NMSE Section 6.1 mentioned, A

Percentage Accurate: 72.6% → 99.2%

Time: 13.2s

Alternatives: 15

Speedup: 0.5×

• 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: 72.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 - \frac{1}{\varepsilon}\right)\\ t_1 := e^{-x}\\ \mathbf{if}\;t_0 \leq 5:\\ \;\;\;\;\frac{\frac{1 + x}{e^{x}} + \left(t_1 + x \cdot t_1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{2}\\ \end{array} \end{array} \]

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)))))
        (t_1 (exp (- x))))
   (if (<= t_0 5.0)
     (/ (+ (/ (+ 1.0 x) (exp x)) (+ t_1 (* x t_1))) 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 t_1 = exp(-x);
	double tmp;
	if (t_0 <= 5.0) {
		tmp = (((1.0 + x) / exp(x)) + (t_1 + (x * t_1))) / 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) :: t_1
    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)))
    t_1 = exp(-x)
    if (t_0 <= 5.0d0) then
        tmp = (((1.0d0 + x) / exp(x)) + (t_1 + (x * t_1))) / 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 t_1 = Math.exp(-x);
	double tmp;
	if (t_0 <= 5.0) {
		tmp = (((1.0 + x) / Math.exp(x)) + (t_1 + (x * t_1))) / 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)))
	t_1 = math.exp(-x)
	tmp = 0
	if t_0 <= 5.0:
		tmp = (((1.0 + x) / math.exp(x)) + (t_1 + (x * t_1))) / 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))))
	t_1 = exp(Float64(-x))
	tmp = 0.0
	if (t_0 <= 5.0)
		tmp = Float64(Float64(Float64(Float64(1.0 + x) / exp(x)) + Float64(t_1 + Float64(x * t_1))) / 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)));
	t_1 = exp(-x);
	tmp = 0.0;
	if (t_0 <= 5.0)
		tmp = (((1.0 + x) / exp(x)) + (t_1 + (x * t_1))) / 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]}, Block[{t$95$1 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[t$95$0, 5.0], N[(N[(N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 + N[(x * t$95$1), $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))))) < 5

   1. Initial program 55.2%

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

   3. Simplified 55.2%

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

   4. Taylor expanded in eps around 0 100.0%

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

      1. distribute-rgt1-in 100.0%

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

      2. neg-mul-1 100.0%

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

      3. rec-exp 100.0%

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

      4. un-div-inv 100.0%

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

   6. Applied egg-rr 100.0%

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

   if 5 < (-.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%

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

Alternative 2: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 - \frac{1}{\varepsilon}\right)\\ t_1 := \left(1 + x\right) \cdot e^{-x}\\ \mathbf{if}\;t_0 \leq 5:\\ \;\;\;\;\frac{t_1 + t_1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{2}\\ \end{array} \end{array} \]

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)))))
        (t_1 (* (+ 1.0 x) (exp (- x)))))
   (if (<= t_0 5.0) (/ (+ t_1 t_1) 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 t_1 = (1.0 + x) * exp(-x);
	double tmp;
	if (t_0 <= 5.0) {
		tmp = (t_1 + t_1) / 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) :: t_1
    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)))
    t_1 = (1.0d0 + x) * exp(-x)
    if (t_0 <= 5.0d0) then
        tmp = (t_1 + t_1) / 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 t_1 = (1.0 + x) * Math.exp(-x);
	double tmp;
	if (t_0 <= 5.0) {
		tmp = (t_1 + t_1) / 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)))
	t_1 = (1.0 + x) * math.exp(-x)
	tmp = 0
	if t_0 <= 5.0:
		tmp = (t_1 + t_1) / 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))))
	t_1 = Float64(Float64(1.0 + x) * exp(Float64(-x)))
	tmp = 0.0
	if (t_0 <= 5.0)
		tmp = Float64(Float64(t_1 + t_1) / 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)));
	t_1 = (1.0 + x) * exp(-x);
	tmp = 0.0;
	if (t_0 <= 5.0)
		tmp = (t_1 + t_1) / 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]}, Block[{t$95$1 = N[(N[(1.0 + x), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5.0], N[(N[(t$95$1 + t$95$1), $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))))) < 5

   1. Initial program 55.2%

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

   3. Simplified 55.2%

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

   4. Taylor expanded in eps around 0 100.0%

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

   6. Simplified 100.0%

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

   if 5 < (-.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%

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

Alternative 3: 67.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ t_1 := \left(1 + x\right) \cdot e^{-x}\\ t_2 := \frac{t_1 + t_1}{2}\\ \mathbf{if}\;x \leq -1.1 \cdot 10^{-216}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 5900:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+33}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+89}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+165}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+239}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+287}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (exp (* x (+ eps -1.0)))) 2.0))
        (t_1 (* (+ 1.0 x) (exp (- x))))
        (t_2 (/ (+ t_1 t_1) 2.0)))
   (if (<= x -1.1e-216)
     (/ (+ 1.0 (exp (* x (- -1.0 eps)))) 2.0)
     (if (<= x 5900.0)
       (/ (+ 1.0 (exp (* eps x))) 2.0)
       (if (<= x 1.4e+33)
         (/ (+ (+ 1.0 (/ 1.0 eps)) (- 1.0 (/ 1.0 eps))) 2.0)
         (if (<= x 5.8e+89)
           t_0
           (if (<= x 2.2e+165)
             t_2
             (if (<= x 6.2e+239)
               t_0
               (if (<= x 1.5e+287) t_2 (/ (/ (expm1 x) eps) 2.0))))))))))

