cdot 7.233796296296296 \cdot 10^{-5} + \color{blue}{x \cdot 1}\right)}^{-4} \]

       2. distribute-lft-out 62.1%

          \[\leadsto {\color{blue}{\left(x \cdot \left(7.233796296296296 \cdot 10^{-5} + 1\right)\right)}}^{-4} \]

       3. metadata-eval 62.1%

          \[\leadsto {\left(x \cdot \color{blue}{1.000072337962963}\right)}^{-4} \]

   13. Simplified 62.1%

       \[\leadsto \color{blue}{{\left(x \cdot 1.000072337962963\right)}^{-4}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 69.6% accurate, 4.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := x \cdot \left(1 + eps\_m\right)\\ t_1 := \frac{t\_0}{eps\_m}\\ t_2 := x \cdot \left(-1 - eps\_m\right)\\ \mathbf{if}\;x \leq -1.05 \cdot 10^{+65}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 7.233796296296296 \cdot 10^{-5} - 0.16666666666666666 \cdot \frac{1}{eps\_m}\right) + \frac{1}{eps\_m} \cdot 0.5\right) + \frac{-1}{eps\_m}\right)}{2}\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{t\_0 \cdot t\_0 + t\_1 \cdot \frac{t\_2}{eps\_m}}{t\_2 - t\_1}}{2}\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+92} \lor \neg \left(x \leq 5.4 \cdot 10^{+168}\right):\\ \;\;\;\;\frac{\left(1 + \frac{-1}{eps\_m}\right) - \left(-1 - \frac{1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* x (+ 1.0 eps_m)))
        (t_1 (/ t_0 eps_m))
        (t_2 (* x (- -1.0 eps_m))))
   (if (<= x -1.05e+65)
     (/
      (*
       x
       (+
        (*
         x
         (+
          (*
           x
           (-
            (* x 7.233796296296296e-5)
            (* 0.16666666666666666 (/ 1.0 eps_m))))
          (* (/ 1.0 eps_m) 0.5)))
        (/ -1.0 eps_m)))
      2.0)
     (if (<= x -4.2e-10)
       (/ (/ (+ (* t_0 t_0) (* t_1 (/ t_2 eps_m))) (- t_2 t_1)) 2.0)
       (if (<= x 1.42)
         (/ (- 2.0 (* x x)) 2.0)
         (if (or (<= x 7.8e+92) (not (<= x 5.4e+168)))
           (/ (- (+ 1.0 (/ -1.0 eps_m)) (- -1.0 (/ 1.0 eps_m))) 2.0)
           (/
            (/
             (*
              x
              (+
               -1.0
               (*
                x
                (+
                 0.5
                 (* x (- (* x 0.041666666666666664) 0.16666666666666666))))))
             eps_m)
            2.0)))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = x * (1.0 + eps_m);
	double t_1 = t_0 / eps_m;
	double t_2 = x * (-1.0 - eps_m);
	double tmp;
	if (x <= -1.05e+65) {
		tmp = (x * ((x * ((x * ((x * 7.233796296296296e-5) - (0.16666666666666666 * (1.0 / eps_m)))) + ((1.0 / eps_m) * 0.5))) + (-1.0 / eps_m))) / 2.0;
	} else if (x <= -4.2e-10) {
		tmp = (((t_0 * t_0) + (t_1 * (t_2 / eps_m))) / (t_2 - t_1)) / 2.0;
	} else if (x <= 1.42) {
		tmp = (2.0 - (x * x)) / 2.0;
	} else if ((x <= 7.8e+92) || !(x <= 5.4e+168)) {
		tmp = ((1.0 + (-1.0 / eps_m)) - (-1.0 - (1.0 / eps_m))) / 2.0;
	} else {
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / 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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = x * (1.0d0 + eps_m)
    t_1 = t_0 / eps_m
    t_2 = x * ((-1.0d0) - eps_m)
    if (x <= (-1.05d+65)) then
        tmp = (x * ((x * ((x * ((x * 7.233796296296296d-5) - (0.16666666666666666d0 * (1.0d0 / eps_m)))) + ((1.0d0 / eps_m) * 0.5d0))) + ((-1.0d0) / eps_m))) / 2.0d0
    else if (x <= (-4.2d-10)) then
        tmp = (((t_0 * t_0) + (t_1 * (t_2 / eps_m))) / (t_2 - t_1)) / 2.0d0
    else if (x <= 1.42d0) then
        tmp = (2.0d0 - (x * x)) / 2.0d0
    else if ((x <= 7.8d+92) .or. (.not. (x <= 5.4d+168))) then
        tmp = ((1.0d0 + ((-1.0d0) / eps_m)) - ((-1.0d0) - (1.0d0 / eps_m))) / 2.0d0
    else
        tmp = ((x * ((-1.0d0) + (x * (0.5d0 + (x * ((x * 0.041666666666666664d0) - 0.16666666666666666d0)))))) / 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 t_0 = x * (1.0 + eps_m);
	double t_1 = t_0 / eps_m;
	double t_2 = x * (-1.0 - eps_m);
	double tmp;
	if (x <= -1.05e+65) {
		tmp = (x * ((x * ((x * ((x * 7.233796296296296e-5) - (0.16666666666666666 * (1.0 / eps_m)))) + ((1.0 / eps_m) * 0.5))) + (-1.0 / eps_m))) / 2.0;
	} else if (x <= -4.2e-10) {
		tmp = (((t_0 * t_0) + (t_1 * (t_2 / eps_m))) / (t_2 - t_1)) / 2.0;
	} else if (x <= 1.42) {
		tmp = (2.0 - (x * x)) / 2.0;
	} else if ((x <= 7.8e+92) || !(x <= 5.4e+168)) {
		tmp = ((1.0 + (-1.0 / eps_m)) - (-1.0 - (1.0 / eps_m))) / 2.0;
	} else {
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = x * (1.0 + eps_m)
	t_1 = t_0 / eps_m
	t_2 = x * (-1.0 - eps_m)
	tmp = 0
	if x <= -1.05e+65:
		tmp = (x * ((x * ((x * ((x * 7.233796296296296e-5) - (0.16666666666666666 * (1.0 / eps_m)))) + ((1.0 / eps_m) * 0.5))) + (-1.0 / eps_m))) / 2.0
	elif x <= -4.2e-10:
		tmp = (((t_0 * t_0) + (t_1 * (t_2 / eps_m))) / (t_2 - t_1)) / 2.0
	elif x <= 1.42:
		tmp = (2.0 - (x * x)) / 2.0
	elif (x <= 7.8e+92) or not (x <= 5.4e+168):
		tmp = ((1.0 + (-1.0 / eps_m)) - (-1.0 - (1.0 / eps_m))) / 2.0
	else:
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(x * Float64(1.0 + eps_m))
	t_1 = Float64(t_0 / eps_m)
	t_2 = Float64(x * Float64(-1.0 - eps_m))
	tmp = 0.0
	if (x <= -1.05e+65)
		tmp = Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * 7.233796296296296e-5) - Float64(0.16666666666666666 * Float64(1.0 / eps_m)))) + Float64(Float64(1.0 / eps_m) * 0.5))) + Float64(-1.0 / eps_m))) / 2.0);
	elseif (x <= -4.2e-10)
		tmp = Float64(Float64(Float64(Float64(t_0 * t_0) + Float64(t_1 * Float64(t_2 / eps_m))) / Float64(t_2 - t_1)) / 2.0);
	elseif (x <= 1.42)
		tmp = Float64(Float64(2.0 - Float64(x * x)) / 2.0);
	elseif ((x <= 7.8e+92) || !(x <= 5.4e+168))
		tmp = Float64(Float64(Float64(1.0 + Float64(-1.0 / eps_m)) - Float64(-1.0 - Float64(1.0 / eps_m))) / 2.0);
	else
		tmp = Float64(Float64(Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(Float64(x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = x * (1.0 + eps_m);
	t_1 = t_0 / eps_m;
	t_2 = x * (-1.0 - eps_m);
	tmp = 0.0;
	if (x <= -1.05e+65)
		tmp = (x * ((x * ((x * ((x * 7.233796296296296e-5) - (0.16666666666666666 * (1.0 / eps_m)))) + ((1.0 / eps_m) * 0.5))) + (-1.0 / eps_m))) / 2.0;
	elseif (x <= -4.2e-10)
		tmp = (((t_0 * t_0) + (t_1 * (t_2 / eps_m))) / (t_2 - t_1)) / 2.0;
	elseif (x <= 1.42)
		tmp = (2.0 - (x * x)) / 2.0;
	elseif ((x <= 7.8e+92) || ~((x <= 5.4e+168)))
		tmp = ((1.0 + (-1.0 / eps_m)) - (-1.0 - (1.0 / eps_m))) / 2.0;
	else
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(x * N[(1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / eps$95$m), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.05e+65], N[(N[(x * N[(N[(x * N[(N[(x * N[(N[(x * 7.233796296296296e-5), $MachinePrecision] - N[(0.16666666666666666 * N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / eps$95$m), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -4.2e-10], N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[(t$95$1 * N[(t$95$2 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 - t$95$1), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.42], N[(N[(2.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 7.8e+92], N[Not[LessEqual[x, 5.4e+168]], $MachinePrecision]], N[(N[(N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] - N[(-1.0 - N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(x * N[(N[(x * 0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -1.04999999999999996e65

   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%

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

   5. Taylor expanded in x around 0 51.6%

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

   6. Taylor expanded in eps around 0 50.0%

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

      1. expm1-define 50.0%

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

      2. neg-mul-1 50.0%

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

   8. Simplified 50.0%

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

   9. Taylor expanded in x around 0 30.2%

      \[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(0.041666666666666664 \cdot \frac{x}{\varepsilon} - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}}{2} \]

   11. Applied egg-rr 88.7%

       \[\leadsto \frac{x \cdot \left(x \cdot \left(x \cdot \left(\color{blue}{\left(\left(-4 + x \cdot 7.233796296296296 \cdot 10^{-5}\right) - -4\right)} - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}{2} \]
   12. Step-by-step derivation

       1. +-commutative 88.7%

          \[\leadsto \frac{x \cdot \left(x \cdot \left(x \cdot \left(\left(\color{blue}{\left(x \cdot 7.233796296296296 \cdot 10^{-5} + -4\right)} - -4\right) - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}{2} \]

       2. associate--l+ 88.7%

          \[\leadsto \frac{x \cdot \left(x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot 7.233796296296296 \cdot 10^{-5} + \left(-4 - -4\right)\right)} - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}{2} \]

