NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.0% → 99.9%

Time: 12.5s

Alternatives: 11

Speedup: 2.0×

• 38.0% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 4 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{x + 1}{e^{x}} + \left(x + 1\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon - x} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= eps 4e-20)
   (/ (+ (/ (+ x 1.0) (exp x)) (* (+ x 1.0) (exp (- x)))) 2.0)
   (/ (+ (exp (- (* x eps) x)) (exp (* x (- eps)))) 2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (eps <= 4e-20) {
		tmp = (((x + 1.0) / exp(x)) + ((x + 1.0) * exp(-x))) / 2.0;
	} else {
		tmp = (exp(((x * eps) - x)) + exp((x * -eps))) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 4d-20) then
        tmp = (((x + 1.0d0) / exp(x)) + ((x + 1.0d0) * exp(-x))) / 2.0d0
    else
        tmp = (exp(((x * eps) - x)) + exp((x * -eps))) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (eps <= 4e-20) {
		tmp = (((x + 1.0) / Math.exp(x)) + ((x + 1.0) * Math.exp(-x))) / 2.0;
	} else {
		tmp = (Math.exp(((x * eps) - x)) + Math.exp((x * -eps))) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if eps <= 4e-20:
		tmp = (((x + 1.0) / math.exp(x)) + ((x + 1.0) * math.exp(-x))) / 2.0
	else:
		tmp = (math.exp(((x * eps) - x)) + math.exp((x * -eps))) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (eps <= 4e-20)
		tmp = Float64(Float64(Float64(Float64(x + 1.0) / exp(x)) + Float64(Float64(x + 1.0) * exp(Float64(-x)))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(Float64(x * eps) - x)) + exp(Float64(x * Float64(-eps)))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 4e-20)
		tmp = (((x + 1.0) / exp(x)) + ((x + 1.0) * exp(-x))) / 2.0;
	else
		tmp = (exp(((x * eps) - x)) + exp((x * -eps))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[eps, 4e-20], N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision] + N[(N[(x + 1.0), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(N[(x * eps), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * (-eps)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 3.99999999999999978e-20

   1. Initial program 62.5%

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

      1. div-sub 62.5%

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

      2. +-rgt-identity 62.5%

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

      3. div-sub 62.5%

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

   3. Simplified 62.5%

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

   4. Taylor expanded in eps around 0 69.1%

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

      1. *-commutative 69.1%

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

      2. distribute-lft1-in 69.1%

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

      3. mul-1-neg 69.1%

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

      4. distribute-lft-out 69.1%

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

      5. mul-1-neg 69.1%

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

      6. *-commutative 69.1%

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

      7. distribute-lft1-in 69.6%

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

      8. mul-1-neg 69.6%

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

   6. Simplified 69.6%

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

      1. exp-neg 69.6%

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

      2. un-div-inv 69.6%

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

   8. Applied egg-rr 69.6%

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

   if 3.99999999999999978e-20 < eps

   1. Initial program 97.0%

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

      1. div-sub 97.0%

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

      2. +-rgt-identity 97.0%

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

      3. div-sub 97.0%

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

   3. Simplified 97.0%

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

   4. Taylor expanded in eps around inf 100.0%

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

      1. neg-mul-1 100.0%

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

      2. *-commutative 100.0%

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

      3. mul-1-neg 100.0%

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

      4. exp-prod 100.0%

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

      5. +-commutative 100.0%

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

      6. *-commutative 100.0%

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

      7. remove-double-neg 100.0%

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

      8. mul-1-neg 100.0%

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

      9. sub-neg 100.0%

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

      10. *-commutative 100.0%

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

      11. exp-prod 100.0%

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

      12. exp-prod 100.0%

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

      13. *-commutative 100.0%

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

      14. cancel-sign-sub-inv 100.0%

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

      15. metadata-eval 100.0%

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

      16. *-lft-identity 100.0%

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

      17. exp-prod 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in eps around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. mul-1-neg 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in x around inf 100.0%

