NMSE Section 6.1 mentioned, A

Percentage Accurate: 72.8% → 99.7%

Time: 10.2s

Alternatives: 8

Speedup: 2.0×

• 38.1% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	double t_0 = ((1.0 + (1.0 / eps)) * exp((x * (eps + -1.0)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (((1.0 + x) / exp(x)) + ((1.0 + x) * exp(-x))) / 2.0;
	} 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))))) + (exp((x * ((-1.0d0) - eps))) * (1.0d0 + ((-1.0d0) / eps)))
    if (t_0 <= 0.0d0) then
        tmp = (((1.0d0 + x) / exp(x)) + ((1.0d0 + x) * exp(-x))) / 2.0d0
    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)))) + (Math.exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (((1.0 + x) / Math.exp(x)) + ((1.0 + x) * Math.exp(-x))) / 2.0;
	} 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)))) + (math.exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)))
	tmp = 0
	if t_0 <= 0.0:
		tmp = (((1.0 + x) / math.exp(x)) + ((1.0 + x) * math.exp(-x))) / 2.0
	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(exp(Float64(x * Float64(-1.0 - eps))) * Float64(1.0 + Float64(-1.0 / eps))))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(1.0 + x) / exp(x)) + Float64(Float64(1.0 + x) * exp(Float64(-x)))) / 2.0);
	else
		tmp = Float64(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)))) + (exp((x * (-1.0 - eps))) * (1.0 + (-1.0 / eps)));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = (((1.0 + x) / exp(x)) + ((1.0 + x) * exp(-x))) / 2.0;
	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[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + x), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(t$95$0 / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 37.8%

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

      1. div-sub 37.8%

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

      2. +-rgt-identity 37.8%

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

      3. div-sub 37.8%

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

   3. Simplified 37.8%

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

   4. Taylor expanded in eps around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. mul-1-neg 100.0%

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

      4. distribute-lft-out 100.0%

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

      5. mul-1-neg 100.0%

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

      6. *-commutative 100.0%

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

      7. distribute-lft1-in 100.0%

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

      8. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

      1. exp-neg 100.0%

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

      2. un-div-inv 100.0%

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

   8. Applied egg-rr 100.0%

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

   if 0.0 < (-.f64 (*.f64 (+.f64 1 (/.f64 1 eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 1 eps) x)))) (*.f64 (-.f64 (/.f64 1 eps) 1) (exp.f64 (neg.f64 (*.f64 (+.f64 1 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. Recombined 2 regimes into one program.

4. Final simplification 100.0%

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

Alternative 2: 78.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{if}\;\varepsilon \leq -31500:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1:\\ \;\;\;\;\frac{\frac{1 + x}{e^{x}} + \left(1 + x\right) \cdot e^{-x}}{2}\\ \mathbf{elif}\;\varepsilon \leq 5 \cdot 10^{+46}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x - x}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (exp (* x (- eps)))) 2.0)))
   (if (<= eps -31500.0)
     t_0
     (if (<= eps 1.0)
       (/ (+ (/ (+ 1.0 x) (exp x)) (* (+ 1.0 x) (exp (- x)))) 2.0)
       (if (<= eps 5e+46) t_0 (/ (+ 1.0 (exp (- (* eps x) x))) 2.0))))))

