NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.3% → 99.6%

Time: 21.9s

Alternatives: 25

Speedup: 6.0×

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

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.3% 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: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \left(-1 - eps\_m\right)}\\ \mathbf{if}\;\left(1 - \frac{-1}{eps\_m}\right) \cdot e^{x \cdot \left(eps\_m + -1\right)} + t\_0 \cdot \left(1 + \frac{-1}{eps\_m}\right) \leq 0:\\ \;\;\;\;\frac{\frac{1 + x}{e^{x}} + \left(1 + x\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(e^{\mathsf{fma}\left(eps\_m, x, x\right)}\right)}^{3}} + t\_0}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (- -1.0 eps_m)))))
   (if (<=
        (+
         (* (- 1.0 (/ -1.0 eps_m)) (exp (* x (+ eps_m -1.0))))
         (* t_0 (+ 1.0 (/ -1.0 eps_m))))
        0.0)
     (/ (+ (/ (+ 1.0 x) (exp x)) (* (+ 1.0 x) (exp (- x)))) 2.0)
     (/ (+ (cbrt (pow (exp (fma eps_m x x)) 3.0)) t_0) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * (-1.0 - eps_m)));
	double tmp;
	if ((((1.0 - (-1.0 / eps_m)) * exp((x * (eps_m + -1.0)))) + (t_0 * (1.0 + (-1.0 / eps_m)))) <= 0.0) {
		tmp = (((1.0 + x) / exp(x)) + ((1.0 + x) * exp(-x))) / 2.0;
	} else {
		tmp = (cbrt(pow(exp(fma(eps_m, x, x)), 3.0)) + t_0) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = exp(Float64(x * Float64(-1.0 - eps_m)))
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 - Float64(-1.0 / eps_m)) * exp(Float64(x * Float64(eps_m + -1.0)))) + Float64(t_0 * Float64(1.0 + Float64(-1.0 / eps_m)))) <= 0.0)
		tmp = Float64(Float64(Float64(Float64(1.0 + x) / exp(x)) + Float64(Float64(1.0 + x) * exp(Float64(-x)))) / 2.0);
	else
		tmp = Float64(Float64(cbrt((exp(fma(eps_m, x, x)) ^ 3.0)) + t_0) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[(1.0 - N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + x), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Power[N[Power[N[Exp[N[(eps$95$m * x + x), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision] + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 0.0

   1. Initial program 35.4%

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

   3. Simplified 35.4%

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

   5. Taylor expanded in eps around 0 100.0%

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

   7. Simplified 100.0%

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

      1. exp-neg 100.0%

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

      2. un-div-inv 100.0%

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

   9. Applied egg-rr 100.0%

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

   if 0.0 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) 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 88.4%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around -inf 100.0%

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

      1. rec-exp 100.0%

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

      2. cancel-sign-sub-inv 100.0%

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

      3. metadata-eval 100.0%

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

      4. distribute-rgt-in 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. unsub-neg 100.0%

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

   8. Simplified 100.0%

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

      1. add-cube-cbrt 100.0%

         \[\leadsto \frac{e^{\color{blue}{\left(\sqrt[3]{x \cdot \left(\varepsilon - 1\right)} \cdot \sqrt[3]{x \cdot \left(\varepsilon - 1\right)}\right) \cdot \sqrt[3]{x \cdot \left(\varepsilon - 1\right)}}} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2} \]

      2. pow3 100.0%

         \[\leadsto \frac{e^{\color{blue}{{\left(\sqrt[3]{x \cdot \left(\varepsilon - 1\right)}\right)}^{3}}} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2} \]

      3. sub-neg 100.0%

         \[\leadsto \frac{e^{{\left(\sqrt[3]{x \cdot \color{blue}{\left(\varepsilon + \left(-1\right)\right)}}\right)}^{3}} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2} \]

      4. metadata-eval 100.0%

         \[\leadsto \frac{e^{{\left(\sqrt[3]{x \cdot \left(\varepsilon + \color{blue}{-1}\right)}\right)}^{3}} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \frac{e^{\color{blue}{{\left(\sqrt[3]{x \cdot \left(\varepsilon + -1\right)}\right)}^{3}}} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2} \]
   11. Step-by-step derivation

       1. rem-cube-cbrt 100.0%

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

       2. add-cbrt-cube 100.0%

          \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(e^{x \cdot \left(\varepsilon + -1\right)} \cdot e^{x \cdot \left(\varepsilon + -1\right)}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)}}} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2} \]

       3. pow3 100.0%

          \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(e^{x \cdot \left(\varepsilon + -1\right)}\right)}^{3}}} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2} \]

       4. distribute-rgt-in 100.0%

          \[\leadsto \frac{\sqrt[3]{{\left(e^{\color{blue}{\varepsilon \cdot x + -1 \cdot x}}\right)}^{3}} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2} \]

       5. neg-mul-1 100.0%

          \[\leadsto \frac{\sqrt[3]{{\left(e^{\varepsilon \cdot x + \color{blue}{\left(-x\right)}}\right)}^{3}} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2} \]

       6. fma-define 100.0%

          \[\leadsto \frac{\sqrt[3]{{\left(e^{\color{blue}{\mathsf{fma}\left(\varepsilon, x, -x\right)}}\right)}^{3}} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2} \]

       7. add-sqr-sqrt 48.7%

          \[\leadsto \frac{\sqrt[3]{{\left(e^{\mathsf{fma}\left(\varepsilon, x, \color{blue}{\sqrt{-x} \cdot \sqrt{-x}}\right)}\right)}^{3}} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2} \]

       8. sqrt-unprod 100.0%

          \[\leadsto \frac{\sqrt[3]{{\left(e^{\mathsf{fma}\left(\varepsilon, x, \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}\right)}\right)}^{3}} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2} \]

       9. sqr-neg 100.0%

          \[\leadsto \frac{\sqrt[3]{{\left(e^{\mathsf{fma}\left(\varepsilon, x, \sqrt{\color{blue}{x \cdot x}}\right)}\right)}^{3}} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2} \]

       10. sqrt-unprod 51.2%

           \[\leadsto \frac{\sqrt[3]{{\left(e^{\mathsf{fma}\left(\varepsilon, x, \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right)}\right)}^{3}} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2} \]

       11. add-sqr-sqrt 100.0%

           \[\leadsto \frac{\sqrt[3]{{\left(e^{\mathsf{fma}\left(\varepsilon, x, \color{blue}{x}\right)}\right)}^{3}} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2} \]

   12. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{-1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 0:\\ \;\;\;\;\frac{\frac{1 + x}{e^{x}} + \left(1 + x\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(e^{\mathsf{fma}\left(\varepsilon, x, x\right)}\right)}^{3}} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := e^{x \cdot \left(-1 - eps\_m\right)}\\ \mathbf{if}\;\left(1 - \frac{-1}{eps\_m}\right) \cdot e^{x \cdot \left(eps\_m + -1\right)} + t\_0 \cdot \left(1 + \frac{-1}{eps\_m}\right) \leq 0:\\ \;\;\;\;\frac{\frac{1 + x}{e^{x}} + \left(1 + x\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 + e^{eps\_m \cdot x}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (exp (* x (- -1.0 eps_m)))))
   (if (<=
        (+
         (* (- 1.0 (/ -1.0 eps_m)) (exp (* x (+ eps_m -1.0))))
         (* t_0 (+ 1.0 (/ -1.0 eps_m))))
        0.0)
     (/ (+ (/ (+ 1.0 x) (exp x)) (* (+ 1.0 x) (exp (- x)))) 2.0)
     (/ (+ t_0 (exp (* eps_m x))) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = exp((x * (-1.0 - eps_m)));
	double tmp;
	if ((((1.0 - (-1.0 / eps_m)) * exp((x * (eps_m + -1.0)))) + (t_0 * (1.0 + (-1.0 / eps_m)))) <= 0.0) {
		tmp = (((1.0 + x) / exp(x)) + ((1.0 + x) * exp(-x))) / 2.0;
	} else {
		tmp = (t_0 + exp((eps_m * x))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((x * ((-1.0d0) - eps_m)))
    if ((((1.0d0 - ((-1.0d0) / eps_m)) * exp((x * (eps_m + (-1.0d0))))) + (t_0 * (1.0d0 + ((-1.0d0) / eps_m)))) <= 0.0d0) then
        tmp = (((1.0d0 + x) / exp(x)) + ((1.0d0 + x) * exp(-x))) / 2.0d0
    else
        tmp = (t_0 + exp((eps_m * x))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = Math.exp((x * (-1.0 - eps_m)));
	double tmp;
	if ((((1.0 - (-1.0 / eps_m)) * Math.exp((x * (eps_m + -1.0)))) + (t_0 * (1.0 + (-1.0 / eps_m)))) <= 0.0) {
		tmp = (((1.0 + x) / Math.exp(x)) + ((1.0 + x) * Math.exp(-x))) / 2.0;
	} else {
		tmp = (t_0 + Math.exp((eps_m * x))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = math.exp((x * (-1.0 - eps_m)))
	tmp = 0
	if (((1.0 - (-1.0 / eps_m)) * math.exp((x * (eps_m + -1.0)))) + (t_0 * (1.0 + (-1.0 / eps_m)))) <= 0.0:
		tmp = (((1.0 + x) / math.exp(x)) + ((1.0 + x) * math.exp(-x))) / 2.0
	else:
		tmp = (t_0 + math.exp((eps_m * x))) / 2.0
	return tmp

Julia

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

MATLAB

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[(1.0 - N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + x), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$0 + N[Exp[N[(eps$95$m * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) x))))) < 0.0

   1. Initial program 35.4%

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

   3. Simplified 35.4%

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

   5. Taylor expanded in eps around 0 100.0%

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

   7. Simplified 100.0%

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

      1. exp-neg 100.0%

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

      2. un-div-inv 100.0%

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

   9. Applied egg-rr 100.0%

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

   if 0.0 < (-.f64 (*.f64 (+.f64 #s(literal 1 binary64) (/.f64 #s(literal 1 binary64) eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 #s(literal 1 binary64) eps) x)))) (*.f64 (-.f64 (/.f64 #s(literal 1 binary64) eps) #s(literal 1 binary64)) (exp.f64 (neg.f64 (*.f64 (+.f64 #s(literal 1 binary64) eps) 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 88.4%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around -inf 100.0%

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

      1. rec-exp 100.0%

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

      2. cancel-sign-sub-inv 100.0%

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

      3. metadata-eval 100.0%

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

      4. distribute-rgt-in 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. distribute-neg-in 100.0%

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

      8. metadata-eval 100.0%

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

      9. unsub-neg 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in eps around inf 100.0%

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

       1. *-commutative 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{-1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right) \leq 0:\\ \;\;\;\;\frac{\frac{1 + x}{e^{x}} + \left(1 + x\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)} + e^{\varepsilon \cdot x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Python