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

       1. associate-*r* 100.0%

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

       2. neg-mul-1 100.0%

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

   12. Simplified 100.0%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 4 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{x + 1}{e^{x}} + \left(x + 1\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \varepsilon - x} + e^{x \cdot \left(-\varepsilon\right)}}{2}\\ \end{array} \]

Alternative 2: 98.6% accurate, 1.1× speedup

Math

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

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (/ (+ (exp (* x (+ eps -1.0))) (exp (* x (- -1.0 eps)))) 2.0))

C

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

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (exp((x * (eps + (-1.0d0)))) + exp((x * ((-1.0d0) - eps)))) / 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := N[(N[(N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 71.2%

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

   1. div-sub 71.2%

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

   2. +-rgt-identity 71.2%

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

   3. div-sub 71.2%

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

3. Simplified 71.2%

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

4. Taylor expanded in eps around inf 98.9%

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

   1. neg-mul-1 98.9%

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

   2. *-commutative 98.9%

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

   3. mul-1-neg 98.9%

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

   4. exp-prod 98.9%

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

   5. +-commutative 98.9%

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

   6. *-commutative 98.9%

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

   7. remove-double-neg 98.9%

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

   8. mul-1-neg 98.9%

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

   9. sub-neg 98.9%

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

   10. *-commutative 98.9%

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

   11. exp-prod 98.9%

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

   12. exp-prod 98.9%

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

   13. *-commutative 98.9%

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

   14. cancel-sign-sub-inv 98.9%

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

   15. metadata-eval 98.9%

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

   16. *-lft-identity 98.9%

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

   17. exp-prod 98.9%

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

6. Simplified 98.9%

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

7. Final simplification 98.9%

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

Alternative 3: 84.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.85 \cdot 10^{+156}:\\ \;\;\;\;\frac{e^{x \cdot \left(-\varepsilon\right)} + e^{x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 2.85e+156) (/ (+ (exp (* x (- eps))) (exp (* x eps))) 2.0) 0.0))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 2.85e+156) {
		tmp = (exp((x * -eps)) + exp((x * eps))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 2.85d+156) then
        tmp = (exp((x * -eps)) + exp((x * eps))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 2.85e+156) {
		tmp = (Math.exp((x * -eps)) + Math.exp((x * eps))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 2.85e+156:
		tmp = (math.exp((x * -eps)) + math.exp((x * eps))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 2.85e+156)
		tmp = Float64(Float64(exp(Float64(x * Float64(-eps))) + exp(Float64(x * eps))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 2.85e+156)
		tmp = (exp((x * -eps)) + exp((x * eps))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 2.85e+156], N[(N[(N[Exp[N[(x * (-eps)), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.84999999999999999e156

   1. Initial program 67.8%

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

      1. div-sub 67.8%

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

      2. +-rgt-identity 67.8%

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

      3. div-sub 67.8%

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

   3. Simplified 67.8%

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

   4. Taylor expanded in eps around inf 98.8%

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

      1. neg-mul-1 98.8%

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

      2. *-commutative 98.8%

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

      3. mul-1-neg 98.8%

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

      4. exp-prod 98.8%

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

      5. +-commutative 98.8%

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

      6. *-commutative 98.8%

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

      7. remove-double-neg 98.8%

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

      8. mul-1-neg 98.8%

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

      9. sub-neg 98.8%

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

      10. *-commutative 98.8%

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

      11. exp-prod 98.8%

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

      12. exp-prod 98.8%

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

      13. *-commutative 98.8%

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

      14. cancel-sign-sub-inv 98.8%

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

      15. metadata-eval 98.8%

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

      16. *-lft-identity 98.8%

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

      17. exp-prod 98.8%

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

   6. Simplified 98.8%

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

   7. Taylor expanded in eps around inf 93.7%

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

      1. associate-*r* 93.7%

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

      2. mul-1-neg 93.7%

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

   9. Simplified 93.7%

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

   10. Taylor expanded in x around inf 93.7%

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

       1. associate-*r* 93.7%

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

       2. neg-mul-1 93.7%

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

   12. Simplified 93.7%

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

   13. Taylor expanded in eps around inf 94.1%

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

       1. *-commutative 94.1%

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

   15. Simplified 94.1%

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

   if 2.84999999999999999e156 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 74.5%

