NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.2% → 99.7%

Time: 14.7s

Alternatives: 20

Speedup: 1.1×

• 38.2% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + \frac{1}{\varepsilon}\right) \cdot e^{x \cdot \left(\varepsilon + -1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)} \cdot \left(1 + \frac{-1}{\varepsilon}\right)\\ t_1 := e^{-x}\\ t_2 := t\_1 + x \cdot t\_1\\ \mathbf{if}\;t\_0 \leq 2:\\ \;\;\;\;\frac{t\_2 + t\_2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2}\\ \end{array} \end{array} \]

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)))
        (t_2 (+ t_1 (* x t_1))))
   (if (<= t_0 2.0) (/ (+ t_2 t_2) 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 t_2 = t_1 + (x * t_1);
	double tmp;
	if (t_0 <= 2.0) {
		tmp = (t_2 + t_2) / 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) :: t_2
    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)
    t_2 = t_1 + (x * t_1)
    if (t_0 <= 2.0d0) then
        tmp = (t_2 + t_2) / 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 t_2 = t_1 + (x * t_1);
	double tmp;
	if (t_0 <= 2.0) {
		tmp = (t_2 + t_2) / 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)
	t_2 = t_1 + (x * t_1)
	tmp = 0
	if t_0 <= 2.0:
		tmp = (t_2 + t_2) / 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))
	t_2 = Float64(t_1 + Float64(x * t_1))
	tmp = 0.0
	if (t_0 <= 2.0)
		tmp = Float64(Float64(t_2 + t_2) / 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);
	t_2 = t_1 + (x * t_1);
	tmp = 0.0;
	if (t_0 <= 2.0)
		tmp = (t_2 + t_2) / 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]}, Block[{t$95$2 = N[(t$95$1 + N[(x * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2.0], N[(N[(t$95$2 + t$95$2), $MachinePrecision] / 2.0), $MachinePrecision], N[(t$95$0 / 2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 49.3%

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

   3. Simplified 49.3%

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

   5. Taylor expanded in eps around 0 100.0%

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

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

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 49.3%

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

   3. Simplified 29.4%

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

   5. Taylor expanded in eps around 0 48.4%

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

      1. associate-+r+ 100.0%

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

      2. mul-1-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-inverses 100.0%

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

      5. distribute-lft-out 100.0%

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

      6. distribute-rgt1-in 100.0%

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

      7. mul-1-neg 100.0%

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

   7. Simplified 100.0%

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

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

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. neg-mul-1 100.0%

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

      3. distribute-rgt-in 100.0%

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

      4. unsub-neg 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 3: 79.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 15:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(2 \cdot \left(e^{-x} \cdot \left(1 + x\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} + \frac{1}{e^{x + \varepsilon \cdot x}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 15.0)
   (/ (/ (* eps (* 2.0 (* (exp (- x)) (+ 1.0 x)))) eps) 2.0)
   (/ (+ (exp (* eps x)) (/ 1.0 (exp (+ x (* eps x))))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 15.0) {
		tmp = ((eps * (2.0 * (exp(-x) * (1.0 + x)))) / eps) / 2.0;
	} else {
		tmp = (exp((eps * x)) + (1.0 / exp((x + (eps * x))))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 15.0d0) then
        tmp = ((eps * (2.0d0 * (exp(-x) * (1.0d0 + x)))) / eps) / 2.0d0
    else
        tmp = (exp((eps * x)) + (1.0d0 / exp((x + (eps * x))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= 15.0) {
		tmp = ((eps * (2.0 * (Math.exp(-x) * (1.0 + x)))) / eps) / 2.0;
	} else {
		tmp = (Math.exp((eps * x)) + (1.0 / Math.exp((x + (eps * x))))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 15.0:
		tmp = ((eps * (2.0 * (math.exp(-x) * (1.0 + x)))) / eps) / 2.0
	else:
		tmp = (math.exp((eps * x)) + (1.0 / math.exp((x + (eps * x))))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 15.0)
		tmp = Float64(Float64(Float64(eps * Float64(2.0 * Float64(exp(Float64(-x)) * Float64(1.0 + x)))) / eps) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(eps * x)) + Float64(1.0 / exp(Float64(x + Float64(eps * x))))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 15.0)
		tmp = ((eps * (2.0 * (exp(-x) * (1.0 + x)))) / eps) / 2.0;
	else
		tmp = (exp((eps * x)) + (1.0 / exp((x + (eps * x))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 15.0], N[(N[(N[(eps * N[(2.0 * N[(N[Exp[(-x)], $MachinePrecision] * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[Exp[N[(x + N[(eps * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 15

   1. Initial program 61.0%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in eps around 0 31.1%

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

      1. associate-+r+ 70.8%

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

      2. mul-1-neg 70.8%

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

      3. sub-neg 70.8%

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

      4. +-inverses 70.8%

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

      5. distribute-lft-out 70.8%

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

      6. distribute-rgt1-in 70.8%

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

      7. mul-1-neg 70.8%

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

   7. Simplified 70.8%

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

   if 15 < eps

   1. Initial program 100.0%

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

   3. Simplified 83.3%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 15:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(2 \cdot \left(e^{-x} \cdot \left(1 + x\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} + \frac{1}{e^{x + \varepsilon \cdot x}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 15:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(2 \cdot \left(e^{-x} \cdot \left(1 + x\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)} + e^{\varepsilon \cdot x}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 15.0)
   (/ (/ (* eps (* 2.0 (* (exp (- x)) (+ 1.0 x)))) eps) 2.0)
   (/ (+ (exp (* x (- -1.0 eps))) (exp (* eps x))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 15.0) {
		tmp = ((eps * (2.0 * (exp(-x) * (1.0 + x)))) / eps) / 2.0;
	} else {
		tmp = (exp((x * (-1.0 - eps))) + exp((eps * x))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 15.0d0) then
        tmp = ((eps * (2.0d0 * (exp(-x) * (1.0d0 + x)))) / eps) / 2.0d0
    else
        tmp = (exp((x * ((-1.0d0) - eps))) + exp((eps * x))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= 15.0) {
		tmp = ((eps * (2.0 * (Math.exp(-x) * (1.0 + x)))) / eps) / 2.0;
	} else {
		tmp = (Math.exp((x * (-1.0 - eps))) + Math.exp((eps * x))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 15.0:
		tmp = ((eps * (2.0 * (math.exp(-x) * (1.0 + x)))) / eps) / 2.0
	else:
		tmp = (math.exp((x * (-1.0 - eps))) + math.exp((eps * x))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 15.0)
		tmp = Float64(Float64(Float64(eps * Float64(2.0 * Float64(exp(Float64(-x)) * Float64(1.0 + x)))) / eps) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 - eps))) + exp(Float64(eps * x))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 15.0)
		tmp = ((eps * (2.0 * (exp(-x) * (1.0 + x)))) / eps) / 2.0;
	else
		tmp = (exp((x * (-1.0 - eps))) + exp((eps * x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 15.0], N[(N[(N[(eps * N[(2.0 * N[(N[Exp[(-x)], $MachinePrecision] * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 15