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

MATLAB

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

Derivation

1. Initial program 75.8%

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

3. Simplified 68.5%

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

5. Taylor expanded in eps around inf 98.2%

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

6. Taylor expanded in x around -inf 98.2%

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

   1. rec-exp 98.2%

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

   2. cancel-sign-sub-inv 98.2%

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

   3. metadata-eval 98.2%

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

   4. distribute-rgt-in 98.2%

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

   5. +-commutative 98.2%

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

   6. distribute-rgt-neg-in 98.2%

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

   7. distribute-neg-in 98.2%

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

   8. metadata-eval 98.2%

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

   9. unsub-neg 98.2%

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

8. Simplified 98.2%

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

9. Final simplification 98.2%

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

Alternative 4: 85.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

MATLAB

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

Derivation

1. Initial program 75.8%

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

3. Simplified 68.5%

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

5. Taylor expanded in eps around inf 98.2%

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

6. Taylor expanded in x around -inf 98.2%

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

   1. rec-exp 98.2%

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

   2. cancel-sign-sub-inv 98.2%

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

   3. metadata-eval 98.2%

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

   4. distribute-rgt-in 98.2%

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

   5. +-commutative 98.2%

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

   6. distribute-rgt-neg-in 98.2%

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

   7. distribute-neg-in 98.2%

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

   8. metadata-eval 98.2%

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

   9. unsub-neg 98.2%

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

8. Simplified 98.2%

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

9. Taylor expanded in eps around inf 89.9%

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

    1. *-commutative 89.9%

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

11. Simplified 89.9%

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

12. Final simplification 89.9%

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

Alternative 5: 84.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-234}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - eps\_m\right)}}{2}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+210} \lor \neg \left(x \leq 8.2 \cdot 10^{+239}\right):\\ \;\;\;\;\frac{e^{x \cdot \left(eps\_m + -1\right)} + \frac{1}{1 + x \cdot \left(1 + eps\_m\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{eps\_m}\right) + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -4.4e-234)
   (/ (+ 1.0 (exp (* x (- -1.0 eps_m)))) 2.0)
   (if (or (<= x 9e+210) (not (<= x 8.2e+239)))
     (/ (+ (exp (* x (+ eps_m -1.0))) (/ 1.0 (+ 1.0 (* x (+ 1.0 eps_m))))) 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 <= -4.4e-234) {
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	} else if ((x <= 9e+210) || !(x <= 8.2e+239)) {
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 / (1.0 + (x * (1.0 + eps_m))))) / 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 <= (-4.4d-234)) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps_m)))) / 2.0d0
    else if ((x <= 9d+210) .or. (.not. (x <= 8.2d+239))) then
        tmp = (exp((x * (eps_m + (-1.0d0)))) + (1.0d0 / (1.0d0 + (x * (1.0d0 + eps_m))))) / 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 <= -4.4e-234) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps_m)))) / 2.0;
	} else if ((x <= 9e+210) || !(x <= 8.2e+239)) {
		tmp = (Math.exp((x * (eps_m + -1.0))) + (1.0 / (1.0 + (x * (1.0 + eps_m))))) / 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 <= -4.4e-234:
		tmp = (1.0 + math.exp((x * (-1.0 - eps_m)))) / 2.0
	elif (x <= 9e+210) or not (x <= 8.2e+239):
		tmp = (math.exp((x * (eps_m + -1.0))) + (1.0 / (1.0 + (x * (1.0 + eps_m))))) / 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 <= -4.4e-234)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	elseif ((x <= 9e+210) || !(x <= 8.2e+239))
		tmp = Float64(Float64(exp(Float64(x * Float64(eps_m + -1.0))) + Float64(1.0 / Float64(1.0 + Float64(x * Float64(1.0 + eps_m))))) / 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 <= -4.4e-234)
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	elseif ((x <= 9e+210) || ~((x <= 8.2e+239)))
		tmp = (exp((x * (eps_m + -1.0))) + (1.0 / (1.0 + (x * (1.0 + eps_m))))) / 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, -4.4e-234], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 9e+210], N[Not[LessEqual[x, 8.2e+239]], $MachinePrecision]], N[(N[(N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(1.0 + N[(x * N[(1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 < -4.3999999999999998e-234

   1. Initial program 69.1%

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

   3. Simplified 69.1%

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

   5. Taylor expanded in x around 0 39.3%

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

   6. Taylor expanded in eps around inf 67.0%

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

      1. sub-neg 67.0%

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

      2. mul-1-neg 67.0%

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

      3. remove-double-neg 67.0%

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

      4. mul-1-neg 67.0%

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

      5. distribute-rgt-neg-in 67.0%

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

      6. distribute-neg-in 67.0%

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

      7. metadata-eval 67.0%

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

      8. unsub-neg 67.0%

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

   8. Simplified 67.0%

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

   if -4.3999999999999998e-234 < x < 9.00000000000000007e210 or 8.2000000000000002e239 < x

   1. Initial program 78.9%

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

   3. Simplified 72.5%

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

   5. Taylor expanded in eps around inf 98.9%

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

   6. Taylor expanded in x around 0 61.3%

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

   if 9.00000000000000007e210 < x < 8.2000000000000002e239

   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 2.2%

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

   6. Taylor expanded in x around 0 100.0%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-234}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+210} \lor \neg \left(x \leq 8.2 \cdot 10^{+239}\right):\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \frac{1}{1 + x \cdot \left(1 + \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-252}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - eps\_m\right)}}{2}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+211} \lor \neg \left(x \leq 9.5 \cdot 10^{+241}\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \left(eps\_m + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{eps\_m}\right) + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -3.3e-252)
   (/ (+ 1.0 (exp (* x (- -1.0 eps_m)))) 2.0)
   (if (or (<= x 1.9e+211) (not (<= x 9.5e+241)))
     (/ (+ 1.0 (exp (* x (+ eps_m -1.0)))) 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 <= -3.3e-252) {
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	} else if ((x <= 1.9e+211) || !(x <= 9.5e+241)) {
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 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 <= (-3.3d-252)) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps_m)))) / 2.0d0
    else if ((x <= 1.9d+211) .or. (.not. (x <= 9.5d+241))) then
        tmp = (1.0d0 + exp((x * (eps_m + (-1.0d0))))) / 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 <= -3.3e-252) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps_m)))) / 2.0;
	} else if ((x <= 1.9e+211) || !(x <= 9.5e+241)) {
		tmp = (1.0 + Math.exp((x * (eps_m + -1.0)))) / 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 <= -3.3e-252:
		tmp = (1.0 + math.exp((x * (-1.0 - eps_m)))) / 2.0
	elif (x <= 1.9e+211) or not (x <= 9.5e+241):
		tmp = (1.0 + math.exp((x * (eps_m + -1.0)))) / 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 <= -3.3e-252)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	elseif ((x <= 1.9e+211) || !(x <= 9.5e+241))
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps_m + -1.0)))) / 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 <= -3.3e-252)
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	elseif ((x <= 1.9e+211) || ~((x <= 9.5e+241)))
		tmp = (1.0 + exp((x * (eps_m + -1.0)))) / 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, -3.3e-252], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 1.9e+211], N[Not[LessEqual[x, 9.5e+241]], $MachinePrecision]], N[(N[(1.0 + N[Exp[N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]], $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 < -3.30000000000000009e-252

   1. Initial program 70.0%

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

   3. Simplified 70.0%

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

   5. Taylor expanded in x around 0 41.4%

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

   6. Taylor expanded in eps around inf 68.5%

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

      1. sub-neg 68.5%

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

      2. mul-1-neg 68.5%

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

      3. remove-double-neg 68.5%

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

      4. mul-1-neg 68.5%

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

      5. distribute-rgt-neg-in 68.5%

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

      6. distribute-neg-in 68.5%

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

      7. metadata-eval 68.5%

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

      8. unsub-neg 68.5%

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

   8. Simplified 68.5%

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

   if -3.30000000000000009e-252 < x < 1.90000000000000008e211 or 9.50000000000000019e241 < x

   1. Initial program 78.7%

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

   3. Simplified 72.4%

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

   5. Taylor expanded in eps around inf 98.9%

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

   6. Taylor expanded in x around 0 60.1%

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

   if 1.90000000000000008e211 < x < 9.50000000000000019e241

   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 2.2%

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

   6. Taylor expanded in x around 0 100.0%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-252}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+211} \lor \neg \left(x \leq 9.5 \cdot 10^{+241}\right):\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := x \cdot \left(eps\_m + -1\right)\\ \mathbf{if}\;x \leq 5 \cdot 10^{-285}:\\ \;\;\;\;\frac{eps\_m \cdot \left(x + \frac{1 + e^{x \cdot \left(-1 - eps\_m\right)}}{eps\_m}\right)}{2}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+211}:\\ \;\;\;\;\frac{1 + {e}^{t\_0}}{2}\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+239}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{eps\_m}\right) + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{t\_0}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* x (+ eps_m -1.0))))
   (if (<= x 5e-285)
     (/ (* eps_m (+ x (/ (+ 1.0 (exp (* x (- -1.0 eps_m)))) eps_m))) 2.0)
     (if (<= x 1.9e+211)
       (/ (+ 1.0 (pow E t_0)) 2.0)
       (if (<= x 8.2e+239)
         (/ (+ (- 1.0 (/ -1.0 eps_m)) (+ 1.0 (/ -1.0 eps_m))) 2.0)
         (/ (+ 1.0 (exp t_0)) 2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = x * (eps_m + -1.0);
	double tmp;
	if (x <= 5e-285) {
		tmp = (eps_m * (x + ((1.0 + exp((x * (-1.0 - eps_m)))) / eps_m))) / 2.0;
	} else if (x <= 1.9e+211) {
		tmp = (1.0 + pow(((double) M_E), t_0)) / 2.0;
	} else if (x <= 8.2e+239) {
		tmp = ((1.0 - (-1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = (1.0 + exp(t_0)) / 2.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = x * (eps_m + -1.0);
	double tmp;
	if (x <= 5e-285) {
		tmp = (eps_m * (x + ((1.0 + Math.exp((x * (-1.0 - eps_m)))) / eps_m))) / 2.0;
	} else if (x <= 1.9e+211) {
		tmp = (1.0 + Math.pow(Math.E, t_0)) / 2.0;
	} else if (x <= 8.2e+239) {
		tmp = ((1.0 - (-1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = (1.0 + Math.exp(t_0)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = x * (eps_m + -1.0)
	tmp = 0
	if x <= 5e-285:
		tmp = (eps_m * (x + ((1.0 + math.exp((x * (-1.0 - eps_m)))) / eps_m))) / 2.0
	elif x <= 1.9e+211:
		tmp = (1.0 + math.pow(math.e, t_0)) / 2.0
	elif x <= 8.2e+239:
		tmp = ((1.0 - (-1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0
	else:
		tmp = (1.0 + math.exp(t_0)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(x * Float64(eps_m + -1.0))
	tmp = 0.0
	if (x <= 5e-285)
		tmp = Float64(Float64(eps_m * Float64(x + Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps_m)))) / eps_m))) / 2.0);
	elseif (x <= 1.9e+211)
		tmp = Float64(Float64(1.0 + (exp(1) ^ t_0)) / 2.0);
	elseif (x <= 8.2e+239)
		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(1.0 + exp(t_0)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = x * (eps_m + -1.0);
	tmp = 0.0;
	if (x <= 5e-285)
		tmp = (eps_m * (x + ((1.0 + exp((x * (-1.0 - eps_m)))) / eps_m))) / 2.0;
	elseif (x <= 1.9e+211)
		tmp = (1.0 + (2.71828182845904523536 ^ t_0)) / 2.0;
	elseif (x <= 8.2e+239)
		tmp = ((1.0 - (-1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	else
		tmp = (1.0 + exp(t_0)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5e-285], N[(N[(eps$95$m * N[(x + N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.9e+211], N[(N[(1.0 + N[Power[E, t$95$0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 8.2e+239], 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[(1.0 + N[Exp[t$95$0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 5.00000000000000018e-285

   1. Initial program 71.4%

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

   3. Simplified 71.4%

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

   5. Taylor expanded in x around 0 46.9%

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

   6. Taylor expanded in eps around inf 81.1%

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

      1. associate--l+ 81.1%

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

      2. associate-*r/ 81.1%

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

      3. div-sub 81.1%

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

      4. sub-neg 81.1%

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

      5. mul-1-neg 81.1%

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

      6. remove-double-neg 81.1%

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

      7. mul-1-neg 81.1%

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

      8. distribute-rgt-neg-in 81.1%

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

      9. distribute-neg-in 81.1%

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

      10. metadata-eval 81.1%

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

      11. unsub-neg 81.1%

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

   8. Simplified 81.1%

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

   if 5.00000000000000018e-285 < x < 1.90000000000000008e211

   1. Initial program 76.8%

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

   3. Simplified 68.9%

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

   5. Taylor expanded in eps around inf 98.6%

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

   6. Taylor expanded in x around 0 54.8%

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

      1. *-un-lft-identity 54.8%

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

      2. exp-prod 54.8%

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

      3. sub-neg 54.8%

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

      4. metadata-eval 54.8%

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

   8. Applied egg-rr 54.8%

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

      1. exp-1-e 54.8%

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

      2. +-commutative 54.8%

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

   10. Simplified 54.8%

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

   if 1.90000000000000008e211 < x < 8.2000000000000002e239

   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 2.2%

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

   6. Taylor expanded in x around 0 100.0%

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

   if 8.2000000000000002e239 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 38.4%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-285}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+211}:\\ \;\;\;\;\frac{1 + {e}^{\left(x \cdot \left(\varepsilon + -1\right)\right)}}{2}\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+239}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := x \cdot \left(eps\_m + -1\right)\\ \mathbf{if}\;x \leq -4 \cdot 10^{-252}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - eps\_m\right)}}{2}\\ \mathbf{elif}\;x \leq 1.46 \cdot 10^{+211}:\\ \;\;\;\;\frac{1 + {e}^{t\_0}}{2}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+240}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{eps\_m}\right) + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{t\_0}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* x (+ eps_m -1.0))))
   (if (<= x -4e-252)
     (/ (+ 1.0 (exp (* x (- -1.0 eps_m)))) 2.0)
     (if (<= x 1.46e+211)
       (/ (+ 1.0 (pow E t_0)) 2.0)
       (if (<= x 2.7e+240)
         (/ (+ (- 1.0 (/ -1.0 eps_m)) (+ 1.0 (/ -1.0 eps_m))) 2.0)
         (/ (+ 1.0 (exp t_0)) 2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = x * (eps_m + -1.0);
	double tmp;
	if (x <= -4e-252) {
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	} else if (x <= 1.46e+211) {
		tmp = (1.0 + pow(((double) M_E), t_0)) / 2.0;
	} else if (x <= 2.7e+240) {
		tmp = ((1.0 - (-1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = (1.0 + exp(t_0)) / 2.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = x * (eps_m + -1.0);
	double tmp;
	if (x <= -4e-252) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps_m)))) / 2.0;
	} else if (x <= 1.46e+211) {
		tmp = (1.0 + Math.pow(Math.E, t_0)) / 2.0;
	} else if (x <= 2.7e+240) {
		tmp = ((1.0 - (-1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = (1.0 + Math.exp(t_0)) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = x * (eps_m + -1.0)
	tmp = 0
	if x <= -4e-252:
		tmp = (1.0 + math.exp((x * (-1.0 - eps_m)))) / 2.0
	elif x <= 1.46e+211:
		tmp = (1.0 + math.pow(math.e, t_0)) / 2.0
	elif x <= 2.7e+240:
		tmp = ((1.0 - (-1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0
	else:
		tmp = (1.0 + math.exp(t_0)) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(x * Float64(eps_m + -1.0))
	tmp = 0.0
	if (x <= -4e-252)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	elseif (x <= 1.46e+211)
		tmp = Float64(Float64(1.0 + (exp(1) ^ t_0)) / 2.0);
	elseif (x <= 2.7e+240)
		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(1.0 + exp(t_0)) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	t_0 = x * (eps_m + -1.0);
	tmp = 0.0;
	if (x <= -4e-252)
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	elseif (x <= 1.46e+211)
		tmp = (1.0 + (2.71828182845904523536 ^ t_0)) / 2.0;
	elseif (x <= 2.7e+240)
		tmp = ((1.0 - (-1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	else
		tmp = (1.0 + exp(t_0)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(x * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4e-252], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.46e+211], N[(N[(1.0 + N[Power[E, t$95$0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.7e+240], 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[(1.0 + N[Exp[t$95$0], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.99999999999999977e-252