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

      1. div-sub 74.5%

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

      2. rec-exp 74.5%

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

      3. mul-1-neg 74.5%

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

      4. +-inverses 74.5%

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

   6. Simplified 74.5%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.85 \cdot 10^{+156}:\\ \;\;\;\;\frac{e^{x \cdot \left(-\varepsilon\right)} + e^{x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4: 91.5% accurate, 1.1× speedup

Math

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

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (/ (+ (exp (- (* x eps) x)) (exp (* x (- eps)))) 2.0))

C

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

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (exp(((x * eps) - x)) + exp((x * -eps))) / 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := N[(N[(N[Exp[N[(N[(x * eps), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(x * (-eps)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 71.2%

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

   1. div-sub 71.2%

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

   2. +-rgt-identity 71.2%

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

   3. div-sub 71.2%

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

3. Simplified 71.2%

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

4. Taylor expanded in eps around inf 98.9%

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

   1. neg-mul-1 98.9%

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

   2. *-commutative 98.9%

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

   3. mul-1-neg 98.9%

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

   4. exp-prod 98.9%

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

   5. +-commutative 98.9%

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

   6. *-commutative 98.9%

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

   7. remove-double-neg 98.9%

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

   8. mul-1-neg 98.9%

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

   9. sub-neg 98.9%

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

   10. *-commutative 98.9%

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

   11. exp-prod 98.9%

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

   12. exp-prod 98.9%

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

   13. *-commutative 98.9%

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

   14. cancel-sign-sub-inv 98.9%

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

   15. metadata-eval 98.9%

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

   16. *-lft-identity 98.9%

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

   17. exp-prod 98.9%

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

6. Simplified 98.9%

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

7. Taylor expanded in eps around inf 89.8%

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

   1. associate-*r* 89.8%

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

   2. mul-1-neg 89.8%

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

9. Simplified 89.8%

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

10. Taylor expanded in x around inf 89.8%

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

    1. associate-*r* 89.8%

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

    2. neg-mul-1 89.8%

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

12. Simplified 89.8%

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

13. Final simplification 89.8%

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

Alternative 5: 77.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+155}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + e^{x \cdot \left(1 + \varepsilon\right)} \cdot \left(\frac{-1}{\varepsilon} - -1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 2e-15)
   (/ (+ 1.0 (exp (* x (- -1.0 eps)))) 2.0)
   (if (<= x 2e+155)
     (/
      (+ (+ 1.0 (/ 1.0 eps)) (* (exp (* x (+ 1.0 eps))) (- (/ -1.0 eps) -1.0)))
      2.0)
     0.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 2e-15) {
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	} else if (x <= 2e+155) {
		tmp = ((1.0 + (1.0 / eps)) + (exp((x * (1.0 + eps))) * ((-1.0 / eps) - -1.0))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 2d-15) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps)))) / 2.0d0
    else if (x <= 2d+155) then
        tmp = ((1.0d0 + (1.0d0 / eps)) + (exp((x * (1.0d0 + eps))) * (((-1.0d0) / eps) - (-1.0d0)))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 2e-15) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps)))) / 2.0;
	} else if (x <= 2e+155) {
		tmp = ((1.0 + (1.0 / eps)) + (Math.exp((x * (1.0 + eps))) * ((-1.0 / eps) - -1.0))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 2e-15:
		tmp = (1.0 + math.exp((x * (-1.0 - eps)))) / 2.0
	elif x <= 2e+155:
		tmp = ((1.0 + (1.0 / eps)) + (math.exp((x * (1.0 + eps))) * ((-1.0 / eps) - -1.0))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 2e-15)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	elseif (x <= 2e+155)
		tmp = Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) + Float64(exp(Float64(x * Float64(1.0 + eps))) * Float64(Float64(-1.0 / eps) - -1.0))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 2e-15)
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	elseif (x <= 2e+155)
		tmp = ((1.0 + (1.0 / eps)) + (exp((x * (1.0 + eps))) * ((-1.0 / eps) - -1.0))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 2e-15], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2e+155], N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] + N[(N[Exp[N[(x * N[(1.0 + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(-1.0 / eps), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 2.0000000000000002e-15