C

double code(double x, double eps) {
	double t_0 = (1.0 + exp((x * -eps))) / 2.0;
	double tmp;
	if (eps <= -31500.0) {
		tmp = t_0;
	} else if (eps <= 1.0) {
		tmp = (((1.0 + x) / exp(x)) + ((1.0 + x) * exp(-x))) / 2.0;
	} else if (eps <= 5e+46) {
		tmp = t_0;
	} else {
		tmp = (1.0 + exp(((eps * x) - x))) / 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 + exp((x * -eps))) / 2.0d0
    if (eps <= (-31500.0d0)) then
        tmp = t_0
    else if (eps <= 1.0d0) then
        tmp = (((1.0d0 + x) / exp(x)) + ((1.0d0 + x) * exp(-x))) / 2.0d0
    else if (eps <= 5d+46) then
        tmp = t_0
    else
        tmp = (1.0d0 + exp(((eps * x) - x))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = (1.0 + Math.exp((x * -eps))) / 2.0;
	double tmp;
	if (eps <= -31500.0) {
		tmp = t_0;
	} else if (eps <= 1.0) {
		tmp = (((1.0 + x) / Math.exp(x)) + ((1.0 + x) * Math.exp(-x))) / 2.0;
	} else if (eps <= 5e+46) {
		tmp = t_0;
	} else {
		tmp = (1.0 + Math.exp(((eps * x) - x))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (1.0 + math.exp((x * -eps))) / 2.0
	tmp = 0
	if eps <= -31500.0:
		tmp = t_0
	elif eps <= 1.0:
		tmp = (((1.0 + x) / math.exp(x)) + ((1.0 + x) * math.exp(-x))) / 2.0
	elif eps <= 5e+46:
		tmp = t_0
	else:
		tmp = (1.0 + math.exp(((eps * x) - x))) / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(1.0 + exp(Float64(x * Float64(-eps)))) / 2.0)
	tmp = 0.0
	if (eps <= -31500.0)
		tmp = t_0;
	elseif (eps <= 1.0)
		tmp = Float64(Float64(Float64(Float64(1.0 + x) / exp(x)) + Float64(Float64(1.0 + x) * exp(Float64(-x)))) / 2.0);
	elseif (eps <= 5e+46)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 + exp(Float64(Float64(eps * x) - x))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (1.0 + exp((x * -eps))) / 2.0;
	tmp = 0.0;
	if (eps <= -31500.0)
		tmp = t_0;
	elseif (eps <= 1.0)
		tmp = (((1.0 + x) / exp(x)) + ((1.0 + x) * exp(-x))) / 2.0;
	elseif (eps <= 5e+46)
		tmp = t_0;
	else
		tmp = (1.0 + exp(((eps * x) - x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(1.0 + N[Exp[N[(x * (-eps)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[eps, -31500.0], t$95$0, If[LessEqual[eps, 1.0], N[(N[(N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + x), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[eps, 5e+46], t$95$0, N[(N[(1.0 + N[Exp[N[(N[(eps * x), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -31500 or 1 < eps < 5.0000000000000002e46

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   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. Taylor expanded in x around 0 47.8%