C

double code(double x, double eps) {
	double t_0 = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	double t_1 = (1.0 + x) * exp(-x);
	double t_2 = (t_1 + t_1) / 2.0;
	double tmp;
	if (x <= -1.1e-216) {
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	} else if (x <= 5900.0) {
		tmp = (1.0 + exp((eps * x))) / 2.0;
	} else if (x <= 1.4e+33) {
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0;
	} else if (x <= 5.8e+89) {
		tmp = t_0;
	} else if (x <= 2.2e+165) {
		tmp = t_2;
	} else if (x <= 6.2e+239) {
		tmp = t_0;
	} else if (x <= 1.5e+287) {
		tmp = t_2;
	} else {
		tmp = (expm1(x) / eps) / 2.0;
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double t_0 = (1.0 + Math.exp((x * (eps + -1.0)))) / 2.0;
	double t_1 = (1.0 + x) * Math.exp(-x);
	double t_2 = (t_1 + t_1) / 2.0;
	double tmp;
	if (x <= -1.1e-216) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps)))) / 2.0;
	} else if (x <= 5900.0) {
		tmp = (1.0 + Math.exp((eps * x))) / 2.0;
	} else if (x <= 1.4e+33) {
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0;
	} else if (x <= 5.8e+89) {
		tmp = t_0;
	} else if (x <= 2.2e+165) {
		tmp = t_2;
	} else if (x <= 6.2e+239) {
		tmp = t_0;
	} else if (x <= 1.5e+287) {
		tmp = t_2;
	} else {
		tmp = (Math.expm1(x) / eps) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (1.0 + math.exp((x * (eps + -1.0)))) / 2.0
	t_1 = (1.0 + x) * math.exp(-x)
	t_2 = (t_1 + t_1) / 2.0
	tmp = 0
	if x <= -1.1e-216:
		tmp = (1.0 + math.exp((x * (-1.0 - eps)))) / 2.0
	elif x <= 5900.0:
		tmp = (1.0 + math.exp((eps * x))) / 2.0
	elif x <= 1.4e+33:
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0
	elif x <= 5.8e+89:
		tmp = t_0
	elif x <= 2.2e+165:
		tmp = t_2
	elif x <= 6.2e+239:
		tmp = t_0
	elif x <= 1.5e+287:
		tmp = t_2
	else:
		tmp = (math.expm1(x) / eps) / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(1.0 + exp(Float64(x * Float64(eps + -1.0)))) / 2.0)
	t_1 = Float64(Float64(1.0 + x) * exp(Float64(-x)))
	t_2 = Float64(Float64(t_1 + t_1) / 2.0)
	tmp = 0.0
	if (x <= -1.1e-216)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	elseif (x <= 5900.0)
		tmp = Float64(Float64(1.0 + exp(Float64(eps * x))) / 2.0);
	elseif (x <= 1.4e+33)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) + Float64(1.0 - Float64(1.0 / eps))) / 2.0);
	elseif (x <= 5.8e+89)
		tmp = t_0;
	elseif (x <= 2.2e+165)
		tmp = t_2;
	elseif (x <= 6.2e+239)
		tmp = t_0;
	elseif (x <= 1.5e+287)
		tmp = t_2;
	else
		tmp = Float64(Float64(expm1(x) / eps) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(1.0 + N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + x), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 + t$95$1), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -1.1e-216], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5900.0], N[(N[(1.0 + N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.4e+33], N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5.8e+89], t$95$0, If[LessEqual[x, 2.2e+165], t$95$2, If[LessEqual[x, 6.2e+239], t$95$0, If[LessEqual[x, 1.5e+287], t$95$2, N[(N[(N[(Exp[x] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < -1.09999999999999995e-216

   1. Initial program 64.9%

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

   3. Simplified 64.9%

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

   4. Taylor expanded in x around 0 38.7%

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

   5. Taylor expanded in eps around inf 73.2%

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

      1. cancel-sign-sub-inv 73.2%

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

      2. metadata-eval 73.2%

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

      3. *-lft-identity 73.2%

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

      4. mul-1-neg 73.2%

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

      5. distribute-rgt-neg-in 73.2%

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

   7. Simplified 73.2%

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

   if -1.09999999999999995e-216 < x < 5900

   1. Initial program 59.8%

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

   3. Simplified 59.8%

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

   4. Taylor expanded in x around 0 50.2%

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

   5. Taylor expanded in eps around inf 89.1%

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

      1. mul-1-neg 89.1%

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

      2. *-commutative 89.1%

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

      3. distribute-lft-neg-in 89.1%

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

   7. Simplified 89.1%

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

   8. Taylor expanded in eps around inf 89.5%

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

   if 5900 < x < 1.4e33

   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%

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

   4. Taylor expanded in x around 0 2.5%

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

   5. Taylor expanded in x around 0 86.2%

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

   if 1.4e33 < x < 5.80000000000000051e89 or 2.1999999999999999e165 < x < 6.20000000000000001e239

   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%

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

   4. Taylor expanded in x around 0 38.3%

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

   5. Taylor expanded in eps around inf 38.6%

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

      1. mul-1-neg 38.6%

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

      2. *-commutative 38.6%

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

      3. distribute-lft-neg-in 38.6%

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

   7. Simplified 38.6%

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

   if 5.80000000000000051e89 < x < 2.1999999999999999e165 or 6.20000000000000001e239 < x < 1.4999999999999999e287

   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%

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

   4. Taylor expanded in eps around 0 72.4%

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

   6. Simplified 72.4%

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

   if 1.4999999999999999e287 < 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%

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

   4. Taylor expanded in x around 0 100.0%

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

   5. Taylor expanded in eps around 0 1.0%

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

      1. expm1-def 1.0%

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

      2. mul-1-neg 1.0%

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

   7. Simplified 1.0%

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

      1. expm1-log1p-u 1.0%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}\right)\right)}}{2} \]

      2. expm1-udef 1.3%

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

      3. expm1-udef 1.3%

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

      4. expm1-udef 1.3%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 100.0%

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

      7. sqr-neg 100.0%

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

      8. sqrt-unprod 100.0%

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

      9. add-sqr-sqrt 100.0%

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

   9. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}\right)} - 1}}{2} \]
   10. Step-by-step derivation

       1. expm1-def 100.0%

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}\right)\right)}}{2} \]

       2. expm1-log1p 100.0%

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

   11. Simplified 100.0%

       \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}}{2} \]
3. Recombined 6 regimes into one program.

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-216}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 5900:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{+33}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+89}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+165}:\\ \;\;\;\;\frac{\left(1 + x\right) \cdot e^{-x} + \left(1 + x\right) \cdot e^{-x}}{2}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+239}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+287}:\\ \;\;\;\;\frac{\left(1 + x\right) \cdot e^{-x} + \left(1 + x\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \end{array} \]

Alternative 4: 67.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ t_1 := e^{-x}\\ t_2 := \frac{\frac{t_1 - t_1}{\varepsilon}}{2}\\ \mathbf{if}\;x \leq -1.1 \cdot 10^{-216}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 16200:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+35}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+94}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.72 \cdot 10^{+164}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+236}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+287}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (/ (+ 1.0 (exp (* x (+ eps -1.0)))) 2.0))
        (t_1 (exp (- x)))
        (t_2 (/ (/ (- t_1 t_1) eps) 2.0)))
   (if (<= x -1.1e-216)
     (/ (+ 1.0 (exp (* x (- -1.0 eps)))) 2.0)
     (if (<= x 16200.0)
       (/ (+ 1.0 (exp (* eps x))) 2.0)
       (if (<= x 1.35e+35)
         (/ (+ (+ 1.0 (/ 1.0 eps)) (- 1.0 (/ 1.0 eps))) 2.0)
         (if (<= x 4.2e+94)
           t_0
           (if (<= x 1.72e+164)
             t_2
             (if (<= x 5e+236)
               t_0
               (if (<= x 1.1e+287) t_2 (/ (/ (expm1 x) eps) 2.0))))))))))