       3. metadata-eval 88.7%

          \[\leadsto \frac{x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot 7.233796296296296 \cdot 10^{-5} + \color{blue}{0}\right) - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}{2} \]

       4. +-rgt-identity 88.7%

          \[\leadsto \frac{x \cdot \left(x \cdot \left(x \cdot \left(\color{blue}{x \cdot 7.233796296296296 \cdot 10^{-5}} - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}{2} \]

   13. Simplified 88.7%

       \[\leadsto \frac{x \cdot \left(x \cdot \left(x \cdot \left(\color{blue}{x \cdot 7.233796296296296 \cdot 10^{-5}} - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}{2} \]

   if -1.04999999999999996e65 < x < -4.2e-10

   1. Initial program 82.5%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in x around 0 37.4%

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

   6. Taylor expanded in x around inf 3.3%

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

      1. mul-1-neg 3.3%

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

      2. distribute-rgt-neg-in 3.3%

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

      3. *-commutative 3.3%

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

      4. distribute-rgt-neg-in 3.3%

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

      5. mul-1-neg 3.3%

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

      6. distribute-lft-in 3.3%

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

      7. metadata-eval 3.3%

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

      8. neg-mul-1 3.3%

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

      9. distribute-neg-frac 3.3%

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

      10. metadata-eval 3.3%

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

   8. Simplified 3.3%

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

   10. Applied egg-rr 12.9%

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

   if -4.2e-10 < x < 1.4199999999999999

   1. Initial program 54.2%

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

   3. Simplified 54.2%

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

   5. Taylor expanded in eps around 0 74.2%

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

   6. Taylor expanded in x around 0 73.8%

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

      1. mul-1-neg 73.8%

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

      2. unsub-neg 73.8%

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

   8. Simplified 73.8%

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

   10. Applied egg-rr 73.8%

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

   if 1.4199999999999999 < x < 7.80000000000000022e92 or 5.40000000000000031e168 < 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%

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

   5. Taylor expanded in x around 0 30.5%

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

   6. Taylor expanded in x around 0 56.9%

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

   if 7.80000000000000022e92 < x < 5.40000000000000031e168

   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%

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

   5. Taylor expanded in x around 0 39.2%

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

   6. Taylor expanded in eps around 0 1.8%

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

      1. expm1-define 1.8%

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

      2. neg-mul-1 1.8%

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

   8. Simplified 1.8%

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

   9. Taylor expanded in x around 0 38.0%

      \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(x \cdot \left(0.5 + x \cdot \left(0.041666666666666664 \cdot x - 0.16666666666666666\right)\right) - 1\right)}}{\varepsilon}}{2} \]
3. Recombined 5 regimes into one program.

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+65}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 7.233796296296296 \cdot 10^{-5} - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + \frac{1}{\varepsilon} \cdot 0.5\right) + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{\left(x \cdot \left(1 + \varepsilon\right)\right) \cdot \left(x \cdot \left(1 + \varepsilon\right)\right) + \frac{x \cdot \left(1 + \varepsilon\right)}{\varepsilon} \cdot \frac{x \cdot \left(-1 - \varepsilon\right)}{\varepsilon}}{x \cdot \left(-1 - \varepsilon\right) - \frac{x \cdot \left(1 + \varepsilon\right)}{\varepsilon}}}{2}\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+92} \lor \neg \left(x \leq 5.4 \cdot 10^{+168}\right):\\ \;\;\;\;\frac{\left(1 + \frac{-1}{\varepsilon}\right) - \left(-1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.9% accurate, 5.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{1 + eps\_m}{eps\_m}\\ t_1 := \frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{eps\_m}}{2}\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{+77}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-11}:\\ \;\;\;\;\frac{x \cdot \frac{\left(1 + eps\_m\right) \cdot \left(1 + eps\_m\right) - t\_0 \cdot t\_0}{\left(-1 - eps\_m\right) - t\_0}}{2}\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+93} \lor \neg \left(x \leq 9.4 \cdot 10^{+170}\right):\\ \;\;\;\;\frac{\left(1 + \frac{-1}{eps\_m}\right) - \left(-1 - \frac{1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 eps_m) eps_m))
        (t_1
         (/
          (/
           (*
            x
            (+
             -1.0
             (*
              x
              (+
               0.5
               (* x (- (* x 0.041666666666666664) 0.16666666666666666))))))
           eps_m)
          2.0)))
   (if (<= x -2.6e+77)
     t_1
     (if (<= x -1e-11)
       (/
        (*
         x
         (/
          (- (* (+ 1.0 eps_m) (+ 1.0 eps_m)) (* t_0 t_0))
          (- (- -1.0 eps_m) t_0)))
        2.0)
       (if (<= x 1.42)
         (/ (- 2.0 (* x x)) 2.0)
         (if (or (<= x 3.4e+93) (not (<= x 9.4e+170)))
           (/ (- (+ 1.0 (/ -1.0 eps_m)) (- -1.0 (/ 1.0 eps_m))) 2.0)
           t_1))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (1.0 + eps_m) / eps_m;
	double t_1 = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0;
	double tmp;
	if (x <= -2.6e+77) {
		tmp = t_1;
	} else if (x <= -1e-11) {
		tmp = (x * ((((1.0 + eps_m) * (1.0 + eps_m)) - (t_0 * t_0)) / ((-1.0 - eps_m) - t_0))) / 2.0;
	} else if (x <= 1.42) {
		tmp = (2.0 - (x * x)) / 2.0;
	} else if ((x <= 3.4e+93) || !(x <= 9.4e+170)) {
		tmp = ((1.0 + (-1.0 / eps_m)) - (-1.0 - (1.0 / eps_m))) / 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (1.0d0 + eps_m) / eps_m
    t_1 = ((x * ((-1.0d0) + (x * (0.5d0 + (x * ((x * 0.041666666666666664d0) - 0.16666666666666666d0)))))) / eps_m) / 2.0d0
    if (x <= (-2.6d+77)) then
        tmp = t_1
    else if (x <= (-1d-11)) then
        tmp = (x * ((((1.0d0 + eps_m) * (1.0d0 + eps_m)) - (t_0 * t_0)) / (((-1.0d0) - eps_m) - t_0))) / 2.0d0
    else if (x <= 1.42d0) then
        tmp = (2.0d0 - (x * x)) / 2.0d0
    else if ((x <= 3.4d+93) .or. (.not. (x <= 9.4d+170))) then
        tmp = ((1.0d0 + ((-1.0d0) / eps_m)) - ((-1.0d0) - (1.0d0 / eps_m))) / 2.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = (1.0 + eps_m) / eps_m;
	double t_1 = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0;
	double tmp;
	if (x <= -2.6e+77) {
		tmp = t_1;
	} else if (x <= -1e-11) {
		tmp = (x * ((((1.0 + eps_m) * (1.0 + eps_m)) - (t_0 * t_0)) / ((-1.0 - eps_m) - t_0))) / 2.0;
	} else if (x <= 1.42) {
		tmp = (2.0 - (x * x)) / 2.0;
	} else if ((x <= 3.4e+93) || !(x <= 9.4e+170)) {
		tmp = ((1.0 + (-1.0 / eps_m)) - (-1.0 - (1.0 / eps_m))) / 2.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (1.0 + eps_m) / eps_m
	t_1 = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0
	tmp = 0
	if x <= -2.6e+77:
		tmp = t_1
	elif x <= -1e-11:
		tmp = (x * ((((1.0 + eps_m) * (1.0 + eps_m)) - (t_0 * t_0)) / ((-1.0 - eps_m) - t_0))) / 2.0
	elif x <= 1.42:
		tmp = (2.0 - (x * x)) / 2.0
	elif (x <= 3.4e+93) or not (x <= 9.4e+170):
		tmp = ((1.0 + (-1.0 / eps_m)) - (-1.0 - (1.0 / eps_m))) / 2.0
	else:
		tmp = t_1
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(1.0 + eps_m) / eps_m)
	t_1 = Float64(Float64(Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(Float64(x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0)
	tmp = 0.0
	if (x <= -2.6e+77)
		tmp = t_1;
	elseif (x <= -1e-11)
		tmp = Float64(Float64(x * Float64(Float64(Float64(Float64(1.0 + eps_m) * Float64(1.0 + eps_m)) - Float64(t_0 * t_0)) / Float64(Float64(-1.0 - eps_m) - t_0))) / 2.0);
	elseif (x <= 1.42)
		tmp = Float64(Float64(2.0 - Float64(x * x)) / 2.0);
	elseif ((x <= 3.4e+93) || !(x <= 9.4e+170))
		tmp = Float64(Float64(Float64(1.0 + Float64(-1.0 / eps_m)) - Float64(-1.0 - Float64(1.0 / eps_m))) / 2.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (1.0 + eps_m) / eps_m;
	t_1 = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0;
	tmp = 0.0;
	if (x <= -2.6e+77)
		tmp = t_1;
	elseif (x <= -1e-11)
		tmp = (x * ((((1.0 + eps_m) * (1.0 + eps_m)) - (t_0 * t_0)) / ((-1.0 - eps_m) - t_0))) / 2.0;
	elseif (x <= 1.42)
		tmp = (2.0 - (x * x)) / 2.0;
	elseif ((x <= 3.4e+93) || ~((x <= 9.4e+170)))
		tmp = ((1.0 + (-1.0 / eps_m)) - (-1.0 - (1.0 / eps_m))) / 2.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(1.0 + eps$95$m), $MachinePrecision] / eps$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(x * N[(N[(x * 0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -2.6e+77], t$95$1, If[LessEqual[x, -1e-11], N[(N[(x * N[(N[(N[(N[(1.0 + eps$95$m), $MachinePrecision] * N[(1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(-1.0 - eps$95$m), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.42], N[(N[(2.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 3.4e+93], N[Not[LessEqual[x, 9.4e+170]], $MachinePrecision]], N[(N[(N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] - N[(-1.0 - N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.6000000000000002e77 or 3.4e93 < x < 9.40000000000000008e170