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

   5. Taylor expanded in eps around inf 47.8%

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

      1. *-commutative 47.8%

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

      2. sub-neg 47.8%

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

      3. mul-1-neg 47.8%

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

      4. *-commutative 47.8%

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

      5. +-commutative 47.8%

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

      6. mul-1-neg 47.8%

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

      7. *-commutative 47.8%

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

      8. mul-1-neg 47.8%

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

      9. sub-neg 47.8%

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

   7. Simplified 47.8%

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

      1. add-sqr-sqrt 10.8%

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

      2. sqrt-unprod 19.1%

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

      3. sqr-neg 19.1%

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

      4. sqrt-unprod 8.4%

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

      5. add-sqr-sqrt 70.9%

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

      6. neg-sub0 70.9%

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

   9. Applied egg-rr 70.9%

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

       1. neg-sub0 70.9%

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

       2. distribute-rgt-neg-in 70.9%

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

       3. sub-neg 70.9%

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

       4. mul-1-neg 70.9%

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

       5. +-commutative 70.9%

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

       6. distribute-neg-in 70.9%

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

       7. mul-1-neg 70.9%

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

       8. remove-double-neg 70.9%

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

       9. metadata-eval 70.9%

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

       10. distribute-rgt-in 70.9%

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

       11. neg-mul-1 70.9%

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

       12. sub-neg 70.9%

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

       13. *-commutative 70.9%

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

   11. Simplified 70.9%

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

   12. Taylor expanded in eps around inf 70.9%

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

   if -31500 < eps < 1

   1. Initial program 39.5%

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

      1. div-sub 39.5%

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

      2. +-rgt-identity 39.5%

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

      3. div-sub 39.5%

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

   3. Simplified 39.5%

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

   4. Taylor expanded in eps around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. mul-1-neg 100.0%

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

      4. distribute-lft-out 100.0%

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

      5. mul-1-neg 100.0%

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

      6. *-commutative 100.0%

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

      7. distribute-lft1-in 100.0%

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

      8. mul-1-neg 100.0%

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

   6. Simplified 100.0%

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

      1. exp-neg 100.0%

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

      2. un-div-inv 100.0%

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

   8. Applied egg-rr 100.0%

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

   if 5.0000000000000002e46 < eps

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   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. Taylor expanded in x around 0 69.2%

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

   5. Taylor expanded in eps around inf 69.2%

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

      1. *-commutative 69.2%

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

      2. sub-neg 69.2%

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

      3. mul-1-neg 69.2%

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

      4. *-commutative 69.2%

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

      5. +-commutative 69.2%

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

      6. mul-1-neg 69.2%

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

      7. *-commutative 69.2%

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

      8. mul-1-neg 69.2%

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

      9. sub-neg 69.2%

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

   7. Simplified 69.2%

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

   8. Taylor expanded in x around inf 69.2%

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

      1. *-commutative 69.2%

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

   10. Simplified 69.2%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -31500:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;\varepsilon \leq 1:\\ \;\;\;\;\frac{\frac{1 + x}{e^{x}} + \left(1 + x\right) \cdot e^{-x}}{2}\\ \mathbf{elif}\;\varepsilon \leq 5 \cdot 10^{+46}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x - x}}{2}\\ \end{array} \]

Alternative 3: 77.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00041:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 700:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+176}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -0.00041)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (<= x 700.0)
     (/ (+ 1.0 (exp (* x (- eps)))) 2.0)
     (if (<= x 2e+176) (/ (+ 1.0 (exp x)) 2.0) 0.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -0.00041) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 700.0) {
		tmp = (1.0 + exp((x * -eps))) / 2.0;
	} else if (x <= 2e+176) {
		tmp = (1.0 + exp(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 <= (-0.00041d0)) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if (x <= 700.0d0) then
        tmp = (1.0d0 + exp((x * -eps))) / 2.0d0
    else if (x <= 2d+176) then
        tmp = (1.0d0 + exp(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 <= -0.00041) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 700.0) {
		tmp = (1.0 + Math.exp((x * -eps))) / 2.0;
	} else if (x <= 2e+176) {
		tmp = (1.0 + Math.exp(x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -0.00041:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 700.0:
		tmp = (1.0 + math.exp((x * -eps))) / 2.0
	elif x <= 2e+176:
		tmp = (1.0 + math.exp(x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -0.00041)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 700.0)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-eps)))) / 2.0);
	elseif (x <= 2e+176)
		tmp = Float64(Float64(1.0 + exp(x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -0.00041)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif (x <= 700.0)
		tmp = (1.0 + exp((x * -eps))) / 2.0;
	elseif (x <= 2e+176)
		tmp = (1.0 + exp(x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -0.00041], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 700.0], N[(N[(1.0 + N[Exp[N[(x * (-eps)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2e+176], N[(N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.0999999999999999e-4

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   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. Taylor expanded in x around 0 49.9%