C

double code(double x, double eps) {
	double t_0 = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	double t_1 = exp(-x);
	double t_2 = ((t_1 - t_1) / eps) / 2.0;
	double tmp;
	if (x <= -1.1e-216) {
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	} else if (x <= 16200.0) {
		tmp = (1.0 + exp((eps * x))) / 2.0;
	} else if (x <= 1.35e+35) {
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0;
	} else if (x <= 4.2e+94) {
		tmp = t_0;
	} else if (x <= 1.72e+164) {
		tmp = t_2;
	} else if (x <= 5e+236) {
		tmp = t_0;
	} else if (x <= 1.1e+287) {
		tmp = t_2;
	} else {
		tmp = (expm1(x) / eps) / 2.0;
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double t_0 = (1.0 + Math.exp((x * (eps + -1.0)))) / 2.0;
	double t_1 = Math.exp(-x);
	double t_2 = ((t_1 - t_1) / eps) / 2.0;
	double tmp;
	if (x <= -1.1e-216) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps)))) / 2.0;
	} else if (x <= 16200.0) {
		tmp = (1.0 + Math.exp((eps * x))) / 2.0;
	} else if (x <= 1.35e+35) {
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0;
	} else if (x <= 4.2e+94) {
		tmp = t_0;
	} else if (x <= 1.72e+164) {
		tmp = t_2;
	} else if (x <= 5e+236) {
		tmp = t_0;
	} else if (x <= 1.1e+287) {
		tmp = t_2;
	} else {
		tmp = (Math.expm1(x) / eps) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (1.0 + math.exp((x * (eps + -1.0)))) / 2.0
	t_1 = math.exp(-x)
	t_2 = ((t_1 - t_1) / eps) / 2.0
	tmp = 0
	if x <= -1.1e-216:
		tmp = (1.0 + math.exp((x * (-1.0 - eps)))) / 2.0
	elif x <= 16200.0:
		tmp = (1.0 + math.exp((eps * x))) / 2.0
	elif x <= 1.35e+35:
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0
	elif x <= 4.2e+94:
		tmp = t_0
	elif x <= 1.72e+164:
		tmp = t_2
	elif x <= 5e+236:
		tmp = t_0
	elif x <= 1.1e+287:
		tmp = t_2
	else:
		tmp = (math.expm1(x) / eps) / 2.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(1.0 + exp(Float64(x * Float64(eps + -1.0)))) / 2.0)
	t_1 = exp(Float64(-x))
	t_2 = Float64(Float64(Float64(t_1 - t_1) / eps) / 2.0)
	tmp = 0.0
	if (x <= -1.1e-216)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	elseif (x <= 16200.0)
		tmp = Float64(Float64(1.0 + exp(Float64(eps * x))) / 2.0);
	elseif (x <= 1.35e+35)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) + Float64(1.0 - Float64(1.0 / eps))) / 2.0);
	elseif (x <= 4.2e+94)
		tmp = t_0;
	elseif (x <= 1.72e+164)
		tmp = t_2;
	elseif (x <= 5e+236)
		tmp = t_0;
	elseif (x <= 1.1e+287)
		tmp = t_2;
	else
		tmp = Float64(Float64(expm1(x) / eps) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(1.0 + N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-x)], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$1 - t$95$1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -1.1e-216], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 16200.0], N[(N[(1.0 + N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.35e+35], N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 4.2e+94], t$95$0, If[LessEqual[x, 1.72e+164], t$95$2, If[LessEqual[x, 5e+236], t$95$0, If[LessEqual[x, 1.1e+287], t$95$2, N[(N[(N[(Exp[x] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < -1.09999999999999995e-216

   1. Initial program 64.9%

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

   3. Simplified 64.9%

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

   4. Taylor expanded in x around 0 38.7%

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

   5. Taylor expanded in eps around inf 73.2%

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

      1. cancel-sign-sub-inv 73.2%

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

      2. metadata-eval 73.2%

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

      3. *-lft-identity 73.2%

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

      4. mul-1-neg 73.2%

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

      5. distribute-rgt-neg-in 73.2%

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

   7. Simplified 73.2%

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

   if -1.09999999999999995e-216 < x < 16200

   1. Initial program 59.8%

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

   3. Simplified 59.8%

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

   4. Taylor expanded in x around 0 50.2%

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

   5. Taylor expanded in eps around inf 89.1%

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

      1. mul-1-neg 89.1%

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

      2. *-commutative 89.1%

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

      3. distribute-lft-neg-in 89.1%

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

   7. Simplified 89.1%

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

   8. Taylor expanded in eps around inf 89.5%

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

   if 16200 < x < 1.35000000000000001e35

   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%

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

   4. Taylor expanded in x around 0 2.5%

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

   5. Taylor expanded in x around 0 86.2%

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

   if 1.35000000000000001e35 < x < 4.19999999999999979e94 or 1.7200000000000001e164 < x < 4.9999999999999997e236

   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%

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

   4. Taylor expanded in x around 0 38.3%

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

   5. Taylor expanded in eps around inf 38.6%

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

      1. mul-1-neg 38.6%

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

      2. *-commutative 38.6%

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

      3. distribute-lft-neg-in 38.6%

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

   7. Simplified 38.6%

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

   if 4.19999999999999979e94 < x < 1.7200000000000001e164 or 4.9999999999999997e236 < x < 1.10000000000000002e287

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 72.4%

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

      1. inv-pow 72.4%

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

   6. Applied egg-rr 72.4%

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

      1. unpow-1 72.4%

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

      2. rec-exp 72.4%

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

   8. Simplified 72.4%

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

   if 1.10000000000000002e287 < 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%

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

   4. Taylor expanded in x around 0 100.0%

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

   5. Taylor expanded in eps around 0 1.0%

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

      1. expm1-def 1.0%

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

      2. mul-1-neg 1.0%

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

   7. Simplified 1.0%

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

      1. expm1-log1p-u 1.0%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}\right)\right)}}{2} \]

      2. expm1-udef 1.3%

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

      3. expm1-udef 1.3%

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

      4. expm1-udef 1.3%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 100.0%

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

      7. sqr-neg 100.0%

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

      8. sqrt-unprod 100.0%

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

      9. add-sqr-sqrt 100.0%

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

   9. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}\right)} - 1}}{2} \]
   10. Step-by-step derivation

       1. expm1-def 100.0%

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}\right)\right)}}{2} \]

       2. expm1-log1p 100.0%

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

   11. Simplified 100.0%

       \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}}{2} \]
3. Recombined 6 regimes into one program.