   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%

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

   5. Taylor expanded in x around 0 48.2%

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

   6. Taylor expanded in eps around 0 21.0%

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

      1. expm1-define 21.0%

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

      2. neg-mul-1 21.0%

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

   8. Simplified 21.0%

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

   9. Taylor expanded in x around 0 40.2%

      \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(x \cdot \left(0.5 + x \cdot \left(0.041666666666666664 \cdot x - 0.16666666666666666\right)\right) - 1\right)}}{\varepsilon}}{2} \]

   if -2.6000000000000002e77 < x < -9.99999999999999939e-12

   1. Initial program 85.2%

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

   3. Simplified 85.2%

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

   5. Taylor expanded in x around 0 36.7%

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

   6. Taylor expanded in x around inf 2.9%

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

      1. mul-1-neg 2.9%

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

      2. distribute-rgt-neg-in 2.9%

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

      3. *-commutative 2.9%

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

      4. distribute-rgt-neg-in 2.9%

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

      5. mul-1-neg 2.9%

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

      6. distribute-lft-in 2.9%

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

      7. metadata-eval 2.9%

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

      8. neg-mul-1 2.9%

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

      9. distribute-neg-frac 2.9%

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

      10. metadata-eval 2.9%

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

   8. Simplified 2.9%

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

      1. distribute-lft-in 2.9%

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

      2. flip-+ 7.2%

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

   10. Applied egg-rr 26.2%

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

       1. *-commutative 26.2%

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

       2. neg-mul-1 26.2%

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

       3. distribute-neg-in 26.2%

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

       4. metadata-eval 26.2%

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

       5. sub-neg 26.2%

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

       6. *-commutative 26.2%

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

       7. neg-mul-1 26.2%

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

       8. distribute-neg-in 26.2%

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

       9. metadata-eval 26.2%

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

       10. sub-neg 26.2%

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

       11. *-lft-identity 26.2%

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

       12. *-lft-identity 26.2%

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

       13. *-lft-identity 26.2%

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

       14. +-commutative 26.2%

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

       15. *-lft-identity 26.2%

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

       16. +-commutative 26.2%

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

   12. Simplified 26.2%

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

   if -9.99999999999999939e-12 < x < 1.4199999999999999

   1. Initial program 54.2%

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

   3. Simplified 54.2%

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

   5. Taylor expanded in eps around 0 74.2%

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

   6. Taylor expanded in x around 0 73.8%

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

      1. mul-1-neg 73.8%

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

      2. unsub-neg 73.8%

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

   8. Simplified 73.8%

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

   10. Applied egg-rr 73.8%

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

   if 1.4199999999999999 < x < 3.4e93 or 9.40000000000000008e170 < 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%

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

   5. Taylor expanded in x around 0 30.5%

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

   6. Taylor expanded in x around 0 56.9%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-11}:\\ \;\;\;\;\frac{x \cdot \frac{\left(1 + \varepsilon\right) \cdot \left(1 + \varepsilon\right) - \frac{1 + \varepsilon}{\varepsilon} \cdot \frac{1 + \varepsilon}{\varepsilon}}{\left(-1 - \varepsilon\right) - \frac{1 + \varepsilon}{\varepsilon}}}{2}\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+93} \lor \neg \left(x \leq 9.4 \cdot 10^{+170}\right):\\ \;\;\;\;\frac{\left(1 + \frac{-1}{\varepsilon}\right) - \left(-1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.1% accurate, 5.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -70000000:\\ \;\;\;\;\frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 7.233796296296296 \cdot 10^{-5} - 0.16666666666666666 \cdot \frac{1}{eps\_m}\right) + \frac{1}{eps\_m} \cdot 0.5\right) + \frac{-1}{eps\_m}\right)}{2}\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+94} \lor \neg \left(x \leq 1.9 \cdot 10^{+171}\right):\\ \;\;\;\;\frac{\left(1 + \frac{-1}{eps\_m}\right) - \left(-1 - \frac{1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -70000000.0)
   (/
    (*
     x
     (+
      (*
       x
       (+
        (*
         x
         (- (* x 7.233796296296296e-5) (* 0.16666666666666666 (/ 1.0 eps_m))))
        (* (/ 1.0 eps_m) 0.5)))
      (/ -1.0 eps_m)))
    2.0)
   (if (<= x 1.42)
     (/ (- 2.0 (* x x)) 2.0)
     (if (or (<= x 1.6e+94) (not (<= x 1.9e+171)))
       (/ (- (+ 1.0 (/ -1.0 eps_m)) (- -1.0 (/ 1.0 eps_m))) 2.0)
       (/
        (/
         (*
          x
          (+
           -1.0
           (*
            x
            (+ 0.5 (* x (- (* x 0.041666666666666664) 0.16666666666666666))))))
         eps_m)
        2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -70000000.0) {
		tmp = (x * ((x * ((x * ((x * 7.233796296296296e-5) - (0.16666666666666666 * (1.0 / eps_m)))) + ((1.0 / eps_m) * 0.5))) + (-1.0 / eps_m))) / 2.0;
	} else if (x <= 1.42) {
		tmp = (2.0 - (x * x)) / 2.0;
	} else if ((x <= 1.6e+94) || !(x <= 1.9e+171)) {
		tmp = ((1.0 + (-1.0 / eps_m)) - (-1.0 - (1.0 / eps_m))) / 2.0;
	} else {
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / 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 <= (-70000000.0d0)) then
        tmp = (x * ((x * ((x * ((x * 7.233796296296296d-5) - (0.16666666666666666d0 * (1.0d0 / eps_m)))) + ((1.0d0 / eps_m) * 0.5d0))) + ((-1.0d0) / eps_m))) / 2.0d0
    else if (x <= 1.42d0) then
        tmp = (2.0d0 - (x * x)) / 2.0d0
    else if ((x <= 1.6d+94) .or. (.not. (x <= 1.9d+171))) then
        tmp = ((1.0d0 + ((-1.0d0) / eps_m)) - ((-1.0d0) - (1.0d0 / eps_m))) / 2.0d0
    else
        tmp = ((x * ((-1.0d0) + (x * (0.5d0 + (x * ((x * 0.041666666666666664d0) - 0.16666666666666666d0)))))) / 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 <= -70000000.0) {
		tmp = (x * ((x * ((x * ((x * 7.233796296296296e-5) - (0.16666666666666666 * (1.0 / eps_m)))) + ((1.0 / eps_m) * 0.5))) + (-1.0 / eps_m))) / 2.0;
	} else if (x <= 1.42) {
		tmp = (2.0 - (x * x)) / 2.0;
	} else if ((x <= 1.6e+94) || !(x <= 1.9e+171)) {
		tmp = ((1.0 + (-1.0 / eps_m)) - (-1.0 - (1.0 / eps_m))) / 2.0;
	} else {
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -70000000.0:
		tmp = (x * ((x * ((x * ((x * 7.233796296296296e-5) - (0.16666666666666666 * (1.0 / eps_m)))) + ((1.0 / eps_m) * 0.5))) + (-1.0 / eps_m))) / 2.0
	elif x <= 1.42:
		tmp = (2.0 - (x * x)) / 2.0
	elif (x <= 1.6e+94) or not (x <= 1.9e+171):
		tmp = ((1.0 + (-1.0 / eps_m)) - (-1.0 - (1.0 / eps_m))) / 2.0
	else:
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -70000000.0)
		tmp = Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * 7.233796296296296e-5) - Float64(0.16666666666666666 * Float64(1.0 / eps_m)))) + Float64(Float64(1.0 / eps_m) * 0.5))) + Float64(-1.0 / eps_m))) / 2.0);
	elseif (x <= 1.42)
		tmp = Float64(Float64(2.0 - Float64(x * x)) / 2.0);
	elseif ((x <= 1.6e+94) || !(x <= 1.9e+171))
		tmp = Float64(Float64(Float64(1.0 + Float64(-1.0 / eps_m)) - Float64(-1.0 - Float64(1.0 / eps_m))) / 2.0);
	else
		tmp = Float64(Float64(Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(Float64(x * 0.041666666666666664) - 0.16666666666666666)))))) / 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 <= -70000000.0)
		tmp = (x * ((x * ((x * ((x * 7.233796296296296e-5) - (0.16666666666666666 * (1.0 / eps_m)))) + ((1.0 / eps_m) * 0.5))) + (-1.0 / eps_m))) / 2.0;
	elseif (x <= 1.42)
		tmp = (2.0 - (x * x)) / 2.0;
	elseif ((x <= 1.6e+94) || ~((x <= 1.9e+171)))
		tmp = ((1.0 + (-1.0 / eps_m)) - (-1.0 - (1.0 / eps_m))) / 2.0;
	else
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -70000000.0], N[(N[(x * N[(N[(x * N[(N[(x * N[(N[(x * 7.233796296296296e-5), $MachinePrecision] - N[(0.16666666666666666 * N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / eps$95$m), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.42], N[(N[(2.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 1.6e+94], N[Not[LessEqual[x, 1.9e+171]], $MachinePrecision]], N[(N[(N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] - N[(-1.0 - N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(x * N[(N[(x * 0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7e7

   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%

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

   5. Taylor expanded in x around 0 54.3%

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

   6. Taylor expanded in eps around 0 47.2%

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

      1. expm1-define 47.2%

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

      2. neg-mul-1 47.2%

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

   8. Simplified 47.2%

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

   9. Taylor expanded in x around 0 20.7%

      \[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(0.041666666666666664 \cdot \frac{x}{\varepsilon} - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}}{2} \]

   11. Applied egg-rr 60.8%

       \[\leadsto \frac{x \cdot \left(x \cdot \left(x \cdot \left(\color{blue}{\left(\left(-4 + x \cdot 7.233796296296296 \cdot 10^{-5}\right) - -4\right)} - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}{2} \]
   12. Step-by-step derivation

       1. +-commutative 60.8%

          \[\leadsto \frac{x \cdot \left(x \cdot \left(x \cdot \left(\left(\color{blue}{\left(x \cdot 7.233796296296296 \cdot 10^{-5} + -4\right)} - -4\right) - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}{2} \]

       2. associate--l+ 60.8%

          \[\leadsto \frac{x \cdot \left(x \cdot \left(x \cdot \left(\color{blue}{\left(x \cdot 7.233796296296296 \cdot 10^{-5} + \left(-4 - -4\right)\right)} - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}{2} \]

       3. metadata-eval 60.8%

          \[\leadsto \frac{x \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot 7.233796296296296 \cdot 10^{-5} + \color{blue}{0}\right) - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}{2} \]

       4. +-rgt-identity 60.8%

          \[\leadsto \frac{x \cdot \left(x \cdot \left(x \cdot \left(\color{blue}{x \cdot 7.233796296296296 \cdot 10^{-5}} - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}{2} \]

