NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.1% → 99.7%

Time: 10.9s

Alternatives: 12

Speedup: 1.8×

• 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: 73.1% 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, 1.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 10^{-40}:\\ \;\;\;\;\frac{2 \cdot \left(e^{-x} \cdot \left(x + 1\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + eps\_m\right)} + e^{x \cdot \left(-eps\_m\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 1e-40)
   (/ (* 2.0 (* (exp (- x)) (+ x 1.0))) 2.0)
   (/ (+ (exp (* x (+ -1.0 eps_m))) (exp (* x (- eps_m)))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1e-40) {
		tmp = (2.0 * (exp(-x) * (x + 1.0))) / 2.0;
	} else {
		tmp = (exp((x * (-1.0 + eps_m))) + exp((x * -eps_m))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 1d-40) then
        tmp = (2.0d0 * (exp(-x) * (x + 1.0d0))) / 2.0d0
    else
        tmp = (exp((x * ((-1.0d0) + eps_m))) + exp((x * -eps_m))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1e-40) {
		tmp = (2.0 * (Math.exp(-x) * (x + 1.0))) / 2.0;
	} else {
		tmp = (Math.exp((x * (-1.0 + eps_m))) + Math.exp((x * -eps_m))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 1e-40:
		tmp = (2.0 * (math.exp(-x) * (x + 1.0))) / 2.0
	else:
		tmp = (math.exp((x * (-1.0 + eps_m))) + math.exp((x * -eps_m))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 1e-40)
		tmp = Float64(Float64(2.0 * Float64(exp(Float64(-x)) * Float64(x + 1.0))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 + eps_m))) + exp(Float64(x * Float64(-eps_m)))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 1e-40)
		tmp = (2.0 * (exp(-x) * (x + 1.0))) / 2.0;
	else
		tmp = (exp((x * (-1.0 + eps_m))) + exp((x * -eps_m))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 1e-40], N[(N[(2.0 * N[(N[Exp[(-x)], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * (-eps$95$m)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 9.9999999999999993e-41

   1. Initial program 62.7%

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

   3. Simplified 62.7%

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

   5. Taylor expanded in eps around 0 73.1%

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

      1. sub-neg 73.1%

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

      2. remove-double-neg 73.1%

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

      3. mul-1-neg 73.1%

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

      4. distribute-lft-out 73.1%

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

      5. *-lft-identity 73.1%

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

      6. distribute-rgt1-in 73.1%

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

      7. metadata-eval 73.1%

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

      8. distribute-lft-out 73.1%

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

      9. mul-1-neg 73.1%

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

      10. remove-double-neg 73.1%

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

      11. distribute-rgt1-in 73.6%

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

      12. *-commutative 73.6%

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

      13. mul-1-neg 73.6%

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

   7. Simplified 73.6%

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

   if 9.9999999999999993e-41 < eps

   1. Initial program 97.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 10^{-40}:\\ \;\;\;\;\frac{2 \cdot \left(e^{-x} \cdot \left(x + 1\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 1.26 \cdot 10^{+245}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + eps\_m\right)} + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + \left(\left(-1 + eps\_m\right) + x \cdot \left(0.5 + 0.5 \cdot {\left(1 - eps\_m\right)}^{2}\right)\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 1.26e+245)
   (/ (+ (exp (* x (+ -1.0 eps_m))) (exp (- x))) 2.0)
   (/
    (+
     2.0
     (*
      x
      (+
       -1.0
       (+ (+ -1.0 eps_m) (* x (+ 0.5 (* 0.5 (pow (- 1.0 eps_m) 2.0))))))))
    2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.26e+245) {
		tmp = (exp((x * (-1.0 + eps_m))) + exp(-x)) / 2.0;
	} else {
		tmp = (2.0 + (x * (-1.0 + ((-1.0 + eps_m) + (x * (0.5 + (0.5 * pow((1.0 - eps_m), 2.0)))))))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 1.26d+245) then
        tmp = (exp((x * ((-1.0d0) + eps_m))) + exp(-x)) / 2.0d0
    else
        tmp = (2.0d0 + (x * ((-1.0d0) + (((-1.0d0) + eps_m) + (x * (0.5d0 + (0.5d0 * ((1.0d0 - eps_m) ** 2.0d0)))))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.26e+245) {
		tmp = (Math.exp((x * (-1.0 + eps_m))) + Math.exp(-x)) / 2.0;
	} else {
		tmp = (2.0 + (x * (-1.0 + ((-1.0 + eps_m) + (x * (0.5 + (0.5 * Math.pow((1.0 - eps_m), 2.0)))))))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 1.26e+245:
		tmp = (math.exp((x * (-1.0 + eps_m))) + math.exp(-x)) / 2.0
	else:
		tmp = (2.0 + (x * (-1.0 + ((-1.0 + eps_m) + (x * (0.5 + (0.5 * math.pow((1.0 - eps_m), 2.0)))))))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 1.26e+245)
		tmp = Float64(Float64(exp(Float64(x * Float64(-1.0 + eps_m))) + exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(Float64(-1.0 + eps_m) + Float64(x * Float64(0.5 + Float64(0.5 * (Float64(1.0 - eps_m) ^ 2.0)))))))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 1.26e+245)
		tmp = (exp((x * (-1.0 + eps_m))) + exp(-x)) / 2.0;
	else
		tmp = (2.0 + (x * (-1.0 + ((-1.0 + eps_m) + (x * (0.5 + (0.5 * ((1.0 - eps_m) ^ 2.0)))))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 1.26e+245], N[(N[(N[Exp[N[(x * N[(-1.0 + eps$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * N[(-1.0 + N[(N[(-1.0 + eps$95$m), $MachinePrecision] + N[(x * N[(0.5 + N[(0.5 * N[Power[N[(1.0 - eps$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.2599999999999999e245