   1. Initial program 61.0%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in eps around 0 31.1%

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

      1. associate-+r+ 70.8%

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

      2. mul-1-neg 70.8%

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

      3. sub-neg 70.8%

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

      4. +-inverses 70.8%

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

      5. distribute-lft-out 70.8%

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

      6. distribute-rgt1-in 70.8%

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

      7. mul-1-neg 70.8%

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

   7. Simplified 70.8%

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

   if 15 < eps

   1. Initial program 100.0%

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

   3. Simplified 83.3%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around -inf 100.0%

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

       1. cancel-sign-sub-inv 100.0%

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

       2. exp-sum 100.0%

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

       3. metadata-eval 100.0%

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

       4. *-lft-identity 100.0%

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

       5. exp-sum 100.0%

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

       6. *-commutative 100.0%

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

       7. fma-undefine 100.0%

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

       8. *-rgt-identity 100.0%

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

       9. exp-neg 100.0%

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

       10. *-rgt-identity 100.0%

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

       11. neg-mul-1 100.0%

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

       12. fma-undefine 100.0%

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

       13. *-commutative 100.0%

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

       14. +-commutative 100.0%

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

       15. distribute-lft-in 100.0%

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

       16. associate-*r* 100.0%

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

       17. neg-mul-1 100.0%

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

       18. distribute-rgt-in 100.0%

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

       19. unsub-neg 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 15:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(2 \cdot \left(e^{-x} \cdot \left(1 + x\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)} + e^{\varepsilon \cdot x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-244}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)} + \left(1 + \varepsilon \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} + \frac{1}{1 + x \cdot \left(1 + \varepsilon\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.6e+77)
   (/
    (/
     (*
      x
      (+
       -1.0
       (* x (+ 0.5 (* x (- (* x 0.041666666666666664) 0.16666666666666666))))))
     eps)
    2.0)
   (if (<= x -1.05e-244)
     (/ (+ (exp (* x (- -1.0 eps))) (+ 1.0 (* eps x))) 2.0)
     (/ (+ (exp (* eps x)) (/ 1.0 (+ 1.0 (* x (+ 1.0 eps))))) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.6e+77) {
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps) / 2.0;
	} else if (x <= -1.05e-244) {
		tmp = (exp((x * (-1.0 - eps))) + (1.0 + (eps * x))) / 2.0;
	} else {
		tmp = (exp((eps * x)) + (1.0 / (1.0 + (x * (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 <= (-2.6d+77)) then
        tmp = ((x * ((-1.0d0) + (x * (0.5d0 + (x * ((x * 0.041666666666666664d0) - 0.16666666666666666d0)))))) / eps) / 2.0d0
    else if (x <= (-1.05d-244)) then
        tmp = (exp((x * ((-1.0d0) - eps))) + (1.0d0 + (eps * x))) / 2.0d0
    else
        tmp = (exp((eps * x)) + (1.0d0 / (1.0d0 + (x * (1.0d0 + eps))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.6e+77) {
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps) / 2.0;
	} else if (x <= -1.05e-244) {
		tmp = (Math.exp((x * (-1.0 - eps))) + (1.0 + (eps * x))) / 2.0;
	} else {
		tmp = (Math.exp((eps * x)) + (1.0 / (1.0 + (x * (1.0 + eps))))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.6e+77:
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps) / 2.0
	elif x <= -1.05e-244:
		tmp = (math.exp((x * (-1.0 - eps))) + (1.0 + (eps * x))) / 2.0
	else:
		tmp = (math.exp((eps * x)) + (1.0 / (1.0 + (x * (1.0 + eps))))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.6e+77)
		tmp = Float64(Float64(Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(Float64(x * 0.041666666666666664) - 0.16666666666666666)))))) / eps) / 2.0);
	elseif (x <= -1.05e-244)
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 - eps))) + Float64(1.0 + Float64(eps * x))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(eps * x)) + Float64(1.0 / Float64(1.0 + Float64(x * Float64(1.0 + eps))))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.6e+77)
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps) / 2.0;
	elseif (x <= -1.05e-244)
		tmp = (exp((x * (-1.0 - eps))) + (1.0 + (eps * x))) / 2.0;
	else
		tmp = (exp((eps * x)) + (1.0 / (1.0 + (x * (1.0 + eps))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.6e+77], N[(N[(N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(x * N[(N[(x * 0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, -1.05e-244], N[(N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 + N[(eps * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(1.0 + N[(x * N[(1.0 + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.6000000000000002e77