   1. Initial program 70.0%

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

   3. Simplified 70.0%

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

   5. Taylor expanded in x around 0 41.4%

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

   6. Taylor expanded in eps around inf 68.5%

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

      1. sub-neg 68.5%

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

      2. mul-1-neg 68.5%

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

      3. remove-double-neg 68.5%

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

      4. mul-1-neg 68.5%

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

      5. distribute-rgt-neg-in 68.5%

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

      6. distribute-neg-in 68.5%

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

      7. metadata-eval 68.5%

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

      8. unsub-neg 68.5%

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

   8. Simplified 68.5%

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

   if -3.99999999999999977e-252 < x < 1.4599999999999999e211

   1. Initial program 77.0%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in eps around inf 98.8%

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

   6. Taylor expanded in x around 0 61.8%

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

      1. *-un-lft-identity 61.8%

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

      2. exp-prod 61.8%

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

      3. sub-neg 61.8%

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

      4. metadata-eval 61.8%

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

   8. Applied egg-rr 61.8%

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

      1. exp-1-e 61.8%

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

      2. +-commutative 61.8%

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

   10. Simplified 61.8%

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

   if 1.4599999999999999e211 < x < 2.6999999999999999e240

   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 2.2%

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

   6. Taylor expanded in x around 0 100.0%

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

   if 2.6999999999999999e240 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 38.4%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-252}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 1.46 \cdot 10^{+211}:\\ \;\;\;\;\frac{1 + {e}^{\left(x \cdot \left(\varepsilon + -1\right)\right)}}{2}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+240}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-252}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - eps\_m\right)}}{2}\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{+208} \lor \neg \left(x \leq 2.1 \cdot 10^{+243}\right):\\ \;\;\;\;\frac{1 + e^{eps\_m \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{eps\_m}\right) + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -3.3e-252)
   (/ (+ 1.0 (exp (* x (- -1.0 eps_m)))) 2.0)
   (if (or (<= x 4.3e+208) (not (<= x 2.1e+243)))
     (/ (+ 1.0 (exp (* eps_m 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 <= -3.3e-252) {
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	} else if ((x <= 4.3e+208) || !(x <= 2.1e+243)) {
		tmp = (1.0 + exp((eps_m * 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 <= (-3.3d-252)) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps_m)))) / 2.0d0
    else if ((x <= 4.3d+208) .or. (.not. (x <= 2.1d+243))) then
        tmp = (1.0d0 + exp((eps_m * 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 <= -3.3e-252) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps_m)))) / 2.0;
	} else if ((x <= 4.3e+208) || !(x <= 2.1e+243)) {
		tmp = (1.0 + Math.exp((eps_m * 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 <= -3.3e-252:
		tmp = (1.0 + math.exp((x * (-1.0 - eps_m)))) / 2.0
	elif (x <= 4.3e+208) or not (x <= 2.1e+243):
		tmp = (1.0 + math.exp((eps_m * 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 <= -3.3e-252)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps_m)))) / 2.0);
	elseif ((x <= 4.3e+208) || !(x <= 2.1e+243))
		tmp = Float64(Float64(1.0 + exp(Float64(eps_m * 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 <= -3.3e-252)
		tmp = (1.0 + exp((x * (-1.0 - eps_m)))) / 2.0;
	elseif ((x <= 4.3e+208) || ~((x <= 2.1e+243)))
		tmp = (1.0 + exp((eps_m * 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, -3.3e-252], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 4.3e+208], N[Not[LessEqual[x, 2.1e+243]], $MachinePrecision]], N[(N[(1.0 + N[Exp[N[(eps$95$m * x), $MachinePrecision]], $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 < -3.30000000000000009e-252

   1. Initial program 70.0%

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

   3. Simplified 70.0%

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

   5. Taylor expanded in x around 0 41.4%

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

   6. Taylor expanded in eps around inf 68.5%

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

      1. sub-neg 68.5%

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

      2. mul-1-neg 68.5%

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

      3. remove-double-neg 68.5%

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

      4. mul-1-neg 68.5%

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

      5. distribute-rgt-neg-in 68.5%

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

      6. distribute-neg-in 68.5%

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

      7. metadata-eval 68.5%

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

      8. unsub-neg 68.5%

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

   8. Simplified 68.5%

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

   if -3.30000000000000009e-252 < x < 4.30000000000000042e208 or 2.0999999999999999e243 < x

   1. Initial program 78.7%

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

   3. Simplified 72.4%

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

   5. Taylor expanded in eps around inf 98.9%

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

   6. Taylor expanded in x around 0 60.1%

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

   7. Taylor expanded in eps around inf 60.1%

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

      1. *-commutative 86.6%

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

   9. Simplified 60.1%

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

   if 4.30000000000000042e208 < x < 2.0999999999999999e243

   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 2.2%

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

   6. Taylor expanded in x around 0 100.0%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{-252}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{+208} \lor \neg \left(x \leq 2.1 \cdot 10^{+243}\right):\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-285}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+211} \lor \neg \left(x \leq 10^{+240}\right):\\ \;\;\;\;\frac{1 + e^{eps\_m \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{eps\_m}\right) + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 5e-285)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (or (<= x 1.45e+211) (not (<= x 1e+240)))
     (/ (+ 1.0 (exp (* eps_m 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 <= 5e-285) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if ((x <= 1.45e+211) || !(x <= 1e+240)) {
		tmp = (1.0 + exp((eps_m * 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 <= 5d-285) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if ((x <= 1.45d+211) .or. (.not. (x <= 1d+240))) then
        tmp = (1.0d0 + exp((eps_m * 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 <= 5e-285) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if ((x <= 1.45e+211) || !(x <= 1e+240)) {
		tmp = (1.0 + Math.exp((eps_m * 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 <= 5e-285:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif (x <= 1.45e+211) or not (x <= 1e+240):
		tmp = (1.0 + math.exp((eps_m * 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 <= 5e-285)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif ((x <= 1.45e+211) || !(x <= 1e+240))
		tmp = Float64(Float64(1.0 + exp(Float64(eps_m * 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 <= 5e-285)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif ((x <= 1.45e+211) || ~((x <= 1e+240)))
		tmp = (1.0 + exp((eps_m * 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, 5e-285], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 1.45e+211], N[Not[LessEqual[x, 1e+240]], $MachinePrecision]], N[(N[(1.0 + N[Exp[N[(eps$95$m * x), $MachinePrecision]], $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 < 5.00000000000000018e-285

   1. Initial program 71.4%

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

   3. Simplified 63.8%

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

   5. Taylor expanded in eps around inf 97.5%

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

   6. Taylor expanded in x around 0 77.2%

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

   7. Taylor expanded in eps around 0 83.9%

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

      1. mul-1-neg 83.9%

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

   9. Simplified 83.9%

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

   if 5.00000000000000018e-285 < x < 1.45e211 or 1.00000000000000001e240 < x

   1. Initial program 78.8%

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

   3. Simplified 71.6%

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

   5. Taylor expanded in eps around inf 98.7%

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

   6. Taylor expanded in x around 0 53.4%

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

   7. Taylor expanded in eps around inf 53.4%

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

      1. *-commutative 84.3%

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

   9. Simplified 53.4%

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

   if 1.45e211 < x < 1.00000000000000001e240

   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 2.2%

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

   6. Taylor expanded in x around 0 100.0%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-285}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+211} \lor \neg \left(x \leq 10^{+240}\right):\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 7.2 \cdot 10^{+37}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{elif}\;eps\_m \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{2 + x \cdot \frac{1 - \left(eps\_m \cdot \left(eps\_m \cdot \left(1 - \left(x \cdot 0.5 + 0.5 \cdot \left(eps\_m \cdot x\right)\right)\right) - x \cdot -0.5\right) - x \cdot -0.5\right)}{eps\_m}}{2}\\ \mathbf{elif}\;eps\_m \leq 2.4 \cdot 10^{+199}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(-1 - \frac{-1}{eps\_m}\right) \cdot \left(1 + eps\_m\right) - -0.5 \cdot \left(x \cdot \frac{-1 + eps\_m \cdot \left(eps\_m + -1\right)}{eps\_m}\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 7.2e+37)
   (/ (+ 1.0 (exp x)) 2.0)
   (if (<= eps_m 1.4e+154)
     (/
      (+
       2.0
       (*
        x
        (/
         (-
          1.0
          (-
           (*
            eps_m
            (- (* eps_m (- 1.0 (+ (* x 0.5) (* 0.5 (* eps_m x))))) (* x -0.5)))
           (* x -0.5)))
         eps_m)))
      2.0)
     (if (<= eps_m 2.4e+199)
       (/ (+ 1.0 (exp (- x))) 2.0)
       (/
        (+
         2.0
         (*
          x
          (-
           (* (- -1.0 (/ -1.0 eps_m)) (+ 1.0 eps_m))
           (* -0.5 (* x (/ (+ -1.0 (* eps_m (+ eps_m -1.0))) eps_m))))))
        2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 7.2e+37) {
		tmp = (1.0 + exp(x)) / 2.0;
	} else if (eps_m <= 1.4e+154) {
		tmp = (2.0 + (x * ((1.0 - ((eps_m * ((eps_m * (1.0 - ((x * 0.5) + (0.5 * (eps_m * x))))) - (x * -0.5))) - (x * -0.5))) / eps_m))) / 2.0;
	} else if (eps_m <= 2.4e+199) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else {
		tmp = (2.0 + (x * (((-1.0 - (-1.0 / eps_m)) * (1.0 + eps_m)) - (-0.5 * (x * ((-1.0 + (eps_m * (eps_m + -1.0))) / eps_m)))))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 7.2d+37) then
        tmp = (1.0d0 + exp(x)) / 2.0d0
    else if (eps_m <= 1.4d+154) then
        tmp = (2.0d0 + (x * ((1.0d0 - ((eps_m * ((eps_m * (1.0d0 - ((x * 0.5d0) + (0.5d0 * (eps_m * x))))) - (x * (-0.5d0)))) - (x * (-0.5d0)))) / eps_m))) / 2.0d0
    else if (eps_m <= 2.4d+199) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else
        tmp = (2.0d0 + (x * ((((-1.0d0) - ((-1.0d0) / eps_m)) * (1.0d0 + eps_m)) - ((-0.5d0) * (x * (((-1.0d0) + (eps_m * (eps_m + (-1.0d0)))) / eps_m)))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 7.2e+37) {
		tmp = (1.0 + Math.exp(x)) / 2.0;
	} else if (eps_m <= 1.4e+154) {
		tmp = (2.0 + (x * ((1.0 - ((eps_m * ((eps_m * (1.0 - ((x * 0.5) + (0.5 * (eps_m * x))))) - (x * -0.5))) - (x * -0.5))) / eps_m))) / 2.0;
	} else if (eps_m <= 2.4e+199) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else {
		tmp = (2.0 + (x * (((-1.0 - (-1.0 / eps_m)) * (1.0 + eps_m)) - (-0.5 * (x * ((-1.0 + (eps_m * (eps_m + -1.0))) / eps_m)))))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 7.2e+37:
		tmp = (1.0 + math.exp(x)) / 2.0
	elif eps_m <= 1.4e+154:
		tmp = (2.0 + (x * ((1.0 - ((eps_m * ((eps_m * (1.0 - ((x * 0.5) + (0.5 * (eps_m * x))))) - (x * -0.5))) - (x * -0.5))) / eps_m))) / 2.0
	elif eps_m <= 2.4e+199:
		tmp = (1.0 + math.exp(-x)) / 2.0
	else:
		tmp = (2.0 + (x * (((-1.0 - (-1.0 / eps_m)) * (1.0 + eps_m)) - (-0.5 * (x * ((-1.0 + (eps_m * (eps_m + -1.0))) / eps_m)))))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 7.2e+37)
		tmp = Float64(Float64(1.0 + exp(x)) / 2.0);
	elseif (eps_m <= 1.4e+154)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(1.0 - Float64(Float64(eps_m * Float64(Float64(eps_m * Float64(1.0 - Float64(Float64(x * 0.5) + Float64(0.5 * Float64(eps_m * x))))) - Float64(x * -0.5))) - Float64(x * -0.5))) / eps_m))) / 2.0);
	elseif (eps_m <= 2.4e+199)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(-1.0 - Float64(-1.0 / eps_m)) * Float64(1.0 + eps_m)) - Float64(-0.5 * Float64(x * Float64(Float64(-1.0 + Float64(eps_m * Float64(eps_m + -1.0))) / eps_m)))))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 7.2e+37)
		tmp = (1.0 + exp(x)) / 2.0;
	elseif (eps_m <= 1.4e+154)
		tmp = (2.0 + (x * ((1.0 - ((eps_m * ((eps_m * (1.0 - ((x * 0.5) + (0.5 * (eps_m * x))))) - (x * -0.5))) - (x * -0.5))) / eps_m))) / 2.0;
	elseif (eps_m <= 2.4e+199)
		tmp = (1.0 + exp(-x)) / 2.0;
	else
		tmp = (2.0 + (x * (((-1.0 - (-1.0 / eps_m)) * (1.0 + eps_m)) - (-0.5 * (x * ((-1.0 + (eps_m * (eps_m + -1.0))) / eps_m)))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 7.2e+37], N[(N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps$95$m, 1.4e+154], N[(N[(2.0 + N[(x * N[(N[(1.0 - N[(N[(eps$95$m * N[(N[(eps$95$m * N[(1.0 - N[(N[(x * 0.5), $MachinePrecision] + N[(0.5 * N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps$95$m, 2.4e+199], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * N[(N[(N[(-1.0 - N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision] - N[(-0.5 * N[(x * N[(N[(-1.0 + N[(eps$95$m * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if eps < 7.19999999999999995e37