   1. Initial program 70.9%

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

   3. Simplified 70.9%

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

   5. Taylor expanded in eps around inf 99.6%

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

   6. Taylor expanded in eps around 0 89.2%

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

   7. Taylor expanded in eps around -inf 89.2%

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

      1. associate-*r* 89.2%

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

      2. mul-1-neg 89.2%

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

      3. mul-1-neg 89.2%

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

      4. sub-neg 89.2%

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

      5. mul-1-neg 89.2%

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

      6. mul-1-neg 89.2%

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

   9. Simplified 89.2%

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

   10. Taylor expanded in x around inf 89.2%

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

       1. +-commutative 89.2%

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

       2. associate-*r* 89.2%

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

       3. sub-neg 89.2%

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

       4. mul-1-neg 89.2%

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

       5. associate-*r* 89.2%

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

       6. associate-*r* 89.2%

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

       7. neg-mul-1 89.2%

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

       8. mul-1-neg 89.2%

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

       9. sub-neg 89.2%

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

   12. Simplified 89.2%

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

   if 1.2599999999999999e245 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around 0 81.8%

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

   7. Taylor expanded in x around 0 100.0%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.26 \cdot 10^{+245}:\\ \;\;\;\;\frac{e^{x \cdot \left(-1 + \varepsilon\right)} + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + \left(\left(-1 + \varepsilon\right) + x \cdot \left(0.5 + 0.5 \cdot {\left(1 - \varepsilon\right)}^{2}\right)\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.7%

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

3. Simplified 72.7%

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

5. Taylor expanded in eps around inf 99.6%

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

6. Final simplification 99.6%

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

Alternative 4: 87.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 6.1 \cdot 10^{+54}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + \left(\left(-1 + eps\_m\right) + x \cdot \left(0.5 + 0.5 \cdot {\left(1 - eps\_m\right)}^{2}\right)\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 6.1e+54)
   (/ (* 2.0 (exp (- x))) 2.0)
   (/
    (+
     2.0
     (*
      x
      (+
       -1.0
       (+ (+ -1.0 eps_m) (* x (+ 0.5 (* 0.5 (pow (- 1.0 eps_m) 2.0))))))))
    2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 6.1e+54) {
		tmp = (2.0 * exp(-x)) / 2.0;
	} else {
		tmp = (2.0 + (x * (-1.0 + ((-1.0 + eps_m) + (x * (0.5 + (0.5 * pow((1.0 - eps_m), 2.0)))))))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 6.1d+54) then
        tmp = (2.0d0 * exp(-x)) / 2.0d0
    else
        tmp = (2.0d0 + (x * ((-1.0d0) + (((-1.0d0) + eps_m) + (x * (0.5d0 + (0.5d0 * ((1.0d0 - eps_m) ** 2.0d0)))))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 6.1e+54) {
		tmp = (2.0 * Math.exp(-x)) / 2.0;
	} else {
		tmp = (2.0 + (x * (-1.0 + ((-1.0 + eps_m) + (x * (0.5 + (0.5 * Math.pow((1.0 - eps_m), 2.0)))))))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 6.1e+54:
		tmp = (2.0 * math.exp(-x)) / 2.0
	else:
		tmp = (2.0 + (x * (-1.0 + ((-1.0 + eps_m) + (x * (0.5 + (0.5 * math.pow((1.0 - eps_m), 2.0)))))))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 6.1e+54)
		tmp = Float64(Float64(2.0 * exp(Float64(-x))) / 2.0);
	else
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(Float64(-1.0 + eps_m) + Float64(x * Float64(0.5 + Float64(0.5 * (Float64(1.0 - eps_m) ^ 2.0)))))))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 6.1e+54)
		tmp = (2.0 * exp(-x)) / 2.0;
	else
		tmp = (2.0 + (x * (-1.0 + ((-1.0 + eps_m) + (x * (0.5 + (0.5 * ((1.0 - eps_m) ^ 2.0)))))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 6.1e+54], N[(N[(2.0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 + N[(x * N[(-1.0 + N[(N[(-1.0 + eps$95$m), $MachinePrecision] + N[(x * N[(0.5 + N[(0.5 * N[Power[N[(1.0 - eps$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 6.0999999999999998e54

   1. Initial program 65.5%

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

   3. Simplified 65.5%

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

   5. Taylor expanded in eps around inf 99.5%

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

   6. Taylor expanded in eps around 0 89.2%

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

   7. Taylor expanded in eps around 0 83.7%

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

      1. cancel-sign-sub-inv 83.7%

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

      2. metadata-eval 83.7%

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

      3. distribute-rgt1-in 83.7%

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

      4. metadata-eval 83.7%

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

      5. mul-1-neg 83.7%

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

   9. Simplified 83.7%

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

   if 6.0999999999999998e54 < eps

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around 0 87.0%

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

   7. Taylor expanded in x around 0 82.8%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 6.1 \cdot 10^{+54}:\\ \;\;\;\;\frac{2 \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + \left(\left(-1 + \varepsilon\right) + x \cdot \left(0.5 + 0.5 \cdot {\left(1 - \varepsilon\right)}^{2}\right)\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \frac{2 \cdot e^{-x}}{2} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.7%