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 51.6%

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

   6. Taylor expanded in eps around 0 50.0%

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

      1. expm1-define 50.0%

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

      2. neg-mul-1 50.0%

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

   8. Simplified 50.0%

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

   9. Taylor expanded in x around 0 50.0%

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

   if -2.6000000000000002e77 < x < -1.05000000000000001e-244

   1. Initial program 55.6%

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

   3. Simplified 47.6%

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

   5. Taylor expanded in eps around inf 94.8%

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

   6. Taylor expanded in eps around inf 94.9%

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

      1. *-commutative 94.9%

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

   8. Simplified 94.9%

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

   9. Taylor expanded in x around -inf 94.9%

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

       1. cancel-sign-sub-inv 94.9%

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

       2. exp-sum 86.9%

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

       3. metadata-eval 86.9%

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

       4. *-lft-identity 86.9%

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

       5. exp-sum 94.9%

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

       6. *-commutative 94.9%

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

       7. fma-undefine 94.9%

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

       8. *-rgt-identity 94.9%

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

       9. exp-neg 94.9%

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

       10. *-rgt-identity 94.9%

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

       11. neg-mul-1 94.9%

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

       12. fma-undefine 94.9%

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

       13. *-commutative 94.9%

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

       14. +-commutative 94.9%

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

       15. distribute-lft-in 94.9%

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

       16. associate-*r* 94.9%

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

       17. neg-mul-1 94.9%

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

       18. distribute-rgt-in 94.9%

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

       19. unsub-neg 94.9%

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

   11. Simplified 94.9%

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

   12. Taylor expanded in x around 0 76.4%

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

   if -1.05000000000000001e-244 < x

   1. Initial program 74.7%

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

   3. Simplified 67.5%

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

   5. Taylor expanded in eps around inf 99.3%

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

   6. Taylor expanded in eps around inf 86.2%

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

      1. *-commutative 86.2%

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

   8. Simplified 86.2%

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

   9. Taylor expanded in x around 0 64.0%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-244}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)} + \left(1 + \varepsilon \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} + \frac{1}{1 + x \cdot \left(1 + \varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 10^{-287}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)} + \left(1 + \varepsilon \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.6e+77)
   (/
    (/
     (*
      x
      (+
       -1.0
       (* x (+ 0.5 (* x (- (* x 0.041666666666666664) 0.16666666666666666))))))
     eps)
    2.0)
   (if (<= x 1e-287)
     (/ (+ (exp (* x (- -1.0 eps))) (+ 1.0 (* eps x))) 2.0)
     (/ (+ 1.0 (exp (* x (+ eps -1.0)))) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.6e+77) {
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps) / 2.0;
	} else if (x <= 1e-287) {
		tmp = (exp((x * (-1.0 - eps))) + (1.0 + (eps * x))) / 2.0;
	} else {
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2.6d+77)) then
        tmp = ((x * ((-1.0d0) + (x * (0.5d0 + (x * ((x * 0.041666666666666664d0) - 0.16666666666666666d0)))))) / eps) / 2.0d0
    else if (x <= 1d-287) then
        tmp = (exp((x * ((-1.0d0) - eps))) + (1.0d0 + (eps * x))) / 2.0d0
    else
        tmp = (1.0d0 + exp((x * (eps + (-1.0d0))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.6e+77) {
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps) / 2.0;
	} else if (x <= 1e-287) {
		tmp = (Math.exp((x * (-1.0 - eps))) + (1.0 + (eps * x))) / 2.0;
	} else {
		tmp = (1.0 + Math.exp((x * (eps + -1.0)))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.6e+77:
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps) / 2.0
	elif x <= 1e-287:
		tmp = (math.exp((x * (-1.0 - eps))) + (1.0 + (eps * x))) / 2.0
	else:
		tmp = (1.0 + math.exp((x * (eps + -1.0)))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.6e+77)
		tmp = Float64(Float64(Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(Float64(x * 0.041666666666666664) - 0.16666666666666666)))))) / eps) / 2.0);
	elseif (x <= 1e-287)
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 - eps))) + Float64(1.0 + Float64(eps * x))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(eps + -1.0)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.6e+77)
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps) / 2.0;
	elseif (x <= 1e-287)
		tmp = (exp((x * (-1.0 - eps))) + (1.0 + (eps * x))) / 2.0;
	else
		tmp = (1.0 + exp((x * (eps + -1.0)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.6e+77], N[(N[(N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(x * N[(N[(x * 0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1e-287], N[(N[(N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 + N[(eps * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.6000000000000002e77

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 51.6%

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

   6. Taylor expanded in eps around 0 50.0%

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

      1. expm1-define 50.0%

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

      2. neg-mul-1 50.0%

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

   8. Simplified 50.0%

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

   9. Taylor expanded in x around 0 50.0%

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

   if -2.6000000000000002e77 < x < 1.00000000000000002e-287

   1. Initial program 55.7%

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

   3. Simplified 49.1%

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

   5. Taylor expanded in eps around inf 96.0%

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

   6. Taylor expanded in eps around inf 96.1%

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

      1. *-commutative 96.1%

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

   8. Simplified 96.1%

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

   9. Taylor expanded in x around -inf 96.1%

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

       1. cancel-sign-sub-inv 96.1%

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

       2. exp-sum 89.9%

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

       3. metadata-eval 89.9%

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

       4. *-lft-identity 89.9%

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

       5. exp-sum 96.1%

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

       6. *-commutative 96.1%

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

       7. fma-undefine 96.1%

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

       8. *-rgt-identity 96.1%

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

       9. exp-neg 96.1%

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

       10. *-rgt-identity 96.1%

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

       11. neg-mul-1 96.1%

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

       12. fma-undefine 96.1%

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

       13. *-commutative 96.1%

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

       14. +-commutative 96.1%

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

       15. distribute-lft-in 96.1%

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

       16. associate-*r* 96.1%

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

       17. neg-mul-1 96.1%

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

       18. distribute-rgt-in 96.1%

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

       19. unsub-neg 96.1%

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

   11. Simplified 96.1%

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

   12. Taylor expanded in x around 0 81.7%

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

   if 1.00000000000000002e-287 < x

   1. Initial program 77.8%

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

   3. Simplified 69.7%

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

   5. Taylor expanded in eps around inf 99.1%

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

   6. Taylor expanded in x around 0 58.1%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 10^{-287}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 - \varepsilon\right)} + \left(1 + \varepsilon \cdot x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(\varepsilon + -1\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-242}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} + \frac{1}{1 + x \cdot \left(1 + \varepsilon\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.1e-242)
   (/ (* eps (+ x (/ (+ 1.0 (exp (* x (- -1.0 eps)))) eps))) 2.0)
   (/ (+ (exp (* eps x)) (/ 1.0 (+ 1.0 (* x (+ 1.0 eps))))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.1e-242) {
		tmp = (eps * (x + ((1.0 + exp((x * (-1.0 - eps)))) / eps))) / 2.0;
	} else {
		tmp = (exp((eps * x)) + (1.0 / (1.0 + (x * (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-242)) then
        tmp = (eps * (x + ((1.0d0 + exp((x * ((-1.0d0) - eps)))) / eps))) / 2.0d0
    else
        tmp = (exp((eps * x)) + (1.0d0 / (1.0d0 + (x * (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-242) {
		tmp = (eps * (x + ((1.0 + Math.exp((x * (-1.0 - eps)))) / eps))) / 2.0;
	} else {
		tmp = (Math.exp((eps * x)) + (1.0 / (1.0 + (x * (1.0 + eps))))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.1e-242:
		tmp = (eps * (x + ((1.0 + math.exp((x * (-1.0 - eps)))) / eps))) / 2.0
	else:
		tmp = (math.exp((eps * x)) + (1.0 / (1.0 + (x * (1.0 + eps))))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.1e-242)
		tmp = Float64(Float64(eps * Float64(x + Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps)))) / eps))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(eps * x)) + Float64(1.0 / Float64(1.0 + Float64(x * Float64(1.0 + eps))))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.1e-242)
		tmp = (eps * (x + ((1.0 + exp((x * (-1.0 - eps)))) / eps))) / 2.0;
	else
		tmp = (exp((eps * x)) + (1.0 / (1.0 + (x * (1.0 + eps))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.1e-242], N[(N[(eps * N[(x + N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[(1.0 + N[(x * N[(1.0 + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.10000000000000001e-242

