NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.0% → 99.5%

Time: 14.8s

Alternatives: 18

Speedup: 0.5×

• 38.3% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{1}{\varepsilon}\\ \mathbf{if}\;t\_0 \cdot e^{x \cdot \left(\varepsilon + -1\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{x \cdot \left(-1 - \varepsilon\right)} \leq 4:\\ \;\;\;\;\frac{1 + x}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\mathsf{fma}\left(t\_0, e^{\mathsf{fma}\left(\varepsilon, x, x\right)}, e^{\mathsf{log1p}\left(\frac{1}{\varepsilon}\right) - \mathsf{fma}\left(\varepsilon, x, x\right)}\right) \cdot 0.5}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ 1.0 eps))))
   (if (<=
        (-
         (* t_0 (exp (* x (+ eps -1.0))))
         (* (+ (/ 1.0 eps) -1.0) (exp (* x (- -1.0 eps)))))
        4.0)
     (/ (+ 1.0 x) (exp x))
     (pow
      (cbrt
       (*
        (fma
         t_0
         (exp (fma eps x x))
         (exp (- (log1p (/ 1.0 eps)) (fma eps x x))))
        0.5))
      3.0))))

C

double code(double x, double eps) {
	double t_0 = 1.0 + (1.0 / eps);
	double tmp;
	if (((t_0 * exp((x * (eps + -1.0)))) - (((1.0 / eps) + -1.0) * exp((x * (-1.0 - eps))))) <= 4.0) {
		tmp = (1.0 + x) / exp(x);
	} else {
		tmp = pow(cbrt((fma(t_0, exp(fma(eps, x, x)), exp((log1p((1.0 / eps)) - fma(eps, x, x)))) * 0.5)), 3.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(1.0 + Float64(1.0 / eps))
	tmp = 0.0
	if (Float64(Float64(t_0 * exp(Float64(x * Float64(eps + -1.0)))) - Float64(Float64(Float64(1.0 / eps) + -1.0) * exp(Float64(x * Float64(-1.0 - eps))))) <= 4.0)
		tmp = Float64(Float64(1.0 + x) / exp(x));
	else
		tmp = cbrt(Float64(fma(t_0, exp(fma(eps, x, x)), exp(Float64(log1p(Float64(1.0 / eps)) - fma(eps, x, x)))) * 0.5)) ^ 3.0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$0 * N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1.0 / eps), $MachinePrecision] + -1.0), $MachinePrecision] * N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4.0], N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(N[(t$95$0 * N[Exp[N[(eps * x + x), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(N[Log[1 + N[(1.0 / eps), $MachinePrecision]], $MachinePrecision] - N[(eps * x + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $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))))) < 4

   1. Initial program 57.7%

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

   3. Simplified 33.9%

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

   5. Taylor expanded in eps around 0 56.4%

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

      1. associate-+r+ 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-inverses 100.0%

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

      5. distribute-lft-out 100.0%

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

      6. distribute-rgt1-in 100.0%

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

      7. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around 0 100.0%

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

      1. +-commutative 100.0%

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

   10. Simplified 100.0%

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

       1. *-commutative 100.0%

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

       2. exp-neg 100.0%

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

       3. un-div-inv 100.0%

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

   12. Applied egg-rr 100.0%

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

   if 4 < (-.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 84.9%

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

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{\mathsf{fma}\left(\varepsilon, x, x\right)}, e^{\mathsf{log1p}\left(\frac{1}{\varepsilon}\right) - \mathsf{fma}\left(\varepsilon, x, x\right)}\right) \cdot 0.5}\right)}^{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{x \cdot \left(-1 - \varepsilon\right)} \leq 4:\\ \;\;\;\;\frac{1 + x}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\mathsf{fma}\left(1 + \frac{1}{\varepsilon}, e^{\mathsf{fma}\left(\varepsilon, x, x\right)}, e^{\mathsf{log1p}\left(\frac{1}{\varepsilon}\right) - \mathsf{fma}\left(\varepsilon, x, x\right)}\right) \cdot 0.5}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{x \cdot \left(-1 - \varepsilon\right)}\\ \mathbf{if}\;t\_0 \leq 4:\\ \;\;\;\;\frac{1 + x}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0
         (-
          (* (+ 1.0 (/ 1.0 eps)) (exp (* x (+ eps -1.0))))
          (* (+ (/ 1.0 eps) -1.0) (exp (* x (- -1.0 eps)))))))
   (if (<= t_0 4.0) (/ (+ 1.0 x) (exp x)) (/ t_0 2.0))))

C

double code(double x, double eps) {
	double t_0 = ((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) - (((1.0 / eps) + -1.0) * exp((x * (-1.0 - eps))));
	double tmp;
	if (t_0 <= 4.0) {
		tmp = (1.0 + x) / exp(x);
	} else {
		tmp = t_0 / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((1.0d0 + (1.0d0 / eps)) * exp((x * (eps + (-1.0d0))))) - (((1.0d0 / eps) + (-1.0d0)) * exp((x * ((-1.0d0) - eps))))
    if (t_0 <= 4.0d0) then
        tmp = (1.0d0 + x) / exp(x)
    else
        tmp = t_0 / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = ((1.0 + (1.0 / eps)) * Math.exp((x * (eps + -1.0)))) - (((1.0 / eps) + -1.0) * Math.exp((x * (-1.0 - eps))));
	double tmp;
	if (t_0 <= 4.0) {
		tmp = (1.0 + x) / Math.exp(x);
	} else {
		tmp = t_0 / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = ((1.0 + (1.0 / eps)) * math.exp((x * (eps + -1.0)))) - (((1.0 / eps) + -1.0) * math.exp((x * (-1.0 - eps))))
	tmp = 0
	if t_0 <= 4.0:
		tmp = (1.0 + x) / math.exp(x)
	else:
		tmp = t_0 / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(x * Float64(eps + -1.0)))) - Float64(Float64(Float64(1.0 / eps) + -1.0) * exp(Float64(x * Float64(-1.0 - eps)))))
	tmp = 0.0
	if (t_0 <= 4.0)
		tmp = Float64(Float64(1.0 + x) / exp(x));
	else
		tmp = Float64(t_0 / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) - (((1.0 / eps) + -1.0) * exp((x * (-1.0 - eps))));
	tmp = 0.0;
	if (t_0 <= 4.0)
		tmp = (1.0 + x) / exp(x);
	else
		tmp = t_0 / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1.0 / eps), $MachinePrecision] + -1.0), $MachinePrecision] * N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4.0], N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(t$95$0 / 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))))) < 4

   1. Initial program 57.7%

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

   3. Simplified 33.9%

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

   5. Taylor expanded in eps around 0 56.4%

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

      1. associate-+r+ 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-inverses 100.0%