   1. Initial program 63.0%

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

      1. div-sub 63.0%

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

      2. +-rgt-identity 63.0%

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

      3. div-sub 63.0%

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

   3. Simplified 63.0%

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

   4. Taylor expanded in x around 0 41.6%

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

   5. Taylor expanded in eps around inf 77.2%

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

      1. mul-1-neg 77.2%

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

      2. associate-*r* 77.2%

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

      3. +-commutative 77.2%

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

      4. associate-*r* 77.2%

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

      5. mul-1-neg 77.2%

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

      6. distribute-rgt-neg-out 77.2%

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

      7. *-commutative 77.2%

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

      8. +-commutative 77.2%

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

   7. Simplified 77.2%

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

   8. Taylor expanded in x around 0 77.2%

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

      1. mul-1-neg 77.2%

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

      2. *-commutative 77.2%

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

      3. distribute-rgt-neg-out 77.2%

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

      4. neg-sub0 77.2%

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

      5. +-commutative 77.2%

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

      6. associate--r+ 77.2%

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

      7. metadata-eval 77.2%

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

   10. Simplified 77.2%

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

   if 2.0000000000000002e-15 < x < 2.00000000000000001e155

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 22.2%

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

      1. add-log-exp 22.2%

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

      2. *-un-lft-identity 22.2%

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

      3. log-prod 22.2%

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

      4. metadata-eval 22.2%

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

      5. add-log-exp 22.2%

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

      6. *-commutative 22.2%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 41.3%

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

      9. sqr-neg 41.3%

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

      10. sqrt-unprod 41.3%

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

      11. add-sqr-sqrt 41.3%

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

   6. Applied egg-rr 41.3%

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

      1. +-lft-identity 41.3%

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

      2. +-commutative 41.3%

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

   8. Simplified 41.3%

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

   if 2.00000000000000001e155 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 74.5%

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

      1. div-sub 74.5%

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

      2. rec-exp 74.5%

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

      3. mul-1-neg 74.5%

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

      4. +-inverses 74.5%

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

   6. Simplified 74.5%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+155}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) + e^{x \cdot \left(1 + \varepsilon\right)} \cdot \left(\frac{-1}{\varepsilon} - -1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6: 76.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 225000:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 225000.0) (/ (+ 1.0 (exp (* x (- -1.0 eps)))) 2.0) 0.0))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 225000.0) {
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 225000.0d0) then
        tmp = (1.0d0 + exp((x * ((-1.0d0) - eps)))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 225000.0) {
		tmp = (1.0 + Math.exp((x * (-1.0 - eps)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 225000.0:
		tmp = (1.0 + math.exp((x * (-1.0 - eps)))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 225000.0)
		tmp = Float64(Float64(1.0 + exp(Float64(x * Float64(-1.0 - eps)))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 225000.0)
		tmp = (1.0 + exp((x * (-1.0 - eps)))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 225000.0], N[(N[(1.0 + N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 225000