   1. Initial program 66.4%

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

   3. Simplified 66.4%

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

   5. Taylor expanded in x around 0 39.4%

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

      1. +-commutative 39.4%

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

      2. mul-1-neg 39.4%

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

      3. unsub-neg 39.4%

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

      4. *-commutative 39.4%

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

   7. Simplified 39.4%

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

   8. Taylor expanded in eps around inf 78.3%

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

   10. Simplified 78.3%

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

   if -1.10000000000000001e-242 < x

   1. Initial program 74.7%

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

   3. Simplified 67.5%

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

   5. Taylor expanded in eps around inf 99.3%

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

   6. Taylor expanded in eps around inf 86.2%

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

      1. *-commutative 86.2%

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

   8. Simplified 86.2%

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

   9. Taylor expanded in x around 0 64.0%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-242}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(x + \frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{\varepsilon}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot x} + \frac{1}{1 + x \cdot \left(1 + \varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10000:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 2900000:\\ \;\;\;\;e^{-x} \cdot \left(1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -10000.0)
   (/ (/ (expm1 (- x)) eps) 2.0)
   (if (<= x 2900000.0) (* (exp (- x)) (+ 1.0 x)) (/ (/ (expm1 x) eps) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -10000.0) {
		tmp = (expm1(-x) / eps) / 2.0;
	} else if (x <= 2900000.0) {
		tmp = exp(-x) * (1.0 + x);
	} else {
		tmp = (expm1(x) / eps) / 2.0;
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -10000.0) {
		tmp = (Math.expm1(-x) / eps) / 2.0;
	} else if (x <= 2900000.0) {
		tmp = Math.exp(-x) * (1.0 + x);
	} else {
		tmp = (Math.expm1(x) / eps) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -10000.0:
		tmp = (math.expm1(-x) / eps) / 2.0
	elif x <= 2900000.0:
		tmp = math.exp(-x) * (1.0 + x)
	else:
		tmp = (math.expm1(x) / eps) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -10000.0)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps) / 2.0);
	elseif (x <= 2900000.0)
		tmp = Float64(exp(Float64(-x)) * Float64(1.0 + x));
	else
		tmp = Float64(Float64(expm1(x) / eps) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -10000.0], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2900000.0], N[(N[Exp[(-x)], $MachinePrecision] * N[(1.0 + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Exp[x] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1e4

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 52.9%

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

   6. Taylor expanded in eps around 0 48.6%

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

      1. expm1-define 48.6%

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

      2. neg-mul-1 48.6%

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

   8. Simplified 48.6%

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

   if -1e4 < x < 2.9e6

   1. Initial program 52.6%

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

   3. Simplified 31.5%

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

   5. Taylor expanded in eps around 0 31.9%

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

      1. associate-+r+ 80.1%

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

      2. mul-1-neg 80.1%

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

      3. sub-neg 80.1%

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

      4. +-inverses 80.1%

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

      5. distribute-lft-out 80.1%

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

      6. distribute-rgt1-in 80.1%

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

      7. mul-1-neg 80.1%

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

   7. Simplified 80.1%

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

   8. Taylor expanded in eps around 0 80.1%

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

   if 2.9e6 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 28.9%

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

   6. Taylor expanded in eps around 0 1.9%

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

      1. expm1-define 1.9%

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

      2. neg-mul-1 1.9%

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

   8. Simplified 1.9%

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

      1. expm1-undefine 1.9%

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

      2. div-sub 1.9%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 27.4%

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

      5. sqr-neg 27.4%

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

      6. sqrt-unprod 27.4%

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

      7. add-sqr-sqrt 27.4%

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

   10. Applied egg-rr 27.4%

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

       1. div-sub 27.4%

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

       2. expm1-define 27.4%

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

   12. Simplified 27.4%

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

Alternative 9: 55.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -16000:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1000000:\\ \;\;\;\;e^{-x} \cdot \left(1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -16000.0)
   (/
    (/
     (*
      x
      (+
       -1.0
       (* x (+ 0.5 (* x (- (* x 0.041666666666666664) 0.16666666666666666))))))
     eps)
    2.0)
   (if (<= x 1000000.0) (* (exp (- x)) (+ 1.0 x)) (/ (/ (expm1 x) eps) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -16000.0) {
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps) / 2.0;
	} else if (x <= 1000000.0) {
		tmp = exp(-x) * (1.0 + x);
	} else {
		tmp = (expm1(x) / eps) / 2.0;
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -16000.0) {
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps) / 2.0;
	} else if (x <= 1000000.0) {
		tmp = Math.exp(-x) * (1.0 + x);
	} else {
		tmp = (Math.expm1(x) / eps) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -16000.0:
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps) / 2.0
	elif x <= 1000000.0:
		tmp = math.exp(-x) * (1.0 + x)
	else:
		tmp = (math.expm1(x) / eps) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -16000.0)
		tmp = Float64(Float64(Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(Float64(x * 0.041666666666666664) - 0.16666666666666666)))))) / eps) / 2.0);
	elseif (x <= 1000000.0)
		tmp = Float64(exp(Float64(-x)) * Float64(1.0 + x));
	else
		tmp = Float64(Float64(expm1(x) / eps) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -16000.0], N[(N[(N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(x * N[(N[(x * 0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1000000.0], N[(N[Exp[(-x)], $MachinePrecision] * N[(1.0 + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Exp[x] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -16000