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

   5. Taylor expanded in eps around inf 49.9%

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

   6. Taylor expanded in eps around 0 93.3%

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

      1. sub-neg 93.3%

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

      2. mul-1-neg 93.3%

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

      3. neg-mul-1 93.3%

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

      4. remove-double-neg 93.3%

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

   8. Simplified 93.3%

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

   if -4.0999999999999999e-4 < x < 700

   1. Initial program 53.9%

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

      1. div-sub 53.9%

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

      2. +-rgt-identity 53.9%

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

      3. div-sub 53.9%

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

   3. Simplified 53.9%

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

   4. Taylor expanded in x around 0 37.9%

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

   5. Taylor expanded in eps around inf 83.2%

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

      1. *-commutative 83.2%

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

      2. sub-neg 83.2%

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

      3. mul-1-neg 83.2%

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

      4. *-commutative 83.2%

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

      5. +-commutative 83.2%

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

      6. mul-1-neg 83.2%

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

      7. *-commutative 83.2%

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

      8. mul-1-neg 83.2%

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

      9. sub-neg 83.2%

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

   7. Simplified 83.2%

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

      1. add-sqr-sqrt 34.5%

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

      2. sqrt-unprod 70.6%

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

      3. sqr-neg 70.6%

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

      4. sqrt-unprod 36.1%

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

      5. add-sqr-sqrt 85.8%

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

      6. neg-sub0 85.8%

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

   9. Applied egg-rr 85.8%

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

       1. neg-sub0 85.8%

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

       2. distribute-rgt-neg-in 85.8%

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

       3. sub-neg 85.8%

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

       4. mul-1-neg 85.8%

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

       5. +-commutative 85.8%

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

       6. distribute-neg-in 85.8%

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

       7. mul-1-neg 85.8%

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

       8. remove-double-neg 85.8%

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

       9. metadata-eval 85.8%

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

       10. distribute-rgt-in 85.8%

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

       11. neg-mul-1 85.8%

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

       12. sub-neg 85.8%

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

       13. *-commutative 85.8%

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

   11. Simplified 85.8%

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

   12. Taylor expanded in eps around inf 86.3%

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

   if 700 < x < 2e176

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   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. Taylor expanded in x around 0 28.0%