   13. Simplified 60.8%

       \[\leadsto \frac{x \cdot \left(x \cdot \left(x \cdot \left(\color{blue}{x \cdot 7.233796296296296 \cdot 10^{-5}} - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}{2} \]

   if -7e7 < x < 1.4199999999999999

   1. Initial program 53.7%

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

   3. Simplified 53.7%

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

   5. Taylor expanded in eps around 0 72.4%

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

   6. Taylor expanded in x around 0 71.7%

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

      1. mul-1-neg 71.7%

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

      2. unsub-neg 71.7%

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

   8. Simplified 71.7%

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

   10. Applied egg-rr 71.7%

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

   if 1.4199999999999999 < x < 1.60000000000000007e94 or 1.9000000000000001e171 < 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%

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

   5. Taylor expanded in x around 0 30.5%

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

   6. Taylor expanded in x around 0 56.9%

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

   if 1.60000000000000007e94 < x < 1.9000000000000001e171

   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%

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

   5. Taylor expanded in x around 0 39.2%

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

   6. Taylor expanded in eps around 0 1.8%

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

      1. expm1-define 1.8%

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

      2. neg-mul-1 1.8%

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

   8. Simplified 1.8%

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

   9. Taylor expanded in x around 0 38.0%

      \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(x \cdot \left(0.5 + x \cdot \left(0.041666666666666664 \cdot x - 0.16666666666666666\right)\right) - 1\right)}}{\varepsilon}}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -70000000:\\ \;\;\;\;\frac{x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 7.233796296296296 \cdot 10^{-5} - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + \frac{1}{\varepsilon} \cdot 0.5\right) + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+94} \lor \neg \left(x \leq 1.9 \cdot 10^{+171}\right):\\ \;\;\;\;\frac{\left(1 + \frac{-1}{\varepsilon}\right) - \left(-1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.1% accurate, 5.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{eps\_m}}{2}\\ \mathbf{if}\;x \leq -12000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+92} \lor \neg \left(x \leq 1.85 \cdot 10^{+171}\right):\\ \;\;\;\;\frac{\left(1 + \frac{-1}{eps\_m}\right) - \left(-1 - \frac{1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0
         (/
          (/
           (*
            x
            (+
             -1.0
             (*
              x
              (+
               0.5
               (* x (- (* x 0.041666666666666664) 0.16666666666666666))))))
           eps_m)
          2.0)))
   (if (<= x -12000000.0)
     t_0
     (if (<= x 1.42)
       (/ (- 2.0 (* x x)) 2.0)
       (if (or (<= x 1.8e+92) (not (<= x 1.85e+171)))
         (/ (- (+ 1.0 (/ -1.0 eps_m)) (- -1.0 (/ 1.0 eps_m))) 2.0)
         t_0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0;
	double tmp;
	if (x <= -12000000.0) {
		tmp = t_0;
	} else if (x <= 1.42) {
		tmp = (2.0 - (x * x)) / 2.0;
	} else if ((x <= 1.8e+92) || !(x <= 1.85e+171)) {
		tmp = ((1.0 + (-1.0 / eps_m)) - (-1.0 - (1.0 / eps_m))) / 2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x * ((-1.0d0) + (x * (0.5d0 + (x * ((x * 0.041666666666666664d0) - 0.16666666666666666d0)))))) / eps_m) / 2.0d0
    if (x <= (-12000000.0d0)) then
        tmp = t_0
    else if (x <= 1.42d0) then
        tmp = (2.0d0 - (x * x)) / 2.0d0
    else if ((x <= 1.8d+92) .or. (.not. (x <= 1.85d+171))) then
        tmp = ((1.0d0 + ((-1.0d0) / eps_m)) - ((-1.0d0) - (1.0d0 / eps_m))) / 2.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0;
	double tmp;
	if (x <= -12000000.0) {
		tmp = t_0;
	} else if (x <= 1.42) {
		tmp = (2.0 - (x * x)) / 2.0;
	} else if ((x <= 1.8e+92) || !(x <= 1.85e+171)) {
		tmp = ((1.0 + (-1.0 / eps_m)) - (-1.0 - (1.0 / eps_m))) / 2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0
	tmp = 0
	if x <= -12000000.0:
		tmp = t_0
	elif x <= 1.42:
		tmp = (2.0 - (x * x)) / 2.0
	elif (x <= 1.8e+92) or not (x <= 1.85e+171):
		tmp = ((1.0 + (-1.0 / eps_m)) - (-1.0 - (1.0 / eps_m))) / 2.0
	else:
		tmp = t_0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(Float64(x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0)
	tmp = 0.0
	if (x <= -12000000.0)
		tmp = t_0;
	elseif (x <= 1.42)
		tmp = Float64(Float64(2.0 - Float64(x * x)) / 2.0);
	elseif ((x <= 1.8e+92) || !(x <= 1.85e+171))
		tmp = Float64(Float64(Float64(1.0 + Float64(-1.0 / eps_m)) - Float64(-1.0 - Float64(1.0 / eps_m))) / 2.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps_m) / 2.0;
	tmp = 0.0;
	if (x <= -12000000.0)
		tmp = t_0;
	elseif (x <= 1.42)
		tmp = (2.0 - (x * x)) / 2.0;
	elseif ((x <= 1.8e+92) || ~((x <= 1.85e+171)))
		tmp = ((1.0 + (-1.0 / eps_m)) - (-1.0 - (1.0 / eps_m))) / 2.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(x * N[(N[(x * 0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -12000000.0], t$95$0, If[LessEqual[x, 1.42], N[(N[(2.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 1.8e+92], N[Not[LessEqual[x, 1.85e+171]], $MachinePrecision]], N[(N[(N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] - N[(-1.0 - N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.2e7 or 1.8e92 < x < 1.84999999999999999e171

   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%

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

   5. Taylor expanded in x around 0 48.2%

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

   6. Taylor expanded in eps around 0 29.1%

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

      1. expm1-define 29.1%

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

      2. neg-mul-1 29.1%

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

   8. Simplified 29.1%

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

   9. Taylor expanded in x around 0 30.7%

      \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(x \cdot \left(0.5 + x \cdot \left(0.041666666666666664 \cdot x - 0.16666666666666666\right)\right) - 1\right)}}{\varepsilon}}{2} \]

   if -1.2e7 < x < 1.4199999999999999

   1. Initial program 53.7%

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

   3. Simplified 53.7%

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

   5. Taylor expanded in eps around 0 72.4%

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

   6. Taylor expanded in x around 0 71.7%

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

      1. mul-1-neg 71.7%

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

      2. unsub-neg 71.7%

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

   8. Simplified 71.7%

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

   10. Applied egg-rr 71.7%

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

   if 1.4199999999999999 < x < 1.8e92 or 1.84999999999999999e171 < 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%

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

   5. Taylor expanded in x around 0 30.5%

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

   6. Taylor expanded in x around 0 56.9%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -12000000:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{+92} \lor \neg \left(x \leq 1.85 \cdot 10^{+171}\right):\\ \;\;\;\;\frac{\left(1 + \frac{-1}{\varepsilon}\right) - \left(-1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.9% accurate, 5.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \frac{x \cdot \frac{-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)}{eps\_m}}{2}\\ \mathbf{if}\;x \leq -16000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+103} \lor \neg \left(x \leq 7.4 \cdot 10^{+167}\right):\\ \;\;\;\;\frac{\left(1 + \frac{-1}{eps\_m}\right) - \left(-1 - \frac{1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0
         (/
          (*
           x
           (/
            (+
             -1.0
             (*
              x
              (+
               0.5
               (* x (- (* x 0.041666666666666664) 0.16666666666666666)))))
            eps_m))
          2.0)))
   (if (<= x -16000000.0)
     t_0
     (if (<= x 1.42)
       (/ (- 2.0 (* x x)) 2.0)
       (if (or (<= x 2.7e+103) (not (<= x 7.4e+167)))
         (/ (- (+ 1.0 (/ -1.0 eps_m)) (- -1.0 (/ 1.0 eps_m))) 2.0)
         t_0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (x * ((-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666))))) / eps_m)) / 2.0;
	double tmp;
	if (x <= -16000000.0) {
		tmp = t_0;
	} else if (x <= 1.42) {
		tmp = (2.0 - (x * x)) / 2.0;
	} else if ((x <= 2.7e+103) || !(x <= 7.4e+167)) {
		tmp = ((1.0 + (-1.0 / eps_m)) - (-1.0 - (1.0 / eps_m))) / 2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * (((-1.0d0) + (x * (0.5d0 + (x * ((x * 0.041666666666666664d0) - 0.16666666666666666d0))))) / eps_m)) / 2.0d0
    if (x <= (-16000000.0d0)) then
        tmp = t_0
    else if (x <= 1.42d0) then
        tmp = (2.0d0 - (x * x)) / 2.0d0
    else if ((x <= 2.7d+103) .or. (.not. (x <= 7.4d+167))) then
        tmp = ((1.0d0 + ((-1.0d0) / eps_m)) - ((-1.0d0) - (1.0d0 / eps_m))) / 2.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = (x * ((-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666))))) / eps_m)) / 2.0;
	double tmp;
	if (x <= -16000000.0) {
		tmp = t_0;
	} else if (x <= 1.42) {
		tmp = (2.0 - (x * x)) / 2.0;
	} else if ((x <= 2.7e+103) || !(x <= 7.4e+167)) {
		tmp = ((1.0 + (-1.0 / eps_m)) - (-1.0 - (1.0 / eps_m))) / 2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (x * ((-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666))))) / eps_m)) / 2.0
	tmp = 0
	if x <= -16000000.0:
		tmp = t_0
	elif x <= 1.42:
		tmp = (2.0 - (x * x)) / 2.0
	elif (x <= 2.7e+103) or not (x <= 7.4e+167):
		tmp = ((1.0 + (-1.0 / eps_m)) - (-1.0 - (1.0 / eps_m))) / 2.0
	else:
		tmp = t_0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(x * Float64(Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(Float64(x * 0.041666666666666664) - 0.16666666666666666))))) / eps_m)) / 2.0)
	tmp = 0.0
	if (x <= -16000000.0)
		tmp = t_0;
	elseif (x <= 1.42)
		tmp = Float64(Float64(2.0 - Float64(x * x)) / 2.0);
	elseif ((x <= 2.7e+103) || !(x <= 7.4e+167))
		tmp = Float64(Float64(Float64(1.0 + Float64(-1.0 / eps_m)) - Float64(-1.0 - Float64(1.0 / eps_m))) / 2.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = (x * ((-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666))))) / eps_m)) / 2.0;
	tmp = 0.0;
	if (x <= -16000000.0)
		tmp = t_0;
	elseif (x <= 1.42)
		tmp = (2.0 - (x * x)) / 2.0;
	elseif ((x <= 2.7e+103) || ~((x <= 7.4e+167)))
		tmp = ((1.0 + (-1.0 / eps_m)) - (-1.0 - (1.0 / eps_m))) / 2.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(x * N[(N[(-1.0 + N[(x * N[(0.5 + N[(x * N[(N[(x * 0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -16000000.0], t$95$0, If[LessEqual[x, 1.42], N[(N[(2.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 2.7e+103], N[Not[LessEqual[x, 7.4e+167]], $MachinePrecision]], N[(N[(N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] - N[(-1.0 - N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6e7 or 2.69999999999999993e103 < x < 7.4000000000000002e167