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

3. Simplified 72.7%

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

5. Taylor expanded in eps around inf 99.6%

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

6. Taylor expanded in eps around 0 88.7%

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

7. Taylor expanded in eps around 0 75.4%

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

   1. cancel-sign-sub-inv 75.4%

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

   2. metadata-eval 75.4%

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

   3. distribute-rgt1-in 75.4%

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

   4. metadata-eval 75.4%

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

   5. mul-1-neg 75.4%

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

9. Simplified 75.4%

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

10. Final simplification 75.4%

    \[\leadsto \frac{2 \cdot e^{-x}}{2} \]
11. Add Preprocessing

Alternative 6: 63.3% accurate, 9.5× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -48:\\ \;\;\;\;\frac{2 \cdot \left(x \cdot \left(1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(1 + x \cdot \left(2 + x \cdot \left(1.5 + x \cdot 0.6666666666666666\right)\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -48.0)
   (/
    (* 2.0 (* x (+ 1.0 (* x (+ 1.0 (* x (+ 0.5 (* x 0.16666666666666666))))))))
    2.0)
   (/
    (* 2.0 (+ 1.0 (* x (+ 2.0 (* x (+ 1.5 (* x 0.6666666666666666)))))))
    2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -48.0) {
		tmp = (2.0 * (x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))))))) / 2.0;
	} else {
		tmp = (2.0 * (1.0 + (x * (2.0 + (x * (1.5 + (x * 0.6666666666666666))))))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-48.0d0)) then
        tmp = (2.0d0 * (x * (1.0d0 + (x * (1.0d0 + (x * (0.5d0 + (x * 0.16666666666666666d0)))))))) / 2.0d0
    else
        tmp = (2.0d0 * (1.0d0 + (x * (2.0d0 + (x * (1.5d0 + (x * 0.6666666666666666d0))))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -48.0) {
		tmp = (2.0 * (x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))))))) / 2.0;
	} else {
		tmp = (2.0 * (1.0 + (x * (2.0 + (x * (1.5 + (x * 0.6666666666666666))))))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -48.0:
		tmp = (2.0 * (x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))))))) / 2.0
	else:
		tmp = (2.0 * (1.0 + (x * (2.0 + (x * (1.5 + (x * 0.6666666666666666))))))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -48.0)
		tmp = Float64(Float64(2.0 * Float64(x * Float64(1.0 + Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * 0.16666666666666666)))))))) / 2.0);
	else
		tmp = Float64(Float64(2.0 * Float64(1.0 + Float64(x * Float64(2.0 + Float64(x * Float64(1.5 + Float64(x * 0.6666666666666666))))))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -48.0)
		tmp = (2.0 * (x * (1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666)))))))) / 2.0;
	else
		tmp = (2.0 * (1.0 + (x * (2.0 + (x * (1.5 + (x * 0.6666666666666666))))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -48.0], N[(N[(2.0 * N[(x * 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] / 2.0), $MachinePrecision], N[(N[(2.0 * N[(1.0 + N[(x * N[(2.0 + N[(x * N[(1.5 + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -48

   1. Initial program 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in eps around 0 0.0%

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

      1. sub-neg 0.0%

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

      2. remove-double-neg 0.0%

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

      3. mul-1-neg 0.0%

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

      4. distribute-lft-out 0.0%

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

      5. *-lft-identity 0.0%

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

      6. distribute-rgt1-in 0.0%

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

      7. metadata-eval 0.0%

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

      8. distribute-lft-out 0.0%

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

      9. mul-1-neg 0.0%

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

      10. remove-double-neg 0.0%

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

      11. distribute-rgt1-in 2.3%

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

      12. *-commutative 2.3%

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

      13. mul-1-neg 2.3%

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

   7. Simplified 2.3%

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

   8. Taylor expanded in x around inf 2.3%

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

      1. exp-neg 2.3%

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

      2. associate-*r/ 2.3%

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

      3. *-rgt-identity 2.3%

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

   10. Simplified 2.3%

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

       1. add-exp-log 0.0%

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

       2. div-inv 0.0%

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

       3. log-prod 0.0%

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

       4. neg-log 0.0%

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

       5. add-log-exp 0.0%

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

       6. mul-1-neg 0.0%

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

       7. add-sqr-sqrt 0.0%

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

       8. sqrt-unprod 0.0%

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

       9. mul-1-neg 0.0%

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

       10. mul-1-neg 0.0%

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

       11. sqr-neg 0.0%

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

       12. sqrt-unprod 0.0%

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

       13. add-sqr-sqrt 0.0%

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

       14. prod-exp 0.0%

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

       15. add-exp-log 1.5%

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

       16. *-commutative 1.5%

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

   12. Applied egg-rr 1.5%

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

   13. Taylor expanded in x around 0 82.0%

       \[\leadsto \frac{2 \cdot \color{blue}{\left(x \cdot \left(1 + x \cdot \left(1 + x \cdot \left(0.5 + 0.16666666666666666 \cdot x\right)\right)\right)\right)}}{2} \]
   14. Step-by-step derivation

       1. *-commutative 82.0%

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

   15. Simplified 82.0%

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

   if -48 < x

   1. Initial program 67.7%

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

   3. Simplified 67.7%

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

   5. Taylor expanded in eps around 0 71.4%

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

      1. sub-neg 71.4%

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

      2. remove-double-neg 71.4%

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

      3. mul-1-neg 71.4%

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

      4. distribute-lft-out 71.4%

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

      5. *-lft-identity 71.4%

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

      6. distribute-rgt1-in 71.4%

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

      7. metadata-eval 71.4%

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

      8. distribute-lft-out 71.4%

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

      9. mul-1-neg 71.4%

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

      10. remove-double-neg 71.4%

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

      11. distribute-rgt1-in 71.4%

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

      12. *-commutative 71.4%

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

      13. mul-1-neg 71.4%

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

   7. Simplified 71.4%

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

      1. +-commutative 71.4%

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

      2. distribute-rgt-in 71.4%

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

      3. *-un-lft-identity 71.4%

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

      4. add-sqr-sqrt 22.4%

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

      5. sqrt-unprod 64.4%

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

      6. sqr-neg 64.4%

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

      7. sqrt-unprod 42.1%

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

      8. add-sqr-sqrt 64.4%

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

      9. add-sqr-sqrt 22.4%

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

      10. sqrt-unprod 64.4%

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

      11. sqr-neg 64.4%

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

      12. sqrt-unprod 42.1%

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

      13. add-sqr-sqrt 64.4%

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

   9. Applied egg-rr 64.4%

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

       1. distribute-rgt1-in 64.4%

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

       2. +-commutative 64.4%

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

   11. Simplified 64.4%

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

   12. Taylor expanded in x around 0 59.6%

       \[\leadsto \frac{2 \cdot \color{blue}{\left(1 + x \cdot \left(2 + x \cdot \left(1.5 + 0.6666666666666666 \cdot x\right)\right)\right)}}{2} \]
   13. Step-by-step derivation

       1. *-commutative 59.6%

          \[\leadsto \frac{2 \cdot \left(1 + x \cdot \left(2 + x \cdot \left(1.5 + \color{blue}{x \cdot 0.6666666666666666}\right)\right)\right)}{2} \]