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

   5. Taylor expanded in eps around inf 28.3%

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

      1. *-commutative 28.3%

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

      2. sub-neg 28.3%

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

      3. mul-1-neg 28.3%

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

      4. *-commutative 28.3%

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

      5. +-commutative 28.3%

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

      6. mul-1-neg 28.3%

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

      7. *-commutative 28.3%

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

      8. mul-1-neg 28.3%

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

      9. sub-neg 28.3%

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

   7. Simplified 28.3%

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

      1. add-sqr-sqrt 2.3%

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

      2. sqrt-unprod 3.1%

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

      3. sqr-neg 3.1%

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

      4. sqrt-unprod 0.8%

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

      5. add-sqr-sqrt 37.4%

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

      6. neg-sub0 37.4%

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

   9. Applied egg-rr 37.4%

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

       1. neg-sub0 37.4%

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

       2. distribute-rgt-neg-in 37.4%

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

       3. sub-neg 37.4%

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

       4. mul-1-neg 37.4%

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

       5. +-commutative 37.4%

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

       6. distribute-neg-in 37.4%

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

       7. mul-1-neg 37.4%

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

       8. remove-double-neg 37.4%

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

       9. metadata-eval 37.4%

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

       10. distribute-rgt-in 37.4%

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

       11. neg-mul-1 37.4%

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

       12. sub-neg 37.4%

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

       13. *-commutative 37.4%

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

   11. Simplified 37.4%

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

   12. Taylor expanded in eps around 0 62.6%

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

   if 2e176 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 67.2%

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

      1. div-sub 67.2%

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

      2. rec-exp 67.2%

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

      3. mul-1-neg 67.2%

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

      4. +-inverses 67.2%

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

   6. Simplified 67.2%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00041:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 700:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+176}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4: 64.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.008:\\ \;\;\;\;\frac{2 + x \cdot x}{2}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+179}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 0.008)
   (/ (+ 2.0 (* x x)) 2.0)
   (if (<= x 6e+179) (/ (+ 1.0 (exp x)) 2.0) 0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 0.008) {
		tmp = (2.0 + (x * x)) / 2.0;
	} else if (x <= 6e+179) {
		tmp = (1.0 + exp(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 <= 0.008d0) then
        tmp = (2.0d0 + (x * x)) / 2.0d0
    else if (x <= 6d+179) then
        tmp = (1.0d0 + exp(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 <= 0.008) {
		tmp = (2.0 + (x * x)) / 2.0;
	} else if (x <= 6e+179) {
		tmp = (1.0 + Math.exp(x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 0.008:
		tmp = (2.0 + (x * x)) / 2.0
	elif x <= 6e+179:
		tmp = (1.0 + math.exp(x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 0.008)
		tmp = Float64(Float64(2.0 + Float64(x * x)) / 2.0);
	elseif (x <= 6e+179)
		tmp = Float64(Float64(1.0 + exp(x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 0.008)
		tmp = (2.0 + (x * x)) / 2.0;
	elseif (x <= 6e+179)
		tmp = (1.0 + exp(x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 0.008], N[(N[(2.0 + N[(x * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 6e+179], N[(N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 0.0080000000000000002

   1. Initial program 61.0%

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

      1. div-sub 61.0%

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

      2. +-rgt-identity 61.0%

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

      3. div-sub 61.0%

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

   3. Simplified 61.0%

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

   4. Taylor expanded in eps around 0 61.1%

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

      1. *-commutative 61.1%

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

      2. distribute-lft1-in 61.1%

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

      3. mul-1-neg 61.1%

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

      4. distribute-lft-out 61.1%

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

      5. mul-1-neg 61.1%

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

      6. *-commutative 61.1%

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

      7. distribute-lft1-in 61.1%

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

      8. mul-1-neg 61.1%

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

   6. Simplified 61.1%

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

   7. Taylor expanded in x around 0 61.0%

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

      1. unpow2 61.0%

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

   9. Simplified 61.0%

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

   10. Taylor expanded in x around 0 61.0%

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

       1. mul-1-neg 61.0%

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

       2. unsub-neg 61.0%

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

       3. unpow2 61.0%

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

   12. Simplified 61.0%

       \[\leadsto \frac{\color{blue}{2 - x \cdot x}}{2} \]
   13. Step-by-step derivation

       1. cancel-sign-sub-inv 61.0%

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

       2. add-sqr-sqrt 29.2%

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

       3. sqrt-unprod 60.9%

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

       4. sqr-neg 60.9%

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

       5. sqrt-prod 31.7%

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

       6. add-sqr-sqrt 68.7%

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

       7. +-commutative 68.7%

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

   14. Applied egg-rr 68.7%

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

   if 0.0080000000000000002 < x < 5.9999999999999996e179

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   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. Taylor expanded in x around 0 28.9%

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

   5. Taylor expanded in eps around inf 29.2%

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

      1. *-commutative 29.2%

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

      2. sub-neg 29.2%

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

      3. mul-1-neg 29.2%

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

      4. *-commutative 29.2%

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

      5. +-commutative 29.2%

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

      6. mul-1-neg 29.2%

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

      7. *-commutative 29.2%

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

      8. mul-1-neg 29.2%

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

      9. sub-neg 29.2%

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

   7. Simplified 29.2%

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

      1. add-sqr-sqrt 2.3%

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

      2. sqrt-unprod 3.1%

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

      3. sqr-neg 3.1%

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

      4. sqrt-unprod 0.8%

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

      5. add-sqr-sqrt 38.0%

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

      6. neg-sub0 38.0%

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

   9. Applied egg-rr 38.0%

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

       1. neg-sub0 38.0%

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

       2. distribute-rgt-neg-in 38.0%

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

       3. sub-neg 38.0%

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

       4. mul-1-neg 38.0%

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

       5. +-commutative 38.0%

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

       6. distribute-neg-in 38.0%

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

       7. mul-1-neg 38.0%

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

       8. remove-double-neg 38.0%

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

       9. metadata-eval 38.0%

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

       10. distribute-rgt-in 38.0%

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

       11. neg-mul-1 38.0%

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

       12. sub-neg 38.0%

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

       13. *-commutative 38.0%

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

   11. Simplified 38.0%

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

   12. Taylor expanded in eps around 0 60.3%

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

   if 5.9999999999999996e179 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 67.2%