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 52.9%

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

   6. Taylor expanded in eps around 0 48.6%

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

      1. expm1-define 48.6%

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

      2. neg-mul-1 48.6%

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

   8. Simplified 48.6%

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

   9. Taylor expanded in x around 0 34.9%

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

   if -16000 < x < 1e6

   1. Initial program 52.6%

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

   3. Simplified 31.5%

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

   5. Taylor expanded in eps around 0 31.9%

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

      1. associate-+r+ 80.1%

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

      2. mul-1-neg 80.1%

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

      3. sub-neg 80.1%

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

      4. +-inverses 80.1%

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

      5. distribute-lft-out 80.1%

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

      6. distribute-rgt1-in 80.1%

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

      7. mul-1-neg 80.1%

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

   7. Simplified 80.1%

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

   8. Taylor expanded in eps around 0 80.1%

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

   if 1e6 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 28.9%

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

   6. Taylor expanded in eps around 0 1.9%

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

      1. expm1-define 1.9%

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

      2. neg-mul-1 1.9%

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

   8. Simplified 1.9%

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

      1. expm1-undefine 1.9%

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

      2. div-sub 1.9%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 27.4%

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

      5. sqr-neg 27.4%

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

      6. sqrt-unprod 27.4%

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

      7. add-sqr-sqrt 27.4%

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

   10. Applied egg-rr 27.4%

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

       1. div-sub 27.4%

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

       2. expm1-define 27.4%

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

   12. Simplified 27.4%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -16000:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 1000000:\\ \;\;\;\;e^{-x} \cdot \left(1 + x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -15000:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 600000:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(2 \cdot \left(\left(1 + x\right) \cdot \left(1 + x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -15000.0)
   (/
    (/
     (*
      x
      (+
       -1.0
       (* x (+ 0.5 (* x (- (* x 0.041666666666666664) 0.16666666666666666))))))
     eps)
    2.0)
   (if (<= x 600000.0)
     (/
      (/
       (*
        eps
        (*
         2.0
         (*
          (+ 1.0 x)
          (+ 1.0 (* x (+ -1.0 (* x (+ 0.5 (* x -0.16666666666666666)))))))))
       eps)
      2.0)
     (/ (/ (expm1 x) eps) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -15000.0) {
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps) / 2.0;
	} else if (x <= 600000.0) {
		tmp = ((eps * (2.0 * ((1.0 + x) * (1.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666))))))))) / 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 <= -15000.0) {
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps) / 2.0;
	} else if (x <= 600000.0) {
		tmp = ((eps * (2.0 * ((1.0 + x) * (1.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666))))))))) / eps) / 2.0;
	} else {
		tmp = (Math.expm1(x) / eps) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -15000.0:
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps) / 2.0
	elif x <= 600000.0:
		tmp = ((eps * (2.0 * ((1.0 + x) * (1.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666))))))))) / eps) / 2.0
	else:
		tmp = (math.expm1(x) / eps) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -15000.0)
		tmp = Float64(Float64(Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(Float64(x * 0.041666666666666664) - 0.16666666666666666)))))) / eps) / 2.0);
	elseif (x <= 600000.0)
		tmp = Float64(Float64(Float64(eps * Float64(2.0 * Float64(Float64(1.0 + x) * Float64(1.0 + Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * -0.16666666666666666))))))))) / eps) / 2.0);
	else
		tmp = Float64(Float64(expm1(x) / eps) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -15000.0], N[(N[(N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(x * N[(N[(x * 0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 600000.0], N[(N[(N[(eps * N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] * N[(1.0 + N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $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 < -15000

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 52.9%

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

   6. Taylor expanded in eps around 0 48.6%

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

      1. expm1-define 48.6%

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

      2. neg-mul-1 48.6%

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

   8. Simplified 48.6%

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

   9. Taylor expanded in x around 0 34.9%

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

   if -15000 < x < 6e5

   1. Initial program 52.6%

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

   3. Simplified 31.5%

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

   5. Taylor expanded in eps around 0 31.9%

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

      1. associate-+r+ 80.1%

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

      2. mul-1-neg 80.1%

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

      3. sub-neg 80.1%

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

      4. +-inverses 80.1%

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

      5. distribute-lft-out 80.1%

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

      6. distribute-rgt1-in 80.1%

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

      7. mul-1-neg 80.1%

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

   7. Simplified 80.1%

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

   8. Taylor expanded in x around 0 77.8%

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

   if 6e5 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 28.9%

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

   6. Taylor expanded in eps around 0 1.9%

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

      1. expm1-define 1.9%

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

      2. neg-mul-1 1.9%

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

   8. Simplified 1.9%

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

      1. expm1-undefine 1.9%

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

      2. div-sub 1.9%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 27.4%

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

      5. sqr-neg 27.4%

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

      6. sqrt-unprod 27.4%

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

      7. add-sqr-sqrt 27.4%

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

   10. Applied egg-rr 27.4%

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

       1. div-sub 27.4%

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

       2. expm1-define 27.4%

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

   12. Simplified 27.4%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -15000:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 600000:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(2 \cdot \left(\left(1 + x\right) \cdot \left(1 + x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(x\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -550:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -550.0)
   (/ (/ (expm1 (- x)) eps) 2.0)
   (/ (+ 1.0 (exp (* eps x))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -550.0) {
		tmp = (expm1(-x) / eps) / 2.0;
	} else {
		tmp = (1.0 + exp((eps * x))) / 2.0;
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -550.0) {
		tmp = (Math.expm1(-x) / eps) / 2.0;
	} else {
		tmp = (1.0 + Math.exp((eps * x))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -550.0:
		tmp = (math.expm1(-x) / eps) / 2.0
	else:
		tmp = (1.0 + math.exp((eps * x))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -550.0)
		tmp = Float64(Float64(expm1(Float64(-x)) / eps) / 2.0);
	else
		tmp = Float64(Float64(1.0 + exp(Float64(eps * x))) / 2.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -550.0], N[(N[(N[(Exp[(-x)] - 1), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[Exp[N[(eps * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -550

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 52.9%

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

   6. Taylor expanded in eps around 0 48.6%

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

      1. expm1-define 48.6%

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

      2. neg-mul-1 48.6%

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

   8. Simplified 48.6%

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

   if -550 < x

   1. Initial program 67.0%

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

   3. Simplified 59.1%

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

   5. Taylor expanded in eps around inf 97.7%

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

   6. Taylor expanded in eps around inf 88.5%

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

      1. *-commutative 88.5%

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

   8. Simplified 88.5%

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

   9. Taylor expanded in x around 0 69.5%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -550:\\ \;\;\;\;\frac{\frac{\mathsf{expm1}\left(-x\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{if}\;x \leq -3 \cdot 10^{+37}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 520:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\frac{1}{\varepsilon} - \left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right) - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+79}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0
         (/
          (/
           (*
            x
            (+
             -1.0
             (*
              x
              (+
               0.5
               (* x (- (* x 0.041666666666666664) 0.16666666666666666))))))
           eps)
          2.0)))
   (if (<= x -3e+37)
     t_0
     (if (<= x 520.0)
       (/
        (+
         2.0
         (* x (- (- (/ 1.0 eps) (* (+ 1.0 (/ 1.0 eps)) (- 1.0 eps))) eps)))
        2.0)
       (if (<= x 2e+79) 0.0 t_0)))))

C

double code(double x, double eps) {
	double t_0 = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps) / 2.0;
	double tmp;
	if (x <= -3e+37) {
		tmp = t_0;
	} else if (x <= 520.0) {
		tmp = (2.0 + (x * (((1.0 / eps) - ((1.0 + (1.0 / eps)) * (1.0 - eps))) - eps))) / 2.0;
	} else if (x <= 2e+79) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x * ((-1.0d0) + (x * (0.5d0 + (x * ((x * 0.041666666666666664d0) - 0.16666666666666666d0)))))) / eps) / 2.0d0
    if (x <= (-3d+37)) then
        tmp = t_0
    else if (x <= 520.0d0) then
        tmp = (2.0d0 + (x * (((1.0d0 / eps) - ((1.0d0 + (1.0d0 / eps)) * (1.0d0 - eps))) - eps))) / 2.0d0
    else if (x <= 2d+79) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps) / 2.0;
	double tmp;
	if (x <= -3e+37) {
		tmp = t_0;
	} else if (x <= 520.0) {
		tmp = (2.0 + (x * (((1.0 / eps) - ((1.0 + (1.0 / eps)) * (1.0 - eps))) - eps))) / 2.0;
	} else if (x <= 2e+79) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps) / 2.0
	tmp = 0
	if x <= -3e+37:
		tmp = t_0
	elif x <= 520.0:
		tmp = (2.0 + (x * (((1.0 / eps) - ((1.0 + (1.0 / eps)) * (1.0 - eps))) - eps))) / 2.0
	elif x <= 2e+79:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(Float64(x * 0.041666666666666664) - 0.16666666666666666)))))) / eps) / 2.0)
	tmp = 0.0
	if (x <= -3e+37)
		tmp = t_0;
	elseif (x <= 520.0)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(Float64(Float64(1.0 / eps) - Float64(Float64(1.0 + Float64(1.0 / eps)) * Float64(1.0 - eps))) - eps))) / 2.0);
	elseif (x <= 2e+79)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps) / 2.0;
	tmp = 0.0;
	if (x <= -3e+37)
		tmp = t_0;
	elseif (x <= 520.0)
		tmp = (2.0 + (x * (((1.0 / eps) - ((1.0 + (1.0 / eps)) * (1.0 - eps))) - eps))) / 2.0;
	elseif (x <= 2e+79)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(x * N[(N[(x * 0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -3e+37], t$95$0, If[LessEqual[x, 520.0], N[(N[(2.0 + N[(x * N[(N[(N[(1.0 / eps), $MachinePrecision] - N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[(1.0 - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2e+79], 0.0, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.00000000000000022e37 or 1.99999999999999993e79 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 39.7%