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

      5. distribute-lft-out 100.0%

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

      6. distribute-rgt1-in 100.0%

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

      7. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in eps around 0 100.0%

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

      1. +-commutative 100.0%

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

   10. Simplified 100.0%

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

       1. *-commutative 100.0%

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

       2. exp-neg 100.0%

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

       3. un-div-inv 100.0%

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

   12. Applied egg-rr 100.0%

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

   if 4 < (-.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%

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

4. Final simplification 100.0%

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

Alternative 3: 62.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{+73}:\\ \;\;\;\;\frac{\frac{x \cdot \left(x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right) + -1\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 0.335:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\frac{1}{\varepsilon} + \frac{-1}{\varepsilon}\right) - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+157}:\\ \;\;\;\;x \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(2 \cdot \left(\left(1 + x\right) \cdot \left(1 + x \cdot \left(x \cdot 0.5 + -1\right)\right)\right)\right)}{\varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.02e+73)
   (/
    (/ (* x (+ (* x (+ 0.5 (* x (* x 0.041666666666666664)))) -1.0)) eps)
    2.0)
   (if (<= x 0.335)
     (/ (+ 2.0 (* x (- (+ (/ 1.0 eps) (/ -1.0 eps)) eps))) 2.0)
     (if (<= x 2e+157)
       (* x (exp (- x)))
       (/
        (/ (* eps (* 2.0 (* (+ 1.0 x) (+ 1.0 (* x (+ (* x 0.5) -1.0)))))) eps)
        2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.02e+73) {
		tmp = ((x * ((x * (0.5 + (x * (x * 0.041666666666666664)))) + -1.0)) / eps) / 2.0;
	} else if (x <= 0.335) {
		tmp = (2.0 + (x * (((1.0 / eps) + (-1.0 / eps)) - eps))) / 2.0;
	} else if (x <= 2e+157) {
		tmp = x * exp(-x);
	} else {
		tmp = ((eps * (2.0 * ((1.0 + x) * (1.0 + (x * ((x * 0.5) + -1.0)))))) / eps) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-1.02d+73)) then
        tmp = ((x * ((x * (0.5d0 + (x * (x * 0.041666666666666664d0)))) + (-1.0d0))) / eps) / 2.0d0
    else if (x <= 0.335d0) then
        tmp = (2.0d0 + (x * (((1.0d0 / eps) + ((-1.0d0) / eps)) - eps))) / 2.0d0
    else if (x <= 2d+157) then
        tmp = x * exp(-x)
    else
        tmp = ((eps * (2.0d0 * ((1.0d0 + x) * (1.0d0 + (x * ((x * 0.5d0) + (-1.0d0))))))) / eps) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.02e+73) {
		tmp = ((x * ((x * (0.5 + (x * (x * 0.041666666666666664)))) + -1.0)) / eps) / 2.0;
	} else if (x <= 0.335) {
		tmp = (2.0 + (x * (((1.0 / eps) + (-1.0 / eps)) - eps))) / 2.0;
	} else if (x <= 2e+157) {
		tmp = x * Math.exp(-x);
	} else {
		tmp = ((eps * (2.0 * ((1.0 + x) * (1.0 + (x * ((x * 0.5) + -1.0)))))) / eps) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.02e+73:
		tmp = ((x * ((x * (0.5 + (x * (x * 0.041666666666666664)))) + -1.0)) / eps) / 2.0
	elif x <= 0.335:
		tmp = (2.0 + (x * (((1.0 / eps) + (-1.0 / eps)) - eps))) / 2.0
	elif x <= 2e+157:
		tmp = x * math.exp(-x)
	else:
		tmp = ((eps * (2.0 * ((1.0 + x) * (1.0 + (x * ((x * 0.5) + -1.0)))))) / eps) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.02e+73)
		tmp = Float64(Float64(Float64(x * Float64(Float64(x * Float64(0.5 + Float64(x * Float64(x * 0.041666666666666664)))) + -1.0)) / eps) / 2.0);
	elseif (x <= 0.335)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(1.0 / eps) + Float64(-1.0 / eps)) - eps))) / 2.0);
	elseif (x <= 2e+157)
		tmp = Float64(x * exp(Float64(-x)));
	else
		tmp = Float64(Float64(Float64(eps * Float64(2.0 * Float64(Float64(1.0 + x) * Float64(1.0 + Float64(x * Float64(Float64(x * 0.5) + -1.0)))))) / eps) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.02e+73)
		tmp = ((x * ((x * (0.5 + (x * (x * 0.041666666666666664)))) + -1.0)) / eps) / 2.0;
	elseif (x <= 0.335)
		tmp = (2.0 + (x * (((1.0 / eps) + (-1.0 / eps)) - eps))) / 2.0;
	elseif (x <= 2e+157)
		tmp = x * exp(-x);
	else
		tmp = ((eps * (2.0 * ((1.0 + x) * (1.0 + (x * ((x * 0.5) + -1.0)))))) / eps) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.02e+73], N[(N[(N[(x * N[(N[(x * N[(0.5 + N[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 0.335], N[(N[(2.0 + N[(x * N[(N[(N[(1.0 / eps), $MachinePrecision] + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2e+157], N[(x * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], N[(N[(N[(eps * N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] * N[(1.0 + N[(x * N[(N[(x * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.01999999999999995e73

   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 53.4%

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

   6. Taylor expanded in eps around 0 48.1%

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

      1. expm1-define 48.1%

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

      2. neg-mul-1 48.1%

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

   8. Simplified 48.1%

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

   9. Taylor expanded in x around 0 44.7%

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

   10. Taylor expanded in x around inf 44.7%

       \[\leadsto \frac{\frac{x \cdot \left(x \cdot \left(0.5 + x \cdot \color{blue}{\left(0.041666666666666664 \cdot x\right)}\right) - 1\right)}{\varepsilon}}{2} \]
   11. Step-by-step derivation

       1. *-commutative 44.7%

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

   12. Simplified 44.7%

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

   if -1.01999999999999995e73 < x < 0.33500000000000002

   1. Initial program 61.2%

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

   3. Simplified 48.8%

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

   5. Taylor expanded in x around 0 69.0%

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

   6. Taylor expanded in eps around 0 69.7%

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

   if 0.33500000000000002 < x < 1.99999999999999997e157

   1. Initial program 94.1%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in eps around 0 51.9%

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

      1. associate-+r+ 57.8%

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

      2. mul-1-neg 57.8%

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

      3. sub-neg 57.8%

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

      4. +-inverses 57.8%

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

      5. distribute-lft-out 57.8%

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

      6. distribute-rgt1-in 57.8%

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

      7. mul-1-neg 57.8%

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

   7. Simplified 57.8%

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

   8. Taylor expanded in x around inf 53.1%

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

   if 1.99999999999999997e157 < 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%

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

   5. Taylor expanded in eps around 0 36.3%

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

      1. associate-+r+ 36.3%

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

      2. mul-1-neg 36.3%

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

      3. sub-neg 36.3%

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

      4. +-inverses 36.3%

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

      5. distribute-lft-out 36.3%

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

      6. distribute-rgt1-in 36.3%

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

      7. mul-1-neg 36.3%

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

   7. Simplified 36.3%

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

   8. Taylor expanded in x around 0 65.3%

      \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot \color{blue}{\left(1 + x \cdot \left(0.5 \cdot x - 1\right)\right)}\right)\right)}{\varepsilon}}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{+73}:\\ \;\;\;\;\frac{\frac{x \cdot \left(x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right) + -1\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 0.335:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\frac{1}{\varepsilon} + \frac{-1}{\varepsilon}\right) - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+157}:\\ \;\;\;\;x \cdot e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(2 \cdot \left(\left(1 + x\right) \cdot \left(1 + x \cdot \left(x \cdot 0.5 + -1\right)\right)\right)\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -50000:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+157}:\\ \;\;\;\;\frac{1 + x}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(2 \cdot \left(\left(1 + x\right) \cdot \left(1 + x \cdot \left(x \cdot 0.5 + -1\right)\right)\right)\right)}{\varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -50000.0)
   (/ (/ (expm1 (- x)) eps) 2.0)
   (if (<= x 2e+157)
     (/ (+ 1.0 x) (exp x))
     (/
      (/ (* eps (* 2.0 (* (+ 1.0 x) (+ 1.0 (* x (+ (* x 0.5) -1.0)))))) eps)
      2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -50000.0) {
		tmp = (expm1(-x) / eps) / 2.0;
	} else if (x <= 2e+157) {
		tmp = (1.0 + x) / exp(x);
	} else {
		tmp = ((eps * (2.0 * ((1.0 + x) * (1.0 + (x * ((x * 0.5) + -1.0)))))) / eps) / 2.0;
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -50000.0) {
		tmp = (Math.expm1(-x) / eps) / 2.0;
	} else if (x <= 2e+157) {
		tmp = (1.0 + x) / Math.exp(x);
	} else {
		tmp = ((eps * (2.0 * ((1.0 + x) * (1.0 + (x * ((x * 0.5) + -1.0)))))) / eps) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -50000.0:
		tmp = (math.expm1(-x) / eps) / 2.0
	elif x <= 2e+157:
		tmp = (1.0 + x) / math.exp(x)
	else:
		tmp = ((eps * (2.0 * ((1.0 + x) * (1.0 + (x * ((x * 0.5) + -1.0)))))) / eps) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -50000.0)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps) / 2.0);
	elseif (x <= 2e+157)
		tmp = Float64(Float64(1.0 + x) / exp(x));
	else
		tmp = Float64(Float64(Float64(eps * Float64(2.0 * Float64(Float64(1.0 + x) * Float64(1.0 + Float64(x * Float64(Float64(x * 0.5) + -1.0)))))) / eps) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -50000.0], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2e+157], N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[(eps * N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] * N[(1.0 + N[(x * N[(N[(x * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5e4

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 47.6%

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

   6. Taylor expanded in eps around 0 54.1%

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

      1. expm1-define 54.1%

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

      2. neg-mul-1 54.1%

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

   8. Simplified 54.1%

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

   if -5e4 < x < 1.99999999999999997e157

   1. Initial program 67.8%

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

   3. Simplified 49.6%

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

   5. Taylor expanded in eps around 0 37.3%

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

      1. associate-+r+ 70.5%

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

      2. mul-1-neg 70.5%

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

      3. sub-neg 70.5%

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

      4. +-inverses 70.5%

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

      5. distribute-lft-out 70.5%

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

      6. distribute-rgt1-in 70.5%

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

      7. mul-1-neg 70.5%

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

   7. Simplified 70.5%

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

   8. Taylor expanded in eps around 0 70.5%

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

      1. +-commutative 70.5%

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

   10. Simplified 70.5%

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

       1. *-commutative 70.5%

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

       2. exp-neg 70.5%

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

       3. un-div-inv 70.5%

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

   12. Applied egg-rr 70.5%

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

   if 1.99999999999999997e157 < 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%