   14. Simplified 59.6%

       \[\leadsto \frac{2 \cdot \color{blue}{\left(1 + x \cdot \left(2 + x \cdot \left(1.5 + x \cdot 0.6666666666666666\right)\right)\right)}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -48:\\ \;\;\;\;\frac{2 \cdot \left(x \cdot \left(1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(1 + x \cdot \left(2 + x \cdot \left(1.5 + x \cdot 0.6666666666666666\right)\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.1% accurate, 10.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+59}:\\ \;\;\;\;\frac{2 \cdot \left(1 + x \cdot \left(x \cdot 1.5\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(1 + x \cdot \left(2 + x \cdot \left(1.5 + x \cdot 0.6666666666666666\right)\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1e+59)
   (/ (* 2.0 (+ 1.0 (* x (* x 1.5)))) 2.0)
   (/
    (* 2.0 (+ 1.0 (* x (+ 2.0 (* x (+ 1.5 (* x 0.6666666666666666)))))))
    2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1e+59) {
		tmp = (2.0 * (1.0 + (x * (x * 1.5)))) / 2.0;
	} else {
		tmp = (2.0 * (1.0 + (x * (2.0 + (x * (1.5 + (x * 0.6666666666666666))))))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 1d+59) then
        tmp = (2.0d0 * (1.0d0 + (x * (x * 1.5d0)))) / 2.0d0
    else
        tmp = (2.0d0 * (1.0d0 + (x * (2.0d0 + (x * (1.5d0 + (x * 0.6666666666666666d0))))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 1e+59) {
		tmp = (2.0 * (1.0 + (x * (x * 1.5)))) / 2.0;
	} else {
		tmp = (2.0 * (1.0 + (x * (2.0 + (x * (1.5 + (x * 0.6666666666666666))))))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 1e+59:
		tmp = (2.0 * (1.0 + (x * (x * 1.5)))) / 2.0
	else:
		tmp = (2.0 * (1.0 + (x * (2.0 + (x * (1.5 + (x * 0.6666666666666666))))))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 1e+59)
		tmp = Float64(Float64(2.0 * Float64(1.0 + Float64(x * Float64(x * 1.5)))) / 2.0);
	else
		tmp = Float64(Float64(2.0 * Float64(1.0 + Float64(x * Float64(2.0 + Float64(x * Float64(1.5 + Float64(x * 0.6666666666666666))))))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 1e+59)
		tmp = (2.0 * (1.0 + (x * (x * 1.5)))) / 2.0;
	else
		tmp = (2.0 * (1.0 + (x * (2.0 + (x * (1.5 + (x * 0.6666666666666666))))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 1e+59], N[(N[(2.0 * N[(1.0 + N[(x * N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 * N[(1.0 + N[(x * N[(2.0 + N[(x * N[(1.5 + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999972e58

   1. Initial program 64.9%

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

   3. Simplified 64.9%

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

   5. Taylor expanded in eps around 0 60.1%

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

      1. sub-neg 60.1%

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

      2. remove-double-neg 60.1%

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

      3. mul-1-neg 60.1%

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

      4. distribute-lft-out 60.1%

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

      5. *-lft-identity 60.1%

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

      6. distribute-rgt1-in 60.1%

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

      7. metadata-eval 60.1%

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

      8. distribute-lft-out 60.1%

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

      9. mul-1-neg 60.1%

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

      10. remove-double-neg 60.1%

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

      11. distribute-rgt1-in 60.6%

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

      12. *-commutative 60.6%

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

      13. mul-1-neg 60.6%

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

   7. Simplified 60.6%

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

      1. +-commutative 60.6%

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

      2. distribute-rgt-in 60.1%

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

      3. *-un-lft-identity 60.1%

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

      4. add-sqr-sqrt 24.0%

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

      5. sqrt-unprod 56.2%

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

      6. sqr-neg 56.2%

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

      7. sqrt-unprod 32.2%

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

      8. add-sqr-sqrt 56.7%

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

      9. add-sqr-sqrt 24.5%

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

      10. sqrt-unprod 56.7%

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

      11. sqr-neg 56.7%

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

      12. sqrt-unprod 32.2%

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

      13. add-sqr-sqrt 56.5%

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

   9. Applied egg-rr 56.5%

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

       1. distribute-rgt1-in 56.5%

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

       2. +-commutative 56.5%

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

   11. Simplified 56.5%

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

   12. Taylor expanded in x around 0 66.3%

       \[\leadsto \frac{2 \cdot \color{blue}{\left(1 + x \cdot \left(2 + 1.5 \cdot x\right)\right)}}{2} \]
   13. Step-by-step derivation

       1. *-commutative 66.3%

          \[\leadsto \frac{2 \cdot \left(1 + x \cdot \left(2 + \color{blue}{x \cdot 1.5}\right)\right)}{2} \]

   14. Simplified 66.3%

       \[\leadsto \frac{2 \cdot \color{blue}{\left(1 + x \cdot \left(2 + x \cdot 1.5\right)\right)}}{2} \]

   15. Taylor expanded in x around inf 66.3%

       \[\leadsto \frac{2 \cdot \left(1 + x \cdot \color{blue}{\left(1.5 \cdot x\right)}\right)}{2} \]
   16. Step-by-step derivation

       1. *-commutative 66.3%

          \[\leadsto \frac{2 \cdot \left(1 + x \cdot \color{blue}{\left(x \cdot 1.5\right)}\right)}{2} \]

   17. Simplified 66.3%

       \[\leadsto \frac{2 \cdot \left(1 + x \cdot \color{blue}{\left(x \cdot 1.5\right)}\right)}{2} \]

   if 9.99999999999999972e58 < 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 56.8%

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

      1. sub-neg 56.8%

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

      2. remove-double-neg 56.8%

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

      3. mul-1-neg 56.8%

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

      4. distribute-lft-out 56.8%

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

      5. *-lft-identity 56.8%

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

      6. distribute-rgt1-in 56.8%

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

      7. metadata-eval 56.8%

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

      8. distribute-lft-out 56.8%

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

      9. mul-1-neg 56.8%

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

      10. remove-double-neg 56.8%

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

      11. distribute-rgt1-in 56.8%

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

      12. *-commutative 56.8%

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

      13. mul-1-neg 56.8%

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

   7. Simplified 56.8%

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

      1. +-commutative 56.8%

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

      2. distribute-rgt-in 56.8%

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

      3. *-un-lft-identity 56.8%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 44.7%