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

   6. Taylor expanded in eps around 0 18.9%

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

      1. expm1-define 18.9%

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

      2. neg-mul-1 18.9%

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

   8. Simplified 18.9%

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

   9. Taylor expanded in x around 0 33.5%

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

   if -3.00000000000000022e37 < x < 520

   1. Initial program 53.5%

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

   3. Simplified 42.5%

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

   5. Taylor expanded in x around 0 75.5%

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

   if 520 < x < 1.99999999999999993e79

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 40.9%

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

      1. div-sub 40.9%

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

      2. mul-1-neg 40.9%

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

      3. rec-exp 40.9%

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

      4. +-inverses 40.9%

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

      5. metadata-eval 40.9%

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

   7. Simplified 40.9%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+37}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 520:\\ \;\;\;\;\frac{2 + x \cdot \left(\left(\frac{1}{\varepsilon} - \left(1 + \frac{1}{\varepsilon}\right) \cdot \left(1 - \varepsilon\right)\right) - \varepsilon\right)}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+79}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.2% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{if}\;x \leq -3.5 \cdot 10^{+37}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 520:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+82}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0
         (/
          (/
           (*
            x
            (+
             -1.0
             (*
              x
              (+
               0.5
               (* x (- (* x 0.041666666666666664) 0.16666666666666666))))))
           eps)
          2.0)))
   (if (<= x -3.5e+37) t_0 (if (<= x 520.0) 1.0 (if (<= x 5e+82) 0.0 t_0)))))

C

double code(double x, double eps) {
	double t_0 = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps) / 2.0;
	double tmp;
	if (x <= -3.5e+37) {
		tmp = t_0;
	} else if (x <= 520.0) {
		tmp = 1.0;
	} else if (x <= 5e+82) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x * ((-1.0d0) + (x * (0.5d0 + (x * ((x * 0.041666666666666664d0) - 0.16666666666666666d0)))))) / eps) / 2.0d0
    if (x <= (-3.5d+37)) then
        tmp = t_0
    else if (x <= 520.0d0) then
        tmp = 1.0d0
    else if (x <= 5d+82) then
        tmp = 0.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps) / 2.0;
	double tmp;
	if (x <= -3.5e+37) {
		tmp = t_0;
	} else if (x <= 520.0) {
		tmp = 1.0;
	} else if (x <= 5e+82) {
		tmp = 0.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps) / 2.0
	tmp = 0
	if x <= -3.5e+37:
		tmp = t_0
	elif x <= 520.0:
		tmp = 1.0
	elif x <= 5e+82:
		tmp = 0.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(Float64(x * 0.041666666666666664) - 0.16666666666666666)))))) / eps) / 2.0)
	tmp = 0.0
	if (x <= -3.5e+37)
		tmp = t_0;
	elseif (x <= 520.0)
		tmp = 1.0;
	elseif (x <= 5e+82)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps) / 2.0;
	tmp = 0.0;
	if (x <= -3.5e+37)
		tmp = t_0;
	elseif (x <= 520.0)
		tmp = 1.0;
	elseif (x <= 5e+82)
		tmp = 0.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(x * N[(N[(x * 0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[x, -3.5e+37], t$95$0, If[LessEqual[x, 520.0], 1.0, If[LessEqual[x, 5e+82], 0.0, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.5e37 or 5.00000000000000015e82 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 39.7%

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

   6. Taylor expanded in eps around 0 18.9%

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

      1. expm1-define 18.9%

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

      2. neg-mul-1 18.9%

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

   8. Simplified 18.9%

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

   9. Taylor expanded in x around 0 33.5%

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

   if -3.5e37 < x < 520

   1. Initial program 53.5%

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

   3. Simplified 42.5%

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

   5. Taylor expanded in eps around inf 96.8%

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

   6. Taylor expanded in eps around inf 96.9%

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

      1. *-commutative 96.9%

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

   8. Simplified 96.9%

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

   9. Taylor expanded in x around 0 75.5%

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

   if 520 < x < 5.00000000000000015e82

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 40.9%

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

      1. div-sub 40.9%

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

      2. mul-1-neg 40.9%

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

      3. rec-exp 40.9%

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

      4. +-inverses 40.9%

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

      5. metadata-eval 40.9%

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

   7. Simplified 40.9%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{+37}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 520:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+82}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.0% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.4 \cdot 10^{+113}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 120:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+152}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot 0.5\right)}{\varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -8.4e+113)
   (/ (/ (* x (+ -1.0 (* x (+ 0.5 (* x -0.16666666666666666))))) eps) 2.0)
   (if (<= x 120.0)
     (/ (- 2.0 (* eps x)) 2.0)
     (if (<= x 2.2e+152) 0.0 (/ (/ (* x (+ -1.0 (* x 0.5))) eps) 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -8.4e+113) {
		tmp = ((x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666))))) / eps) / 2.0;
	} else if (x <= 120.0) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else if (x <= 2.2e+152) {
		tmp = 0.0;
	} else {
		tmp = ((x * (-1.0 + (x * 0.5))) / 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 <= (-8.4d+113)) then
        tmp = ((x * ((-1.0d0) + (x * (0.5d0 + (x * (-0.16666666666666666d0)))))) / eps) / 2.0d0
    else if (x <= 120.0d0) then
        tmp = (2.0d0 - (eps * x)) / 2.0d0
    else if (x <= 2.2d+152) then
        tmp = 0.0d0
    else
        tmp = ((x * ((-1.0d0) + (x * 0.5d0))) / eps) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -8.4e+113) {
		tmp = ((x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666))))) / eps) / 2.0;
	} else if (x <= 120.0) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else if (x <= 2.2e+152) {
		tmp = 0.0;
	} else {
		tmp = ((x * (-1.0 + (x * 0.5))) / eps) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -8.4e+113:
		tmp = ((x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666))))) / eps) / 2.0
	elif x <= 120.0:
		tmp = (2.0 - (eps * x)) / 2.0
	elif x <= 2.2e+152:
		tmp = 0.0
	else:
		tmp = ((x * (-1.0 + (x * 0.5))) / eps) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -8.4e+113)
		tmp = Float64(Float64(Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * -0.16666666666666666))))) / eps) / 2.0);
	elseif (x <= 120.0)
		tmp = Float64(Float64(2.0 - Float64(eps * x)) / 2.0);
	elseif (x <= 2.2e+152)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(x * Float64(-1.0 + Float64(x * 0.5))) / eps) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -8.4e+113)
		tmp = ((x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666))))) / eps) / 2.0;
	elseif (x <= 120.0)
		tmp = (2.0 - (eps * x)) / 2.0;
	elseif (x <= 2.2e+152)
		tmp = 0.0;
	else
		tmp = ((x * (-1.0 + (x * 0.5))) / eps) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -8.4e+113], N[(N[(N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 120.0], N[(N[(2.0 - N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.2e+152], 0.0, N[(N[(N[(x * N[(-1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.3999999999999996e113