   1. Initial program 67.7%

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

   3. Simplified 62.6%

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

   5. Taylor expanded in eps around inf 97.5%

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

   6. Taylor expanded in x around 0 63.1%

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

   7. Taylor expanded in eps around 0 57.2%

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

      1. mul-1-neg 57.2%

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

   9. Simplified 57.2%

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

       1. *-un-lft-identity 57.2%

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

       2. +-commutative 57.2%

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

       3. add-sqr-sqrt 35.5%

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

       4. sqrt-unprod 70.6%

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

       5. sqr-neg 70.6%

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

       6. sqrt-unprod 35.0%

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

       7. add-sqr-sqrt 61.0%

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

   11. Applied egg-rr 61.0%

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

       1. *-lft-identity 61.0%

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

   13. Simplified 61.0%

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

   if 7.19999999999999995e37 < eps < 1.4e154

   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 68.9%

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

   6. Taylor expanded in x around 0 72.8%

      \[\leadsto \frac{\color{blue}{2 + x \cdot \left(-0.5 \cdot \left(x \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)}}{2} \]
   7. Step-by-step derivation

      1. sub-neg 72.8%

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

      2. metadata-eval 72.8%

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

      3. +-commutative 72.8%

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

      4. associate-*r* 72.8%

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

      5. distribute-lft-in 72.8%

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

      6. metadata-eval 72.8%

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

      7. neg-mul-1 72.8%

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

      8. sub-neg 72.8%

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

      9. metadata-eval 72.8%

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

      10. +-commutative 72.8%

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

   8. Simplified 72.8%

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

   9. Taylor expanded in eps around 0 79.5%

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

   if 1.4e154 < eps < 2.40000000000000015e199

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 63.7%

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

   7. Taylor expanded in eps around 0 87.9%

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

      1. mul-1-neg 87.9%

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

   9. Simplified 87.9%

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

   if 2.40000000000000015e199 < 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 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 55.0%

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

   6. Taylor expanded in x around 0 89.6%

      \[\leadsto \frac{\color{blue}{2 + x \cdot \left(-0.5 \cdot \left(x \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)}}{2} \]
   7. Step-by-step derivation

      1. sub-neg 89.6%

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

      2. metadata-eval 89.6%

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

      3. +-commutative 89.6%

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

      4. associate-*r* 89.6%

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

      5. distribute-lft-in 89.6%

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

      6. metadata-eval 89.6%

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

      7. neg-mul-1 89.6%

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

      8. sub-neg 89.6%

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

      9. metadata-eval 89.6%

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

      10. +-commutative 89.6%

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

   8. Simplified 89.6%

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

   9. Taylor expanded in eps around 0 89.6%

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

       1. mul-1-neg 89.6%

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

       2. sub-neg 89.6%

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

   11. Simplified 89.6%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 7.2 \cdot 10^{+37}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{2 + x \cdot \frac{1 - \left(\varepsilon \cdot \left(\varepsilon \cdot \left(1 - \left(x \cdot 0.5 + 0.5 \cdot \left(\varepsilon \cdot x\right)\right)\right) - x \cdot -0.5\right) - x \cdot -0.5\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 2.4 \cdot 10^{+199}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(-1 - \frac{-1}{\varepsilon}\right) \cdot \left(1 + \varepsilon\right) - -0.5 \cdot \left(x \cdot \frac{-1 + \varepsilon \cdot \left(\varepsilon + -1\right)}{\varepsilon}\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 1.8 \cdot 10^{+37}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \frac{1 - \left(eps\_m \cdot \left(eps\_m \cdot \left(1 - \left(x \cdot 0.5 + 0.5 \cdot \left(eps\_m \cdot x\right)\right)\right) - x \cdot -0.5\right) - x \cdot -0.5\right)}{eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 1.8e+37)
   (/ (+ 1.0 (exp x)) 2.0)
   (/
    (+
     2.0
     (*
      x
      (/
       (-
        1.0
        (-
         (*
          eps_m
          (- (* eps_m (- 1.0 (+ (* x 0.5) (* 0.5 (* eps_m x))))) (* x -0.5)))
         (* x -0.5)))
       eps_m)))
    2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.8e+37) {
		tmp = (1.0 + exp(x)) / 2.0;
	} else {
		tmp = (2.0 + (x * ((1.0 - ((eps_m * ((eps_m * (1.0 - ((x * 0.5) + (0.5 * (eps_m * x))))) - (x * -0.5))) - (x * -0.5))) / eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 1.8d+37) then
        tmp = (1.0d0 + exp(x)) / 2.0d0
    else
        tmp = (2.0d0 + (x * ((1.0d0 - ((eps_m * ((eps_m * (1.0d0 - ((x * 0.5d0) + (0.5d0 * (eps_m * x))))) - (x * (-0.5d0)))) - (x * (-0.5d0)))) / eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.8e+37) {
		tmp = (1.0 + Math.exp(x)) / 2.0;
	} else {
		tmp = (2.0 + (x * ((1.0 - ((eps_m * ((eps_m * (1.0 - ((x * 0.5) + (0.5 * (eps_m * x))))) - (x * -0.5))) - (x * -0.5))) / eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 1.8e+37:
		tmp = (1.0 + math.exp(x)) / 2.0
	else:
		tmp = (2.0 + (x * ((1.0 - ((eps_m * ((eps_m * (1.0 - ((x * 0.5) + (0.5 * (eps_m * x))))) - (x * -0.5))) - (x * -0.5))) / eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 1.8e+37)
		tmp = Float64(Float64(1.0 + exp(x)) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(1.0 - Float64(Float64(eps_m * Float64(Float64(eps_m * Float64(1.0 - Float64(Float64(x * 0.5) + Float64(0.5 * Float64(eps_m * x))))) - Float64(x * -0.5))) - Float64(x * -0.5))) / eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 1.8e+37)
		tmp = (1.0 + exp(x)) / 2.0;
	else
		tmp = (2.0 + (x * ((1.0 - ((eps_m * ((eps_m * (1.0 - ((x * 0.5) + (0.5 * (eps_m * x))))) - (x * -0.5))) - (x * -0.5))) / eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 1.8e+37], N[(N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * N[(N[(1.0 - N[(N[(eps$95$m * N[(N[(eps$95$m * N[(1.0 - N[(N[(x * 0.5), $MachinePrecision] + N[(0.5 * N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.79999999999999999e37

   1. Initial program 67.7%

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

   3. Simplified 62.6%

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

   5. Taylor expanded in eps around inf 97.5%

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

   6. Taylor expanded in x around 0 63.1%

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

   7. Taylor expanded in eps around 0 57.2%

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

      1. mul-1-neg 57.2%

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

   9. Simplified 57.2%

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

       1. *-un-lft-identity 57.2%

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

       2. +-commutative 57.2%

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

       3. add-sqr-sqrt 35.5%

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

       4. sqrt-unprod 70.6%

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

       5. sqr-neg 70.6%

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

       6. sqrt-unprod 35.0%

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

       7. add-sqr-sqrt 61.0%

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

   11. Applied egg-rr 61.0%

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

       1. *-lft-identity 61.0%

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

   13. Simplified 61.0%

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

   if 1.79999999999999999e37 < 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 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 66.7%

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

   6. Taylor expanded in x around 0 76.0%

      \[\leadsto \frac{\color{blue}{2 + x \cdot \left(-0.5 \cdot \left(x \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)}}{2} \]
   7. Step-by-step derivation