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

   5. Taylor expanded in eps around 0 36.3%

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

      1. associate-+r+ 36.3%

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

      2. mul-1-neg 36.3%

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

      3. sub-neg 36.3%

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

      4. +-inverses 36.3%

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

      5. distribute-lft-out 36.3%

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

      6. distribute-rgt1-in 36.3%

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

      7. mul-1-neg 36.3%

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

   7. Simplified 36.3%

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

   8. Taylor expanded in x around 0 65.3%

      \[\leadsto \frac{\frac{0 + \varepsilon \cdot \left(2 \cdot \left(\left(x + 1\right) \cdot \color{blue}{\left(1 + x \cdot \left(0.5 \cdot x - 1\right)\right)}\right)\right)}{\varepsilon}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -50000:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+157}:\\ \;\;\;\;\frac{1 + x}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(2 \cdot \left(\left(1 + x\right) \cdot \left(1 + x \cdot \left(x \cdot 0.5 + -1\right)\right)\right)\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-236}:\\ \;\;\;\;\frac{1 + e^{\left(1 - \varepsilon\right) \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.5e-236)
   (/ (+ 1.0 (exp (* (- 1.0 eps) x))) 2.0)
   (/ (+ 1.0 (exp (* x (+ eps -1.0)))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.5e-236) {
		tmp = (1.0 + exp(((1.0 - eps) * x))) / 2.0;
	} else {
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-3.5d-236)) then
        tmp = (1.0d0 + exp(((1.0d0 - eps) * x))) / 2.0d0
    else
        tmp = (1.0d0 + exp((x * (eps + (-1.0d0))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -3.5e-236) {
		tmp = (1.0 + Math.exp(((1.0 - eps) * x))) / 2.0;
	} else {
		tmp = (1.0 + Math.exp((x * (eps + -1.0)))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -3.5e-236:
		tmp = (1.0 + math.exp(((1.0 - eps) * x))) / 2.0
	else:
		tmp = (1.0 + math.exp((x * (eps + -1.0)))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.5e-236)
		tmp = Float64(Float64(1.0 + exp(Float64(Float64(1.0 - eps) * x))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps + -1.0)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -3.5e-236)
		tmp = (1.0 + exp(((1.0 - eps) * x))) / 2.0;
	else
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.5e-236], N[(N[(1.0 + N[Exp[N[(N[(1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.49999999999999994e-236

   1. Initial program 75.5%

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

   3. Simplified 75.5%

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

   5. Taylor expanded in x around 0 42.2%

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

   6. Taylor expanded in eps around inf 64.8%

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

      1. associate-*r* 64.8%

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

      2. neg-mul-1 64.8%

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

   8. Simplified 64.8%

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

      1. add-sqr-sqrt 64.8%

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

      2. sqrt-unprod 63.7%

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

      3. sqr-neg 63.7%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 71.3%

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

      6. sub-neg 71.3%

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

      7. distribute-lft-in 71.3%

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

      8. *-rgt-identity 71.3%

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

   10. Applied egg-rr 71.3%

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

       1. *-rgt-identity 71.3%

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

       2. distribute-lft-in 71.3%

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

       3. sub-neg 71.3%

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

   12. Simplified 71.3%

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

   if -3.49999999999999994e-236 < x

   1. Initial program 77.4%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in x around 0 39.3%

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

   6. Taylor expanded in eps around inf 60.1%

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

      1. associate-*r* 60.1%

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

      2. neg-mul-1 60.1%

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

   8. Simplified 60.1%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-236}:\\ \;\;\;\;\frac{1 + e^{\left(1 - \varepsilon\right) \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-236}:\\ \;\;\;\;\frac{1 + e^{\left(1 - \varepsilon\right) \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.5e-236)
   (/ (+ 1.0 (exp (* (- 1.0 eps) x))) 2.0)
   (/ (+ 1.0 (exp (* eps x))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.5e-236) {
		tmp = (1.0 + exp(((1.0 - eps) * x))) / 2.0;
	} else {
		tmp = (1.0 + exp((eps * x))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-3.5d-236)) then
        tmp = (1.0d0 + exp(((1.0d0 - eps) * x))) / 2.0d0
    else
        tmp = (1.0d0 + exp((eps * x))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -3.5e-236) {
		tmp = (1.0 + Math.exp(((1.0 - eps) * x))) / 2.0;
	} else {
		tmp = (1.0 + Math.exp((eps * x))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -3.5e-236:
		tmp = (1.0 + math.exp(((1.0 - eps) * x))) / 2.0
	else:
		tmp = (1.0 + math.exp((eps * x))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.5e-236)
		tmp = Float64(Float64(1.0 + exp(Float64(Float64(1.0 - eps) * x))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(Float64(eps * x))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -3.5e-236)
		tmp = (1.0 + exp(((1.0 - eps) * x))) / 2.0;
	else
		tmp = (1.0 + exp((eps * x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.5e-236], N[(N[(1.0 + N[Exp[N[(N[(1.0 - eps), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.49999999999999994e-236

   1. Initial program 75.5%

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

   3. Simplified 75.5%

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

   5. Taylor expanded in x around 0 42.2%

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

   6. Taylor expanded in eps around inf 64.8%

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

      1. associate-*r* 64.8%

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

      2. neg-mul-1 64.8%

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

   8. Simplified 64.8%

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

      1. add-sqr-sqrt 64.8%

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

      2. sqrt-unprod 63.7%

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

      3. sqr-neg 63.7%

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

      4. sqrt-unprod 0.0%

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

      5. add-sqr-sqrt 71.3%

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

      6. sub-neg 71.3%

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

      7. distribute-lft-in 71.3%

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

      8. *-rgt-identity 71.3%

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

   10. Applied egg-rr 71.3%

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

       1. *-rgt-identity 71.3%

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

       2. distribute-lft-in 71.3%

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

       3. sub-neg 71.3%

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

   12. Simplified 71.3%

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

   if -3.49999999999999994e-236 < x

   1. Initial program 77.4%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in x around 0 39.3%

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

   6. Taylor expanded in eps around inf 60.1%

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

      1. associate-*r* 60.1%

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

      2. neg-mul-1 60.1%

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

   8. Simplified 60.1%

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

   9. Taylor expanded in eps around inf 60.1%

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

       1. *-commutative 60.1%

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

   11. Simplified 60.1%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-236}:\\ \;\;\;\;\frac{1 + e^{\left(1 - \varepsilon\right) \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.005:\\ \;\;\;\;\frac{1 + x}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 0.005) (/ (+ 1.0 x) (exp x)) (/ (+ 1.0 (exp (* eps x))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 0.005) {
		tmp = (1.0 + x) / exp(x);
	} else {
		tmp = (1.0 + exp((eps * x))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 0.005d0) then
        tmp = (1.0d0 + x) / exp(x)
    else
        tmp = (1.0d0 + exp((eps * x))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= 0.005) {
		tmp = (1.0 + x) / Math.exp(x);
	} else {
		tmp = (1.0 + Math.exp((eps * x))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 0.005:
		tmp = (1.0 + x) / math.exp(x)
	else:
		tmp = (1.0 + math.exp((eps * x))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 0.005)
		tmp = Float64(Float64(1.0 + x) / exp(x));
	else
		tmp = Float64(Float64(1.0 + exp(Float64(eps * x))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 0.005)
		tmp = (1.0 + x) / exp(x);
	else
		tmp = (1.0 + exp((eps * x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 0.005], N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 0.0050000000000000001

   1. Initial program 65.6%

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

   3. Simplified 55.6%

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

   5. Taylor expanded in eps around 0 34.9%

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

      1. associate-+r+ 70.3%

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

      2. mul-1-neg 70.3%

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

      3. sub-neg 70.3%

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

      4. +-inverses 70.3%

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

      5. distribute-lft-out 70.3%

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

      6. distribute-rgt1-in 70.3%

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

      7. mul-1-neg 70.3%

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

   7. Simplified 70.3%

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

   8. Taylor expanded in eps around 0 70.3%

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

      1. +-commutative 70.3%

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

   10. Simplified 70.3%

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

       1. *-commutative 70.3%

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

       2. exp-neg 70.3%

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

       3. un-div-inv 70.4%

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

   12. Applied egg-rr 70.4%

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

   if 0.0050000000000000001 < 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 64.4%