   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%

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

   5. Taylor expanded in x around 0 47.2%

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

   6. Taylor expanded in eps around 0 30.5%

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

      1. expm1-define 30.5%

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

      2. neg-mul-1 30.5%

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

   8. Simplified 30.5%

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

   9. Taylor expanded in x around 0 23.9%

      \[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(0.041666666666666664 \cdot \frac{x}{\varepsilon} - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}}{2} \]

   10. Taylor expanded in eps around 0 27.1%

       \[\leadsto \frac{x \cdot \color{blue}{\frac{x \cdot \left(0.5 + x \cdot \left(0.041666666666666664 \cdot x - 0.16666666666666666\right)\right) - 1}{\varepsilon}}}{2} \]

   if -1.6e7 < x < 1.4199999999999999

   1. Initial program 53.7%

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

   3. Simplified 53.7%

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

   5. Taylor expanded in eps around 0 72.4%

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

   6. Taylor expanded in x around 0 71.7%

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

      1. mul-1-neg 71.7%

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

      2. unsub-neg 71.7%

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

   8. Simplified 71.7%

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

   10. Applied egg-rr 71.7%

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

   if 1.4199999999999999 < x < 2.69999999999999993e103 or 7.4000000000000002e167 < 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%

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

   5. Taylor expanded in x around 0 32.9%

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

   6. Taylor expanded in x around 0 55.6%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -16000000:\\ \;\;\;\;\frac{x \cdot \frac{-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+103} \lor \neg \left(x \leq 7.4 \cdot 10^{+167}\right):\\ \;\;\;\;\frac{\left(1 + \frac{-1}{\varepsilon}\right) - \left(-1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 67.3% accurate, 8.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;\frac{x \cdot \left(\left(1 + \frac{1}{eps\_m}\right) + \left(-1 - eps\_m\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{-1}{eps\_m}\right) - \left(-1 - \frac{1}{eps\_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1e+103)
   (/ (/ (* x (+ -1.0 (* x (+ 0.5 (* x -0.16666666666666666))))) eps_m) 2.0)
   (if (<= x -1.0)
     (/ (* x (+ (+ 1.0 (/ 1.0 eps_m)) (- -1.0 eps_m))) 2.0)
     (if (<= x 1.42)
       (/ (- 2.0 (* x x)) 2.0)
       (/ (- (+ 1.0 (/ -1.0 eps_m)) (- -1.0 (/ 1.0 eps_m))) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1e+103) {
		tmp = ((x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666))))) / eps_m) / 2.0;
	} else if (x <= -1.0) {
		tmp = (x * ((1.0 + (1.0 / eps_m)) + (-1.0 - eps_m))) / 2.0;
	} else if (x <= 1.42) {
		tmp = (2.0 - (x * x)) / 2.0;
	} else {
		tmp = ((1.0 + (-1.0 / eps_m)) - (-1.0 - (1.0 / eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-1d+103)) then
        tmp = ((x * ((-1.0d0) + (x * (0.5d0 + (x * (-0.16666666666666666d0)))))) / eps_m) / 2.0d0
    else if (x <= (-1.0d0)) then
        tmp = (x * ((1.0d0 + (1.0d0 / eps_m)) + ((-1.0d0) - eps_m))) / 2.0d0
    else if (x <= 1.42d0) then
        tmp = (2.0d0 - (x * x)) / 2.0d0
    else
        tmp = ((1.0d0 + ((-1.0d0) / eps_m)) - ((-1.0d0) - (1.0d0 / eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -1e+103) {
		tmp = ((x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666))))) / eps_m) / 2.0;
	} else if (x <= -1.0) {
		tmp = (x * ((1.0 + (1.0 / eps_m)) + (-1.0 - eps_m))) / 2.0;
	} else if (x <= 1.42) {
		tmp = (2.0 - (x * x)) / 2.0;
	} else {
		tmp = ((1.0 + (-1.0 / eps_m)) - (-1.0 - (1.0 / eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1e+103:
		tmp = ((x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666))))) / eps_m) / 2.0
	elif x <= -1.0:
		tmp = (x * ((1.0 + (1.0 / eps_m)) + (-1.0 - eps_m))) / 2.0
	elif x <= 1.42:
		tmp = (2.0 - (x * x)) / 2.0
	else:
		tmp = ((1.0 + (-1.0 / eps_m)) - (-1.0 - (1.0 / eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1e+103)
		tmp = Float64(Float64(Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * -0.16666666666666666))))) / eps_m) / 2.0);
	elseif (x <= -1.0)
		tmp = Float64(Float64(x * Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(-1.0 - eps_m))) / 2.0);
	elseif (x <= 1.42)
		tmp = Float64(Float64(2.0 - Float64(x * x)) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(-1.0 / eps_m)) - Float64(-1.0 - Float64(1.0 / eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1e+103)
		tmp = ((x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666))))) / eps_m) / 2.0;
	elseif (x <= -1.0)
		tmp = (x * ((1.0 + (1.0 / eps_m)) + (-1.0 - eps_m))) / 2.0;
	elseif (x <= 1.42)
		tmp = (2.0 - (x * x)) / 2.0;
	else
		tmp = ((1.0 + (-1.0 / eps_m)) - (-1.0 - (1.0 / eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1e+103], N[(N[(N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -1.0], N[(N[(x * N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.42], N[(N[(2.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] - N[(-1.0 - N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1e103

   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%

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

   5. Taylor expanded in x around 0 56.9%

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

   6. Taylor expanded in eps around 0 44.4%

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

      1. expm1-define 44.4%

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

      2. neg-mul-1 44.4%

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

   8. Simplified 44.4%

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

   9. Taylor expanded in x around 0 44.4%

      \[\leadsto \frac{\frac{\color{blue}{x \cdot \left(x \cdot \left(0.5 + -0.16666666666666666 \cdot x\right) - 1\right)}}{\varepsilon}}{2} \]