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

      1. div-sub 67.2%

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

      2. rec-exp 67.2%

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

      3. mul-1-neg 67.2%

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

      4. +-inverses 67.2%

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

   6. Simplified 67.2%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.008:\\ \;\;\;\;\frac{2 + x \cdot x}{2}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+179}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5: 70.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-16}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+167}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -9e-16)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (<= x 5e+167) (/ (+ 1.0 (exp x)) 2.0) 0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -9e-16) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 5e+167) {
		tmp = (1.0 + exp(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 <= (-9d-16)) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if (x <= 5d+167) then
        tmp = (1.0d0 + exp(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 <= -9e-16) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 5e+167) {
		tmp = (1.0 + Math.exp(x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -9e-16:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 5e+167:
		tmp = (1.0 + math.exp(x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -9e-16)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 5e+167)
		tmp = Float64(Float64(1.0 + exp(x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -9e-16)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif (x <= 5e+167)
		tmp = (1.0 + exp(x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -9e-16], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5e+167], N[(N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -9.0000000000000003e-16

   1. Initial program 90.9%

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

      1. div-sub 90.9%

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

      2. +-rgt-identity 90.9%

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

      3. div-sub 90.9%

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

   3. Simplified 90.9%

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

   4. Taylor expanded in x around 0 45.8%

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

   5. Taylor expanded in eps around inf 52.9%

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

   6. Taylor expanded in eps around 0 92.2%

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

      1. sub-neg 92.2%

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

      2. mul-1-neg 92.2%

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

      3. neg-mul-1 92.2%

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

      4. remove-double-neg 92.2%

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

   8. Simplified 92.2%

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

   if -9.0000000000000003e-16 < x < 4.9999999999999997e167

   1. Initial program 66.2%

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

      1. div-sub 66.2%

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

      2. +-rgt-identity 66.2%

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

      3. div-sub 66.2%

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

   3. Simplified 66.2%

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

   4. Taylor expanded in x around 0 35.7%

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

   5. Taylor expanded in eps around inf 69.2%

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

      1. *-commutative 69.2%

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

      2. sub-neg 69.2%

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

      3. mul-1-neg 69.2%

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

      4. *-commutative 69.2%

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

      5. +-commutative 69.2%

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

      6. mul-1-neg 69.2%

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

      7. *-commutative 69.2%

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

      8. mul-1-neg 69.2%

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

      9. sub-neg 69.2%

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

   7. Simplified 69.2%

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

      1. add-sqr-sqrt 27.0%

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

      2. sqrt-unprod 53.6%

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

      3. sqr-neg 53.6%

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

      4. sqrt-unprod 26.6%

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

      5. add-sqr-sqrt 74.1%

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

      6. neg-sub0 74.1%

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

   9. Applied egg-rr 74.1%

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

       1. neg-sub0 74.1%

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

       2. distribute-rgt-neg-in 74.1%

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

       3. sub-neg 74.1%

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

       4. mul-1-neg 74.1%

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

       5. +-commutative 74.1%

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

       6. distribute-neg-in 74.1%

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

       7. mul-1-neg 74.1%

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

       8. remove-double-neg 74.1%

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

       9. metadata-eval 74.1%

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

       10. distribute-rgt-in 74.1%

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

       11. neg-mul-1 74.1%

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

       12. sub-neg 74.1%

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

       13. *-commutative 74.1%

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

   11. Simplified 74.1%

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

   12. Taylor expanded in eps around 0 69.2%

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

   if 4.9999999999999997e167 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 66.2%