      1. sub-neg 76.0%

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

      2. metadata-eval 76.0%

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

      3. +-commutative 76.0%

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

      4. associate-*r* 76.0%

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

      5. distribute-lft-in 76.0%

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

      6. metadata-eval 76.0%

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

      7. neg-mul-1 76.0%

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

      8. sub-neg 76.0%

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

      9. metadata-eval 76.0%

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

      10. +-commutative 76.0%

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

   8. Simplified 76.0%

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

   9. Taylor expanded in eps around 0 78.8%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.8 \cdot 10^{+37}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \frac{1 - \left(\varepsilon \cdot \left(\varepsilon \cdot \left(1 - \left(x \cdot 0.5 + 0.5 \cdot \left(\varepsilon \cdot x\right)\right)\right) - x \cdot -0.5\right) - x \cdot -0.5\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 71.4% accurate, 5.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 7.2 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{elif}\;eps\_m \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{2 - x \cdot \frac{-1 + \left(eps\_m \cdot \left(eps\_m \cdot \left(1 - x \cdot 0.5\right) - x \cdot -0.5\right) - x \cdot -0.5\right)}{eps\_m}}{2}\\ \mathbf{elif}\;eps\_m \leq 8.5 \cdot 10^{+202}:\\ \;\;\;\;\frac{x \cdot 0.5 + eps\_m \cdot \left(1 + -0.5 \cdot \left(eps\_m \cdot x\right)\right)}{eps\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(-1 - \frac{-1}{eps\_m}\right) \cdot \left(1 + eps\_m\right) - -0.5 \cdot \left(x \cdot \frac{-1 + eps\_m \cdot \left(eps\_m + -1\right)}{eps\_m}\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 7.2e-10)
   1.0
   (if (<= eps_m 1.4e+154)
     (/
      (-
       2.0
       (*
        x
        (/
         (+
          -1.0
          (- (* eps_m (- (* eps_m (- 1.0 (* x 0.5))) (* x -0.5))) (* x -0.5)))
         eps_m)))
      2.0)
     (if (<= eps_m 8.5e+202)
       (/ (+ (* x 0.5) (* eps_m (+ 1.0 (* -0.5 (* eps_m x))))) eps_m)
       (/
        (+
         2.0
         (*
          x
          (-
           (* (- -1.0 (/ -1.0 eps_m)) (+ 1.0 eps_m))
           (* -0.5 (* x (/ (+ -1.0 (* eps_m (+ eps_m -1.0))) eps_m))))))
        2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 7.2e-10) {
		tmp = 1.0;
	} else if (eps_m <= 1.4e+154) {
		tmp = (2.0 - (x * ((-1.0 + ((eps_m * ((eps_m * (1.0 - (x * 0.5))) - (x * -0.5))) - (x * -0.5))) / eps_m))) / 2.0;
	} else if (eps_m <= 8.5e+202) {
		tmp = ((x * 0.5) + (eps_m * (1.0 + (-0.5 * (eps_m * x))))) / eps_m;
	} else {
		tmp = (2.0 + (x * (((-1.0 - (-1.0 / eps_m)) * (1.0 + eps_m)) - (-0.5 * (x * ((-1.0 + (eps_m * (eps_m + -1.0))) / eps_m)))))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 7.2d-10) then
        tmp = 1.0d0
    else if (eps_m <= 1.4d+154) then
        tmp = (2.0d0 - (x * (((-1.0d0) + ((eps_m * ((eps_m * (1.0d0 - (x * 0.5d0))) - (x * (-0.5d0)))) - (x * (-0.5d0)))) / eps_m))) / 2.0d0
    else if (eps_m <= 8.5d+202) then
        tmp = ((x * 0.5d0) + (eps_m * (1.0d0 + ((-0.5d0) * (eps_m * x))))) / eps_m
    else
        tmp = (2.0d0 + (x * ((((-1.0d0) - ((-1.0d0) / eps_m)) * (1.0d0 + eps_m)) - ((-0.5d0) * (x * (((-1.0d0) + (eps_m * (eps_m + (-1.0d0)))) / eps_m)))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 7.2e-10) {
		tmp = 1.0;
	} else if (eps_m <= 1.4e+154) {
		tmp = (2.0 - (x * ((-1.0 + ((eps_m * ((eps_m * (1.0 - (x * 0.5))) - (x * -0.5))) - (x * -0.5))) / eps_m))) / 2.0;
	} else if (eps_m <= 8.5e+202) {
		tmp = ((x * 0.5) + (eps_m * (1.0 + (-0.5 * (eps_m * x))))) / eps_m;
	} else {
		tmp = (2.0 + (x * (((-1.0 - (-1.0 / eps_m)) * (1.0 + eps_m)) - (-0.5 * (x * ((-1.0 + (eps_m * (eps_m + -1.0))) / eps_m)))))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 7.2e-10:
		tmp = 1.0
	elif eps_m <= 1.4e+154:
		tmp = (2.0 - (x * ((-1.0 + ((eps_m * ((eps_m * (1.0 - (x * 0.5))) - (x * -0.5))) - (x * -0.5))) / eps_m))) / 2.0
	elif eps_m <= 8.5e+202:
		tmp = ((x * 0.5) + (eps_m * (1.0 + (-0.5 * (eps_m * x))))) / eps_m
	else:
		tmp = (2.0 + (x * (((-1.0 - (-1.0 / eps_m)) * (1.0 + eps_m)) - (-0.5 * (x * ((-1.0 + (eps_m * (eps_m + -1.0))) / eps_m)))))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 7.2e-10)
		tmp = 1.0;
	elseif (eps_m <= 1.4e+154)
		tmp = Float64(Float64(2.0 - Float64(x * Float64(Float64(-1.0 + Float64(Float64(eps_m * Float64(Float64(eps_m * Float64(1.0 - Float64(x * 0.5))) - Float64(x * -0.5))) - Float64(x * -0.5))) / eps_m))) / 2.0);
	elseif (eps_m <= 8.5e+202)
		tmp = Float64(Float64(Float64(x * 0.5) + Float64(eps_m * Float64(1.0 + Float64(-0.5 * Float64(eps_m * x))))) / eps_m);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(-1.0 - Float64(-1.0 / eps_m)) * Float64(1.0 + eps_m)) - Float64(-0.5 * Float64(x * Float64(Float64(-1.0 + Float64(eps_m * Float64(eps_m + -1.0))) / eps_m)))))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 7.2e-10)
		tmp = 1.0;
	elseif (eps_m <= 1.4e+154)
		tmp = (2.0 - (x * ((-1.0 + ((eps_m * ((eps_m * (1.0 - (x * 0.5))) - (x * -0.5))) - (x * -0.5))) / eps_m))) / 2.0;
	elseif (eps_m <= 8.5e+202)
		tmp = ((x * 0.5) + (eps_m * (1.0 + (-0.5 * (eps_m * x))))) / eps_m;
	else
		tmp = (2.0 + (x * (((-1.0 - (-1.0 / eps_m)) * (1.0 + eps_m)) - (-0.5 * (x * ((-1.0 + (eps_m * (eps_m + -1.0))) / eps_m)))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 7.2e-10], 1.0, If[LessEqual[eps$95$m, 1.4e+154], N[(N[(2.0 - N[(x * N[(N[(-1.0 + N[(N[(eps$95$m * N[(N[(eps$95$m * N[(1.0 - N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps$95$m, 8.5e+202], N[(N[(N[(x * 0.5), $MachinePrecision] + N[(eps$95$m * N[(1.0 + N[(-0.5 * N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision], N[(N[(2.0 + N[(x * N[(N[(N[(-1.0 - N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision] - N[(-0.5 * N[(x * N[(N[(-1.0 + N[(eps$95$m * N[(eps$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if eps < 7.2e-10

   1. Initial program 65.3%

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

   3. Simplified 65.3%

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

   5. Taylor expanded in x around 0 46.2%

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

   if 7.2e-10 < eps < 1.4e154

   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 71.6%

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

   6. Taylor expanded in x around 0 74.5%

      \[\leadsto \frac{\color{blue}{2 + x \cdot \left(-0.5 \cdot \left(x \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)}}{2} \]
   7. Step-by-step derivation

      1. sub-neg 74.5%

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

      2. metadata-eval 74.5%

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

      3. +-commutative 74.5%

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

      4. associate-*r* 74.5%

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

      5. distribute-lft-in 74.5%

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

      6. metadata-eval 74.5%

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

      7. neg-mul-1 74.5%

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

      8. sub-neg 74.5%

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

      9. metadata-eval 74.5%

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

      10. +-commutative 74.5%

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

   8. Simplified 74.5%

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

   9. Taylor expanded in eps around 0 71.8%

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

   if 1.4e154 < eps < 8.5000000000000003e202

   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 0 100.0%

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

   6. Taylor expanded in x around 0 64.4%

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

   7. Taylor expanded in x around inf 64.4%

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

   8. Taylor expanded in eps around 0 88.0%

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

   if 8.5000000000000003e202 < 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 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 55.0%

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

   6. Taylor expanded in x around 0 89.6%

      \[\leadsto \frac{\color{blue}{2 + x \cdot \left(-0.5 \cdot \left(x \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)}}{2} \]
   7. Step-by-step derivation

      1. sub-neg 89.6%

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

      2. metadata-eval 89.6%

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

      3. +-commutative 89.6%

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

      4. associate-*r* 89.6%

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

      5. distribute-lft-in 89.6%

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

      6. metadata-eval 89.6%

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

      7. neg-mul-1 89.6%

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

      8. sub-neg 89.6%

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

      9. metadata-eval 89.6%

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

      10. +-commutative 89.6%

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

   8. Simplified 89.6%

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

   9. Taylor expanded in eps around 0 89.6%

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

       1. mul-1-neg 89.6%

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

       2. sub-neg 89.6%

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

   11. Simplified 89.6%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 7.2 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{elif}\;\varepsilon \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{2 - x \cdot \frac{-1 + \left(\varepsilon \cdot \left(\varepsilon \cdot \left(1 - x \cdot 0.5\right) - x \cdot -0.5\right) - x \cdot -0.5\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;\varepsilon \leq 8.5 \cdot 10^{+202}:\\ \;\;\;\;\frac{x \cdot 0.5 + \varepsilon \cdot \left(1 + -0.5 \cdot \left(\varepsilon \cdot x\right)\right)}{\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(-1 - \frac{-1}{\varepsilon}\right) \cdot \left(1 + \varepsilon\right) - -0.5 \cdot \left(x \cdot \frac{-1 + \varepsilon \cdot \left(\varepsilon + -1\right)}{\varepsilon}\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 63.5% accurate, 5.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{-214}:\\ \;\;\;\;eps\_m \cdot \left(\frac{1}{eps\_m} + x \cdot -0.5\right)\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-155}:\\ \;\;\;\;x \cdot \frac{0.5 + \frac{eps\_m}{x}}{eps\_m}\\ \mathbf{elif}\;x \leq 360:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+164} \lor \neg \left(x \leq 1.32 \cdot 10^{+211}\right) \land x \leq 3.2 \cdot 10^{+241}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{eps\_m}\right) + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot 0.5\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1.7e-214)
   (* eps_m (+ (/ 1.0 eps_m) (* x -0.5)))
   (if (<= x 3.8e-155)
     (* x (/ (+ 0.5 (/ eps_m x)) eps_m))
     (if (<= x 360.0)
       1.0
       (if (or (<= x 1.35e+164) (and (not (<= x 1.32e+211)) (<= x 3.2e+241)))
         (/ (+ (- 1.0 (/ -1.0 eps_m)) (+ 1.0 (/ -1.0 eps_m))) 2.0)
         (/ (+ 2.0 (* x (+ -1.0 (* x 0.5)))) 2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.7e-214) {
		tmp = eps_m * ((1.0 / eps_m) + (x * -0.5));
	} else if (x <= 3.8e-155) {
		tmp = x * ((0.5 + (eps_m / x)) / eps_m);
	} else if (x <= 360.0) {
		tmp = 1.0;
	} else if ((x <= 1.35e+164) || (!(x <= 1.32e+211) && (x <= 3.2e+241))) {
		tmp = ((1.0 - (-1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 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 <= 1.7d-214) then
        tmp = eps_m * ((1.0d0 / eps_m) + (x * (-0.5d0)))
    else if (x <= 3.8d-155) then
        tmp = x * ((0.5d0 + (eps_m / x)) / eps_m)
    else if (x <= 360.0d0) then
        tmp = 1.0d0
    else if ((x <= 1.35d+164) .or. (.not. (x <= 1.32d+211)) .and. (x <= 3.2d+241)) then
        tmp = ((1.0d0 - ((-1.0d0) / eps_m)) + (1.0d0 + ((-1.0d0) / eps_m))) / 2.0d0
    else
        tmp = (2.0d0 + (x * ((-1.0d0) + (x * 0.5d0)))) / 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 <= 1.7e-214) {
		tmp = eps_m * ((1.0 / eps_m) + (x * -0.5));
	} else if (x <= 3.8e-155) {
		tmp = x * ((0.5 + (eps_m / x)) / eps_m);
	} else if (x <= 360.0) {
		tmp = 1.0;
	} else if ((x <= 1.35e+164) || (!(x <= 1.32e+211) && (x <= 3.2e+241))) {
		tmp = ((1.0 - (-1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 1.7e-214:
		tmp = eps_m * ((1.0 / eps_m) + (x * -0.5))
	elif x <= 3.8e-155:
		tmp = x * ((0.5 + (eps_m / x)) / eps_m)
	elif x <= 360.0:
		tmp = 1.0
	elif (x <= 1.35e+164) or (not (x <= 1.32e+211) and (x <= 3.2e+241)):
		tmp = ((1.0 - (-1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0
	else:
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 1.7e-214)
		tmp = Float64(eps_m * Float64(Float64(1.0 / eps_m) + Float64(x * -0.5)));
	elseif (x <= 3.8e-155)
		tmp = Float64(x * Float64(Float64(0.5 + Float64(eps_m / x)) / eps_m));
	elseif (x <= 360.0)
		tmp = 1.0;
	elseif ((x <= 1.35e+164) || (!(x <= 1.32e+211) && (x <= 3.2e+241)))
		tmp = Float64(Float64(Float64(1.0 - Float64(-1.0 / eps_m)) + Float64(1.0 + Float64(-1.0 / eps_m))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(x * 0.5)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 1.7e-214)
		tmp = eps_m * ((1.0 / eps_m) + (x * -0.5));
	elseif (x <= 3.8e-155)
		tmp = x * ((0.5 + (eps_m / x)) / eps_m);
	elseif (x <= 360.0)
		tmp = 1.0;
	elseif ((x <= 1.35e+164) || (~((x <= 1.32e+211)) && (x <= 3.2e+241)))
		tmp = ((1.0 - (-1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	else
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 1.7e-214], N[(eps$95$m * N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.8e-155], N[(x * N[(N[(0.5 + N[(eps$95$m / x), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 360.0], 1.0, If[Or[LessEqual[x, 1.35e+164], And[N[Not[LessEqual[x, 1.32e+211]], $MachinePrecision], LessEqual[x, 3.2e+241]]], N[(N[(N[(1.0 - N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * N[(-1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 1.7e-214

   1. Initial program 68.6%

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

   3. Simplified 68.6%

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

   5. Taylor expanded in x around 0 46.6%

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

   6. Taylor expanded in x around 0 35.7%

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

   7. Taylor expanded in x around inf 49.1%

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

   8. Taylor expanded in eps around inf 65.0%

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

   if 1.7e-214 < x < 3.7999999999999998e-155

   1. Initial program 61.3%

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

   3. Simplified 61.3%

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

   5. Taylor expanded in x around 0 48.3%

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

   6. Taylor expanded in x around 0 22.2%

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

   7. Taylor expanded in x around inf 48.0%

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

   8. Taylor expanded in eps around 0 87.0%

      \[\leadsto x \cdot \color{blue}{\frac{0.5 + \frac{\varepsilon}{x}}{\varepsilon}} \]

   if 3.7999999999999998e-155 < x < 360

   1. Initial program 58.0%

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

   3. Simplified 58.0%

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

   5. Taylor expanded in x around 0 62.7%

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

   if 360 < x < 1.35000000000000003e164 or 1.31999999999999998e211 < x < 3.20000000000000004e241