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

      6. sqr-neg 44.7%

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

      7. sqrt-unprod 44.7%

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

      8. add-sqr-sqrt 44.7%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 44.7%

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

      11. sqr-neg 44.7%

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

      12. sqrt-unprod 44.7%

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

      13. add-sqr-sqrt 44.7%

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

   9. Applied egg-rr 44.7%

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

       1. distribute-rgt1-in 44.7%

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

       2. +-commutative 44.7%

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

   11. Simplified 44.7%

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

   12. Taylor expanded in x around 0 35.0%

       \[\leadsto \frac{2 \cdot \color{blue}{\left(1 + x \cdot \left(2 + x \cdot \left(1.5 + 0.6666666666666666 \cdot x\right)\right)\right)}}{2} \]
   13. Step-by-step derivation

       1. *-commutative 35.0%

          \[\leadsto \frac{2 \cdot \left(1 + x \cdot \left(2 + x \cdot \left(1.5 + \color{blue}{x \cdot 0.6666666666666666}\right)\right)\right)}{2} \]

   14. Simplified 35.0%

       \[\leadsto \frac{2 \cdot \color{blue}{\left(1 + x \cdot \left(2 + x \cdot \left(1.5 + x \cdot 0.6666666666666666\right)\right)\right)}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+59}:\\ \;\;\;\;\frac{2 \cdot \left(1 + x \cdot \left(x \cdot 1.5\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(1 + x \cdot \left(2 + x \cdot \left(1.5 + x \cdot 0.6666666666666666\right)\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.2% accurate, 10.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 6500:\\ \;\;\;\;\frac{2 \cdot \left(1 + x \cdot \left(2 + x \cdot \left(1.5 - x \cdot 0.6666666666666666\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(1 + x \cdot \left(2 + x \cdot \left(1.5 + x \cdot 0.6666666666666666\right)\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 6500.0)
   (/ (* 2.0 (+ 1.0 (* x (+ 2.0 (* x (- 1.5 (* x 0.6666666666666666))))))) 2.0)
   (/
    (* 2.0 (+ 1.0 (* x (+ 2.0 (* x (+ 1.5 (* x 0.6666666666666666)))))))
    2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 6500.0) {
		tmp = (2.0 * (1.0 + (x * (2.0 + (x * (1.5 - (x * 0.6666666666666666))))))) / 2.0;
	} else {
		tmp = (2.0 * (1.0 + (x * (2.0 + (x * (1.5 + (x * 0.6666666666666666))))))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 6500.0d0) then
        tmp = (2.0d0 * (1.0d0 + (x * (2.0d0 + (x * (1.5d0 - (x * 0.6666666666666666d0))))))) / 2.0d0
    else
        tmp = (2.0d0 * (1.0d0 + (x * (2.0d0 + (x * (1.5d0 + (x * 0.6666666666666666d0))))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 6500.0) {
		tmp = (2.0 * (1.0 + (x * (2.0 + (x * (1.5 - (x * 0.6666666666666666))))))) / 2.0;
	} else {
		tmp = (2.0 * (1.0 + (x * (2.0 + (x * (1.5 + (x * 0.6666666666666666))))))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 6500.0:
		tmp = (2.0 * (1.0 + (x * (2.0 + (x * (1.5 - (x * 0.6666666666666666))))))) / 2.0
	else:
		tmp = (2.0 * (1.0 + (x * (2.0 + (x * (1.5 + (x * 0.6666666666666666))))))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 6500.0)
		tmp = Float64(Float64(2.0 * Float64(1.0 + Float64(x * Float64(2.0 + Float64(x * Float64(1.5 - Float64(x * 0.6666666666666666))))))) / 2.0);
	else
		tmp = Float64(Float64(2.0 * Float64(1.0 + Float64(x * Float64(2.0 + Float64(x * Float64(1.5 + Float64(x * 0.6666666666666666))))))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 6500.0)
		tmp = (2.0 * (1.0 + (x * (2.0 + (x * (1.5 - (x * 0.6666666666666666))))))) / 2.0;
	else
		tmp = (2.0 * (1.0 + (x * (2.0 + (x * (1.5 + (x * 0.6666666666666666))))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 6500.0], N[(N[(2.0 * N[(1.0 + N[(x * N[(2.0 + N[(x * N[(1.5 - N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 * N[(1.0 + N[(x * N[(2.0 + N[(x * N[(1.5 + N[(x * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6500

   1. Initial program 61.7%

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

   3. Simplified 61.7%

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

   5. Taylor expanded in eps around 0 59.1%

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

      1. sub-neg 59.1%

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

      2. remove-double-neg 59.1%

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

      3. mul-1-neg 59.1%

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

      4. distribute-lft-out 59.1%

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

      5. *-lft-identity 59.1%

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

      6. distribute-rgt1-in 59.1%

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

      7. metadata-eval 59.1%

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

      8. distribute-lft-out 59.1%

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

      9. mul-1-neg 59.1%

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

      10. remove-double-neg 59.1%

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

      11. distribute-rgt1-in 59.6%

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

      12. *-commutative 59.6%

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

      13. mul-1-neg 59.6%

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

   7. Simplified 59.6%

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

      1. +-commutative 59.6%

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

      2. distribute-rgt-in 59.1%

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

      3. *-un-lft-identity 59.1%

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

      4. add-sqr-sqrt 26.2%

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

      5. sqrt-unprod 58.6%

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

      6. sqr-neg 58.6%

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

      7. sqrt-unprod 32.4%

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

      8. add-sqr-sqrt 59.1%

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

      9. add-sqr-sqrt 26.7%

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

      10. sqrt-unprod 59.1%

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

      11. sqr-neg 59.1%

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

      12. sqrt-unprod 32.4%

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

      13. add-sqr-sqrt 58.9%

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

   9. Applied egg-rr 58.9%

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

       1. distribute-rgt1-in 58.9%

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

       2. +-commutative 58.9%

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

   11. Simplified 58.9%

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

   12. Taylor expanded in x around 0 58.6%

       \[\leadsto \frac{2 \cdot \color{blue}{\left(1 + x \cdot \left(2 + x \cdot \left(1.5 + 0.6666666666666666 \cdot x\right)\right)\right)}}{2} \]
   13. Step-by-step derivation

       1. *-commutative 58.6%

          \[\leadsto \frac{2 \cdot \left(1 + x \cdot \left(2 + x \cdot \left(1.5 + \color{blue}{x \cdot 0.6666666666666666}\right)\right)\right)}{2} \]