   if -1e103 < x < -1

   1. Initial program 90.0%

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

   3. Simplified 90.0%

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

   5. Taylor expanded in x around 0 32.0%

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

   6. Taylor expanded in x around inf 2.7%

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

      1. mul-1-neg 2.7%

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

      2. distribute-rgt-neg-in 2.7%

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

      3. *-commutative 2.7%

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

      4. distribute-rgt-neg-in 2.7%

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

      5. mul-1-neg 2.7%

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

      6. distribute-lft-in 2.7%

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

      7. metadata-eval 2.7%

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

      8. neg-mul-1 2.7%

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

      9. distribute-neg-frac 2.7%

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

      10. metadata-eval 2.7%

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

   8. Simplified 2.7%

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

      1. *-commutative 2.7%

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

      2. sub-neg 2.7%

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

      3. distribute-rgt-in 2.7%

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

      4. *-un-lft-identity 2.7%

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

      5. metadata-eval 2.7%

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

      6. add-sqr-sqrt 0.1%

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

      7. sqrt-unprod 2.1%

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

      8. sqr-neg 2.1%

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

      9. sqrt-unprod 2.0%

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

      10. add-sqr-sqrt 2.3%

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

      11. frac-2neg 2.3%

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

      12. add-sqr-sqrt 2.0%

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

      13. sqrt-unprod 27.9%

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

      14. sqr-neg 27.9%

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

      15. sqrt-unprod 21.3%

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

      16. add-sqr-sqrt 21.5%

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

      17. metadata-eval 21.5%

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

      18. add-sqr-sqrt 21.3%

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

      19. sqrt-unprod 26.5%

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

      20. sqr-neg 26.5%

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

      21. sqrt-unprod 0.2%

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

      22. add-sqr-sqrt 21.5%

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

      23. frac-2neg 21.5%

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

   10. Applied egg-rr 21.5%

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

       1. distribute-rgt1-in 21.5%

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

       2. +-commutative 21.5%

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

       3. +-commutative 21.5%

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

       4. distribute-rgt-in 21.5%

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

       5. associate-*l/ 21.5%

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

       6. *-lft-identity 21.5%

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

       7. neg-mul-1 21.5%

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

       8. unsub-neg 21.5%

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

       9. *-lft-identity 21.5%

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

       10. associate-*l/ 21.5%

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

       11. +-commutative 21.5%

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

       12. distribute-lft-in 21.5%

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

       13. lft-mult-inverse 21.5%

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

       14. *-rgt-identity 21.5%

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

       15. +-commutative 21.5%

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

   12. Simplified 21.5%

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

   if -1 < x < 1.4199999999999999

   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 54.4%

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

   5. Taylor expanded in eps around 0 73.4%

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

   6. Taylor expanded in x around 0 72.6%

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

      1. mul-1-neg 72.6%

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

      2. unsub-neg 72.6%

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

   8. Simplified 72.6%

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

   10. Applied egg-rr 72.6%

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

   if 1.4199999999999999 < 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%

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

   5. Taylor expanded in x around 0 33.6%

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

   6. Taylor expanded in x around 0 46.6%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;\frac{x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) + \left(-1 - \varepsilon\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{-1}{\varepsilon}\right) - \left(-1 - \frac{1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.5% accurate, 9.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -6.6:\\ \;\;\;\;\frac{x \cdot \left(\left(1 + \frac{1}{eps\_m}\right) + \left(-1 - eps\_m\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{-1}{eps\_m}\right) - \left(-1 - \frac{1}{eps\_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -6.6)
   (/ (* x (+ (+ 1.0 (/ 1.0 eps_m)) (- -1.0 eps_m))) 2.0)
   (if (<= x 1.42)
     (/ (- 2.0 (* x x)) 2.0)
     (/ (- (+ 1.0 (/ -1.0 eps_m)) (- -1.0 (/ 1.0 eps_m))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -6.6) {
		tmp = (x * ((1.0 + (1.0 / eps_m)) + (-1.0 - eps_m))) / 2.0;
	} else if (x <= 1.42) {
		tmp = (2.0 - (x * x)) / 2.0;
	} else {
		tmp = ((1.0 + (-1.0 / eps_m)) - (-1.0 - (1.0 / eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-6.6d0)) then
        tmp = (x * ((1.0d0 + (1.0d0 / eps_m)) + ((-1.0d0) - eps_m))) / 2.0d0
    else if (x <= 1.42d0) then
        tmp = (2.0d0 - (x * x)) / 2.0d0
    else
        tmp = ((1.0d0 + ((-1.0d0) / eps_m)) - ((-1.0d0) - (1.0d0 / eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -6.6) {
		tmp = (x * ((1.0 + (1.0 / eps_m)) + (-1.0 - eps_m))) / 2.0;
	} else if (x <= 1.42) {
		tmp = (2.0 - (x * x)) / 2.0;
	} else {
		tmp = ((1.0 + (-1.0 / eps_m)) - (-1.0 - (1.0 / eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -6.6:
		tmp = (x * ((1.0 + (1.0 / eps_m)) + (-1.0 - eps_m))) / 2.0
	elif x <= 1.42:
		tmp = (2.0 - (x * x)) / 2.0
	else:
		tmp = ((1.0 + (-1.0 / eps_m)) - (-1.0 - (1.0 / eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -6.6)
		tmp = Float64(Float64(x * Float64(Float64(1.0 + Float64(1.0 / eps_m)) + Float64(-1.0 - eps_m))) / 2.0);
	elseif (x <= 1.42)
		tmp = Float64(Float64(2.0 - Float64(x * x)) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(-1.0 / eps_m)) - Float64(-1.0 - Float64(1.0 / eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -6.6)
		tmp = (x * ((1.0 + (1.0 / eps_m)) + (-1.0 - eps_m))) / 2.0;
	elseif (x <= 1.42)
		tmp = (2.0 - (x * x)) / 2.0;
	else
		tmp = ((1.0 + (-1.0 / eps_m)) - (-1.0 - (1.0 / eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -6.6], N[(N[(x * N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.42], N[(N[(2.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] - N[(-1.0 - N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.5999999999999996

   1. Initial program 94.7%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in x around 0 46.3%

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

   6. Taylor expanded in x around inf 20.4%

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

      1. mul-1-neg 20.4%

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

      2. distribute-rgt-neg-in 20.4%

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

      3. *-commutative 20.4%

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

      4. distribute-rgt-neg-in 20.4%

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

      5. mul-1-neg 20.4%

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

      6. distribute-lft-in 20.4%

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

      7. metadata-eval 20.4%

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

      8. neg-mul-1 20.4%

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

      9. distribute-neg-frac 20.4%

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

      10. metadata-eval 20.4%

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

   8. Simplified 20.4%

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

      1. *-commutative 20.4%

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

      2. sub-neg 20.4%

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

      3. distribute-rgt-in 20.4%

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

      4. *-un-lft-identity 20.4%

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

      5. metadata-eval 20.4%

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

      6. add-sqr-sqrt 0.1%

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

      7. sqrt-unprod 20.0%

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

      8. sqr-neg 20.0%

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

      9. sqrt-unprod 19.9%

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

      10. add-sqr-sqrt 20.1%

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

      11. frac-2neg 20.1%

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

      12. add-sqr-sqrt 19.9%

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

      13. sqrt-unprod 47.3%

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

      14. sqr-neg 47.3%

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

      15. sqrt-unprod 22.5%

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

      16. add-sqr-sqrt 22.6%

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

      17. metadata-eval 22.6%

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

      18. add-sqr-sqrt 22.5%

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

      19. sqrt-unprod 25.2%

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

      20. sqr-neg 25.2%

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

      21. sqrt-unprod 0.1%

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

      22. add-sqr-sqrt 22.6%

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

      23. frac-2neg 22.6%

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

   10. Applied egg-rr 22.6%

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

       1. distribute-rgt1-in 22.6%

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

       2. +-commutative 22.6%

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

       3. +-commutative 22.6%

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

       4. distribute-rgt-in 22.6%

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

       5. associate-*l/ 22.6%

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

       6. *-lft-identity 22.6%

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

       7. neg-mul-1 22.6%

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

       8. unsub-neg 22.6%

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

       9. *-lft-identity 22.6%

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

       10. associate-*l/ 22.6%

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

       11. +-commutative 22.6%

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

       12. distribute-lft-in 22.6%

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

       13. lft-mult-inverse 22.6%

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

       14. *-rgt-identity 22.6%

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

       15. +-commutative 22.6%

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

   12. Simplified 22.6%

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

   if -6.5999999999999996 < x < 1.4199999999999999

   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 54.4%

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

   5. Taylor expanded in eps around 0 73.4%

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

   6. Taylor expanded in x around 0 72.6%

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

      1. mul-1-neg 72.6%

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

      2. unsub-neg 72.6%

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

   8. Simplified 72.6%

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

   10. Applied egg-rr 72.6%

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

   if 1.4199999999999999 < 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%

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

   5. Taylor expanded in x around 0 33.6%

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

   6. Taylor expanded in x around 0 46.6%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.6:\\ \;\;\;\;\frac{x \cdot \left(\left(1 + \frac{1}{\varepsilon}\right) + \left(-1 - \varepsilon\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{-1}{\varepsilon}\right) - \left(-1 - \frac{1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 58.0% accurate, 12.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 94.7%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in x around 0 46.3%

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

   6. Taylor expanded in x around inf 20.4%

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

      1. mul-1-neg 20.4%

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

      2. distribute-rgt-neg-in 20.4%

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

      3. *-commutative 20.4%

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

      4. distribute-rgt-neg-in 20.4%

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

      5. mul-1-neg 20.4%

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

      6. distribute-lft-in 20.4%

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

      7. metadata-eval 20.4%

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

      8. neg-mul-1 20.4%

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

      9. distribute-neg-frac 20.4%

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

      10. metadata-eval 20.4%

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

   8. Simplified 20.4%

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

      1. *-commutative 20.4%

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

      2. sub-neg 20.4%

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

      3. distribute-rgt-in 20.4%

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

      4. *-un-lft-identity 20.4%

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

      5. metadata-eval 20.4%

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

      6. add-sqr-sqrt 0.1%

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

      7. sqrt-unprod 20.0%

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

      8. sqr-neg 20.0%

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

      9. sqrt-unprod 19.9%

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

      10. add-sqr-sqrt 20.1%

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

      11. frac-2neg 20.1%

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

      12. add-sqr-sqrt 19.9%

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

      13. sqrt-unprod 47.3%

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

      14. sqr-neg 47.3%

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

      15. sqrt-unprod 22.5%

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

      16. add-sqr-sqrt 22.6%

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

      17. metadata-eval 22.6%

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

      18. add-sqr-sqrt 22.5%

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

      19. sqrt-unprod 25.2%

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

      20. sqr-neg 25.2%

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

      21. sqrt-unprod 0.1%

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

      22. add-sqr-sqrt 22.6%

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

      23. frac-2neg 22.6%

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

   10. Applied egg-rr 22.6%

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

       1. distribute-rgt1-in 22.6%

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

       2. +-commutative 22.6%

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

       3. +-commutative 22.6%

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

       4. distribute-rgt-in 22.6%

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

       5. associate-*l/ 22.6%

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

       6. *-lft-identity 22.6%

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

       7. neg-mul-1 22.6%

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

       8. unsub-neg 22.6%

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

       9. *-lft-identity 22.6%

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

       10. associate-*l/ 22.6%

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

       11. +-commutative 22.6%

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

       12. distribute-lft-in 22.6%

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

       13. lft-mult-inverse 22.6%

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

       14. *-rgt-identity 22.6%

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

       15. +-commutative 22.6%

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

   12. Simplified 22.6%

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

   if -1 < x

   1. Initial program 68.9%

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

   3. Simplified 68.9%

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

   5. Taylor expanded in x around 0 37.0%

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

   6. Taylor expanded in eps around inf 67.2%

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

      1. associate-*r* 67.2%

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

      2. neg-mul-1 67.2%

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

      3. *-commutative 67.2%

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

   8. Simplified 67.2%

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

   9. Taylor expanded in x around 0 54.2%

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

       1. mul-1-neg 54.2%

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

       2. distribute-rgt-neg-in 54.2%

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

       3. sub-neg 54.2%

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

       4. metadata-eval 54.2%

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

       5. distribute-neg-in 54.2%

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

       6. +-commutative 54.2%

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

       7. remove-double-neg 54.2%

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

       8. +-commutative 54.2%

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

   11. Simplified 54.2%

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

4. Final simplification 49.5%

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

Alternative 16: 59.0% accurate, 13.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -3500000:\\ \;\;\;\;\left(x \cdot 1.000072337962963\right) \cdot \left(x \cdot 1.000072337962963\right)\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -3500000.0)
   (* (* x 1.000072337962963) (* x 1.000072337962963))
   (if (<= x 1.42) (/ (- 2.0 (* x x)) 2.0) (/ (* x eps_m) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -3500000.0) {
		tmp = (x * 1.000072337962963) * (x * 1.000072337962963);
	} else if (x <= 1.42) {
		tmp = (2.0 - (x * x)) / 2.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-3500000.0d0)) then
        tmp = (x * 1.000072337962963d0) * (x * 1.000072337962963d0)
    else if (x <= 1.42d0) then
        tmp = (2.0d0 - (x * x)) / 2.0d0
    else
        tmp = (x * eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -3500000.0) {
		tmp = (x * 1.000072337962963) * (x * 1.000072337962963);
	} else if (x <= 1.42) {
		tmp = (2.0 - (x * x)) / 2.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -3500000.0:
		tmp = (x * 1.000072337962963) * (x * 1.000072337962963)
	elif x <= 1.42:
		tmp = (2.0 - (x * x)) / 2.0
	else:
		tmp = (x * eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -3500000.0)
		tmp = Float64(Float64(x * 1.000072337962963) * Float64(x * 1.000072337962963));
	elseif (x <= 1.42)
		tmp = Float64(Float64(2.0 - Float64(x * x)) / 2.0);
	else
		tmp = Float64(Float64(x * eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -3500000.0)
		tmp = (x * 1.000072337962963) * (x * 1.000072337962963);
	elseif (x <= 1.42)
		tmp = (2.0 - (x * x)) / 2.0;
	else
		tmp = (x * eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -3500000.0], N[(N[(x * 1.000072337962963), $MachinePrecision] * N[(x * 1.000072337962963), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.42], N[(N[(2.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.5e6

   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%

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

   5. Taylor expanded in x around 0 54.3%

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

   6. Taylor expanded in eps around 0 47.2%

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

      1. expm1-define 47.2%

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

      2. neg-mul-1 47.2%

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

   8. Simplified 47.2%

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

   9. Taylor expanded in x around 0 20.7%

      \[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(0.041666666666666664 \cdot \frac{x}{\varepsilon} - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}}{2} \]