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-216}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 16200:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+35}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{+94}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{elif}\;x \leq 1.72 \cdot 10^{+164}:\\ \;\;\;\;\frac{\frac{e^{-x} - e^{-x}}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+236}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+287}:\\ \;\;\;\;\frac{\frac{e^{-x} - e^{-x}}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \end{array} \]

Alternative 5: 62.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -490:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 0.19:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+165} \lor \neg \left(x \leq 4.1 \cdot 10^{+222}\right) \land x \leq 1.25 \cdot 10^{+287}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -490.0)
   (/ (/ (expm1 (- x)) eps) 2.0)
   (if (<= x 0.19)
     1.0
     (if (or (<= x 2.25e+165) (and (not (<= x 4.1e+222)) (<= x 1.25e+287)))
       (/ (+ (+ 1.0 (/ 1.0 eps)) (- 1.0 (/ 1.0 eps))) 2.0)
       (/ (/ (expm1 x) eps) 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -490.0) {
		tmp = (expm1(-x) / eps) / 2.0;
	} else if (x <= 0.19) {
		tmp = 1.0;
	} else if ((x <= 2.25e+165) || (!(x <= 4.1e+222) && (x <= 1.25e+287))) {
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0;
	} else {
		tmp = (expm1(x) / eps) / 2.0;
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -490.0) {
		tmp = (Math.expm1(-x) / eps) / 2.0;
	} else if (x <= 0.19) {
		tmp = 1.0;
	} else if ((x <= 2.25e+165) || (!(x <= 4.1e+222) && (x <= 1.25e+287))) {
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0;
	} else {
		tmp = (Math.expm1(x) / eps) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -490.0:
		tmp = (math.expm1(-x) / eps) / 2.0
	elif x <= 0.19:
		tmp = 1.0
	elif (x <= 2.25e+165) or (not (x <= 4.1e+222) and (x <= 1.25e+287)):
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0
	else:
		tmp = (math.expm1(x) / eps) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -490.0)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps) / 2.0);
	elseif (x <= 0.19)
		tmp = 1.0;
	elseif ((x <= 2.25e+165) || (!(x <= 4.1e+222) && (x <= 1.25e+287)))
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) + Float64(1.0 - Float64(1.0 / eps))) / 2.0);
	else
		tmp = Float64(Float64(expm1(x) / eps) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -490.0], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 0.19], 1.0, If[Or[LessEqual[x, 2.25e+165], And[N[Not[LessEqual[x, 4.1e+222]], $MachinePrecision], LessEqual[x, 1.25e+287]]], N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(Exp[x] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -490

   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%

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

   4. Taylor expanded in x around 0 49.9%

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

   5. Taylor expanded in eps around 0 51.7%

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

      1. expm1-def 51.7%

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

      2. mul-1-neg 51.7%

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

   7. Simplified 51.7%

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

   if -490 < x < 0.19

   1. Initial program 54.6%

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

   3. Simplified 54.6%

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

   4. Taylor expanded in x around 0 76.1%

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

   if 0.19 < x < 2.2499999999999998e165 or 4.09999999999999987e222 < x < 1.25e287

   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%

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

   4. Taylor expanded in x around 0 20.2%

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

   5. Taylor expanded in x around 0 57.7%

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

   if 2.2499999999999998e165 < x < 4.09999999999999987e222 or 1.25e287 < 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%

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

   4. Taylor expanded in x around 0 53.5%

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

   5. Taylor expanded in eps around 0 1.7%

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

      1. expm1-def 1.7%

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

      2. mul-1-neg 1.7%

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

   7. Simplified 1.7%

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

      1. expm1-log1p-u 1.5%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}\right)\right)}}{2} \]

      2. expm1-udef 1.5%

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

      3. expm1-udef 1.5%

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

      4. expm1-udef 1.5%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 52.3%

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

      7. sqr-neg 52.3%

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

      8. sqrt-unprod 52.3%

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

      9. add-sqr-sqrt 52.3%

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

   9. Applied egg-rr 52.3%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}\right)} - 1}}{2} \]
   10. Step-by-step derivation

       1. expm1-def 52.3%

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}\right)\right)}}{2} \]

       2. expm1-log1p 52.5%

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

   11. Simplified 52.5%

       \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}}{2} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -490:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 0.19:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.25 \cdot 10^{+165} \lor \neg \left(x \leq 4.1 \cdot 10^{+222}\right) \land x \leq 1.25 \cdot 10^{+287}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \end{array} \]

Alternative 6: 67.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-216}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 1050:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 1.72 \cdot 10^{+164}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.1e-216)
   (/ (+ 1.0 (exp (* x (- -1.0 eps)))) 2.0)
   (if (<= x 1050.0)
     (/ (+ 1.0 (exp (* eps x))) 2.0)
     (if (<= x 1.72e+164)
       (/ (+ (+ 1.0 (/ 1.0 eps)) (- 1.0 (/ 1.0 eps))) 2.0)
       (/ (+ 1.0 (exp (* x (+ eps -1.0)))) 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.1e-216) {
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	} else if (x <= 1050.0) {
		tmp = (1.0 + exp((eps * x))) / 2.0;
	} else if (x <= 1.72e+164) {
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0;
	} else {
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-1.1d-216)) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps)))) / 2.0d0
    else if (x <= 1050.0d0) then
        tmp = (1.0d0 + exp((eps * x))) / 2.0d0
    else if (x <= 1.72d+164) then
        tmp = ((1.0d0 + (1.0d0 / eps)) + (1.0d0 - (1.0d0 / eps))) / 2.0d0
    else
        tmp = (1.0d0 + exp((x * (eps + (-1.0d0))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.1e-216) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps)))) / 2.0;
	} else if (x <= 1050.0) {
		tmp = (1.0 + Math.exp((eps * x))) / 2.0;
	} else if (x <= 1.72e+164) {
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0;
	} else {
		tmp = (1.0 + Math.exp((x * (eps + -1.0)))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.1e-216:
		tmp = (1.0 + math.exp((x * (-1.0 - eps)))) / 2.0
	elif x <= 1050.0:
		tmp = (1.0 + math.exp((eps * x))) / 2.0
	elif x <= 1.72e+164:
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0
	else:
		tmp = (1.0 + math.exp((x * (eps + -1.0)))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.1e-216)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	elseif (x <= 1050.0)
		tmp = Float64(Float64(1.0 + exp(Float64(eps * x))) / 2.0);
	elseif (x <= 1.72e+164)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) + Float64(1.0 - Float64(1.0 / eps))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps + -1.0)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.1e-216)
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	elseif (x <= 1050.0)
		tmp = (1.0 + exp((eps * x))) / 2.0;
	elseif (x <= 1.72e+164)
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0;
	else
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.1e-216], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1050.0], N[(N[(1.0 + N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.72e+164], N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.09999999999999995e-216