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

   6. Taylor expanded in eps around inf 64.4%

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

      1. associate-*r* 64.4%

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

      2. neg-mul-1 64.4%

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

   8. Simplified 64.4%

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

   9. Taylor expanded in eps around inf 64.4%

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

       1. *-commutative 64.4%

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

   11. Simplified 64.4%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.005:\\ \;\;\;\;\frac{1 + x}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.005:\\ \;\;\;\;\frac{1 + x}{e^{x}}\\ \mathbf{elif}\;\varepsilon \leq 1.8 \cdot 10^{+249}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(2 \cdot \left(\left(1 + x\right) \cdot \left(1 + x \cdot \left(x \cdot 0.5 + -1\right)\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\frac{1}{\varepsilon} + \frac{-1}{\varepsilon}\right) - \varepsilon\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 0.005)
   (/ (+ 1.0 x) (exp x))
   (if (<= eps 1.8e+249)
     (/
      (/ (* eps (* 2.0 (* (+ 1.0 x) (+ 1.0 (* x (+ (* x 0.5) -1.0)))))) eps)
      2.0)
     (/ (+ 2.0 (* x (- (+ (/ 1.0 eps) (/ -1.0 eps)) eps))) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 0.005) {
		tmp = (1.0 + x) / exp(x);
	} else if (eps <= 1.8e+249) {
		tmp = ((eps * (2.0 * ((1.0 + x) * (1.0 + (x * ((x * 0.5) + -1.0)))))) / eps) / 2.0;
	} else {
		tmp = (2.0 + (x * (((1.0 / eps) + (-1.0 / eps)) - eps))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 0.005d0) then
        tmp = (1.0d0 + x) / exp(x)
    else if (eps <= 1.8d+249) then
        tmp = ((eps * (2.0d0 * ((1.0d0 + x) * (1.0d0 + (x * ((x * 0.5d0) + (-1.0d0))))))) / eps) / 2.0d0
    else
        tmp = (2.0d0 + (x * (((1.0d0 / eps) + ((-1.0d0) / eps)) - eps))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= 0.005) {
		tmp = (1.0 + x) / Math.exp(x);
	} else if (eps <= 1.8e+249) {
		tmp = ((eps * (2.0 * ((1.0 + x) * (1.0 + (x * ((x * 0.5) + -1.0)))))) / eps) / 2.0;
	} else {
		tmp = (2.0 + (x * (((1.0 / eps) + (-1.0 / eps)) - eps))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 0.005:
		tmp = (1.0 + x) / math.exp(x)
	elif eps <= 1.8e+249:
		tmp = ((eps * (2.0 * ((1.0 + x) * (1.0 + (x * ((x * 0.5) + -1.0)))))) / eps) / 2.0
	else:
		tmp = (2.0 + (x * (((1.0 / eps) + (-1.0 / eps)) - eps))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 0.005)
		tmp = Float64(Float64(1.0 + x) / exp(x));
	elseif (eps <= 1.8e+249)
		tmp = Float64(Float64(Float64(eps * Float64(2.0 * Float64(Float64(1.0 + x) * Float64(1.0 + Float64(x * Float64(Float64(x * 0.5) + -1.0)))))) / eps) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(1.0 / eps) + Float64(-1.0 / eps)) - eps))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 0.005)
		tmp = (1.0 + x) / exp(x);
	elseif (eps <= 1.8e+249)
		tmp = ((eps * (2.0 * ((1.0 + x) * (1.0 + (x * ((x * 0.5) + -1.0)))))) / eps) / 2.0;
	else
		tmp = (2.0 + (x * (((1.0 / eps) + (-1.0 / eps)) - eps))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 0.005], N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 1.8e+249], N[(N[(N[(eps * N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] * N[(1.0 + N[(x * N[(N[(x * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * N[(N[(N[(1.0 / eps), $MachinePrecision] + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < 0.0050000000000000001

   1. Initial program 65.6%

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

   3. Simplified 55.6%

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

   5. Taylor expanded in eps around 0 34.9%

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

      1. associate-+r+ 70.3%

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

      2. mul-1-neg 70.3%

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

      3. sub-neg 70.3%

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

      4. +-inverses 70.3%

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

      5. distribute-lft-out 70.3%

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

      6. distribute-rgt1-in 70.3%

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

      7. mul-1-neg 70.3%

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

   7. Simplified 70.3%

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

   8. Taylor expanded in eps around 0 70.3%

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

      1. +-commutative 70.3%

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

   10. Simplified 70.3%

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

       1. *-commutative 70.3%

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

       2. exp-neg 70.3%

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

       3. un-div-inv 70.4%

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

   12. Applied egg-rr 70.4%

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

   if 0.0050000000000000001 < eps < 1.7999999999999999e249

   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 76.6%

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

   5. Taylor expanded in eps around 0 29.6%

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

      1. associate-+r+ 29.7%

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

      2. mul-1-neg 29.7%

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

      3. sub-neg 29.7%

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

      4. +-inverses 29.7%

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

      5. distribute-lft-out 29.7%

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

      6. distribute-rgt1-in 29.7%

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

      7. mul-1-neg 29.7%

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

   7. Simplified 29.7%

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

   8. Taylor expanded in x around 0 50.2%

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

   if 1.7999999999999999e249 < eps

   1. Initial program 99.9%

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

   3. Simplified 89.0%

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

   5. Taylor expanded in x around 0 13.0%

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

   6. Taylor expanded in eps around 0 39.9%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 0.005:\\ \;\;\;\;\frac{1 + x}{e^{x}}\\ \mathbf{elif}\;\varepsilon \leq 1.8 \cdot 10^{+249}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(2 \cdot \left(\left(1 + x\right) \cdot \left(1 + x \cdot \left(x \cdot 0.5 + -1\right)\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\frac{1}{\varepsilon} + \frac{-1}{\varepsilon}\right) - \varepsilon\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.2% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -350000:\\ \;\;\;\;\frac{\frac{x \cdot \left(x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right) + -1\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+59} \lor \neg \left(x \leq 1.5 \cdot 10^{+157}\right):\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(2 \cdot \left(\left(1 + x\right) \cdot \left(1 + x \cdot \left(x \cdot 0.5 + -1\right)\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -350000.0)
   (/
    (/
     (*
      x
      (+
       (* x (+ 0.5 (* x (- (* x 0.041666666666666664) 0.16666666666666666))))
       -1.0))
     eps)
    2.0)
   (if (or (<= x 1.55e+59) (not (<= x 1.5e+157)))
     (/
      (/ (* eps (* 2.0 (* (+ 1.0 x) (+ 1.0 (* x (+ (* x 0.5) -1.0)))))) eps)
      2.0)
     0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -350000.0) {
		tmp = ((x * ((x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))) + -1.0)) / eps) / 2.0;
	} else if ((x <= 1.55e+59) || !(x <= 1.5e+157)) {
		tmp = ((eps * (2.0 * ((1.0 + x) * (1.0 + (x * ((x * 0.5) + -1.0)))))) / eps) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-350000.0d0)) then
        tmp = ((x * ((x * (0.5d0 + (x * ((x * 0.041666666666666664d0) - 0.16666666666666666d0)))) + (-1.0d0))) / eps) / 2.0d0
    else if ((x <= 1.55d+59) .or. (.not. (x <= 1.5d+157))) then
        tmp = ((eps * (2.0d0 * ((1.0d0 + x) * (1.0d0 + (x * ((x * 0.5d0) + (-1.0d0))))))) / eps) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -350000.0) {
		tmp = ((x * ((x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))) + -1.0)) / eps) / 2.0;
	} else if ((x <= 1.55e+59) || !(x <= 1.5e+157)) {
		tmp = ((eps * (2.0 * ((1.0 + x) * (1.0 + (x * ((x * 0.5) + -1.0)))))) / eps) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -350000.0:
		tmp = ((x * ((x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))) + -1.0)) / eps) / 2.0
	elif (x <= 1.55e+59) or not (x <= 1.5e+157):
		tmp = ((eps * (2.0 * ((1.0 + x) * (1.0 + (x * ((x * 0.5) + -1.0)))))) / eps) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -350000.0)
		tmp = Float64(Float64(Float64(x * Float64(Float64(x * Float64(0.5 + Float64(x * Float64(Float64(x * 0.041666666666666664) - 0.16666666666666666)))) + -1.0)) / eps) / 2.0);
	elseif ((x <= 1.55e+59) || !(x <= 1.5e+157))
		tmp = Float64(Float64(Float64(eps * Float64(2.0 * Float64(Float64(1.0 + x) * Float64(1.0 + Float64(x * Float64(Float64(x * 0.5) + -1.0)))))) / eps) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -350000.0)
		tmp = ((x * ((x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))) + -1.0)) / eps) / 2.0;
	elseif ((x <= 1.55e+59) || ~((x <= 1.5e+157)))
		tmp = ((eps * (2.0 * ((1.0 + x) * (1.0 + (x * ((x * 0.5) + -1.0)))))) / eps) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -350000.0], N[(N[(N[(x * N[(N[(x * N[(0.5 + N[(x * N[(N[(x * 0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 1.55e+59], N[Not[LessEqual[x, 1.5e+157]], $MachinePrecision]], N[(N[(N[(eps * N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] * N[(1.0 + N[(x * N[(N[(x * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -3.5e5