   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 29.6%

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

   6. Taylor expanded in x around 0 51.6%

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

   if 1.35000000000000003e164 < x < 1.31999999999999998e211 or 3.20000000000000004e241 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 35.4%

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

   7. Taylor expanded in eps around 0 3.1%

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

      1. mul-1-neg 3.1%

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

   9. Simplified 3.1%

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

   10. Taylor expanded in x around 0 83.6%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.7 \cdot 10^{-214}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{1}{\varepsilon} + x \cdot -0.5\right)\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-155}:\\ \;\;\;\;x \cdot \frac{0.5 + \frac{\varepsilon}{x}}{\varepsilon}\\ \mathbf{elif}\;x \leq 360:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+164} \lor \neg \left(x \leq 1.32 \cdot 10^{+211}\right) \land x \leq 3.2 \cdot 10^{+241}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot 0.5\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 68.5% accurate, 6.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-8}:\\ \;\;\;\;\frac{2 - x \cdot \frac{-1 + \left(eps\_m \cdot \left(eps\_m \cdot \left(1 - x \cdot 0.5\right) - x \cdot -0.5\right) - x \cdot -0.5\right)}{eps\_m}}{2}\\ \mathbf{elif}\;x \leq 360:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+164} \lor \neg \left(x \leq 3.6 \cdot 10^{+210}\right) \land x \leq 5.2 \cdot 10^{+240}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{eps\_m}\right) + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot 0.5\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -1.55e-8)
   (/
    (-
     2.0
     (*
      x
      (/
       (+
        -1.0
        (- (* eps_m (- (* eps_m (- 1.0 (* x 0.5))) (* x -0.5))) (* x -0.5)))
       eps_m)))
    2.0)
   (if (<= x 360.0)
     1.0
     (if (or (<= x 1.25e+164) (and (not (<= x 3.6e+210)) (<= x 5.2e+240)))
       (/ (+ (- 1.0 (/ -1.0 eps_m)) (+ 1.0 (/ -1.0 eps_m))) 2.0)
       (/ (+ 2.0 (* x (+ -1.0 (* x 0.5)))) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -1.55e-8) {
		tmp = (2.0 - (x * ((-1.0 + ((eps_m * ((eps_m * (1.0 - (x * 0.5))) - (x * -0.5))) - (x * -0.5))) / eps_m))) / 2.0;
	} else if (x <= 360.0) {
		tmp = 1.0;
	} else if ((x <= 1.25e+164) || (!(x <= 3.6e+210) && (x <= 5.2e+240))) {
		tmp = ((1.0 - (-1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 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 <= (-1.55d-8)) then
        tmp = (2.0d0 - (x * (((-1.0d0) + ((eps_m * ((eps_m * (1.0d0 - (x * 0.5d0))) - (x * (-0.5d0)))) - (x * (-0.5d0)))) / eps_m))) / 2.0d0
    else if (x <= 360.0d0) then
        tmp = 1.0d0
    else if ((x <= 1.25d+164) .or. (.not. (x <= 3.6d+210)) .and. (x <= 5.2d+240)) then
        tmp = ((1.0d0 - ((-1.0d0) / eps_m)) + (1.0d0 + ((-1.0d0) / eps_m))) / 2.0d0
    else
        tmp = (2.0d0 + (x * ((-1.0d0) + (x * 0.5d0)))) / 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 <= -1.55e-8) {
		tmp = (2.0 - (x * ((-1.0 + ((eps_m * ((eps_m * (1.0 - (x * 0.5))) - (x * -0.5))) - (x * -0.5))) / eps_m))) / 2.0;
	} else if (x <= 360.0) {
		tmp = 1.0;
	} else if ((x <= 1.25e+164) || (!(x <= 3.6e+210) && (x <= 5.2e+240))) {
		tmp = ((1.0 - (-1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -1.55e-8:
		tmp = (2.0 - (x * ((-1.0 + ((eps_m * ((eps_m * (1.0 - (x * 0.5))) - (x * -0.5))) - (x * -0.5))) / eps_m))) / 2.0
	elif x <= 360.0:
		tmp = 1.0
	elif (x <= 1.25e+164) or (not (x <= 3.6e+210) and (x <= 5.2e+240)):
		tmp = ((1.0 - (-1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0
	else:
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -1.55e-8)
		tmp = Float64(Float64(2.0 - Float64(x * Float64(Float64(-1.0 + Float64(Float64(eps_m * Float64(Float64(eps_m * Float64(1.0 - Float64(x * 0.5))) - Float64(x * -0.5))) - Float64(x * -0.5))) / eps_m))) / 2.0);
	elseif (x <= 360.0)
		tmp = 1.0;
	elseif ((x <= 1.25e+164) || (!(x <= 3.6e+210) && (x <= 5.2e+240)))
		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(2.0 + Float64(x * Float64(-1.0 + Float64(x * 0.5)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -1.55e-8)
		tmp = (2.0 - (x * ((-1.0 + ((eps_m * ((eps_m * (1.0 - (x * 0.5))) - (x * -0.5))) - (x * -0.5))) / eps_m))) / 2.0;
	elseif (x <= 360.0)
		tmp = 1.0;
	elseif ((x <= 1.25e+164) || (~((x <= 3.6e+210)) && (x <= 5.2e+240)))
		tmp = ((1.0 - (-1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	else
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -1.55e-8], N[(N[(2.0 - N[(x * N[(N[(-1.0 + N[(N[(eps$95$m * N[(N[(eps$95$m * N[(1.0 - N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 360.0], 1.0, If[Or[LessEqual[x, 1.25e+164], And[N[Not[LessEqual[x, 3.6e+210]], $MachinePrecision], LessEqual[x, 5.2e+240]]], 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[(2.0 + N[(x * N[(-1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.55e-8

   1. Initial program 92.9%

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

   3. Simplified 92.9%

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

   5. Taylor expanded in x around 0 47.8%

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

   6. Taylor expanded in x around 0 81.2%

      \[\leadsto \frac{\color{blue}{2 + x \cdot \left(-0.5 \cdot \left(x \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)}}{2} \]
   7. Step-by-step derivation

      1. sub-neg 81.2%

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

      2. metadata-eval 81.2%

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

      3. +-commutative 81.2%

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

      4. associate-*r* 81.2%

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

      5. distribute-lft-in 81.2%

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

      6. metadata-eval 81.2%

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

      7. neg-mul-1 81.2%

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

      8. sub-neg 81.2%

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

      9. metadata-eval 81.2%

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

      10. +-commutative 81.2%

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

   8. Simplified 81.2%

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

   9. Taylor expanded in eps around 0 39.4%

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

   if -1.55e-8 < x < 360

   1. Initial program 58.7%

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

   3. Simplified 58.7%

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

   5. Taylor expanded in x around 0 75.8%

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

   if 360 < x < 1.24999999999999987e164 or 3.6000000000000003e210 < x < 5.2e240

   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 29.6%

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

   6. Taylor expanded in x around 0 51.6%

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

   if 1.24999999999999987e164 < x < 3.6000000000000003e210 or 5.2e240 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 35.4%

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

   7. Taylor expanded in eps around 0 3.1%

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

      1. mul-1-neg 3.1%

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

   9. Simplified 3.1%

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

   10. Taylor expanded in x around 0 83.6%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-8}:\\ \;\;\;\;\frac{2 - x \cdot \frac{-1 + \left(\varepsilon \cdot \left(\varepsilon \cdot \left(1 - x \cdot 0.5\right) - x \cdot -0.5\right) - x \cdot -0.5\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 360:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+164} \lor \neg \left(x \leq 3.6 \cdot 10^{+210}\right) \land x \leq 5.2 \cdot 10^{+240}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot 0.5\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 67.6% accurate, 6.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-9}:\\ \;\;\;\;\frac{x \cdot 0.5 + eps\_m \cdot \left(1 + -0.5 \cdot \left(eps\_m \cdot x\right)\right)}{eps\_m}\\ \mathbf{elif}\;x \leq 360:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+162} \lor \neg \left(x \leq 1.9 \cdot 10^{+211}\right) \land x \leq 1.25 \cdot 10^{+241}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{eps\_m}\right) + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot 0.5\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -3.7e-9)
   (/ (+ (* x 0.5) (* eps_m (+ 1.0 (* -0.5 (* eps_m x))))) eps_m)
   (if (<= x 360.0)
     1.0
     (if (or (<= x 1.3e+162) (and (not (<= x 1.9e+211)) (<= x 1.25e+241)))
       (/ (+ (- 1.0 (/ -1.0 eps_m)) (+ 1.0 (/ -1.0 eps_m))) 2.0)
       (/ (+ 2.0 (* x (+ -1.0 (* x 0.5)))) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -3.7e-9) {
		tmp = ((x * 0.5) + (eps_m * (1.0 + (-0.5 * (eps_m * x))))) / eps_m;
	} else if (x <= 360.0) {
		tmp = 1.0;
	} else if ((x <= 1.3e+162) || (!(x <= 1.9e+211) && (x <= 1.25e+241))) {
		tmp = ((1.0 - (-1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 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 <= (-3.7d-9)) then
        tmp = ((x * 0.5d0) + (eps_m * (1.0d0 + ((-0.5d0) * (eps_m * x))))) / eps_m
    else if (x <= 360.0d0) then
        tmp = 1.0d0
    else if ((x <= 1.3d+162) .or. (.not. (x <= 1.9d+211)) .and. (x <= 1.25d+241)) then
        tmp = ((1.0d0 - ((-1.0d0) / eps_m)) + (1.0d0 + ((-1.0d0) / eps_m))) / 2.0d0
    else
        tmp = (2.0d0 + (x * ((-1.0d0) + (x * 0.5d0)))) / 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 <= -3.7e-9) {
		tmp = ((x * 0.5) + (eps_m * (1.0 + (-0.5 * (eps_m * x))))) / eps_m;
	} else if (x <= 360.0) {
		tmp = 1.0;
	} else if ((x <= 1.3e+162) || (!(x <= 1.9e+211) && (x <= 1.25e+241))) {
		tmp = ((1.0 - (-1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -3.7e-9:
		tmp = ((x * 0.5) + (eps_m * (1.0 + (-0.5 * (eps_m * x))))) / eps_m
	elif x <= 360.0:
		tmp = 1.0
	elif (x <= 1.3e+162) or (not (x <= 1.9e+211) and (x <= 1.25e+241)):
		tmp = ((1.0 - (-1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0
	else:
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -3.7e-9)
		tmp = Float64(Float64(Float64(x * 0.5) + Float64(eps_m * Float64(1.0 + Float64(-0.5 * Float64(eps_m * x))))) / eps_m);
	elseif (x <= 360.0)
		tmp = 1.0;
	elseif ((x <= 1.3e+162) || (!(x <= 1.9e+211) && (x <= 1.25e+241)))
		tmp = Float64(Float64(Float64(1.0 - Float64(-1.0 / eps_m)) + Float64(1.0 + Float64(-1.0 / eps_m))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(x * 0.5)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -3.7e-9)
		tmp = ((x * 0.5) + (eps_m * (1.0 + (-0.5 * (eps_m * x))))) / eps_m;
	elseif (x <= 360.0)
		tmp = 1.0;
	elseif ((x <= 1.3e+162) || (~((x <= 1.9e+211)) && (x <= 1.25e+241)))
		tmp = ((1.0 - (-1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	else
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -3.7e-9], N[(N[(N[(x * 0.5), $MachinePrecision] + N[(eps$95$m * N[(1.0 + N[(-0.5 * N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision], If[LessEqual[x, 360.0], 1.0, If[Or[LessEqual[x, 1.3e+162], And[N[Not[LessEqual[x, 1.9e+211]], $MachinePrecision], LessEqual[x, 1.25e+241]]], N[(N[(N[(1.0 - N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * N[(-1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.7e-9

   1. Initial program 92.9%

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

   3. Simplified 92.9%

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

   5. Taylor expanded in x around 0 47.8%

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

   6. Taylor expanded in x around 0 25.8%

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

   7. Taylor expanded in x around inf 25.8%

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

   8. Taylor expanded in eps around 0 39.4%

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

   if -3.7e-9 < x < 360

   1. Initial program 58.7%

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

   3. Simplified 58.7%

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

   5. Taylor expanded in x around 0 75.8%

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

   if 360 < x < 1.3e162 or 1.90000000000000008e211 < x < 1.25000000000000006e241

   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 29.6%

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

   6. Taylor expanded in x around 0 51.6%

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

   if 1.3e162 < x < 1.90000000000000008e211 or 1.25000000000000006e241 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 35.4%