   11. Applied egg-rr 39.2%

       \[\leadsto \color{blue}{\left(x \cdot 7.233796296296296 \cdot 10^{-5} + x\right) \cdot \left(x \cdot 7.233796296296296 \cdot 10^{-5} + x\right)} \]
   12. Step-by-step derivation

       1. *-commutative 39.2%

          \[\leadsto \left(\color{blue}{7.233796296296296 \cdot 10^{-5} \cdot x} + x\right) \cdot \left(x \cdot 7.233796296296296 \cdot 10^{-5} + x\right) \]

       2. distribute-lft1-in 39.2%

          \[\leadsto \color{blue}{\left(\left(7.233796296296296 \cdot 10^{-5} + 1\right) \cdot x\right)} \cdot \left(x \cdot 7.233796296296296 \cdot 10^{-5} + x\right) \]

       3. metadata-eval 39.2%

          \[\leadsto \left(\color{blue}{1.000072337962963} \cdot x\right) \cdot \left(x \cdot 7.233796296296296 \cdot 10^{-5} + x\right) \]

       4. *-commutative 39.2%

          \[\leadsto \left(1.000072337962963 \cdot x\right) \cdot \left(\color{blue}{7.233796296296296 \cdot 10^{-5} \cdot x} + x\right) \]

       5. distribute-lft1-in 39.2%

          \[\leadsto \left(1.000072337962963 \cdot x\right) \cdot \color{blue}{\left(\left(7.233796296296296 \cdot 10^{-5} + 1\right) \cdot x\right)} \]

       6. metadata-eval 39.2%

          \[\leadsto \left(1.000072337962963 \cdot x\right) \cdot \left(\color{blue}{1.000072337962963} \cdot x\right) \]

   13. Simplified 39.2%

       \[\leadsto \color{blue}{\left(1.000072337962963 \cdot x\right) \cdot \left(1.000072337962963 \cdot x\right)} \]

   if -3.5e6 < x < 1.4199999999999999

   1. Initial program 53.7%

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

   3. Simplified 53.7%

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

   5. Taylor expanded in eps around 0 72.4%

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

   6. Taylor expanded in x around 0 71.7%

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

      1. mul-1-neg 71.7%

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

      2. unsub-neg 71.7%

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

   8. Simplified 71.7%

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

   10. Applied egg-rr 71.7%

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

   if 1.4199999999999999 < 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%

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

   5. Taylor expanded in x around 0 26.3%

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

   6. Taylor expanded in x around inf 17.6%

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

      1. mul-1-neg 17.6%

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

      2. distribute-rgt-neg-in 17.6%

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

      3. *-commutative 17.6%

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

      4. distribute-rgt-neg-in 17.6%

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

      5. mul-1-neg 17.6%

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

      6. distribute-lft-in 17.6%

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

      7. metadata-eval 17.6%

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

      8. neg-mul-1 17.6%

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

      9. distribute-neg-frac 17.6%

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

      10. metadata-eval 17.6%

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

   8. Simplified 17.6%

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

   9. Taylor expanded in eps around inf 18.5%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3500000:\\ \;\;\;\;\left(x \cdot 1.000072337962963\right) \cdot \left(x \cdot 1.000072337962963\right)\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 58.4% accurate, 15.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2900000 \lor \neg \left(x \leq 6.6 \cdot 10^{+69}\right):\\ \;\;\;\;x \cdot \left(x \cdot 1.000072337962963\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (or (<= x -2900000.0) (not (<= x 6.6e+69)))
   (* x (* x 1.000072337962963))
   1.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if ((x <= -2900000.0) || !(x <= 6.6e+69)) {
		tmp = x * (x * 1.000072337962963);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if ((x <= (-2900000.0d0)) .or. (.not. (x <= 6.6d+69))) then
        tmp = x * (x * 1.000072337962963d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if ((x <= -2900000.0) || !(x <= 6.6e+69)) {
		tmp = x * (x * 1.000072337962963);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if (x <= -2900000.0) or not (x <= 6.6e+69):
		tmp = x * (x * 1.000072337962963)
	else:
		tmp = 1.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if ((x <= -2900000.0) || !(x <= 6.6e+69))
		tmp = Float64(x * Float64(x * 1.000072337962963));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if ((x <= -2900000.0) || ~((x <= 6.6e+69)))
		tmp = x * (x * 1.000072337962963);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[Or[LessEqual[x, -2900000.0], N[Not[LessEqual[x, 6.6e+69]], $MachinePrecision]], N[(x * N[(x * 1.000072337962963), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -2.9e6 or 6.5999999999999997e69 < 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%

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

   5. Taylor expanded in x around 0 40.1%

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

   6. Taylor expanded in eps around 0 19.8%

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

      1. expm1-define 19.8%

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

      2. neg-mul-1 19.8%

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

   8. Simplified 19.8%

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

   9. Taylor expanded in x around 0 23.1%

      \[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(0.041666666666666664 \cdot \frac{x}{\varepsilon} - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}}{2} \]

   11. Applied egg-rr 33.4%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot 7.233796296296296 \cdot 10^{-5} + x\right)} \]
   12. Step-by-step derivation

       1. *-commutative 33.4%

          \[\leadsto x \cdot \left(\color{blue}{7.233796296296296 \cdot 10^{-5} \cdot x} + x\right) \]

       2. distribute-lft1-in 33.4%

          \[\leadsto x \cdot \color{blue}{\left(\left(7.233796296296296 \cdot 10^{-5} + 1\right) \cdot x\right)} \]

       3. metadata-eval 33.4%

          \[\leadsto x \cdot \left(\color{blue}{1.000072337962963} \cdot x\right) \]

   13. Simplified 33.4%

       \[\leadsto \color{blue}{x \cdot \left(1.000072337962963 \cdot x\right)} \]

   if -2.9e6 < x < 6.5999999999999997e69

   1. Initial program 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in x around 0 39.2%

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

   6. Taylor expanded in eps around inf 78.3%

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

      1. associate-*r* 78.3%

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

      2. neg-mul-1 78.3%

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

      3. *-commutative 78.3%

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

   8. Simplified 78.3%

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

   9. Taylor expanded in x around 0 65.6%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2900000 \lor \neg \left(x \leq 6.6 \cdot 10^{+69}\right):\\ \;\;\;\;x \cdot \left(x \cdot 1.000072337962963\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 58.9% accurate, 15.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2900000:\\ \;\;\;\;\left(x \cdot 1.000072337962963\right) \cdot \left(x \cdot 1.000072337962963\right)\\ \mathbf{elif}\;x \leq 36:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2900000.0)
   (* (* x 1.000072337962963) (* x 1.000072337962963))
   (if (<= x 36.0) 1.0 (/ (* x eps_m) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -2900000.0) {
		tmp = (x * 1.000072337962963) * (x * 1.000072337962963);
	} else if (x <= 36.0) {
		tmp = 1.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-2900000.0d0)) then
        tmp = (x * 1.000072337962963d0) * (x * 1.000072337962963d0)
    else if (x <= 36.0d0) then
        tmp = 1.0d0
    else
        tmp = (x * eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -2900000.0) {
		tmp = (x * 1.000072337962963) * (x * 1.000072337962963);
	} else if (x <= 36.0) {
		tmp = 1.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -2900000.0:
		tmp = (x * 1.000072337962963) * (x * 1.000072337962963)
	elif x <= 36.0:
		tmp = 1.0
	else:
		tmp = (x * eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -2900000.0)
		tmp = Float64(Float64(x * 1.000072337962963) * Float64(x * 1.000072337962963));
	elseif (x <= 36.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(x * eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -2900000.0)
		tmp = (x * 1.000072337962963) * (x * 1.000072337962963);
	elseif (x <= 36.0)
		tmp = 1.0;
	else
		tmp = (x * eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -2900000.0], N[(N[(x * 1.000072337962963), $MachinePrecision] * N[(x * 1.000072337962963), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 36.0], 1.0, N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.9e6

   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%

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

   5. Taylor expanded in x around 0 54.3%

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

   6. Taylor expanded in eps around 0 47.2%

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

      1. expm1-define 47.2%

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

      2. neg-mul-1 47.2%

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

   8. Simplified 47.2%

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

   9. Taylor expanded in x around 0 20.7%

      \[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(0.041666666666666664 \cdot \frac{x}{\varepsilon} - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}}{2} \]

   11. Applied egg-rr 39.2%

       \[\leadsto \color{blue}{\left(x \cdot 7.233796296296296 \cdot 10^{-5} + x\right) \cdot \left(x \cdot 7.233796296296296 \cdot 10^{-5} + x\right)} \]
   12. Step-by-step derivation

       1. *-commutative 39.2%

          \[\leadsto \left(\color{blue}{7.233796296296296 \cdot 10^{-5} \cdot x} + x\right) \cdot \left(x \cdot 7.233796296296296 \cdot 10^{-5} + x\right) \]

       2. distribute-lft1-in 39.2%

          \[\leadsto \color{blue}{\left(\left(7.233796296296296 \cdot 10^{-5} + 1\right) \cdot x\right)} \cdot \left(x \cdot 7.233796296296296 \cdot 10^{-5} + x\right) \]

       3. metadata-eval 39.2%

          \[\leadsto \left(\color{blue}{1.000072337962963} \cdot x\right) \cdot \left(x \cdot 7.233796296296296 \cdot 10^{-5} + x\right) \]

       4. *-commutative 39.2%

          \[\leadsto \left(1.000072337962963 \cdot x\right) \cdot \left(\color{blue}{7.233796296296296 \cdot 10^{-5} \cdot x} + x\right) \]

       5. distribute-lft1-in 39.2%

          \[\leadsto \left(1.000072337962963 \cdot x\right) \cdot \color{blue}{\left(\left(7.233796296296296 \cdot 10^{-5} + 1\right) \cdot x\right)} \]

       6. metadata-eval 39.2%

          \[\leadsto \left(1.000072337962963 \cdot x\right) \cdot \left(\color{blue}{1.000072337962963} \cdot x\right) \]