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 47.6%

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

   6. Taylor expanded in eps around 0 54.1%

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

      1. expm1-define 54.1%

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

      2. neg-mul-1 54.1%

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

   8. Simplified 54.1%

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

   9. Taylor expanded in x around 0 33.3%

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

   if -3.5e5 < x < 1.55000000000000007e59 or 1.50000000000000005e157 < x

   1. Initial program 68.1%

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

   3. Simplified 50.2%

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

   5. Taylor expanded in eps around 0 32.7%

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

      1. associate-+r+ 65.6%

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

      2. mul-1-neg 65.6%

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

      3. sub-neg 65.6%

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

      4. +-inverses 65.6%

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

      5. distribute-lft-out 65.6%

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

      6. distribute-rgt1-in 65.6%

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

      7. mul-1-neg 65.6%

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

   7. Simplified 65.6%

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

   8. Taylor expanded in x around 0 69.4%

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

   if 1.55000000000000007e59 < x < 1.50000000000000005e157

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 63.1%

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

      1. div-sub 63.1%

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

      2. mul-1-neg 63.1%

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

      3. rec-exp 63.1%

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

      4. +-inverses 63.1%

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

      5. metadata-eval 63.1%

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

   7. Simplified 63.1%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -350000:\\ \;\;\;\;\frac{\frac{x \cdot \left(x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right) + -1\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+59} \lor \neg \left(x \leq 1.5 \cdot 10^{+157}\right):\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(2 \cdot \left(\left(1 + x\right) \cdot \left(1 + x \cdot \left(x \cdot 0.5 + -1\right)\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.0% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{+73}:\\ \;\;\;\;\frac{\frac{x \cdot \left(x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right) + -1\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 240:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\frac{1}{\varepsilon} + \frac{-1}{\varepsilon}\right) - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+157}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot 0.25 - 0.5\right)}{\varepsilon}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.02e+73)
   (/
    (/ (* x (+ (* x (+ 0.5 (* x (* x 0.041666666666666664)))) -1.0)) eps)
    2.0)
   (if (<= x 240.0)
     (/ (+ 2.0 (* x (- (+ (/ 1.0 eps) (/ -1.0 eps)) eps))) 2.0)
     (if (<= x 1.5e+157) 0.0 (/ (* x (- (* x 0.25) 0.5)) eps)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.02e+73) {
		tmp = ((x * ((x * (0.5 + (x * (x * 0.041666666666666664)))) + -1.0)) / eps) / 2.0;
	} else if (x <= 240.0) {
		tmp = (2.0 + (x * (((1.0 / eps) + (-1.0 / eps)) - eps))) / 2.0;
	} else if (x <= 1.5e+157) {
		tmp = 0.0;
	} else {
		tmp = (x * ((x * 0.25) - 0.5)) / eps;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-1.02d+73)) then
        tmp = ((x * ((x * (0.5d0 + (x * (x * 0.041666666666666664d0)))) + (-1.0d0))) / eps) / 2.0d0
    else if (x <= 240.0d0) then
        tmp = (2.0d0 + (x * (((1.0d0 / eps) + ((-1.0d0) / eps)) - eps))) / 2.0d0
    else if (x <= 1.5d+157) then
        tmp = 0.0d0
    else
        tmp = (x * ((x * 0.25d0) - 0.5d0)) / eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.02e+73) {
		tmp = ((x * ((x * (0.5 + (x * (x * 0.041666666666666664)))) + -1.0)) / eps) / 2.0;
	} else if (x <= 240.0) {
		tmp = (2.0 + (x * (((1.0 / eps) + (-1.0 / eps)) - eps))) / 2.0;
	} else if (x <= 1.5e+157) {
		tmp = 0.0;
	} else {
		tmp = (x * ((x * 0.25) - 0.5)) / eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.02e+73:
		tmp = ((x * ((x * (0.5 + (x * (x * 0.041666666666666664)))) + -1.0)) / eps) / 2.0
	elif x <= 240.0:
		tmp = (2.0 + (x * (((1.0 / eps) + (-1.0 / eps)) - eps))) / 2.0
	elif x <= 1.5e+157:
		tmp = 0.0
	else:
		tmp = (x * ((x * 0.25) - 0.5)) / eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.02e+73)
		tmp = Float64(Float64(Float64(x * Float64(Float64(x * Float64(0.5 + Float64(x * Float64(x * 0.041666666666666664)))) + -1.0)) / eps) / 2.0);
	elseif (x <= 240.0)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(1.0 / eps) + Float64(-1.0 / eps)) - eps))) / 2.0);
	elseif (x <= 1.5e+157)
		tmp = 0.0;
	else
		tmp = Float64(Float64(x * Float64(Float64(x * 0.25) - 0.5)) / eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.02e+73)
		tmp = ((x * ((x * (0.5 + (x * (x * 0.041666666666666664)))) + -1.0)) / eps) / 2.0;
	elseif (x <= 240.0)
		tmp = (2.0 + (x * (((1.0 / eps) + (-1.0 / eps)) - eps))) / 2.0;
	elseif (x <= 1.5e+157)
		tmp = 0.0;
	else
		tmp = (x * ((x * 0.25) - 0.5)) / eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.02e+73], N[(N[(N[(x * N[(N[(x * N[(0.5 + N[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 240.0], N[(N[(2.0 + N[(x * N[(N[(N[(1.0 / eps), $MachinePrecision] + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.5e+157], 0.0, N[(N[(x * N[(N[(x * 0.25), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.01999999999999995e73

   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 53.4%

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

   6. Taylor expanded in eps around 0 48.1%

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

      1. expm1-define 48.1%

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

      2. neg-mul-1 48.1%

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

   8. Simplified 48.1%

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

   9. Taylor expanded in x around 0 44.7%

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

   10. Taylor expanded in x around inf 44.7%

       \[\leadsto \frac{\frac{x \cdot \left(x \cdot \left(0.5 + x \cdot \color{blue}{\left(0.041666666666666664 \cdot x\right)}\right) - 1\right)}{\varepsilon}}{2} \]
   11. Step-by-step derivation

       1. *-commutative 44.7%

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

   12. Simplified 44.7%

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

   if -1.01999999999999995e73 < x < 240

   1. Initial program 60.3%

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

   3. Simplified 48.2%

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

   5. Taylor expanded in x around 0 67.4%

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

   6. Taylor expanded in eps around 0 68.1%

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

   if 240 < x < 1.50000000000000005e157

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 56.3%

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

      1. div-sub 56.3%

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

      2. mul-1-neg 56.3%

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

      3. rec-exp 56.3%

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

      4. +-inverses 56.3%

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

      5. metadata-eval 56.3%

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

   7. Simplified 56.3%

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

   if 1.50000000000000005e157 < 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 28.4%

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

   6. Taylor expanded in eps around 0 1.9%

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

      1. expm1-define 1.9%

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

      2. neg-mul-1 1.9%

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

   8. Simplified 1.9%

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

   9. Taylor expanded in x around 0 15.9%

      \[\leadsto \color{blue}{x \cdot \left(0.25 \cdot \frac{x}{\varepsilon} - 0.5 \cdot \frac{1}{\varepsilon}\right)} \]
   10. Step-by-step derivation