   14. Simplified 58.6%

       \[\leadsto \frac{2 \cdot \color{blue}{\left(1 + x \cdot \left(2 + x \cdot \left(1.5 + x \cdot 0.6666666666666666\right)\right)\right)}}{2} \]
   15. Step-by-step derivation

       1. add-sqr-sqrt 32.4%

          \[\leadsto \frac{2 \cdot \left(1 + x \cdot \left(2 + x \cdot \left(1.5 + \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot 0.6666666666666666\right)\right)\right)}{2} \]

       2. sqrt-unprod 75.4%

          \[\leadsto \frac{2 \cdot \left(1 + x \cdot \left(2 + x \cdot \left(1.5 + \color{blue}{\sqrt{x \cdot x}} \cdot 0.6666666666666666\right)\right)\right)}{2} \]

       3. sqr-neg 75.4%

          \[\leadsto \frac{2 \cdot \left(1 + x \cdot \left(2 + x \cdot \left(1.5 + \sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}} \cdot 0.6666666666666666\right)\right)\right)}{2} \]

       4. sqrt-unprod 43.0%

          \[\leadsto \frac{2 \cdot \left(1 + x \cdot \left(2 + x \cdot \left(1.5 + \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)} \cdot 0.6666666666666666\right)\right)\right)}{2} \]

       5. add-sqr-sqrt 75.4%

          \[\leadsto \frac{2 \cdot \left(1 + x \cdot \left(2 + x \cdot \left(1.5 + \color{blue}{\left(-x\right)} \cdot 0.6666666666666666\right)\right)\right)}{2} \]

       6. cancel-sign-sub-inv 75.4%

          \[\leadsto \frac{2 \cdot \left(1 + x \cdot \left(2 + x \cdot \color{blue}{\left(1.5 - x \cdot 0.6666666666666666\right)}\right)\right)}{2} \]

   16. Applied egg-rr 75.4%

       \[\leadsto \frac{2 \cdot \left(1 + x \cdot \left(2 + x \cdot \color{blue}{\left(1.5 - x \cdot 0.6666666666666666\right)}\right)\right)}{2} \]

   if 6500 < 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 60.1%

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

      1. sub-neg 60.1%

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

      2. remove-double-neg 60.1%

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

      3. mul-1-neg 60.1%

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

      4. distribute-lft-out 60.1%

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

      5. *-lft-identity 60.1%

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

      6. distribute-rgt1-in 60.1%

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

      7. metadata-eval 60.1%

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

      8. distribute-lft-out 60.1%

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

      9. mul-1-neg 60.1%

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

      10. remove-double-neg 60.1%

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

      11. distribute-rgt1-in 60.1%

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

      12. *-commutative 60.1%

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

      13. mul-1-neg 60.1%

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

   7. Simplified 60.1%

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

      1. +-commutative 60.1%

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

      2. distribute-rgt-in 60.1%

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

      3. *-un-lft-identity 60.1%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 41.5%

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

      6. sqr-neg 41.5%

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

      7. sqrt-unprod 41.5%

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

      8. add-sqr-sqrt 41.5%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 41.5%

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

      11. sqr-neg 41.5%

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

      12. sqrt-unprod 41.5%

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

      13. add-sqr-sqrt 41.5%

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

   9. Applied egg-rr 41.5%

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

       1. distribute-rgt1-in 41.5%

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

       2. +-commutative 41.5%

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

   11. Simplified 41.5%

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

   12. Taylor expanded in x around 0 27.6%

       \[\leadsto \frac{2 \cdot \color{blue}{\left(1 + x \cdot \left(2 + x \cdot \left(1.5 + 0.6666666666666666 \cdot x\right)\right)\right)}}{2} \]
   13. Step-by-step derivation

       1. *-commutative 27.6%

          \[\leadsto \frac{2 \cdot \left(1 + x \cdot \left(2 + x \cdot \left(1.5 + \color{blue}{x \cdot 0.6666666666666666}\right)\right)\right)}{2} \]

   14. Simplified 27.6%

       \[\leadsto \frac{2 \cdot \color{blue}{\left(1 + x \cdot \left(2 + x \cdot \left(1.5 + x \cdot 0.6666666666666666\right)\right)\right)}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6500:\\ \;\;\;\;\frac{2 \cdot \left(1 + x \cdot \left(2 + x \cdot \left(1.5 - x \cdot 0.6666666666666666\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(1 + x \cdot \left(2 + x \cdot \left(1.5 + x \cdot 0.6666666666666666\right)\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.1% accurate, 11.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+59}:\\ \;\;\;\;\frac{2 \cdot \left(1 + x \cdot \left(x \cdot 1.5\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(x \cdot \left(1 + x \cdot \left(-1 + x \cdot 0.5\right)\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 1.35e+59)
   (/ (* 2.0 (+ 1.0 (* x (* x 1.5)))) 2.0)
   (/ (* 2.0 (* x (+ 1.0 (* x (+ -1.0 (* x 0.5)))))) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.35e+59) {
		tmp = (2.0 * (1.0 + (x * (x * 1.5)))) / 2.0;
	} else {
		tmp = (2.0 * (x * (1.0 + (x * (-1.0 + (x * 0.5)))))) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 1.35d+59) then
        tmp = (2.0d0 * (1.0d0 + (x * (x * 1.5d0)))) / 2.0d0
    else
        tmp = (2.0d0 * (x * (1.0d0 + (x * ((-1.0d0) + (x * 0.5d0)))))) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 1.35e+59) {
		tmp = (2.0 * (1.0 + (x * (x * 1.5)))) / 2.0;
	} else {
		tmp = (2.0 * (x * (1.0 + (x * (-1.0 + (x * 0.5)))))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 1.35e+59:
		tmp = (2.0 * (1.0 + (x * (x * 1.5)))) / 2.0
	else:
		tmp = (2.0 * (x * (1.0 + (x * (-1.0 + (x * 0.5)))))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 1.35e+59)
		tmp = Float64(Float64(2.0 * Float64(1.0 + Float64(x * Float64(x * 1.5)))) / 2.0);
	else
		tmp = Float64(Float64(2.0 * Float64(x * Float64(1.0 + Float64(x * Float64(-1.0 + Float64(x * 0.5)))))) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 1.35e+59)
		tmp = (2.0 * (1.0 + (x * (x * 1.5)))) / 2.0;
	else
		tmp = (2.0 * (x * (1.0 + (x * (-1.0 + (x * 0.5)))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 1.35e+59], N[(N[(2.0 * N[(1.0 + N[(x * N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(2.0 * N[(x * N[(1.0 + N[(x * N[(-1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3500000000000001e59