   13. Simplified 39.2%

       \[\leadsto \color{blue}{\left(1.000072337962963 \cdot x\right) \cdot \left(1.000072337962963 \cdot x\right)} \]

   if -2.9e6 < x < 36

   1. Initial program 53.7%

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

   3. Simplified 53.7%

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

   5. Taylor expanded in x around 0 38.8%

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

   6. Taylor expanded in eps around inf 81.5%

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

      1. associate-*r* 81.5%

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

      2. neg-mul-1 81.5%

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

      3. *-commutative 81.5%

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

   8. Simplified 81.5%

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

   9. Taylor expanded in x around 0 71.3%

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

   if 36 < 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%

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

   5. Taylor expanded in x around 0 26.3%

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

   6. Taylor expanded in x around inf 17.6%

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

      1. mul-1-neg 17.6%

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

      2. distribute-rgt-neg-in 17.6%

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

      3. *-commutative 17.6%

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

      4. distribute-rgt-neg-in 17.6%

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

      5. mul-1-neg 17.6%

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

      6. distribute-lft-in 17.6%

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

      7. metadata-eval 17.6%

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

      8. neg-mul-1 17.6%

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

      9. distribute-neg-frac 17.6%

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

      10. metadata-eval 17.6%

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

   8. Simplified 17.6%

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

   9. Taylor expanded in eps around inf 18.5%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2900000:\\ \;\;\;\;\left(x \cdot 1.000072337962963\right) \cdot \left(x \cdot 1.000072337962963\right)\\ \mathbf{elif}\;x \leq 36:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 58.9% accurate, 15.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2900000:\\ \;\;\;\;x \cdot \left(x \cdot 1.000072337962963\right)\\ \mathbf{elif}\;x \leq 39:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot eps\_m}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2900000.0)
   (* x (* x 1.000072337962963))
   (if (<= x 39.0) 1.0 (/ (* x eps_m) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -2900000.0) {
		tmp = x * (x * 1.000072337962963);
	} else if (x <= 39.0) {
		tmp = 1.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-2900000.0d0)) then
        tmp = x * (x * 1.000072337962963d0)
    else if (x <= 39.0d0) then
        tmp = 1.0d0
    else
        tmp = (x * eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -2900000.0) {
		tmp = x * (x * 1.000072337962963);
	} else if (x <= 39.0) {
		tmp = 1.0;
	} else {
		tmp = (x * eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -2900000.0:
		tmp = x * (x * 1.000072337962963)
	elif x <= 39.0:
		tmp = 1.0
	else:
		tmp = (x * eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -2900000.0)
		tmp = Float64(x * Float64(x * 1.000072337962963));
	elseif (x <= 39.0)
		tmp = 1.0;
	else
		tmp = Float64(Float64(x * eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -2900000.0)
		tmp = x * (x * 1.000072337962963);
	elseif (x <= 39.0)
		tmp = 1.0;
	else
		tmp = (x * eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -2900000.0], N[(x * N[(x * 1.000072337962963), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 39.0], 1.0, N[(N[(x * eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.9e6

   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%

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

   5. Taylor expanded in x around 0 54.3%

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

   6. Taylor expanded in eps around 0 47.2%

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

      1. expm1-define 47.2%

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

      2. neg-mul-1 47.2%

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

   8. Simplified 47.2%

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

   9. Taylor expanded in x around 0 20.7%

      \[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(0.041666666666666664 \cdot \frac{x}{\varepsilon} - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}}{2} \]

   11. Applied egg-rr 39.2%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot 7.233796296296296 \cdot 10^{-5} + x\right)} \]
   12. Step-by-step derivation

       1. *-commutative 39.2%

          \[\leadsto x \cdot \left(\color{blue}{7.233796296296296 \cdot 10^{-5} \cdot x} + x\right) \]

       2. distribute-lft1-in 39.2%

          \[\leadsto x \cdot \color{blue}{\left(\left(7.233796296296296 \cdot 10^{-5} + 1\right) \cdot x\right)} \]

       3. metadata-eval 39.2%

          \[\leadsto x \cdot \left(\color{blue}{1.000072337962963} \cdot x\right) \]

   13. Simplified 39.2%

       \[\leadsto \color{blue}{x \cdot \left(1.000072337962963 \cdot x\right)} \]

   if -2.9e6 < x < 39

   1. Initial program 53.7%

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

   3. Simplified 53.7%

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

   5. Taylor expanded in x around 0 38.8%

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

   6. Taylor expanded in eps around inf 81.5%

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

      1. associate-*r* 81.5%

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

      2. neg-mul-1 81.5%

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

      3. *-commutative 81.5%

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

   8. Simplified 81.5%

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

   9. Taylor expanded in x around 0 71.3%

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

   if 39 < 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%

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

   5. Taylor expanded in x around 0 26.3%

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

   6. Taylor expanded in x around inf 17.6%

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

      1. mul-1-neg 17.6%

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

      2. distribute-rgt-neg-in 17.6%

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

      3. *-commutative 17.6%

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

      4. distribute-rgt-neg-in 17.6%

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

      5. mul-1-neg 17.6%

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

      6. distribute-lft-in 17.6%

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

      7. metadata-eval 17.6%

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

      8. neg-mul-1 17.6%

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

      9. distribute-neg-frac 17.6%

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

      10. metadata-eval 17.6%

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

   8. Simplified 17.6%

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

   9. Taylor expanded in eps around inf 18.5%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2900000:\\ \;\;\;\;x \cdot \left(x \cdot 1.000072337962963\right)\\ \mathbf{elif}\;x \leq 39:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \varepsilon}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 58.1% accurate, 16.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2900000:\\ \;\;\;\;\left(x \cdot 1.000072337962963\right) \cdot \left(x \cdot 1.000072337962963\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + eps\_m\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2900000.0)
   (* (* x 1.000072337962963) (* x 1.000072337962963))
   (/ (+ 2.0 (* x (+ -1.0 eps_m))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -2900000.0) {
		tmp = (x * 1.000072337962963) * (x * 1.000072337962963);
	} else {
		tmp = (2.0 + (x * (-1.0 + eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-2900000.0d0)) then
        tmp = (x * 1.000072337962963d0) * (x * 1.000072337962963d0)
    else
        tmp = (2.0d0 + (x * ((-1.0d0) + eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -2900000.0) {
		tmp = (x * 1.000072337962963) * (x * 1.000072337962963);
	} else {
		tmp = (2.0 + (x * (-1.0 + eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -2900000.0:
		tmp = (x * 1.000072337962963) * (x * 1.000072337962963)
	else:
		tmp = (2.0 + (x * (-1.0 + eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -2900000.0)
		tmp = Float64(Float64(x * 1.000072337962963) * Float64(x * 1.000072337962963));
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 + eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -2900000.0)
		tmp = (x * 1.000072337962963) * (x * 1.000072337962963);
	else
		tmp = (2.0 + (x * (-1.0 + eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -2900000.0], N[(N[(x * 1.000072337962963), $MachinePrecision] * N[(x * 1.000072337962963), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.9e6

   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%

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

   5. Taylor expanded in x around 0 54.3%

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

   6. Taylor expanded in eps around 0 47.2%

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

      1. expm1-define 47.2%

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

      2. neg-mul-1 47.2%

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

   8. Simplified 47.2%

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

   9. Taylor expanded in x around 0 20.7%

      \[\leadsto \frac{\color{blue}{x \cdot \left(x \cdot \left(x \cdot \left(0.041666666666666664 \cdot \frac{x}{\varepsilon} - 0.16666666666666666 \cdot \frac{1}{\varepsilon}\right) + 0.5 \cdot \frac{1}{\varepsilon}\right) - \frac{1}{\varepsilon}\right)}}{2} \]

   11. Applied egg-rr 39.2%

       \[\leadsto \color{blue}{\left(x \cdot 7.233796296296296 \cdot 10^{-5} + x\right) \cdot \left(x \cdot 7.233796296296296 \cdot 10^{-5} + x\right)} \]
   12. Step-by-step derivation

       1. *-commutative 39.2%

          \[\leadsto \left(\color{blue}{7.233796296296296 \cdot 10^{-5} \cdot x} + x\right) \cdot \left(x \cdot 7.233796296296296 \cdot 10^{-5} + x\right) \]

       2. distribute-lft1-in 39.2%

          \[\leadsto \color{blue}{\left(\left(7.233796296296296 \cdot 10^{-5} + 1\right) \cdot x\right)} \cdot \left(x \cdot 7.233796296296296 \cdot 10^{-5} + x\right) \]

       3. metadata-eval 39.2%

          \[\leadsto \left(\color{blue}{1.000072337962963} \cdot x\right) \cdot \left(x \cdot 7.233796296296296 \cdot 10^{-5} + x\right) \]

       4. *-commutative 39.2%

          \[\leadsto \left(1.000072337962963 \cdot x\right) \cdot \left(\color{blue}{7.233796296296296 \cdot 10^{-5} \cdot x} + x\right) \]

       5. distribute-lft1-in 39.2%

          \[\leadsto \left(1.000072337962963 \cdot x\right) \cdot \color{blue}{\left(\left(7.233796296296296 \cdot 10^{-5} + 1\right) \cdot x\right)} \]

       6. metadata-eval 39.2%

          \[\leadsto \left(1.000072337962963 \cdot x\right) \cdot \left(\color{blue}{1.000072337962963} \cdot x\right) \]

   13. Simplified 39.2%

       \[\leadsto \color{blue}{\left(1.000072337962963 \cdot x\right) \cdot \left(1.000072337962963 \cdot x\right)} \]

   if -2.9e6 < x

   1. Initial program 68.2%

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

   3. Simplified 68.2%

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

   5. Taylor expanded in x around 0 37.1%

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

   6. Taylor expanded in eps around inf 66.6%

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

      1. associate-*r* 66.6%

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

      2. neg-mul-1 66.6%

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

      3. *-commutative 66.6%

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

   8. Simplified 66.6%

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

   9. Taylor expanded in x around 0 53.7%

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

       1. mul-1-neg 53.7%

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

       2. distribute-rgt-neg-in 53.7%

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

       3. sub-neg 53.7%

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

       4. metadata-eval 53.7%

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

       5. distribute-neg-in 53.7%

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

       6. +-commutative 53.7%

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

       7. remove-double-neg 53.7%

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

       8. +-commutative 53.7%

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

   11. Simplified 53.7%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2900000:\\ \;\;\;\;\left(x \cdot 1.000072337962963\right) \cdot \left(x \cdot 1.000072337962963\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + \varepsilon\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 44.2% accurate, 227.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.7%

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

3. Simplified 72.7%

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

5. Taylor expanded in x around 0 39.5%

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

6. Taylor expanded in eps around inf 64.8%

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

   1. associate-*r* 64.8%

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

   2. neg-mul-1 64.8%

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

   3. *-commutative 64.8%

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

8. Simplified 64.8%

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

9. Taylor expanded in x around 0 43.4%

   \[\leadsto \color{blue}{1} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024150 
(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))