       1. *-commutative 15.9%

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

       2. associate-*r/ 15.9%

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

       3. metadata-eval 15.9%

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

   11. Simplified 15.9%

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

   12. Taylor expanded in eps around 0 27.0%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{+73}:\\ \;\;\;\;\frac{\frac{x \cdot \left(x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664\right)\right) + -1\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 240:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\frac{1}{\varepsilon} + \frac{-1}{\varepsilon}\right) - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+157}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot 0.25 - 0.5\right)}{\varepsilon}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.4% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+120}:\\ \;\;\;\;\frac{\frac{x \cdot \left(x \cdot \left(0.5 + x \cdot -0.16666666666666666\right) + -1\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 195:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\frac{1}{\varepsilon} + \frac{-1}{\varepsilon}\right) - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+157}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot 0.25 - 0.5\right)}{\varepsilon}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.4e+120)
   (/ (/ (* x (+ (* x (+ 0.5 (* x -0.16666666666666666))) -1.0)) eps) 2.0)
   (if (<= x 195.0)
     (/ (+ 2.0 (* x (- (+ (/ 1.0 eps) (/ -1.0 eps)) eps))) 2.0)
     (if (<= x 5e+157) 0.0 (/ (* x (- (* x 0.25) 0.5)) eps)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.4e+120) {
		tmp = ((x * ((x * (0.5 + (x * -0.16666666666666666))) + -1.0)) / eps) / 2.0;
	} else if (x <= 195.0) {
		tmp = (2.0 + (x * (((1.0 / eps) + (-1.0 / eps)) - eps))) / 2.0;
	} else if (x <= 5e+157) {
		tmp = 0.0;
	} else {
		tmp = (x * ((x * 0.25) - 0.5)) / eps;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2.4d+120)) then
        tmp = ((x * ((x * (0.5d0 + (x * (-0.16666666666666666d0)))) + (-1.0d0))) / eps) / 2.0d0
    else if (x <= 195.0d0) then
        tmp = (2.0d0 + (x * (((1.0d0 / eps) + ((-1.0d0) / eps)) - eps))) / 2.0d0
    else if (x <= 5d+157) then
        tmp = 0.0d0
    else
        tmp = (x * ((x * 0.25d0) - 0.5d0)) / eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.4e+120) {
		tmp = ((x * ((x * (0.5 + (x * -0.16666666666666666))) + -1.0)) / eps) / 2.0;
	} else if (x <= 195.0) {
		tmp = (2.0 + (x * (((1.0 / eps) + (-1.0 / eps)) - eps))) / 2.0;
	} else if (x <= 5e+157) {
		tmp = 0.0;
	} else {
		tmp = (x * ((x * 0.25) - 0.5)) / eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.4e+120:
		tmp = ((x * ((x * (0.5 + (x * -0.16666666666666666))) + -1.0)) / eps) / 2.0
	elif x <= 195.0:
		tmp = (2.0 + (x * (((1.0 / eps) + (-1.0 / eps)) - eps))) / 2.0
	elif x <= 5e+157:
		tmp = 0.0
	else:
		tmp = (x * ((x * 0.25) - 0.5)) / eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.4e+120)
		tmp = Float64(Float64(Float64(x * Float64(Float64(x * Float64(0.5 + Float64(x * -0.16666666666666666))) + -1.0)) / eps) / 2.0);
	elseif (x <= 195.0)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(1.0 / eps) + Float64(-1.0 / eps)) - eps))) / 2.0);
	elseif (x <= 5e+157)
		tmp = 0.0;
	else
		tmp = Float64(Float64(x * Float64(Float64(x * 0.25) - 0.5)) / eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.4e+120)
		tmp = ((x * ((x * (0.5 + (x * -0.16666666666666666))) + -1.0)) / eps) / 2.0;
	elseif (x <= 195.0)
		tmp = (2.0 + (x * (((1.0 / eps) + (-1.0 / eps)) - eps))) / 2.0;
	elseif (x <= 5e+157)
		tmp = 0.0;
	else
		tmp = (x * ((x * 0.25) - 0.5)) / eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.4e+120], N[(N[(N[(x * N[(N[(x * N[(0.5 + N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 195.0], N[(N[(2.0 + N[(x * N[(N[(N[(1.0 / eps), $MachinePrecision] + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5e+157], 0.0, N[(N[(x * N[(N[(x * 0.25), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.40000000000000001e120

   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 63.1%

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

   6. Taylor expanded in eps around 0 38.1%

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

      1. expm1-define 38.1%

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

      2. neg-mul-1 38.1%

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

   8. Simplified 38.1%

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

   9. Taylor expanded in x around 0 38.1%

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

   if -2.40000000000000001e120 < x < 195

   1. Initial program 61.8%

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

   3. Simplified 50.2%

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

   5. Taylor expanded in x around 0 65.0%

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

   6. Taylor expanded in eps around 0 68.1%

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

   if 195 < x < 4.99999999999999976e157

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 56.3%

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

      1. div-sub 56.3%

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

      2. mul-1-neg 56.3%

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

      3. rec-exp 56.3%

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

      4. +-inverses 56.3%

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

      5. metadata-eval 56.3%

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

   7. Simplified 56.3%

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

   if 4.99999999999999976e157 < 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 28.4%

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

   6. Taylor expanded in eps around 0 1.9%

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

      1. expm1-define 1.9%

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

      2. neg-mul-1 1.9%

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

   8. Simplified 1.9%

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

   9. Taylor expanded in x around 0 15.9%

      \[\leadsto \color{blue}{x \cdot \left(0.25 \cdot \frac{x}{\varepsilon} - 0.5 \cdot \frac{1}{\varepsilon}\right)} \]
   10. Step-by-step derivation

       1. *-commutative 15.9%

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

       2. associate-*r/ 15.9%

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

       3. metadata-eval 15.9%

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

   11. Simplified 15.9%

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

   12. Taylor expanded in eps around 0 27.0%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+120}:\\ \;\;\;\;\frac{\frac{x \cdot \left(x \cdot \left(0.5 + x \cdot -0.16666666666666666\right) + -1\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 195:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\frac{1}{\varepsilon} + \frac{-1}{\varepsilon}\right) - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+157}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot 0.25 - 0.5\right)}{\varepsilon}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.9% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(x \cdot 0.25 - 0.5\right)}{\varepsilon}\\ \mathbf{if}\;x \leq -190:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{+19}:\\ \;\;\;\;\left(1 + x\right) \cdot \left(1 + x \cdot \left(x \cdot 0.5 + -1\right)\right)\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+157}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (* x (- (* x 0.25) 0.5)) eps)))
   (if (<= x -190.0)
     t_0
     (if (<= x 7.4e+19)
       (* (+ 1.0 x) (+ 1.0 (* x (+ (* x 0.5) -1.0))))
       (if (<= x 1.15e+157) 0.0 t_0)))))

C

double code(double x, double eps) {
	double t_0 = (x * ((x * 0.25) - 0.5)) / eps;
	double tmp;
	if (x <= -190.0) {
		tmp = t_0;
	} else if (x <= 7.4e+19) {
		tmp = (1.0 + x) * (1.0 + (x * ((x * 0.5) + -1.0)));
	} else if (x <= 1.15e+157) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * ((x * 0.25d0) - 0.5d0)) / eps
    if (x <= (-190.0d0)) then
        tmp = t_0
    else if (x <= 7.4d+19) then
        tmp = (1.0d0 + x) * (1.0d0 + (x * ((x * 0.5d0) + (-1.0d0))))
    else if (x <= 1.15d+157) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = (x * ((x * 0.25) - 0.5)) / eps;
	double tmp;
	if (x <= -190.0) {
		tmp = t_0;
	} else if (x <= 7.4e+19) {
		tmp = (1.0 + x) * (1.0 + (x * ((x * 0.5) + -1.0)));
	} else if (x <= 1.15e+157) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (x * ((x * 0.25) - 0.5)) / eps
	tmp = 0
	if x <= -190.0:
		tmp = t_0
	elif x <= 7.4e+19:
		tmp = (1.0 + x) * (1.0 + (x * ((x * 0.5) + -1.0)))
	elif x <= 1.15e+157:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(x * Float64(Float64(x * 0.25) - 0.5)) / eps)
	tmp = 0.0
	if (x <= -190.0)
		tmp = t_0;
	elseif (x <= 7.4e+19)
		tmp = Float64(Float64(1.0 + x) * Float64(1.0 + Float64(x * Float64(Float64(x * 0.5) + -1.0))));
	elseif (x <= 1.15e+157)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (x * ((x * 0.25) - 0.5)) / eps;
	tmp = 0.0;
	if (x <= -190.0)
		tmp = t_0;
	elseif (x <= 7.4e+19)
		tmp = (1.0 + x) * (1.0 + (x * ((x * 0.5) + -1.0)));
	elseif (x <= 1.15e+157)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(x * N[(N[(x * 0.25), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]}, If[LessEqual[x, -190.0], t$95$0, If[LessEqual[x, 7.4e+19], N[(N[(1.0 + x), $MachinePrecision] * N[(1.0 + N[(x * N[(N[(x * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.15e+157], 0.0, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -190 or 1.15000000000000002e157 < 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 38.4%

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

   6. Taylor expanded in eps around 0 29.1%

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

      1. expm1-define 29.1%

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

      2. neg-mul-1 29.1%

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

   8. Simplified 29.1%

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

   9. Taylor expanded in x around 0 14.1%

      \[\leadsto \color{blue}{x \cdot \left(0.25 \cdot \frac{x}{\varepsilon} - 0.5 \cdot \frac{1}{\varepsilon}\right)} \]
   10. Step-by-step derivation