   1. Initial program 64.9%

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

   3. Simplified 64.9%

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

   5. Taylor expanded in eps around 0 60.1%

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

      1. sub-neg 60.1%

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

      2. remove-double-neg 60.1%

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

      3. mul-1-neg 60.1%

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

      4. distribute-lft-out 60.1%

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

      5. *-lft-identity 60.1%

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

      6. distribute-rgt1-in 60.1%

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

      7. metadata-eval 60.1%

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

      8. distribute-lft-out 60.1%

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

      9. mul-1-neg 60.1%

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

      10. remove-double-neg 60.1%

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

      11. distribute-rgt1-in 60.6%

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

      12. *-commutative 60.6%

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

      13. mul-1-neg 60.6%

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

   7. Simplified 60.6%

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

      1. +-commutative 60.6%

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

      2. distribute-rgt-in 60.1%

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

      3. *-un-lft-identity 60.1%

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

      4. add-sqr-sqrt 24.0%

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

      5. sqrt-unprod 56.2%

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

      6. sqr-neg 56.2%

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

      7. sqrt-unprod 32.2%

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

      8. add-sqr-sqrt 56.7%

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

      9. add-sqr-sqrt 24.5%

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

      10. sqrt-unprod 56.7%

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

      11. sqr-neg 56.7%

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

      12. sqrt-unprod 32.2%

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

      13. add-sqr-sqrt 56.5%

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

   9. Applied egg-rr 56.5%

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

       1. distribute-rgt1-in 56.5%

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

       2. +-commutative 56.5%

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

   11. Simplified 56.5%

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

   12. Taylor expanded in x around 0 66.3%

       \[\leadsto \frac{2 \cdot \color{blue}{\left(1 + x \cdot \left(2 + 1.5 \cdot x\right)\right)}}{2} \]
   13. Step-by-step derivation

       1. *-commutative 66.3%

          \[\leadsto \frac{2 \cdot \left(1 + x \cdot \left(2 + \color{blue}{x \cdot 1.5}\right)\right)}{2} \]

   14. Simplified 66.3%

       \[\leadsto \frac{2 \cdot \color{blue}{\left(1 + x \cdot \left(2 + x \cdot 1.5\right)\right)}}{2} \]

   15. Taylor expanded in x around inf 66.3%

       \[\leadsto \frac{2 \cdot \left(1 + x \cdot \color{blue}{\left(1.5 \cdot x\right)}\right)}{2} \]
   16. Step-by-step derivation

       1. *-commutative 66.3%

          \[\leadsto \frac{2 \cdot \left(1 + x \cdot \color{blue}{\left(x \cdot 1.5\right)}\right)}{2} \]

   17. Simplified 66.3%

       \[\leadsto \frac{2 \cdot \left(1 + x \cdot \color{blue}{\left(x \cdot 1.5\right)}\right)}{2} \]

   if 1.3500000000000001e59 < 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 56.8%

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

      1. sub-neg 56.8%

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

      2. remove-double-neg 56.8%

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

      3. mul-1-neg 56.8%

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

      4. distribute-lft-out 56.8%

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

      5. *-lft-identity 56.8%

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

      6. distribute-rgt1-in 56.8%

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

      7. metadata-eval 56.8%

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

      8. distribute-lft-out 56.8%

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

      9. mul-1-neg 56.8%

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

      10. remove-double-neg 56.8%

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

      11. distribute-rgt1-in 56.8%

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

      12. *-commutative 56.8%

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

      13. mul-1-neg 56.8%

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

   7. Simplified 56.8%

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

   8. Taylor expanded in x around inf 56.8%

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

      1. exp-neg 56.8%

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

      2. associate-*r/ 56.8%

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

      3. *-rgt-identity 56.8%

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

   10. Simplified 56.8%

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

   11. Taylor expanded in x around 0 35.0%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+59}:\\ \;\;\;\;\frac{2 \cdot \left(1 + x \cdot \left(x \cdot 1.5\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(x \cdot \left(1 + x \cdot \left(-1 + x \cdot 0.5\right)\right)\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.1% accurate, 11.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \lor \neg \left(x \leq 10^{+59}\right):\\ \;\;\;\;\frac{2 \cdot \left(x \cdot \left(x + 1\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (or (<= x -1.6) (not (<= x 1e+59))) (/ (* 2.0 (* x (+ x 1.0))) 2.0) 1.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if ((x <= -1.6) || !(x <= 1e+59)) {
		tmp = (2.0 * (x * (x + 1.0))) / 2.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if ((x <= (-1.6d0)) .or. (.not. (x <= 1d+59))) then
        tmp = (2.0d0 * (x * (x + 1.0d0))) / 2.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if ((x <= -1.6) || !(x <= 1e+59)) {
		tmp = (2.0 * (x * (x + 1.0))) / 2.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if (x <= -1.6) or not (x <= 1e+59):
		tmp = (2.0 * (x * (x + 1.0))) / 2.0
	else:
		tmp = 1.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if ((x <= -1.6) || !(x <= 1e+59))
		tmp = Float64(Float64(2.0 * Float64(x * Float64(x + 1.0))) / 2.0);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if ((x <= -1.6) || ~((x <= 1e+59)))
		tmp = (2.0 * (x * (x + 1.0))) / 2.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[Or[LessEqual[x, -1.6], N[Not[LessEqual[x, 1e+59]], $MachinePrecision]], N[(N[(2.0 * N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -1.6000000000000001 or 9.99999999999999972e58 < x