   1. Initial program 63.5%

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

      1. div-sub 63.5%

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

      2. +-rgt-identity 63.5%

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

      3. div-sub 63.5%

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

   3. Simplified 63.5%

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

   4. Taylor expanded in x around 0 42.0%

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

   5. Taylor expanded in eps around inf 77.1%

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

      1. mul-1-neg 77.1%

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

      2. associate-*r* 77.1%

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

      3. +-commutative 77.1%

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

      4. associate-*r* 77.1%

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

      5. mul-1-neg 77.1%

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

      6. distribute-rgt-neg-out 77.1%

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

      7. *-commutative 77.1%

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

      8. +-commutative 77.1%

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

   7. Simplified 77.1%

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

   8. Taylor expanded in x around 0 77.1%

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

      1. mul-1-neg 77.1%

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

      2. *-commutative 77.1%

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

      3. distribute-rgt-neg-out 77.1%

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

      4. neg-sub0 77.1%

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

      5. +-commutative 77.1%

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

      6. associate--r+ 77.1%

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

      7. metadata-eval 77.1%

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

   10. Simplified 77.1%

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

   if 225000 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 59.9%

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

      1. div-sub 59.9%

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

      2. rec-exp 59.9%

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

      3. mul-1-neg 59.9%

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

      4. +-inverses 59.9%

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

   6. Simplified 59.9%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 225000:\\ \;\;\;\;\frac{1 + e^{x \cdot \left(-1 - \varepsilon\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7: 63.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-14}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{-x}}{2}\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 6e-14) (/ (- 2.0 (* x eps)) 2.0) (/ (* x (exp (- x))) 2.0)))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 6e-14) {
		tmp = (2.0 - (x * eps)) / 2.0;
	} else {
		tmp = (x * exp(-x)) / 2.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 6d-14) then
        tmp = (2.0d0 - (x * eps)) / 2.0d0
    else
        tmp = (x * exp(-x)) / 2.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 6e-14) {
		tmp = (2.0 - (x * eps)) / 2.0;
	} else {
		tmp = (x * Math.exp(-x)) / 2.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 6e-14:
		tmp = (2.0 - (x * eps)) / 2.0
	else:
		tmp = (x * math.exp(-x)) / 2.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 6e-14)
		tmp = Float64(Float64(2.0 - Float64(x * eps)) / 2.0);
	else
		tmp = Float64(Float64(x * exp(Float64(-x))) / 2.0);
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 6e-14)
		tmp = (2.0 - (x * eps)) / 2.0;
	else
		tmp = (x * exp(-x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 6e-14], N[(N[(2.0 - N[(x * eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.9999999999999997e-14

   1. Initial program 63.0%

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

      1. div-sub 63.0%

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

      2. +-rgt-identity 63.0%

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

      3. div-sub 63.0%

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

   3. Simplified 63.0%

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

   4. Taylor expanded in x around 0 41.6%

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

   5. Taylor expanded in x around 0 47.4%

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

   6. Taylor expanded in eps around inf 63.6%

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

      1. associate-*r* 63.6%

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

      2. mul-1-neg 63.6%

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

   8. Simplified 63.6%

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

   if 5.9999999999999997e-14 < x

   1. Initial program 100.0%

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

      1. div-sub 100.0%

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

      2. +-rgt-identity 100.0%

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

      3. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 56.9%

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

   5. Taylor expanded in x around 0 3.1%

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

   6. Taylor expanded in x around inf 56.9%

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

      1. *-commutative 56.9%

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

      2. neg-mul-1 56.9%

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

   8. Simplified 56.9%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-14}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{-x}}{2}\\ \end{array} \]

Alternative 8: 70.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 225000:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 225000.0) (/ (+ 1.0 (exp (- x))) 2.0) 0.0))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 225000.0) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 225000.0d0) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 225000.0) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 225000.0:
		tmp = (1.0 + math.exp(-x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 225000.0)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 225000.0)
		tmp = (1.0 + exp(-x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 225000.0], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 225000