       1. *-commutative 14.1%

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

       2. associate-*r/ 14.1%

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

       3. metadata-eval 14.1%

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

   11. Simplified 14.1%

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

   12. Taylor expanded in eps around 0 19.4%

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

   if -190 < x < 7.4e19

   1. Initial program 58.3%

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

   3. Simplified 34.8%

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

   5. Taylor expanded in eps around 0 30.6%

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

      1. associate-+r+ 73.5%

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

      2. mul-1-neg 73.5%

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

      3. sub-neg 73.5%

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

      4. +-inverses 73.5%

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

      5. distribute-lft-out 73.5%

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

      6. distribute-rgt1-in 73.5%

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

      7. mul-1-neg 73.5%

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

   7. Simplified 73.5%

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

   8. Taylor expanded in eps around 0 73.5%

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

      1. +-commutative 73.5%

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

   10. Simplified 73.5%

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

   11. Taylor expanded in x around 0 71.0%

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

   if 7.4e19 < x < 1.15000000000000002e157

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 60.2%

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

      1. div-sub 60.2%

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

      2. mul-1-neg 60.2%

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

      3. rec-exp 60.2%

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

      4. +-inverses 60.2%

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

      5. metadata-eval 60.2%

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

   7. Simplified 60.2%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -190:\\ \;\;\;\;\frac{x \cdot \left(x \cdot 0.25 - 0.5\right)}{\varepsilon}\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{+19}:\\ \;\;\;\;\left(1 + x\right) \cdot \left(1 + x \cdot \left(x \cdot 0.5 + -1\right)\right)\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+157}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot 0.25 - 0.5\right)}{\varepsilon}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.3% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot \left(x \cdot 0.25 - 0.5\right)}{\varepsilon}\\ \mathbf{if}\;x \leq -0.0074:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{+19}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+157}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (* x (- (* x 0.25) 0.5)) eps)))
   (if (<= x -0.0074)
     t_0
     (if (<= x 7.4e+19)
       (/ (+ 2.0 (* eps x)) 2.0)
       (if (<= x 2e+157) 0.0 t_0)))))

C

double code(double x, double eps) {
	double t_0 = (x * ((x * 0.25) - 0.5)) / eps;
	double tmp;
	if (x <= -0.0074) {
		tmp = t_0;
	} else if (x <= 7.4e+19) {
		tmp = (2.0 + (eps * x)) / 2.0;
	} else if (x <= 2e+157) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * ((x * 0.25d0) - 0.5d0)) / eps
    if (x <= (-0.0074d0)) then
        tmp = t_0
    else if (x <= 7.4d+19) then
        tmp = (2.0d0 + (eps * x)) / 2.0d0
    else if (x <= 2d+157) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = (x * ((x * 0.25) - 0.5)) / eps;
	double tmp;
	if (x <= -0.0074) {
		tmp = t_0;
	} else if (x <= 7.4e+19) {
		tmp = (2.0 + (eps * x)) / 2.0;
	} else if (x <= 2e+157) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (x * ((x * 0.25) - 0.5)) / eps
	tmp = 0
	if x <= -0.0074:
		tmp = t_0
	elif x <= 7.4e+19:
		tmp = (2.0 + (eps * x)) / 2.0
	elif x <= 2e+157:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(x * Float64(Float64(x * 0.25) - 0.5)) / eps)
	tmp = 0.0
	if (x <= -0.0074)
		tmp = t_0;
	elseif (x <= 7.4e+19)
		tmp = Float64(Float64(2.0 + Float64(eps * x)) / 2.0);
	elseif (x <= 2e+157)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (x * ((x * 0.25) - 0.5)) / eps;
	tmp = 0.0;
	if (x <= -0.0074)
		tmp = t_0;
	elseif (x <= 7.4e+19)
		tmp = (2.0 + (eps * x)) / 2.0;
	elseif (x <= 2e+157)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(x * N[(N[(x * 0.25), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]}, If[LessEqual[x, -0.0074], t$95$0, If[LessEqual[x, 7.4e+19], N[(N[(2.0 + N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2e+157], 0.0, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0074000000000000003 or 1.99999999999999997e157 < 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 39.3%

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

   6. Taylor expanded in eps around 0 28.7%

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

      1. expm1-define 28.7%

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

      2. neg-mul-1 28.7%

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

   8. Simplified 28.7%

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

   9. Taylor expanded in x around 0 13.9%

      \[\leadsto \color{blue}{x \cdot \left(0.25 \cdot \frac{x}{\varepsilon} - 0.5 \cdot \frac{1}{\varepsilon}\right)} \]
   10. Step-by-step derivation

       1. *-commutative 13.9%

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

       2. associate-*r/ 13.9%

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

       3. metadata-eval 13.9%

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

   11. Simplified 13.9%

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

   12. Taylor expanded in eps around 0 19.1%

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

   if -0.0074000000000000003 < x < 7.4e19

   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 46.5%

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

   6. Taylor expanded in eps around inf 84.9%

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

      1. associate-*r* 84.9%

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

      2. neg-mul-1 84.9%

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

   8. Simplified 84.9%

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

   9. Taylor expanded in eps around inf 85.7%

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

       1. *-commutative 85.7%

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

   11. Simplified 85.7%

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

   12. Taylor expanded in x around 0 70.6%

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

   if 7.4e19 < x < 1.99999999999999997e157

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 60.2%

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

      1. div-sub 60.2%

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

      2. mul-1-neg 60.2%

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

      3. rec-exp 60.2%

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

      4. +-inverses 60.2%

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

      5. metadata-eval 60.2%

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

   7. Simplified 60.2%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0074:\\ \;\;\;\;\frac{x \cdot \left(x \cdot 0.25 - 0.5\right)}{\varepsilon}\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{+19}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+157}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot 0.25 - 0.5\right)}{\varepsilon}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 55.7% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0074:\\ \;\;\;\;x \cdot \frac{x \cdot 0.25 - 0.5}{\varepsilon}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+19} \lor \neg \left(x \leq 1.22 \cdot 10^{+222}\right):\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -0.0074)
   (* x (/ (- (* x 0.25) 0.5) eps))
   (if (or (<= x 7.5e+19) (not (<= x 1.22e+222)))
     (/ (+ 2.0 (* eps x)) 2.0)
     0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -0.0074) {
		tmp = x * (((x * 0.25) - 0.5) / eps);
	} else if ((x <= 7.5e+19) || !(x <= 1.22e+222)) {
		tmp = (2.0 + (eps * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-0.0074d0)) then
        tmp = x * (((x * 0.25d0) - 0.5d0) / eps)
    else if ((x <= 7.5d+19) .or. (.not. (x <= 1.22d+222))) then
        tmp = (2.0d0 + (eps * x)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -0.0074) {
		tmp = x * (((x * 0.25) - 0.5) / eps);
	} else if ((x <= 7.5e+19) || !(x <= 1.22e+222)) {
		tmp = (2.0 + (eps * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -0.0074:
		tmp = x * (((x * 0.25) - 0.5) / eps)
	elif (x <= 7.5e+19) or not (x <= 1.22e+222):
		tmp = (2.0 + (eps * x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -0.0074)
		tmp = Float64(x * Float64(Float64(Float64(x * 0.25) - 0.5) / eps));
	elseif ((x <= 7.5e+19) || !(x <= 1.22e+222))
		tmp = Float64(Float64(2.0 + Float64(eps * x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -0.0074)
		tmp = x * (((x * 0.25) - 0.5) / eps);
	elseif ((x <= 7.5e+19) || ~((x <= 1.22e+222)))
		tmp = (2.0 + (eps * x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -0.0074], N[(x * N[(N[(N[(x * 0.25), $MachinePrecision] - 0.5), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, 7.5e+19], N[Not[LessEqual[x, 1.22e+222]], $MachinePrecision]], N[(N[(2.0 + N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0074000000000000003

   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 49.0%

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

   6. Taylor expanded in eps around 0 52.7%

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

      1. expm1-define 52.7%

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

      2. neg-mul-1 52.7%

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

   8. Simplified 52.7%

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

   9. Taylor expanded in x around 0 12.1%

      \[\leadsto \color{blue}{x \cdot \left(0.25 \cdot \frac{x}{\varepsilon} - 0.5 \cdot \frac{1}{\varepsilon}\right)} \]
   10. Step-by-step derivation

       1. *-commutative 12.1%

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

       2. associate-*r/ 12.1%

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

       3. metadata-eval 12.1%

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

   11. Simplified 12.1%

       \[\leadsto \color{blue}{x \cdot \left(\frac{x}{\varepsilon} \cdot 0.25 - \frac{0.5}{\varepsilon}\right)} \]
   12. Step-by-step derivation

       1. associate-*l/ 12.1%

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

       2. sub-div 12.1%

          \[\leadsto x \cdot \color{blue}{\frac{x \cdot 0.25 - 0.5}{\varepsilon}} \]