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 51.6%

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

   6. Taylor expanded in eps around 0 50.0%

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

      1. expm1-define 50.0%

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

      2. neg-mul-1 50.0%

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

   8. Simplified 50.0%

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

   9. Taylor expanded in x around 0 50.0%

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

   if -8.3999999999999996e113 < x < 120

   1. Initial program 56.3%

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

   3. Simplified 45.9%

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

   5. Taylor expanded in x around 0 71.2%

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

   6. Taylor expanded in eps around 0 71.1%

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

   7. Taylor expanded in x around 0 71.1%

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

      1. mul-1-neg 71.1%

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

      2. *-commutative 71.1%

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

      3. unsub-neg 71.1%

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

      4. *-commutative 71.1%

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

   9. Simplified 71.1%

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

   if 120 < x < 2.1999999999999998e152

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 38.5%

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

      1. div-sub 38.5%

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

      2. mul-1-neg 38.5%

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

      3. rec-exp 38.5%

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

      4. +-inverses 38.5%

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

      5. metadata-eval 38.5%

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

   7. Simplified 38.5%

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

   if 2.1999999999999998e152 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 31.6%

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

   6. Taylor expanded in eps around 0 1.9%

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

      1. expm1-define 1.9%

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

      2. neg-mul-1 1.9%

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

   8. Simplified 1.9%

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

   9. Taylor expanded in x around 0 30.4%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.4 \cdot 10^{+113}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)}{\varepsilon}}{2}\\ \mathbf{elif}\;x \leq 120:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+152}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot 0.5\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 59.6% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9500:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(2 \cdot \left(\left(1 + x\right) \cdot \left(1 + x \cdot \left(-1 + x \cdot 0.5\right)\right)\right)\right)}{\varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -9500.0)
   (/
    (/
     (*
      x
      (+
       -1.0
       (* x (+ 0.5 (* x (- (* x 0.041666666666666664) 0.16666666666666666))))))
     eps)
    2.0)
   (/
    (/ (* eps (* 2.0 (* (+ 1.0 x) (+ 1.0 (* x (+ -1.0 (* x 0.5))))))) eps)
    2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -9500.0) {
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps) / 2.0;
	} else {
		tmp = ((eps * (2.0 * ((1.0 + x) * (1.0 + (x * (-1.0 + (x * 0.5))))))) / 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 <= (-9500.0d0)) then
        tmp = ((x * ((-1.0d0) + (x * (0.5d0 + (x * ((x * 0.041666666666666664d0) - 0.16666666666666666d0)))))) / eps) / 2.0d0
    else
        tmp = ((eps * (2.0d0 * ((1.0d0 + x) * (1.0d0 + (x * ((-1.0d0) + (x * 0.5d0))))))) / eps) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -9500.0) {
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps) / 2.0;
	} else {
		tmp = ((eps * (2.0 * ((1.0 + x) * (1.0 + (x * (-1.0 + (x * 0.5))))))) / eps) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -9500.0:
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps) / 2.0
	else:
		tmp = ((eps * (2.0 * ((1.0 + x) * (1.0 + (x * (-1.0 + (x * 0.5))))))) / eps) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -9500.0)
		tmp = Float64(Float64(Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * Float64(Float64(x * 0.041666666666666664) - 0.16666666666666666)))))) / eps) / 2.0);
	else
		tmp = Float64(Float64(Float64(eps * Float64(2.0 * Float64(Float64(1.0 + x) * Float64(1.0 + Float64(x * Float64(-1.0 + Float64(x * 0.5))))))) / eps) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -9500.0)
		tmp = ((x * (-1.0 + (x * (0.5 + (x * ((x * 0.041666666666666664) - 0.16666666666666666)))))) / eps) / 2.0;
	else
		tmp = ((eps * (2.0 * ((1.0 + x) * (1.0 + (x * (-1.0 + (x * 0.5))))))) / eps) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -9500.0], N[(N[(N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(x * N[(N[(x * 0.041666666666666664), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(eps * N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] * N[(1.0 + N[(x * N[(-1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9500