   1. Initial program 64.9%

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

   3. Simplified 64.9%

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

   4. Taylor expanded in x around 0 38.7%

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

   5. Taylor expanded in eps around inf 73.2%

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

      1. cancel-sign-sub-inv 73.2%

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

      2. metadata-eval 73.2%

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

      3. *-lft-identity 73.2%

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

      4. mul-1-neg 73.2%

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

      5. distribute-rgt-neg-in 73.2%

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

   7. Simplified 73.2%

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

   if -1.09999999999999995e-216 < x < 1050

   1. Initial program 59.8%

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

   3. Simplified 59.8%

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

   4. Taylor expanded in x around 0 50.2%

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

   5. Taylor expanded in eps around inf 89.1%

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

      1. mul-1-neg 89.1%

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

      2. *-commutative 89.1%

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

      3. distribute-lft-neg-in 89.1%

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

   7. Simplified 89.1%

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

   8. Taylor expanded in eps around inf 89.5%

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

   if 1050 < x < 1.7200000000000001e164

   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%

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

   4. Taylor expanded in x around 0 17.3%

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

   5. Taylor expanded in x around 0 58.9%

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

   if 1.7200000000000001e164 < 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%

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

   4. Taylor expanded in x around 0 38.0%

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

   5. Taylor expanded in eps around inf 38.4%

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

      1. mul-1-neg 38.4%

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

      2. *-commutative 38.4%

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

      3. distribute-lft-neg-in 38.4%

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

   7. Simplified 38.4%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-216}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 1050:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{elif}\;x \leq 1.72 \cdot 10^{+164}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]

Alternative 7: 58.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.19:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 3.25 \cdot 10^{+165} \lor \neg \left(x \leq 2.9 \cdot 10^{+222}\right) \land x \leq 1.5 \cdot 10^{+287}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 0.19)
   (/ (+ 2.0 (* x (- -1.0 eps))) 2.0)
   (if (or (<= x 3.25e+165) (and (not (<= x 2.9e+222)) (<= x 1.5e+287)))
     (/ (+ (+ 1.0 (/ 1.0 eps)) (- 1.0 (/ 1.0 eps))) 2.0)
     (/ (/ (expm1 x) eps) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 0.19) {
		tmp = (2.0 + (x * (-1.0 - eps))) / 2.0;
	} else if ((x <= 3.25e+165) || (!(x <= 2.9e+222) && (x <= 1.5e+287))) {
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0;
	} else {
		tmp = (expm1(x) / eps) / 2.0;
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 0.19) {
		tmp = (2.0 + (x * (-1.0 - eps))) / 2.0;
	} else if ((x <= 3.25e+165) || (!(x <= 2.9e+222) && (x <= 1.5e+287))) {
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0;
	} else {
		tmp = (Math.expm1(x) / eps) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 0.19:
		tmp = (2.0 + (x * (-1.0 - eps))) / 2.0
	elif (x <= 3.25e+165) or (not (x <= 2.9e+222) and (x <= 1.5e+287)):
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0
	else:
		tmp = (math.expm1(x) / eps) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 0.19)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 - eps))) / 2.0);
	elseif ((x <= 3.25e+165) || (!(x <= 2.9e+222) && (x <= 1.5e+287)))
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) + Float64(1.0 - Float64(1.0 / eps))) / 2.0);
	else
		tmp = Float64(Float64(expm1(x) / eps) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 0.19], N[(N[(2.0 + N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 3.25e+165], And[N[Not[LessEqual[x, 2.9e+222]], $MachinePrecision], LessEqual[x, 1.5e+287]]], N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(Exp[x] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.19

   1. Initial program 62.0%

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

   3. Simplified 62.0%

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

   4. Taylor expanded in x around 0 42.2%

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

   5. Taylor expanded in eps around inf 79.3%

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

      1. cancel-sign-sub-inv 79.3%

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

      2. metadata-eval 79.3%

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

      3. *-lft-identity 79.3%

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

      4. mul-1-neg 79.3%

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

      5. distribute-rgt-neg-in 79.3%

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

   7. Simplified 79.3%

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

   8. Taylor expanded in x around 0 67.0%

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

   if 0.19 < x < 3.2499999999999999e165 or 2.89999999999999981e222 < x < 1.4999999999999999e287

   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%

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

   4. Taylor expanded in x around 0 20.2%

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

   5. Taylor expanded in x around 0 57.7%

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

   if 3.2499999999999999e165 < x < 2.89999999999999981e222 or 1.4999999999999999e287 < 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%

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

   4. Taylor expanded in x around 0 53.5%

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

   5. Taylor expanded in eps around 0 1.7%

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

      1. expm1-def 1.7%

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

      2. mul-1-neg 1.7%

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

   7. Simplified 1.7%

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

      1. expm1-log1p-u 1.5%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}\right)\right)}}{2} \]

      2. expm1-udef 1.5%

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

      3. expm1-udef 1.5%

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

      4. expm1-udef 1.5%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 52.3%

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

      7. sqr-neg 52.3%

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

      8. sqrt-unprod 52.3%

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

      9. add-sqr-sqrt 52.3%

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

   9. Applied egg-rr 52.3%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}\right)} - 1}}{2} \]
   10. Step-by-step derivation

       1. expm1-def 52.3%

          \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}\right)\right)}}{2} \]

       2. expm1-log1p 52.5%

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

   11. Simplified 52.5%

       \[\leadsto \frac{\color{blue}{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}}{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.19:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 3.25 \cdot 10^{+165} \lor \neg \left(x \leq 2.9 \cdot 10^{+222}\right) \land x \leq 1.5 \cdot 10^{+287}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \end{array} \]