   1. Initial program 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in eps around 0 32.4%

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

      1. sub-neg 32.4%

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

      2. remove-double-neg 32.4%

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

      3. mul-1-neg 32.4%

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

      4. distribute-lft-out 32.4%

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

      5. *-lft-identity 32.4%

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

      6. distribute-rgt1-in 32.4%

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

      7. metadata-eval 32.4%

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

      8. distribute-lft-out 32.4%

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

      9. mul-1-neg 32.4%

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

      10. remove-double-neg 32.4%

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

      11. distribute-rgt1-in 33.4%

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

      12. *-commutative 33.4%

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

      13. mul-1-neg 33.4%

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

   7. Simplified 33.4%

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

   8. Taylor expanded in x around inf 33.4%

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

      1. exp-neg 33.4%

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

      2. associate-*r/ 33.4%

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

      3. *-rgt-identity 33.4%

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

   10. Simplified 33.4%

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

       1. add-exp-log 32.4%

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

       2. div-inv 32.4%

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

       3. log-prod 32.4%

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

       4. neg-log 32.4%

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

       5. add-log-exp 32.4%

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

       6. mul-1-neg 32.4%

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

       7. add-sqr-sqrt 0.0%

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

       8. sqrt-unprod 25.5%

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

       9. mul-1-neg 25.5%

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

       10. mul-1-neg 25.5%

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

       11. sqr-neg 25.5%

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

       12. sqrt-unprod 25.5%

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

       13. add-sqr-sqrt 25.5%

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

       14. prod-exp 25.5%

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

       15. add-exp-log 26.2%

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

       16. *-commutative 26.2%

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

   12. Applied egg-rr 26.2%

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

   13. Taylor expanded in x around 0 40.2%

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

       1. +-commutative 40.2%

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

   15. Simplified 40.2%

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

   if -1.6000000000000001 < x < 9.99999999999999972e58

   1. Initial program 55.9%

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

   3. Simplified 55.9%

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

   5. Taylor expanded in x around 0 68.7%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \lor \neg \left(x \leq 10^{+59}\right):\\ \;\;\;\;\frac{2 \cdot \left(x \cdot \left(x + 1\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.1% accurate, 20.6× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \frac{2 \cdot \left(1 + x \cdot \left(x \cdot 1.5\right)\right)}{2} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (/ (* 2.0 (+ 1.0 (* x (* x 1.5)))) 2.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return (2.0 * (1.0 + (x * (x * 1.5)))) / 2.0;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    code = (2.0d0 * (1.0d0 + (x * (x * 1.5d0)))) / 2.0d0
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return (2.0 * (1.0 + (x * (x * 1.5)))) / 2.0;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return (2.0 * (1.0 + (x * (x * 1.5)))) / 2.0

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return Float64(Float64(2.0 * Float64(1.0 + Float64(x * Float64(x * 1.5)))) / 2.0)
end

MATLAB

eps_m = abs(eps);
function tmp = code(x, eps_m)
	tmp = (2.0 * (1.0 + (x * (x * 1.5)))) / 2.0;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := N[(N[(2.0 * N[(1.0 + N[(x * N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 72.7%

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

3. Simplified 72.7%

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

5. Taylor expanded in eps around 0 59.4%

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

   1. sub-neg 59.4%

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

   2. remove-double-neg 59.4%

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

   3. mul-1-neg 59.4%

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

   4. distribute-lft-out 59.4%

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

   5. *-lft-identity 59.4%

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

   6. distribute-rgt1-in 59.4%

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

   7. metadata-eval 59.4%

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

   8. distribute-lft-out 59.4%

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

   9. mul-1-neg 59.4%

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

   10. remove-double-neg 59.4%

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

   11. distribute-rgt1-in 59.8%

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

   12. *-commutative 59.8%

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

   13. mul-1-neg 59.8%

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

7. Simplified 59.8%

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

   1. +-commutative 59.8%

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

   2. distribute-rgt-in 59.4%

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

   3. *-un-lft-identity 59.4%

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

   4. add-sqr-sqrt 18.6%

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

   5. sqrt-unprod 53.6%

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

   6. sqr-neg 53.6%

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

   7. sqrt-unprod 35.0%

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

   8. add-sqr-sqrt 54.0%

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

   9. add-sqr-sqrt 19.0%

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

   10. sqrt-unprod 54.0%

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

   11. sqr-neg 54.0%

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

   12. sqrt-unprod 35.0%

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

   13. add-sqr-sqrt 53.9%

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

9. Applied egg-rr 53.9%

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

    1. distribute-rgt1-in 53.9%

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

    2. +-commutative 53.9%

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

11. Simplified 53.9%

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

12. Taylor expanded in x around 0 57.5%

    \[\leadsto \frac{2 \cdot \color{blue}{\left(1 + x \cdot \left(2 + 1.5 \cdot x\right)\right)}}{2} \]
13. Step-by-step derivation

    1. *-commutative 57.5%

       \[\leadsto \frac{2 \cdot \left(1 + x \cdot \left(2 + \color{blue}{x \cdot 1.5}\right)\right)}{2} \]

14. Simplified 57.5%

    \[\leadsto \frac{2 \cdot \color{blue}{\left(1 + x \cdot \left(2 + x \cdot 1.5\right)\right)}}{2} \]

15. Taylor expanded in x around inf 57.5%

    \[\leadsto \frac{2 \cdot \left(1 + x \cdot \color{blue}{\left(1.5 \cdot x\right)}\right)}{2} \]
16. Step-by-step derivation

    1. *-commutative 57.5%

       \[\leadsto \frac{2 \cdot \left(1 + x \cdot \color{blue}{\left(x \cdot 1.5\right)}\right)}{2} \]

17. Simplified 57.5%

    \[\leadsto \frac{2 \cdot \left(1 + x \cdot \color{blue}{\left(x \cdot 1.5\right)}\right)}{2} \]

18. Final simplification 57.5%

    \[\leadsto \frac{2 \cdot \left(1 + x \cdot \left(x \cdot 1.5\right)\right)}{2} \]
19. Add Preprocessing

Alternative 12: 44.6% accurate, 227.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ 1 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 1.0)

C

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

Fortran

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

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	return 1.0;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return 1.0

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return 1.0
end

MATLAB

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := 1.0

Derivation

1. Initial program 72.7%

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

3. Simplified 72.7%

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

5. Taylor expanded in x around 0 43.0%

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

6. Final simplification 43.0%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024130 
(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))