   1. Initial program 63.5%

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

      1. div-sub 63.5%

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

      2. +-rgt-identity 63.5%

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

      3. div-sub 63.5%

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

   3. Simplified 63.5%

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

   4. Taylor expanded in eps around inf 98.6%

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

      1. neg-mul-1 98.6%

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

      2. *-commutative 98.6%

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

      3. mul-1-neg 98.6%

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

      4. exp-prod 98.6%

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

      5. +-commutative 98.6%

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

      6. *-commutative 98.6%

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

      7. remove-double-neg 98.6%

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

      8. mul-1-neg 98.6%

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

      9. sub-neg 98.6%

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

      10. *-commutative 98.6%

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

      11. exp-prod 98.6%

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

      12. exp-prod 98.6%

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

      13. *-commutative 98.6%

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

      14. cancel-sign-sub-inv 98.6%

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

      15. metadata-eval 98.6%

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

      16. *-lft-identity 98.6%

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

      17. exp-prod 98.6%

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

   6. Simplified 98.6%

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

   7. Taylor expanded in eps around inf 98.7%

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

      1. associate-*r* 98.7%

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

      2. mul-1-neg 98.7%

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

   9. Simplified 98.7%

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

   10. Taylor expanded in eps around 0 77.4%

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

   if 225000 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 59.9%

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

      1. div-sub 59.9%

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

      2. rec-exp 59.9%

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

      3. mul-1-neg 59.9%

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

      4. +-inverses 59.9%

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

   6. Simplified 59.9%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 225000:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9: 63.4% accurate, 25.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-14}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps)
 :precision binary64
 (if (<= x 6e-14) (/ (- 2.0 (* x eps)) 2.0) 0.0))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 6e-14) {
		tmp = (2.0 - (x * eps)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 6d-14) then
        tmp = (2.0d0 - (x * eps)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 6e-14) {
		tmp = (2.0 - (x * eps)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 6e-14:
		tmp = (2.0 - (x * eps)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 6e-14)
		tmp = Float64(Float64(2.0 - Float64(x * eps)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 6e-14)
		tmp = (2.0 - (x * eps)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 6e-14], N[(N[(2.0 - N[(x * eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 5.9999999999999997e-14

   1. Initial program 63.0%

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

      1. div-sub 63.0%

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

      2. +-rgt-identity 63.0%

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

      3. div-sub 63.0%

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

   3. Simplified 63.0%

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

   4. Taylor expanded in x around 0 41.6%

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

   5. Taylor expanded in x around 0 47.4%

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

   6. Taylor expanded in eps around inf 63.6%

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

      1. associate-*r* 63.6%

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

      2. mul-1-neg 63.6%

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

   8. Simplified 63.6%

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

   if 5.9999999999999997e-14 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 56.8%

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

      1. div-sub 56.8%

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

      2. rec-exp 56.8%

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

      3. mul-1-neg 56.8%

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

      4. +-inverses 56.8%

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

   6. Simplified 56.8%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-14}:\\ \;\;\;\;\frac{2 - x \cdot \varepsilon}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 10: 57.6% accurate, 74.1× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 225000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps) :precision binary64 (if (<= x 225000.0) 1.0 0.0))

C

eps = abs(eps);
double code(double x, double eps) {
	double tmp;
	if (x <= 225000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 225000.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	double tmp;
	if (x <= 225000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps = abs(eps)
def code(x, eps):
	tmp = 0
	if x <= 225000.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps = abs(eps)
function code(x, eps)
	tmp = 0.0
	if (x <= 225000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps = abs(eps)
function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 225000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := If[LessEqual[x, 225000.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 225000

   1. Initial program 63.5%

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

      1. div-sub 63.5%

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

      2. +-rgt-identity 63.5%

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

      3. div-sub 63.5%

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

   3. Simplified 63.5%

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

   4. Taylor expanded in x around 0 59.5%

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

   if 225000 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in eps around 0 59.9%

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

      1. div-sub 59.9%

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

      2. rec-exp 59.9%

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

      3. mul-1-neg 59.9%

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

      4. +-inverses 59.9%

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

   6. Simplified 59.9%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 225000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11: 16.1% accurate, 227.0× speedup

Math

\[\begin{array}{l} eps = |eps|\\ \\ 0 \end{array} \]

FPCore

NOTE: eps should be positive before calling this function
(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

NOTE: eps should be positive before calling this function
real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

eps = Math.abs(eps);
public static double code(double x, double eps) {
	return 0.0;
}

Python

eps = abs(eps)
def code(x, eps):
	return 0.0

Julia

eps = abs(eps)
function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

NOTE: eps should be positive before calling this function
code[x_, eps_] := 0.0

Derivation

1. Initial program 71.2%

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

3. Simplified 59.0%

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

4. Taylor expanded in eps around 0 14.4%

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

   1. div-sub 14.4%

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

   2. rec-exp 14.4%

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

   3. mul-1-neg 14.4%

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

   4. +-inverses 14.6%

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

6. Simplified 14.6%

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

7. Final simplification 14.6%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023216 
(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))