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

   7. Taylor expanded in eps around 0 3.1%

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

      1. mul-1-neg 3.1%

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

   9. Simplified 3.1%

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

   10. Taylor expanded in x around 0 83.6%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-9}:\\ \;\;\;\;\frac{x \cdot 0.5 + \varepsilon \cdot \left(1 + -0.5 \cdot \left(\varepsilon \cdot x\right)\right)}{\varepsilon}\\ \mathbf{elif}\;x \leq 360:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+162} \lor \neg \left(x \leq 1.9 \cdot 10^{+211}\right) \land x \leq 1.25 \cdot 10^{+241}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot 0.5\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 66.1% accurate, 6.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \frac{0.5 - eps\_m \cdot \left(\frac{-1}{x} - eps\_m \cdot -0.5\right)}{eps\_m}\\ \mathbf{elif}\;x \leq 360:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+161} \lor \neg \left(x \leq 9 \cdot 10^{+210}\right) \land x \leq 6.5 \cdot 10^{+242}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{eps\_m}\right) + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot 0.5\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2.9e-7)
   (* x (/ (- 0.5 (* eps_m (- (/ -1.0 x) (* eps_m -0.5)))) eps_m))
   (if (<= x 360.0)
     1.0
     (if (or (<= x 2.3e+161) (and (not (<= x 9e+210)) (<= x 6.5e+242)))
       (/ (+ (- 1.0 (/ -1.0 eps_m)) (+ 1.0 (/ -1.0 eps_m))) 2.0)
       (/ (+ 2.0 (* x (+ -1.0 (* x 0.5)))) 2.0)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -2.9e-7) {
		tmp = x * ((0.5 - (eps_m * ((-1.0 / x) - (eps_m * -0.5)))) / eps_m);
	} else if (x <= 360.0) {
		tmp = 1.0;
	} else if ((x <= 2.3e+161) || (!(x <= 9e+210) && (x <= 6.5e+242))) {
		tmp = ((1.0 - (-1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 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 <= (-2.9d-7)) then
        tmp = x * ((0.5d0 - (eps_m * (((-1.0d0) / x) - (eps_m * (-0.5d0))))) / eps_m)
    else if (x <= 360.0d0) then
        tmp = 1.0d0
    else if ((x <= 2.3d+161) .or. (.not. (x <= 9d+210)) .and. (x <= 6.5d+242)) then
        tmp = ((1.0d0 - ((-1.0d0) / eps_m)) + (1.0d0 + ((-1.0d0) / eps_m))) / 2.0d0
    else
        tmp = (2.0d0 + (x * ((-1.0d0) + (x * 0.5d0)))) / 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 <= -2.9e-7) {
		tmp = x * ((0.5 - (eps_m * ((-1.0 / x) - (eps_m * -0.5)))) / eps_m);
	} else if (x <= 360.0) {
		tmp = 1.0;
	} else if ((x <= 2.3e+161) || (!(x <= 9e+210) && (x <= 6.5e+242))) {
		tmp = ((1.0 - (-1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -2.9e-7:
		tmp = x * ((0.5 - (eps_m * ((-1.0 / x) - (eps_m * -0.5)))) / eps_m)
	elif x <= 360.0:
		tmp = 1.0
	elif (x <= 2.3e+161) or (not (x <= 9e+210) and (x <= 6.5e+242)):
		tmp = ((1.0 - (-1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0
	else:
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -2.9e-7)
		tmp = Float64(x * Float64(Float64(0.5 - Float64(eps_m * Float64(Float64(-1.0 / x) - Float64(eps_m * -0.5)))) / eps_m));
	elseif (x <= 360.0)
		tmp = 1.0;
	elseif ((x <= 2.3e+161) || (!(x <= 9e+210) && (x <= 6.5e+242)))
		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(2.0 + Float64(x * Float64(-1.0 + Float64(x * 0.5)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -2.9e-7)
		tmp = x * ((0.5 - (eps_m * ((-1.0 / x) - (eps_m * -0.5)))) / eps_m);
	elseif (x <= 360.0)
		tmp = 1.0;
	elseif ((x <= 2.3e+161) || (~((x <= 9e+210)) && (x <= 6.5e+242)))
		tmp = ((1.0 - (-1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	else
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -2.9e-7], N[(x * N[(N[(0.5 - N[(eps$95$m * N[(N[(-1.0 / x), $MachinePrecision] - N[(eps$95$m * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 360.0], 1.0, If[Or[LessEqual[x, 2.3e+161], And[N[Not[LessEqual[x, 9e+210]], $MachinePrecision], LessEqual[x, 6.5e+242]]], 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[(2.0 + N[(x * N[(-1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.8999999999999998e-7

   1. Initial program 92.9%

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

   3. Simplified 92.9%

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

   5. Taylor expanded in x around 0 47.8%

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

   6. Taylor expanded in x around 0 25.8%

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

   7. Taylor expanded in x around inf 25.8%

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

   8. Taylor expanded in eps around 0 34.9%

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

   if -2.8999999999999998e-7 < x < 360

   1. Initial program 58.7%

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

   3. Simplified 58.7%

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

   5. Taylor expanded in x around 0 75.8%

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

   if 360 < x < 2.2999999999999999e161 or 9.00000000000000007e210 < x < 6.49999999999999992e242

   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 29.6%

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

   6. Taylor expanded in x around 0 51.6%

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

   if 2.2999999999999999e161 < x < 9.00000000000000007e210 or 6.49999999999999992e242 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 35.4%

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

   7. Taylor expanded in eps around 0 3.1%

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

      1. mul-1-neg 3.1%

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

   9. Simplified 3.1%

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

   10. Taylor expanded in x around 0 83.6%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \frac{0.5 - \varepsilon \cdot \left(\frac{-1}{x} - \varepsilon \cdot -0.5\right)}{\varepsilon}\\ \mathbf{elif}\;x \leq 360:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+161} \lor \neg \left(x \leq 9 \cdot 10^{+210}\right) \land x \leq 6.5 \cdot 10^{+242}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot 0.5\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 76.6% accurate, 6.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 1.05 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \frac{1 - \left(eps\_m \cdot \left(eps\_m \cdot \left(1 - \left(x \cdot 0.5 + 0.5 \cdot \left(eps\_m \cdot x\right)\right)\right) - x \cdot -0.5\right) - x \cdot -0.5\right)}{eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 1.05e-7)
   1.0
   (/
    (+
     2.0
     (*
      x
      (/
       (-
        1.0
        (-
         (*
          eps_m
          (- (* eps_m (- 1.0 (+ (* x 0.5) (* 0.5 (* eps_m x))))) (* x -0.5)))
         (* x -0.5)))
       eps_m)))
    2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.05e-7) {
		tmp = 1.0;
	} else {
		tmp = (2.0 + (x * ((1.0 - ((eps_m * ((eps_m * (1.0 - ((x * 0.5) + (0.5 * (eps_m * x))))) - (x * -0.5))) - (x * -0.5))) / eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 1.05d-7) then
        tmp = 1.0d0
    else
        tmp = (2.0d0 + (x * ((1.0d0 - ((eps_m * ((eps_m * (1.0d0 - ((x * 0.5d0) + (0.5d0 * (eps_m * x))))) - (x * (-0.5d0)))) - (x * (-0.5d0)))) / eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.05e-7) {
		tmp = 1.0;
	} else {
		tmp = (2.0 + (x * ((1.0 - ((eps_m * ((eps_m * (1.0 - ((x * 0.5) + (0.5 * (eps_m * x))))) - (x * -0.5))) - (x * -0.5))) / eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 1.05e-7:
		tmp = 1.0
	else:
		tmp = (2.0 + (x * ((1.0 - ((eps_m * ((eps_m * (1.0 - ((x * 0.5) + (0.5 * (eps_m * x))))) - (x * -0.5))) - (x * -0.5))) / eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 1.05e-7)
		tmp = 1.0;
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(1.0 - Float64(Float64(eps_m * Float64(Float64(eps_m * Float64(1.0 - Float64(Float64(x * 0.5) + Float64(0.5 * Float64(eps_m * x))))) - Float64(x * -0.5))) - Float64(x * -0.5))) / eps_m))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 1.05e-7)
		tmp = 1.0;
	else
		tmp = (2.0 + (x * ((1.0 - ((eps_m * ((eps_m * (1.0 - ((x * 0.5) + (0.5 * (eps_m * x))))) - (x * -0.5))) - (x * -0.5))) / eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 1.05e-7], 1.0, N[(N[(2.0 + N[(x * N[(N[(1.0 - N[(N[(eps$95$m * N[(N[(eps$95$m * N[(1.0 - N[(N[(x * 0.5), $MachinePrecision] + N[(0.5 * N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.05e-7

   1. Initial program 65.3%

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

   3. Simplified 65.3%

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

   5. Taylor expanded in x around 0 46.2%

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

   if 1.05e-7 < eps

   1. Initial program 100.0%

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

   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 68.5%

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

   6. Taylor expanded in x around 0 76.4%

      \[\leadsto \frac{\color{blue}{2 + x \cdot \left(-0.5 \cdot \left(x \cdot \left({\left(1 + \varepsilon\right)}^{2} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right) - -1 \cdot \left(\left(1 + \varepsilon\right) \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)\right)}}{2} \]
   7. Step-by-step derivation

      1. sub-neg 76.4%

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

      2. metadata-eval 76.4%

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

      3. +-commutative 76.4%

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

      4. associate-*r* 76.4%

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

      5. distribute-lft-in 76.4%

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

      6. metadata-eval 76.4%

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

      7. neg-mul-1 76.4%

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

      8. sub-neg 76.4%

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

      9. metadata-eval 76.4%

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

      10. +-commutative 76.4%

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

   8. Simplified 76.4%

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

   9. Taylor expanded in eps around 0 78.7%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.05 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \frac{1 - \left(\varepsilon \cdot \left(\varepsilon \cdot \left(1 - \left(x \cdot 0.5 + 0.5 \cdot \left(\varepsilon \cdot x\right)\right)\right) - x \cdot -0.5\right) - x \cdot -0.5\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 67.6% accurate, 6.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -2.95 \cdot 10^{+79}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)}{2}\\ \mathbf{elif}\;x \leq -1.46 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \frac{0.5 - eps\_m \cdot \left(\frac{-1}{x} - eps\_m \cdot -0.5\right)}{eps\_m}\\ \mathbf{elif}\;x \leq 360:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+241}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{eps\_m}\right) + \left(1 + \frac{-1}{eps\_m}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot 0.5\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -2.95e+79)
   (/ (+ 2.0 (* x (+ -1.0 (* x (+ 0.5 (* x -0.16666666666666666)))))) 2.0)
   (if (<= x -1.46e-8)
     (* x (/ (- 0.5 (* eps_m (- (/ -1.0 x) (* eps_m -0.5)))) eps_m))
     (if (<= x 360.0)
       1.0
       (if (<= x 9.5e+241)
         (/ (+ (- 1.0 (/ -1.0 eps_m)) (+ 1.0 (/ -1.0 eps_m))) 2.0)
         (/ (+ 2.0 (* x (+ -1.0 (* x 0.5)))) 2.0))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -2.95e+79) {
		tmp = (2.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))) / 2.0;
	} else if (x <= -1.46e-8) {
		tmp = x * ((0.5 - (eps_m * ((-1.0 / x) - (eps_m * -0.5)))) / eps_m);
	} else if (x <= 360.0) {
		tmp = 1.0;
	} else if (x <= 9.5e+241) {
		tmp = ((1.0 - (-1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 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 <= (-2.95d+79)) then
        tmp = (2.0d0 + (x * ((-1.0d0) + (x * (0.5d0 + (x * (-0.16666666666666666d0))))))) / 2.0d0
    else if (x <= (-1.46d-8)) then
        tmp = x * ((0.5d0 - (eps_m * (((-1.0d0) / x) - (eps_m * (-0.5d0))))) / eps_m)
    else if (x <= 360.0d0) then
        tmp = 1.0d0
    else if (x <= 9.5d+241) then
        tmp = ((1.0d0 - ((-1.0d0) / eps_m)) + (1.0d0 + ((-1.0d0) / eps_m))) / 2.0d0
    else
        tmp = (2.0d0 + (x * ((-1.0d0) + (x * 0.5d0)))) / 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 <= -2.95e+79) {
		tmp = (2.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))) / 2.0;
	} else if (x <= -1.46e-8) {
		tmp = x * ((0.5 - (eps_m * ((-1.0 / x) - (eps_m * -0.5)))) / eps_m);
	} else if (x <= 360.0) {
		tmp = 1.0;
	} else if (x <= 9.5e+241) {
		tmp = ((1.0 - (-1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	} else {
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -2.95e+79:
		tmp = (2.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))) / 2.0
	elif x <= -1.46e-8:
		tmp = x * ((0.5 - (eps_m * ((-1.0 / x) - (eps_m * -0.5)))) / eps_m)
	elif x <= 360.0:
		tmp = 1.0
	elif x <= 9.5e+241:
		tmp = ((1.0 - (-1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0
	else:
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -2.95e+79)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * -0.16666666666666666)))))) / 2.0);
	elseif (x <= -1.46e-8)
		tmp = Float64(x * Float64(Float64(0.5 - Float64(eps_m * Float64(Float64(-1.0 / x) - Float64(eps_m * -0.5)))) / eps_m));
	elseif (x <= 360.0)
		tmp = 1.0;
	elseif (x <= 9.5e+241)
		tmp = Float64(Float64(Float64(1.0 - Float64(-1.0 / eps_m)) + Float64(1.0 + Float64(-1.0 / eps_m))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(x * 0.5)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -2.95e+79)
		tmp = (2.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))) / 2.0;
	elseif (x <= -1.46e-8)
		tmp = x * ((0.5 - (eps_m * ((-1.0 / x) - (eps_m * -0.5)))) / eps_m);
	elseif (x <= 360.0)
		tmp = 1.0;
	elseif (x <= 9.5e+241)
		tmp = ((1.0 - (-1.0 / eps_m)) + (1.0 + (-1.0 / eps_m))) / 2.0;
	else
		tmp = (2.0 + (x * (-1.0 + (x * 0.5)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -2.95e+79], N[(N[(2.0 + N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -1.46e-8], N[(x * N[(N[(0.5 - N[(eps$95$m * N[(N[(-1.0 / x), $MachinePrecision] - N[(eps$95$m * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 360.0], 1.0, If[LessEqual[x, 9.5e+241], N[(N[(N[(1.0 - N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * N[(-1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -2.95e79