Alternative 8: 67.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -420:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 3350 \lor \neg \left(x \leq 1.72 \cdot 10^{+164}\right):\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -420.0)
   (/ (/ (expm1 (- x)) eps) 2.0)
   (if (or (<= x 3350.0) (not (<= x 1.72e+164)))
     (/ (+ 1.0 (exp (* eps x))) 2.0)
     (/ (+ (+ 1.0 (/ 1.0 eps)) (- 1.0 (/ 1.0 eps))) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -420.0) {
		tmp = (expm1(-x) / eps) / 2.0;
	} else if ((x <= 3350.0) || !(x <= 1.72e+164)) {
		tmp = (1.0 + exp((eps * x))) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0;
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -420.0) {
		tmp = (Math.expm1(-x) / eps) / 2.0;
	} else if ((x <= 3350.0) || !(x <= 1.72e+164)) {
		tmp = (1.0 + Math.exp((eps * x))) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -420.0:
		tmp = (math.expm1(-x) / eps) / 2.0
	elif (x <= 3350.0) or not (x <= 1.72e+164):
		tmp = (1.0 + math.exp((eps * x))) / 2.0
	else:
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -420.0)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps) / 2.0);
	elseif ((x <= 3350.0) || !(x <= 1.72e+164))
		tmp = Float64(Float64(1.0 + exp(Float64(eps * x))) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) + Float64(1.0 - Float64(1.0 / eps))) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -420.0], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 3350.0], N[Not[LessEqual[x, 1.72e+164]], $MachinePrecision]], N[(N[(1.0 + N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -420

   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%

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

   4. Taylor expanded in x around 0 49.9%

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

   5. Taylor expanded in eps around 0 51.7%

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

      1. expm1-def 51.7%

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

      2. mul-1-neg 51.7%

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

   7. Simplified 51.7%

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

   if -420 < x < 3350 or 1.7200000000000001e164 < x

   1. Initial program 65.1%

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

   3. Simplified 65.1%

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

   4. Taylor expanded in x around 0 43.9%

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

   5. Taylor expanded in eps around inf 78.0%

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

      1. mul-1-neg 78.0%

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

      2. *-commutative 78.0%

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

      3. distribute-lft-neg-in 78.0%

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

   7. Simplified 78.0%

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

   8. Taylor expanded in eps around inf 78.4%

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

   if 3350 < x < 1.7200000000000001e164

   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%

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

   4. Taylor expanded in x around 0 17.3%

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

   5. Taylor expanded in x around 0 58.9%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -420:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 3350 \lor \neg \left(x \leq 1.72 \cdot 10^{+164}\right):\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \end{array} \]

Alternative 9: 67.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-216}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 12400 \lor \neg \left(x \leq 2.25 \cdot 10^{+164}\right):\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.1e-216)
   (/ (+ 1.0 (exp (* x (- -1.0 eps)))) 2.0)
   (if (or (<= x 12400.0) (not (<= x 2.25e+164)))
     (/ (+ 1.0 (exp (* eps x))) 2.0)
     (/ (+ (+ 1.0 (/ 1.0 eps)) (- 1.0 (/ 1.0 eps))) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.1e-216) {
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	} else if ((x <= 12400.0) || !(x <= 2.25e+164)) {
		tmp = (1.0 + exp((eps * x))) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-1.1d-216)) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps)))) / 2.0d0
    else if ((x <= 12400.0d0) .or. (.not. (x <= 2.25d+164))) then
        tmp = (1.0d0 + exp((eps * x))) / 2.0d0
    else
        tmp = ((1.0d0 + (1.0d0 / eps)) + (1.0d0 - (1.0d0 / eps))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.1e-216) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps)))) / 2.0;
	} else if ((x <= 12400.0) || !(x <= 2.25e+164)) {
		tmp = (1.0 + Math.exp((eps * x))) / 2.0;
	} else {
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.1e-216:
		tmp = (1.0 + math.exp((x * (-1.0 - eps)))) / 2.0
	elif (x <= 12400.0) or not (x <= 2.25e+164):
		tmp = (1.0 + math.exp((eps * x))) / 2.0
	else:
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.1e-216)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	elseif ((x <= 12400.0) || !(x <= 2.25e+164))
		tmp = Float64(Float64(1.0 + exp(Float64(eps * x))) / 2.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) + Float64(1.0 - Float64(1.0 / eps))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.1e-216)
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	elseif ((x <= 12400.0) || ~((x <= 2.25e+164)))
		tmp = (1.0 + exp((eps * x))) / 2.0;
	else
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.1e-216], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 12400.0], N[Not[LessEqual[x, 2.25e+164]], $MachinePrecision]], N[(N[(1.0 + N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.09999999999999995e-216

   1. Initial program 64.9%

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

   3. Simplified 64.9%

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

   4. Taylor expanded in x around 0 38.7%

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

   5. Taylor expanded in eps around inf 73.2%

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

      1. cancel-sign-sub-inv 73.2%

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

      2. metadata-eval 73.2%

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

      3. *-lft-identity 73.2%

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

      4. mul-1-neg 73.2%

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

      5. distribute-rgt-neg-in 73.2%

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

   7. Simplified 73.2%

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

   if -1.09999999999999995e-216 < x < 12400 or 2.24999999999999988e164 < x

   1. Initial program 72.6%

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

   3. Simplified 72.6%

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

   4. Taylor expanded in x around 0 46.3%

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

   5. Taylor expanded in eps around inf 72.9%

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

      1. mul-1-neg 72.9%

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

      2. *-commutative 72.9%

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

      3. distribute-lft-neg-in 72.9%

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

   7. Simplified 72.9%

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

   8. Taylor expanded in eps around inf 73.2%

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

   if 12400 < x < 2.24999999999999988e164

   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%

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

   4. Taylor expanded in x around 0 17.3%

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

   5. Taylor expanded in x around 0 58.9%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-216}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 12400 \lor \neg \left(x \leq 2.25 \cdot 10^{+164}\right):\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \end{array} \]