   13. Applied egg-rr 12.1%

       \[\leadsto x \cdot \color{blue}{\frac{x \cdot 0.25 - 0.5}{\varepsilon}} \]

   if -0.0074000000000000003 < x < 7.5e19 or 1.21999999999999996e222 < x

   1. Initial program 63.0%

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

   3. Simplified 63.0%

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

   5. Taylor expanded in x around 0 44.9%

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

   6. Taylor expanded in eps around inf 78.9%

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

      1. associate-*r* 78.9%

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

      2. neg-mul-1 78.9%

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

   8. Simplified 78.9%

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

   9. Taylor expanded in eps around inf 79.5%

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

       1. *-commutative 79.5%

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

   11. Simplified 79.5%

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

   12. Taylor expanded in x around 0 65.5%

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

   if 7.5e19 < x < 1.21999999999999996e222

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 56.8%

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

      1. div-sub 56.8%

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

      2. mul-1-neg 56.8%

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

      3. rec-exp 56.8%

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

      4. +-inverses 56.8%

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

      5. metadata-eval 56.8%

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

   7. Simplified 56.8%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0074:\\ \;\;\;\;x \cdot \frac{x \cdot 0.25 - 0.5}{\varepsilon}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+19} \lor \neg \left(x \leq 1.22 \cdot 10^{+222}\right):\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 57.1% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 155:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\frac{1}{\varepsilon} + \frac{-1}{\varepsilon}\right) - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+157}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot 0.25 - 0.5\right)}{\varepsilon}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 155.0)
   (/ (+ 2.0 (* x (- (+ (/ 1.0 eps) (/ -1.0 eps)) eps))) 2.0)
   (if (<= x 1.5e+157) 0.0 (/ (* x (- (* x 0.25) 0.5)) eps))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 155.0) {
		tmp = (2.0 + (x * (((1.0 / eps) + (-1.0 / eps)) - eps))) / 2.0;
	} else if (x <= 1.5e+157) {
		tmp = 0.0;
	} else {
		tmp = (x * ((x * 0.25) - 0.5)) / eps;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 155.0d0) then
        tmp = (2.0d0 + (x * (((1.0d0 / eps) + ((-1.0d0) / eps)) - eps))) / 2.0d0
    else if (x <= 1.5d+157) then
        tmp = 0.0d0
    else
        tmp = (x * ((x * 0.25d0) - 0.5d0)) / eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 155.0) {
		tmp = (2.0 + (x * (((1.0 / eps) + (-1.0 / eps)) - eps))) / 2.0;
	} else if (x <= 1.5e+157) {
		tmp = 0.0;
	} else {
		tmp = (x * ((x * 0.25) - 0.5)) / eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 155.0:
		tmp = (2.0 + (x * (((1.0 / eps) + (-1.0 / eps)) - eps))) / 2.0
	elif x <= 1.5e+157:
		tmp = 0.0
	else:
		tmp = (x * ((x * 0.25) - 0.5)) / eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 155.0)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(1.0 / eps) + Float64(-1.0 / eps)) - eps))) / 2.0);
	elseif (x <= 1.5e+157)
		tmp = 0.0;
	else
		tmp = Float64(Float64(x * Float64(Float64(x * 0.25) - 0.5)) / eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 155.0)
		tmp = (2.0 + (x * (((1.0 / eps) + (-1.0 / eps)) - eps))) / 2.0;
	elseif (x <= 1.5e+157)
		tmp = 0.0;
	else
		tmp = (x * ((x * 0.25) - 0.5)) / eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 155.0], N[(N[(2.0 + N[(x * N[(N[(N[(1.0 / eps), $MachinePrecision] + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.5e+157], 0.0, N[(N[(x * N[(N[(x * 0.25), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 155

   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 56.1%

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

   5. Taylor expanded in x around 0 57.6%

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

   6. Taylor expanded in eps around 0 63.0%

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

   if 155 < x < 1.50000000000000005e157

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 56.3%

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

      1. div-sub 56.3%

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

      2. mul-1-neg 56.3%

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

      3. rec-exp 56.3%

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

      4. +-inverses 56.3%

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

      5. metadata-eval 56.3%

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

   7. Simplified 56.3%

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

   if 1.50000000000000005e157 < 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 28.4%

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

   6. Taylor expanded in eps around 0 1.9%

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

      1. expm1-define 1.9%

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

      2. neg-mul-1 1.9%

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

   8. Simplified 1.9%

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

   9. Taylor expanded in x around 0 15.9%

      \[\leadsto \color{blue}{x \cdot \left(0.25 \cdot \frac{x}{\varepsilon} - 0.5 \cdot \frac{1}{\varepsilon}\right)} \]
   10. Step-by-step derivation

       1. *-commutative 15.9%

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

       2. associate-*r/ 15.9%

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

       3. metadata-eval 15.9%

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

   11. Simplified 15.9%

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

   12. Taylor expanded in eps around 0 27.0%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 155:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\frac{1}{\varepsilon} + \frac{-1}{\varepsilon}\right) - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+157}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot 0.25 - 0.5\right)}{\varepsilon}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 54.2% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.4 \cdot 10^{+19}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+224}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 7.4e+19) 1.0 (if (<= x 2.3e+224) 0.0 (/ (+ 2.0 (* eps x)) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 7.4e+19) {
		tmp = 1.0;
	} else if (x <= 2.3e+224) {
		tmp = 0.0;
	} else {
		tmp = (2.0 + (eps * x)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 7.4d+19) then
        tmp = 1.0d0
    else if (x <= 2.3d+224) then
        tmp = 0.0d0
    else
        tmp = (2.0d0 + (eps * x)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 7.4e+19) {
		tmp = 1.0;
	} else if (x <= 2.3e+224) {
		tmp = 0.0;
	} else {
		tmp = (2.0 + (eps * x)) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 7.4e+19:
		tmp = 1.0
	elif x <= 2.3e+224:
		tmp = 0.0
	else:
		tmp = (2.0 + (eps * x)) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 7.4e+19)
		tmp = 1.0;
	elseif (x <= 2.3e+224)
		tmp = 0.0;
	else
		tmp = Float64(Float64(2.0 + Float64(eps * x)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 7.4e+19)
		tmp = 1.0;
	elseif (x <= 2.3e+224)
		tmp = 0.0;
	else
		tmp = (2.0 + (eps * x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 7.4e+19], 1.0, If[LessEqual[x, 2.3e+224], 0.0, N[(N[(2.0 + N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 7.4e19

   1. Initial program 66.9%

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

   3. Simplified 56.8%

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

   5. Taylor expanded in x around 0 56.7%

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

   6. Taylor expanded in x around 0 56.7%

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

   if 7.4e19 < x < 2.3000000000000002e224

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 56.8%

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

      1. div-sub 56.8%

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

      2. mul-1-neg 56.8%

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

      3. rec-exp 56.8%

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

      4. +-inverses 56.8%

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

      5. metadata-eval 56.8%

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

   7. Simplified 56.8%

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

   if 2.3000000000000002e224 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 33.5%

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

   6. Taylor expanded in eps around inf 33.7%

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

      1. associate-*r* 33.7%

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

      2. neg-mul-1 33.7%

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

   8. Simplified 33.7%

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

   9. Taylor expanded in eps around inf 33.4%

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

       1. *-commutative 33.4%

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

   11. Simplified 33.4%

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

   12. Taylor expanded in x around 0 27.7%

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

Alternative 17: 56.7% accurate, 37.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.4 \cdot 10^{+19}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (if (<= x 7.4e+19) 1.0 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 7.4e+19) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 7.4d+19) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 7.4e+19) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 7.4e+19:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 7.4e+19)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 7.4e+19)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 7.4e+19], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 7.4e19

   1. Initial program 66.9%

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

   3. Simplified 56.8%

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

   5. Taylor expanded in x around 0 56.7%

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

   6. Taylor expanded in x around 0 56.7%

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

   if 7.4e19 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 49.5%

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

      1. div-sub 49.5%

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

      2. mul-1-neg 49.5%

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

      3. rec-exp 49.5%

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

      4. +-inverses 49.5%

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

      5. metadata-eval 49.5%

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

   7. Simplified 49.5%

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

Alternative 18: 16.0% accurate, 227.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

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

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 76.7%

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

3. Simplified 69.6%

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

5. Taylor expanded in eps around 0 16.2%

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

   1. div-sub 16.2%

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

   2. mul-1-neg 16.2%

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

   3. rec-exp 16.2%

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

   4. +-inverses 16.4%

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

   5. metadata-eval 16.4%

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

7. Simplified 16.4%

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

Reproduce

herbie shell --seed 2024167 
(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))