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 52.9%

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

   6. Taylor expanded in eps around 0 48.6%

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

      1. expm1-define 48.6%

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

      2. neg-mul-1 48.6%

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

   8. Simplified 48.6%

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

   9. Taylor expanded in x around 0 34.9%

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

   if -9500 < x

   1. Initial program 67.0%

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

   3. Simplified 52.3%

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

   5. Taylor expanded in eps around 0 33.8%

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

      1. associate-+r+ 67.4%

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

      2. mul-1-neg 67.4%

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

      3. sub-neg 67.4%

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

      4. +-inverses 67.4%

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

      5. distribute-lft-out 67.4%

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

      6. distribute-rgt1-in 67.4%

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

      7. mul-1-neg 67.4%

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

   7. Simplified 67.4%

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

   8. Taylor expanded in x around 0 70.9%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9500:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot \left(x \cdot 0.041666666666666664 - 0.16666666666666666\right)\right)\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(2 \cdot \left(\left(1 + x\right) \cdot \left(1 + x \cdot \left(-1 + x \cdot 0.5\right)\right)\right)\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 56.7% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 210:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+152}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot 0.5\right)}{\varepsilon}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 210.0)
   (/ (- 2.0 (* eps x)) 2.0)
   (if (<= x 2.2e+152) 0.0 (/ (/ (* x (+ -1.0 (* x 0.5))) eps) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 210.0) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else if (x <= 2.2e+152) {
		tmp = 0.0;
	} else {
		tmp = ((x * (-1.0 + (x * 0.5))) / 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 <= 210.0d0) then
        tmp = (2.0d0 - (eps * x)) / 2.0d0
    else if (x <= 2.2d+152) then
        tmp = 0.0d0
    else
        tmp = ((x * ((-1.0d0) + (x * 0.5d0))) / eps) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 210.0) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else if (x <= 2.2e+152) {
		tmp = 0.0;
	} else {
		tmp = ((x * (-1.0 + (x * 0.5))) / eps) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 210.0:
		tmp = (2.0 - (eps * x)) / 2.0
	elif x <= 2.2e+152:
		tmp = 0.0
	else:
		tmp = ((x * (-1.0 + (x * 0.5))) / eps) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 210.0)
		tmp = Float64(Float64(2.0 - Float64(eps * x)) / 2.0);
	elseif (x <= 2.2e+152)
		tmp = 0.0;
	else
		tmp = Float64(Float64(Float64(x * Float64(-1.0 + Float64(x * 0.5))) / eps) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 210.0)
		tmp = (2.0 - (eps * x)) / 2.0;
	elseif (x <= 2.2e+152)
		tmp = 0.0;
	else
		tmp = ((x * (-1.0 + (x * 0.5))) / eps) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 210.0], N[(N[(2.0 - N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.2e+152], 0.0, N[(N[(N[(x * N[(-1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 210

   1. Initial program 60.9%

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

   3. Simplified 51.7%

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

   5. Taylor expanded in x around 0 63.9%

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

   6. Taylor expanded in eps around 0 67.3%

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

   7. Taylor expanded in x around 0 67.3%

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

      1. mul-1-neg 67.3%

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

      2. *-commutative 67.3%

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

      3. unsub-neg 67.3%

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

      4. *-commutative 67.3%

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

   9. Simplified 67.3%

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

   if 210 < x < 2.1999999999999998e152

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 38.5%

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

      1. div-sub 38.5%

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

      2. mul-1-neg 38.5%

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

      3. rec-exp 38.5%

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

      4. +-inverses 38.5%

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

      5. metadata-eval 38.5%

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

   7. Simplified 38.5%

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

   if 2.1999999999999998e152 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 31.6%

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

   6. Taylor expanded in eps around 0 1.9%

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

      1. expm1-define 1.9%

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

      2. neg-mul-1 1.9%

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

   8. Simplified 1.9%

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

   9. Taylor expanded in x around 0 30.4%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 210:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+152}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \left(-1 + x \cdot 0.5\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 60.7% accurate, 18.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 200:\\ \;\;\;\;\frac{2 - \varepsilon \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 200.0) (/ (- 2.0 (* eps x)) 2.0) 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 200.0) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 200.0d0) then
        tmp = (2.0d0 - (eps * x)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 200.0) {
		tmp = (2.0 - (eps * x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 200.0:
		tmp = (2.0 - (eps * x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 200.0)
		tmp = Float64(Float64(2.0 - Float64(eps * x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 200.0)
		tmp = (2.0 - (eps * x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 200.0], N[(N[(2.0 - N[(eps * x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 200

   1. Initial program 60.9%

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

   3. Simplified 51.7%

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

   5. Taylor expanded in x around 0 63.9%

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

   6. Taylor expanded in eps around 0 67.3%

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

   7. Taylor expanded in x around 0 67.3%

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

      1. mul-1-neg 67.3%

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

      2. *-commutative 67.3%

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

      3. unsub-neg 67.3%

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

      4. *-commutative 67.3%

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

   9. Simplified 67.3%

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

   if 200 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 40.1%

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

      1. div-sub 40.1%

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

      2. mul-1-neg 40.1%

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

      3. rec-exp 40.1%

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

      4. +-inverses 40.1%

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

      5. metadata-eval 40.1%

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

   7. Simplified 40.1%

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

Alternative 18: 61.1% accurate, 20.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.32:\\ \;\;\;\;x \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{elif}\;x \leq 600:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -0.32) (* x (* eps -0.5)) (if (<= x 600.0) 1.0 0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -0.32) {
		tmp = x * (eps * -0.5);
	} else if (x <= 600.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-0.32d0)) then
        tmp = x * (eps * (-0.5d0))
    else if (x <= 600.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -0.32) {
		tmp = x * (eps * -0.5);
	} else if (x <= 600.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -0.32:
		tmp = x * (eps * -0.5)
	elif x <= 600.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -0.32)
		tmp = Float64(x * Float64(eps * -0.5));
	elseif (x <= 600.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -0.32)
		tmp = x * (eps * -0.5);
	elseif (x <= 600.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -0.32], N[(x * N[(eps * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 600.0], 1.0, 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -0.320000000000000007

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 3.1%

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

   6. Taylor expanded in eps around 0 26.2%

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

   7. Taylor expanded in x around 0 26.2%

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

      1. mul-1-neg 26.2%

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

      2. *-commutative 26.2%

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

      3. unsub-neg 26.2%

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

      4. *-commutative 26.2%

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

   9. Simplified 26.2%

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

   10. Taylor expanded in eps around inf 26.2%

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

       1. associate-*r* 26.2%

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

   12. Simplified 26.2%

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

   if -0.320000000000000007 < x < 600

   1. Initial program 51.6%

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

   3. Simplified 40.2%

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

   5. Taylor expanded in eps around inf 96.6%

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

   6. Taylor expanded in eps around inf 96.7%

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

      1. *-commutative 96.7%

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

   8. Simplified 96.7%

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

   9. Taylor expanded in x around 0 78.4%

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

   if 600 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 40.1%

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

      1. div-sub 40.1%

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

      2. mul-1-neg 40.1%

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

      3. rec-exp 40.1%

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

      4. +-inverses 40.1%

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

      5. metadata-eval 40.1%

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

   7. Simplified 40.1%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.32:\\ \;\;\;\;x \cdot \left(\varepsilon \cdot -0.5\right)\\ \mathbf{elif}\;x \leq 600:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 57.7% accurate, 37.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 520:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (if (<= x 520.0) 1.0 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 520.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 520.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 520.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 520.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 520.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 520.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 520.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 520

   1. Initial program 60.9%

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

   3. Simplified 51.7%

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

   5. Taylor expanded in eps around inf 97.3%

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

   6. Taylor expanded in eps around inf 97.4%

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

      1. *-commutative 97.4%

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

   8. Simplified 97.4%

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

   9. Taylor expanded in x around 0 63.9%

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

   if 520 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 40.1%

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

      1. div-sub 40.1%

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

      2. mul-1-neg 40.1%

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

      3. rec-exp 40.1%

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

      4. +-inverses 40.1%

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

      5. metadata-eval 40.1%

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

   7. Simplified 40.1%

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

Alternative 20: 16.9% accurate, 227.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 71.5%

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

3. Simplified 64.7%

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

5. Taylor expanded in eps around 0 12.5%

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

   1. div-sub 12.5%

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

   2. mul-1-neg 12.5%

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

   3. rec-exp 12.5%

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

   4. +-inverses 12.7%

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

   5. metadata-eval 12.7%

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

7. Simplified 12.7%

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

Reproduce

herbie shell --seed 2024152 
(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))