Alternative 10: 58.9% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.19:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{+194} \lor \neg \left(x \leq 1.9 \cdot 10^{+222}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 0.19)
   (/ (+ 2.0 (* x (- -1.0 eps))) 2.0)
   (if (or (<= x 2.75e+194) (not (<= x 1.9e+222)))
     (/ (+ (+ 1.0 (/ 1.0 eps)) (- 1.0 (/ 1.0 eps))) 2.0)
     (/ (+ 2.0 (* eps x)) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 0.19) {
		tmp = (2.0 + (x * (-1.0 - eps))) / 2.0;
	} else if ((x <= 2.75e+194) || !(x <= 1.9e+222)) {
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0;
	} else {
		tmp = (2.0 + (eps * x)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 0.19d0) then
        tmp = (2.0d0 + (x * ((-1.0d0) - eps))) / 2.0d0
    else if ((x <= 2.75d+194) .or. (.not. (x <= 1.9d+222))) then
        tmp = ((1.0d0 + (1.0d0 / eps)) + (1.0d0 - (1.0d0 / eps))) / 2.0d0
    else
        tmp = (2.0d0 + (eps * x)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 0.19) {
		tmp = (2.0 + (x * (-1.0 - eps))) / 2.0;
	} else if ((x <= 2.75e+194) || !(x <= 1.9e+222)) {
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0;
	} else {
		tmp = (2.0 + (eps * x)) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 0.19:
		tmp = (2.0 + (x * (-1.0 - eps))) / 2.0
	elif (x <= 2.75e+194) or not (x <= 1.9e+222):
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0
	else:
		tmp = (2.0 + (eps * x)) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 0.19)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 - eps))) / 2.0);
	elseif ((x <= 2.75e+194) || !(x <= 1.9e+222))
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) + Float64(1.0 - Float64(1.0 / eps))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(eps * x)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 0.19)
		tmp = (2.0 + (x * (-1.0 - eps))) / 2.0;
	elseif ((x <= 2.75e+194) || ~((x <= 1.9e+222)))
		tmp = ((1.0 + (1.0 / eps)) + (1.0 - (1.0 / eps))) / 2.0;
	else
		tmp = (2.0 + (eps * x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 0.19], N[(N[(2.0 + N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 2.75e+194], N[Not[LessEqual[x, 1.9e+222]], $MachinePrecision]], N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] + N[(1.0 - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.19

   1. Initial program 62.0%

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

   3. Simplified 62.0%

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

   4. Taylor expanded in x around 0 42.2%

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

   5. Taylor expanded in eps around inf 79.3%

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

      1. cancel-sign-sub-inv 79.3%

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

      2. metadata-eval 79.3%

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

      3. *-lft-identity 79.3%

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

      4. mul-1-neg 79.3%

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

      5. distribute-rgt-neg-in 79.3%

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

   7. Simplified 79.3%

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

   8. Taylor expanded in x around 0 67.0%

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

   if 0.19 < x < 2.75e194 or 1.90000000000000009e222 < 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%

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

   4. Taylor expanded in x around 0 26.2%

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

   5. Taylor expanded in x around 0 51.6%

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

   if 2.75e194 < x < 1.90000000000000009e222

   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%

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

   4. Taylor expanded in x around 0 63.7%

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

   5. Taylor expanded in eps around inf 63.7%

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

      1. mul-1-neg 63.7%

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

      2. *-commutative 63.7%

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

      3. distribute-lft-neg-in 63.7%

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

   7. Simplified 63.7%

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

   8. Taylor expanded in eps around inf 63.7%

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

   9. Taylor expanded in eps around 0 27.7%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.19:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{+194} \lor \neg \left(x \leq 1.9 \cdot 10^{+222}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \left(1 - \frac{1}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \end{array} \]

Alternative 11: 59.0% accurate, 13.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.19:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+193} \lor \neg \left(x \leq 2.8 \cdot 10^{+220}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \frac{-1}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 0.19)
   (/ (+ 2.0 (* x (- -1.0 eps))) 2.0)
   (if (or (<= x 1.1e+193) (not (<= x 2.8e+220)))
     (/ (+ (+ 1.0 (/ 1.0 eps)) (/ -1.0 eps)) 2.0)
     (/ (+ 2.0 (* eps x)) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 0.19) {
		tmp = (2.0 + (x * (-1.0 - eps))) / 2.0;
	} else if ((x <= 1.1e+193) || !(x <= 2.8e+220)) {
		tmp = ((1.0 + (1.0 / eps)) + (-1.0 / eps)) / 2.0;
	} else {
		tmp = (2.0 + (eps * x)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 0.19d0) then
        tmp = (2.0d0 + (x * ((-1.0d0) - eps))) / 2.0d0
    else if ((x <= 1.1d+193) .or. (.not. (x <= 2.8d+220))) then
        tmp = ((1.0d0 + (1.0d0 / eps)) + ((-1.0d0) / eps)) / 2.0d0
    else
        tmp = (2.0d0 + (eps * x)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 0.19) {
		tmp = (2.0 + (x * (-1.0 - eps))) / 2.0;
	} else if ((x <= 1.1e+193) || !(x <= 2.8e+220)) {
		tmp = ((1.0 + (1.0 / eps)) + (-1.0 / eps)) / 2.0;
	} else {
		tmp = (2.0 + (eps * x)) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 0.19:
		tmp = (2.0 + (x * (-1.0 - eps))) / 2.0
	elif (x <= 1.1e+193) or not (x <= 2.8e+220):
		tmp = ((1.0 + (1.0 / eps)) + (-1.0 / eps)) / 2.0
	else:
		tmp = (2.0 + (eps * x)) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 0.19)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 - eps))) / 2.0);
	elseif ((x <= 1.1e+193) || !(x <= 2.8e+220))
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) + Float64(-1.0 / eps)) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(eps * x)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 0.19)
		tmp = (2.0 + (x * (-1.0 - eps))) / 2.0;
	elseif ((x <= 1.1e+193) || ~((x <= 2.8e+220)))
		tmp = ((1.0 + (1.0 / eps)) + (-1.0 / eps)) / 2.0;
	else
		tmp = (2.0 + (eps * x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 0.19], N[(N[(2.0 + N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[x, 1.1e+193], N[Not[LessEqual[x, 2.8e+220]], $MachinePrecision]], N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.19

   1. Initial program 62.0%

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

   3. Simplified 62.0%

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

   4. Taylor expanded in x around 0 42.2%

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

   5. Taylor expanded in eps around inf 79.3%

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

      1. cancel-sign-sub-inv 79.3%

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

      2. metadata-eval 79.3%

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

      3. *-lft-identity 79.3%

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

      4. mul-1-neg 79.3%

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

      5. distribute-rgt-neg-in 79.3%

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

   7. Simplified 79.3%

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

   8. Taylor expanded in x around 0 67.0%

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

   if 0.19 < x < 1.09999999999999993e193 or 2.8000000000000001e220 < 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%

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

   4. Taylor expanded in x around 0 26.2%

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

   5. Taylor expanded in x around 0 51.6%

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

   6. Taylor expanded in eps around 0 51.6%

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

   if 1.09999999999999993e193 < x < 2.8000000000000001e220

   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%

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

   4. Taylor expanded in x around 0 63.7%

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

   5. Taylor expanded in eps around inf 63.7%

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

      1. mul-1-neg 63.7%

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

      2. *-commutative 63.7%

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

      3. distribute-lft-neg-in 63.7%

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

   7. Simplified 63.7%

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

   8. Taylor expanded in eps around inf 63.7%

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

   9. Taylor expanded in eps around 0 27.7%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.19:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+193} \lor \neg \left(x \leq 2.8 \cdot 10^{+220}\right):\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + \frac{-1}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \end{array} \]