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

      1. div-sub 66.2%

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

      2. rec-exp 66.2%

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

      3. mul-1-neg 66.2%

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

      4. +-inverses 66.2%

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

   6. Simplified 66.2%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-16}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+167}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6: 64.5% accurate, 25.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 54000:\\ \;\;\;\;\frac{2 + x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 54000.0) (/ (+ 2.0 (* x x)) 2.0) 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 54000.0) {
		tmp = (2.0 + (x * 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 <= 54000.0d0) then
        tmp = (2.0d0 + (x * 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 <= 54000.0) {
		tmp = (2.0 + (x * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 54000.0:
		tmp = (2.0 + (x * x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 54000.0)
		tmp = Float64(Float64(2.0 + Float64(x * x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 54000.0)
		tmp = (2.0 + (x * x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 54000.0], N[(N[(2.0 + N[(x * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 54000

   1. Initial program 62.1%

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

      1. div-sub 62.1%

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

      2. +-rgt-identity 62.1%

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

      3. div-sub 62.1%

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

   3. Simplified 62.1%

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

   4. Taylor expanded in eps around 0 59.5%

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

      1. *-commutative 59.5%

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

      2. distribute-lft1-in 59.5%

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

      3. mul-1-neg 59.5%

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

      4. distribute-lft-out 59.5%

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

      5. mul-1-neg 59.5%

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

      6. *-commutative 59.5%

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

      7. distribute-lft1-in 59.5%

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

      8. mul-1-neg 59.5%

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

   6. Simplified 59.5%

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

   7. Taylor expanded in x around 0 59.3%

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

      1. unpow2 59.3%

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

   9. Simplified 59.3%

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

   10. Taylor expanded in x around 0 59.4%

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

       1. mul-1-neg 59.4%

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

       2. unsub-neg 59.4%

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

       3. unpow2 59.4%

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

   12. Simplified 59.4%

       \[\leadsto \frac{\color{blue}{2 - x \cdot x}}{2} \]
   13. Step-by-step derivation

       1. cancel-sign-sub-inv 59.4%

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

       2. add-sqr-sqrt 28.3%

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

       3. sqrt-unprod 59.3%

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

       4. sqr-neg 59.3%

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

       5. sqrt-prod 30.9%

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

       6. add-sqr-sqrt 66.9%

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

       7. +-commutative 66.9%

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

   14. Applied egg-rr 66.9%

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

   if 54000 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 51.4%

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

      1. div-sub 51.4%

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

      2. rec-exp 51.4%

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

      3. mul-1-neg 51.4%

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

      4. +-inverses 51.4%

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

   6. Simplified 51.4%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 54000:\\ \;\;\;\;\frac{2 + x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7: 57.3% accurate, 74.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 54000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (if (<= x 54000.0) 1.0 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 54000.0) {
		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 <= 54000.0d0) 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 <= 54000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 54000.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 54000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 54000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 54000.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 54000

   1. Initial program 62.1%

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

      1. div-sub 62.1%

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

      2. +-rgt-identity 62.1%

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

      3. div-sub 62.1%

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

   3. Simplified 62.1%

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

   4. Taylor expanded in x around 0 59.7%

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

   if 54000 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 51.4%

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

      1. div-sub 51.4%

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

      2. rec-exp 51.4%

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

      3. mul-1-neg 51.4%

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

      4. +-inverses 51.4%

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

   6. Simplified 51.4%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 54000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8: 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 73.5%

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

3. Simplified 62.0%

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

4. Taylor expanded in eps around 0 17.0%

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

   1. div-sub 17.0%

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

   2. rec-exp 17.0%

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

   3. mul-1-neg 17.0%

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

   4. +-inverses 17.2%

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

6. Simplified 17.2%

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

7. Final simplification 17.2%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023187 
(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))