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 61.2%

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

   7. Taylor expanded in eps around 0 100.0%

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

      1. mul-1-neg 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in x around 0 96.3%

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

   if -2.95e79 < x < -1.46e-8

   1. Initial program 81.8%

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

   3. Simplified 81.8%

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

   5. Taylor expanded in x around 0 57.2%

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

   6. Taylor expanded in x around 0 15.2%

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

   7. Taylor expanded in x around inf 15.2%

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

   8. Taylor expanded in eps around 0 38.5%

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

   if -1.46e-8 < x < 360

   1. Initial program 58.7%

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

   3. Simplified 58.7%

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

   5. Taylor expanded in x around 0 75.8%

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

   if 360 < x < 9.50000000000000019e241

   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%

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

   6. Taylor expanded in x around 0 47.6%

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

   if 9.50000000000000019e241 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 38.4%

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

   7. Taylor expanded in eps around 0 3.1%

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

      1. mul-1-neg 3.1%

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

   9. Simplified 3.1%

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

   10. Taylor expanded in x around 0 82.1%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.95 \cdot 10^{+79}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)}{2}\\ \mathbf{elif}\;x \leq -1.46 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \frac{0.5 - \varepsilon \cdot \left(\frac{-1}{x} - \varepsilon \cdot -0.5\right)}{\varepsilon}\\ \mathbf{elif}\;x \leq 360:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+241}:\\ \;\;\;\;\frac{\left(1 - \frac{-1}{\varepsilon}\right) + \left(1 + \frac{-1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + x \cdot 0.5\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 57.3% accurate, 11.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-216}:\\ \;\;\;\;eps\_m \cdot \left(\frac{1}{eps\_m} + x \cdot -0.5\right)\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-155}:\\ \;\;\;\;x \cdot \frac{0.5 + \frac{eps\_m}{x}}{eps\_m}\\ \mathbf{elif}\;x \leq 4.9:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(eps\_m \cdot x\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 2e-216)
   (* eps_m (+ (/ 1.0 eps_m) (* x -0.5)))
   (if (<= x 6e-155)
     (* x (/ (+ 0.5 (/ eps_m x)) eps_m))
     (if (<= x 4.9) 1.0 (* 0.5 (* eps_m x))))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 2e-216) {
		tmp = eps_m * ((1.0 / eps_m) + (x * -0.5));
	} else if (x <= 6e-155) {
		tmp = x * ((0.5 + (eps_m / x)) / eps_m);
	} else if (x <= 4.9) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (eps_m * x);
	}
	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 <= 2d-216) then
        tmp = eps_m * ((1.0d0 / eps_m) + (x * (-0.5d0)))
    else if (x <= 6d-155) then
        tmp = x * ((0.5d0 + (eps_m / x)) / eps_m)
    else if (x <= 4.9d0) then
        tmp = 1.0d0
    else
        tmp = 0.5d0 * (eps_m * x)
    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 <= 2e-216) {
		tmp = eps_m * ((1.0 / eps_m) + (x * -0.5));
	} else if (x <= 6e-155) {
		tmp = x * ((0.5 + (eps_m / x)) / eps_m);
	} else if (x <= 4.9) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (eps_m * x);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 2e-216:
		tmp = eps_m * ((1.0 / eps_m) + (x * -0.5))
	elif x <= 6e-155:
		tmp = x * ((0.5 + (eps_m / x)) / eps_m)
	elif x <= 4.9:
		tmp = 1.0
	else:
		tmp = 0.5 * (eps_m * x)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 2e-216)
		tmp = Float64(eps_m * Float64(Float64(1.0 / eps_m) + Float64(x * -0.5)));
	elseif (x <= 6e-155)
		tmp = Float64(x * Float64(Float64(0.5 + Float64(eps_m / x)) / eps_m));
	elseif (x <= 4.9)
		tmp = 1.0;
	else
		tmp = Float64(0.5 * Float64(eps_m * x));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 2e-216)
		tmp = eps_m * ((1.0 / eps_m) + (x * -0.5));
	elseif (x <= 6e-155)
		tmp = x * ((0.5 + (eps_m / x)) / eps_m);
	elseif (x <= 4.9)
		tmp = 1.0;
	else
		tmp = 0.5 * (eps_m * x);
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 2e-216], N[(eps$95$m * N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6e-155], N[(x * N[(N[(0.5 + N[(eps$95$m / x), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.9], 1.0, N[(0.5 * N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 2.0000000000000001e-216

   1. Initial program 68.6%

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

   3. Simplified 68.6%

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

   5. Taylor expanded in x around 0 46.6%

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

   6. Taylor expanded in x around 0 35.7%

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

   7. Taylor expanded in x around inf 49.1%

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

   8. Taylor expanded in eps around inf 65.0%

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

   if 2.0000000000000001e-216 < x < 5.99999999999999967e-155

   1. Initial program 61.3%

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

   3. Simplified 61.3%

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

   5. Taylor expanded in x around 0 48.3%

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

   6. Taylor expanded in x around 0 22.2%

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

   7. Taylor expanded in x around inf 48.0%

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

   8. Taylor expanded in eps around 0 87.0%

      \[\leadsto x \cdot \color{blue}{\frac{0.5 + \frac{\varepsilon}{x}}{\varepsilon}} \]

   if 5.99999999999999967e-155 < x < 4.9000000000000004

   1. Initial program 58.0%

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

   3. Simplified 58.0%

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

   5. Taylor expanded in x around 0 62.7%

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

   if 4.9000000000000004 < 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 32.0%

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

   6. Taylor expanded in eps around inf 13.4%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-216}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{1}{\varepsilon} + x \cdot -0.5\right)\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-155}:\\ \;\;\;\;x \cdot \frac{0.5 + \frac{\varepsilon}{x}}{\varepsilon}\\ \mathbf{elif}\;x \leq 4.9:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\varepsilon \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 58.2% accurate, 15.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -0.78:\\ \;\;\;\;x \cdot \left(eps\_m \cdot -0.5\right)\\ \mathbf{elif}\;x \leq 56:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(eps\_m \cdot x\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -0.78)
   (* x (* eps_m -0.5))
   (if (<= x 56.0) 1.0 (* 0.5 (* eps_m x)))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -0.78) {
		tmp = x * (eps_m * -0.5);
	} else if (x <= 56.0) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (eps_m * x);
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-0.78d0)) then
        tmp = x * (eps_m * (-0.5d0))
    else if (x <= 56.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.5d0 * (eps_m * x)
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -0.78) {
		tmp = x * (eps_m * -0.5);
	} else if (x <= 56.0) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (eps_m * x);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -0.78:
		tmp = x * (eps_m * -0.5)
	elif x <= 56.0:
		tmp = 1.0
	else:
		tmp = 0.5 * (eps_m * x)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -0.78)
		tmp = Float64(x * Float64(eps_m * -0.5));
	elseif (x <= 56.0)
		tmp = 1.0;
	else
		tmp = Float64(0.5 * Float64(eps_m * x));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -0.78)
		tmp = x * (eps_m * -0.5);
	elseif (x <= 56.0)
		tmp = 1.0;
	else
		tmp = 0.5 * (eps_m * x);
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -0.78], N[(x * N[(eps$95$m * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 56.0], 1.0, N[(0.5 * N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.78000000000000003

   1. Initial program 94.9%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in x around 0 48.9%

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

   6. Taylor expanded in x around 0 27.6%

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

   7. Taylor expanded in eps around inf 27.7%

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

      1. *-commutative 27.7%

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

      2. *-commutative 27.7%

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

      3. associate-*l* 27.7%

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

   9. Simplified 27.7%

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

   if -0.78000000000000003 < x < 56

   1. Initial program 58.8%

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

   3. Simplified 58.8%

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

   5. Taylor expanded in x around 0 74.6%

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

   if 56 < 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 32.0%

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

   6. Taylor expanded in eps around inf 13.4%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.78:\\ \;\;\;\;x \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{elif}\;x \leq 56:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\varepsilon \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 57.6% accurate, 16.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 27:\\ \;\;\;\;eps\_m \cdot \left(\frac{1}{eps\_m} + x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(eps\_m \cdot x\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 27.0) (* eps_m (+ (/ 1.0 eps_m) (* x -0.5))) (* 0.5 (* eps_m x))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 27.0) {
		tmp = eps_m * ((1.0 / eps_m) + (x * -0.5));
	} else {
		tmp = 0.5 * (eps_m * x);
	}
	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 <= 27.0d0) then
        tmp = eps_m * ((1.0d0 / eps_m) + (x * (-0.5d0)))
    else
        tmp = 0.5d0 * (eps_m * x)
    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 <= 27.0) {
		tmp = eps_m * ((1.0 / eps_m) + (x * -0.5));
	} else {
		tmp = 0.5 * (eps_m * x);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 27.0:
		tmp = eps_m * ((1.0 / eps_m) + (x * -0.5))
	else:
		tmp = 0.5 * (eps_m * x)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 27.0)
		tmp = Float64(eps_m * Float64(Float64(1.0 / eps_m) + Float64(x * -0.5)));
	else
		tmp = Float64(0.5 * Float64(eps_m * x));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 27.0)
		tmp = eps_m * ((1.0 / eps_m) + (x * -0.5));
	else
		tmp = 0.5 * (eps_m * x);
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 27.0], N[(eps$95$m * N[(N[(1.0 / eps$95$m), $MachinePrecision] + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 27

   1. Initial program 66.3%

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

   3. Simplified 66.3%

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

   5. Taylor expanded in x around 0 45.7%

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

   6. Taylor expanded in x around 0 33.0%

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

   7. Taylor expanded in x around inf 45.8%

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

   8. Taylor expanded in eps around inf 64.3%

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

   if 27 < 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 32.0%

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

   6. Taylor expanded in eps around inf 13.4%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 27:\\ \;\;\;\;\varepsilon \cdot \left(\frac{1}{\varepsilon} + x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\varepsilon \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 57.4% accurate, 20.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(x * 0.5)))) / 2.0)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.8%

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

3. Simplified 68.5%

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

5. Taylor expanded in eps around inf 98.2%

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

6. Taylor expanded in x around 0 63.6%

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

7. Taylor expanded in eps around 0 57.3%

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

   1. mul-1-neg 57.3%

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

9. Simplified 57.3%

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

10. Taylor expanded in x around 0 56.0%

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

11. Final simplification 56.0%

    \[\leadsto \frac{2 + x \cdot \left(-1 + x \cdot 0.5\right)}{2} \]
12. Add Preprocessing

Alternative 24: 50.8% accurate, 22.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 38:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(eps\_m \cdot x\right)\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 38.0) 1.0 (* 0.5 (* eps_m x))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 38.0) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (eps_m * x);
	}
	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 <= 38.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.5d0 * (eps_m * x)
    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 <= 38.0) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * (eps_m * x);
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 38.0:
		tmp = 1.0
	else:
		tmp = 0.5 * (eps_m * x)
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 38.0)
		tmp = 1.0;
	else
		tmp = Float64(0.5 * Float64(eps_m * x));
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 38.0)
		tmp = 1.0;
	else
		tmp = 0.5 * (eps_m * x);
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 38.0], 1.0, N[(0.5 * N[(eps$95$m * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 38

   1. Initial program 66.3%

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

   3. Simplified 66.3%

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

   5. Taylor expanded in x around 0 59.8%

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

   if 38 < 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 32.0%

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

   6. Taylor expanded in eps around inf 13.4%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 38:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\varepsilon \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 44.1% 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 75.8%

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

3. Simplified 75.8%

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

5. Taylor expanded in x around 0 43.9%

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

6. Final simplification 43.9%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024084 
(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))