Alternative 12: 47.5% accurate, 25.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00064:\\ \;\;\;\;\frac{x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -0.00064) (/ (* x (- -1.0 eps)) 2.0) 1.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -0.00064) {
		tmp = (x * (-1.0 - eps)) / 2.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-0.00064d0)) then
        tmp = (x * ((-1.0d0) - eps)) / 2.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -0.00064) {
		tmp = (x * (-1.0 - eps)) / 2.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -0.00064:
		tmp = (x * (-1.0 - eps)) / 2.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -0.00064)
		tmp = Float64(Float64(x * Float64(-1.0 - eps)) / 2.0);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -0.00064)
		tmp = (x * (-1.0 - eps)) / 2.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -0.00064], N[(N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -6.40000000000000052e-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

   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 50.1%

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

   5. Taylor expanded in eps around inf 50.1%

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

      1. cancel-sign-sub-inv 50.1%

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

      2. metadata-eval 50.1%

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

      3. *-lft-identity 50.1%

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

      4. mul-1-neg 50.1%

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

      5. distribute-rgt-neg-in 50.1%

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

   7. Simplified 50.1%

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

   8. Taylor expanded in x around 0 23.1%

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

   9. Taylor expanded in x around inf 23.1%

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

       1. neg-mul-1 23.1%

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

       2. distribute-rgt-neg-in 23.1%

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

       3. distribute-neg-in 23.1%

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

       4. metadata-eval 23.1%

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

       5. unsub-neg 23.1%

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

   11. Simplified 23.1%

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

   if -6.40000000000000052e-4 < x

   1. Initial program 69.7%

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

   3. Simplified 69.7%

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

   4. Taylor expanded in x around 0 51.9%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00064:\\ \;\;\;\;\frac{x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13: 50.1% accurate, 25.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00064:\\ \;\;\;\;\frac{x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -0.00064) (/ (* x (- -1.0 eps)) 2.0) (/ (+ 2.0 (* eps x)) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -0.00064) {
		tmp = (x * (-1.0 - eps)) / 2.0;
	} else {
		tmp = (2.0 + (eps * x)) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-0.00064d0)) then
        tmp = (x * ((-1.0d0) - eps)) / 2.0d0
    else
        tmp = (2.0d0 + (eps * x)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -0.00064) {
		tmp = (x * (-1.0 - eps)) / 2.0;
	} else {
		tmp = (2.0 + (eps * x)) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -0.00064:
		tmp = (x * (-1.0 - eps)) / 2.0
	else:
		tmp = (2.0 + (eps * x)) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -0.00064)
		tmp = Float64(Float64(x * Float64(-1.0 - eps)) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(eps * x)) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -0.00064)
		tmp = (x * (-1.0 - eps)) / 2.0;
	else
		tmp = (2.0 + (eps * x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -0.00064], N[(N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.40000000000000052e-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

   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 50.1%

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

   5. Taylor expanded in eps around inf 50.1%

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

      1. cancel-sign-sub-inv 50.1%

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

      2. metadata-eval 50.1%

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

      3. *-lft-identity 50.1%

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

      4. mul-1-neg 50.1%

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

      5. distribute-rgt-neg-in 50.1%

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

   7. Simplified 50.1%

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

   8. Taylor expanded in x around 0 23.1%

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

   9. Taylor expanded in x around inf 23.1%

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

       1. neg-mul-1 23.1%

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

       2. distribute-rgt-neg-in 23.1%

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

       3. distribute-neg-in 23.1%

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

       4. metadata-eval 23.1%

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

       5. unsub-neg 23.1%

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

   11. Simplified 23.1%

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

   if -6.40000000000000052e-4 < x

   1. Initial program 69.7%

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

   3. Simplified 69.7%

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

   4. Taylor expanded in x around 0 39.4%

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

   5. Taylor expanded in eps around inf 69.1%

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

      1. mul-1-neg 69.1%

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

      2. *-commutative 69.1%

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

      3. distribute-lft-neg-in 69.1%

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

   7. Simplified 69.1%

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

   8. Taylor expanded in eps around inf 69.4%

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

   9. Taylor expanded in eps around 0 55.5%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00064:\\ \;\;\;\;\frac{x \cdot \left(-1 - \varepsilon\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + \varepsilon \cdot x}{2}\\ \end{array} \]

Alternative 14: 47.5% accurate, 28.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00064:\\ \;\;\;\;\frac{\varepsilon \cdot \left(-x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -0.00064) (/ (* eps (- x)) 2.0) 1.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -0.00064) {
		tmp = (eps * -x) / 2.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-0.00064d0)) then
        tmp = (eps * -x) / 2.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -0.00064) {
		tmp = (eps * -x) / 2.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -0.00064:
		tmp = (eps * -x) / 2.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -0.00064)
		tmp = Float64(Float64(eps * Float64(-x)) / 2.0);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -0.00064)
		tmp = (eps * -x) / 2.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -0.00064], N[(N[(eps * (-x)), $MachinePrecision] / 2.0), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -6.40000000000000052e-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

   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 50.1%

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

   5. Taylor expanded in eps around inf 50.1%

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

      1. cancel-sign-sub-inv 50.1%

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

      2. metadata-eval 50.1%

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

      3. *-lft-identity 50.1%

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

      4. mul-1-neg 50.1%

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

      5. distribute-rgt-neg-in 50.1%

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

   7. Simplified 50.1%

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

   8. Taylor expanded in x around 0 23.1%

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

   9. Taylor expanded in eps around inf 23.1%

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

       1. neg-mul-1 23.1%

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

       2. distribute-rgt-neg-in 23.1%

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

   11. Simplified 23.1%

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

   if -6.40000000000000052e-4 < x

   1. Initial program 69.7%

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

   3. Simplified 69.7%

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

   4. Taylor expanded in x around 0 51.9%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00064:\\ \;\;\;\;\frac{\varepsilon \cdot \left(-x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 15: 44.1% accurate, 227.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 1.0)

C

double code(double x, double eps) {
	return 1.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 1.0d0
end function

Java

public static double code(double x, double eps) {
	return 1.0;
}

Python

def code(x, eps):
	return 1.0

Julia

function code(x, eps)
	return 1.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 1.0;
end

Wolfram

code[x_, eps_] := 1.0

Derivation

1. Initial program 73.6%

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

3. Simplified 73.6%

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

4. Taylor expanded in x around 0 45.6%

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

5. Final simplification 45.6%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023331 
(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))