NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.4% → 99.9%

Time: 14.0s

Alternatives: 9

Speedup: 2.1×

• 37.6% 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: 73.4% 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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if eps < 1

   1. Initial program 58.3%

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

   3. Simplified 47.3%

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

   5. Taylor expanded in eps around 0 34.6%

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

      1. associate-+r+ 75.7%

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

      2. mul-1-neg 75.7%

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

      3. sub-neg 75.7%

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

      4. +-inverses 75.7%

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

      5. distribute-lft-out 75.7%

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

      6. distribute-rgt1-in 76.2%

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

      7. mul-1-neg 76.2%

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

   7. Simplified 76.2%

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

   8. Taylor expanded in eps around 0 76.2%

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

      1. +-commutative 76.2%

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

   10. Simplified 76.2%

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

   if 1 < eps

   1. Initial program 100.0%

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

   3. Simplified 89.5%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around -inf 100.0%

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

      1. rec-exp 100.0%

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

      2. *-commutative 100.0%

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

      3. fma-neg 100.0%

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

      4. neg-mul-1 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. exp-prod 100.0%

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

      8. fma-define 100.0%

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

      9. +-commutative 100.0%

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

      10. *-rgt-identity 100.0%

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

      11. distribute-lft-in 100.0%

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

      12. exp-prod 100.0%

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

      13. mul-1-neg 100.0%

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

      14. distribute-rgt-neg-in 100.0%

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

      15. distribute-neg-in 100.0%

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

      16. metadata-eval 100.0%

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

      17. unsub-neg 100.0%

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

   8. Simplified 100.0%

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

   10. Applied egg-rr 100.0%

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

       1. count-2 100.0%

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

   12. Simplified 100.0%

       \[\leadsto \frac{e^{\color{blue}{2 \cdot \log \left(\sqrt{e^{\mathsf{fma}\left(x, \varepsilon, x\right)}}\right)}} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2} \]
   13. Step-by-step derivation

       1. expm1-log1p-u 70.6%

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

       2. expm1-undefine 70.6%

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

       3. pow1/2 70.6%

          \[\leadsto \frac{e^{2 \cdot \left(e^{\mathsf{log1p}\left(\log \color{blue}{\left({\left(e^{\mathsf{fma}\left(x, \varepsilon, x\right)}\right)}^{0.5}\right)}\right)} - 1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2} \]

       4. log-pow 70.6%

          \[\leadsto \frac{e^{2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{0.5 \cdot \log \left(e^{\mathsf{fma}\left(x, \varepsilon, x\right)}\right)}\right)} - 1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2} \]

       5. add-log-exp 70.6%

          \[\leadsto \frac{e^{2 \cdot \left(e^{\mathsf{log1p}\left(0.5 \cdot \color{blue}{\mathsf{fma}\left(x, \varepsilon, x\right)}\right)} - 1\right)} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2} \]

   14. Applied egg-rr 70.6%

       \[\leadsto \frac{e^{2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(0.5 \cdot \mathsf{fma}\left(x, \varepsilon, x\right)\right)} - 1\right)}} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2} \]
   15. Step-by-step derivation

       1. sub-neg 70.6%

          \[\leadsto \frac{e^{2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(0.5 \cdot \mathsf{fma}\left(x, \varepsilon, x\right)\right)} + \left(-1\right)\right)}} + e^{x \cdot \left(-1 - \varepsilon\right)}}{2} \]

       2. log1p-undefine 70.6%

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

       3. rem-exp-log 100.0%

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

       4. *-rgt-identity 100.0%

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

       5. distribute-rgt1-in 100.0%

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

       6. distribute-lft1-in 100.0%

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

       7. *-rgt-identity 100.0%

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

       8. fma-define 100.0%

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

       9. metadata-eval 100.0%

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

   16. Simplified 100.0%

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

4. Final simplification 82.5%

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

Alternative 2: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if eps < 1

   1. Initial program 58.3%

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

   3. Simplified 47.3%

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

   5. Taylor expanded in eps around 0 34.6%

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

      1. associate-+r+ 75.7%

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

      2. mul-1-neg 75.7%

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

      3. sub-neg 75.7%

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

      4. +-inverses 75.7%

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

      5. distribute-lft-out 75.7%

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

      6. distribute-rgt1-in 76.2%

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

      7. mul-1-neg 76.2%

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

   7. Simplified 76.2%

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

   8. Taylor expanded in eps around 0 76.2%

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

      1. +-commutative 76.2%

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

   10. Simplified 76.2%

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

   if 1 < eps

   1. Initial program 100.0%

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

   3. Simplified 89.5%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around -inf 100.0%

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

      1. rec-exp 100.0%

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

      2. *-commutative 100.0%

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

      3. fma-neg 100.0%

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

      4. neg-mul-1 100.0%

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

      5. remove-double-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. exp-prod 100.0%

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

      8. fma-define 100.0%

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

      9. +-commutative 100.0%

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

      10. *-rgt-identity 100.0%

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

      11. distribute-lft-in 100.0%

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

      12. exp-prod 100.0%

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

      13. mul-1-neg 100.0%

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

      14. distribute-rgt-neg-in 100.0%

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

      15. distribute-neg-in 100.0%

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

      16. metadata-eval 100.0%

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

      17. unsub-neg 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in eps around inf 100.0%

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

       1. *-commutative 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 82.5%

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

Alternative 3: 84.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-257}:\\ \;\;\;\;\frac{1 + e^{eps\_m \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 75000000000000:\\ \;\;\;\;\frac{1 + e^{eps\_m \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x -6.5e-257)
   (/ (+ 1.0 (exp (* eps_m (- x)))) 2.0)
   (if (<= x 75000000000000.0) (/ (+ 1.0 (exp (* eps_m x))) 2.0) 0.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= -6.5e-257) {
		tmp = (1.0 + exp((eps_m * -x))) / 2.0;
	} else if (x <= 75000000000000.0) {
		tmp = (1.0 + exp((eps_m * x))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= (-6.5d-257)) then
        tmp = (1.0d0 + exp((eps_m * -x))) / 2.0d0
    else if (x <= 75000000000000.0d0) then
        tmp = (1.0d0 + exp((eps_m * x))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= -6.5e-257) {
		tmp = (1.0 + Math.exp((eps_m * -x))) / 2.0;
	} else if (x <= 75000000000000.0) {
		tmp = (1.0 + Math.exp((eps_m * x))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= -6.5e-257:
		tmp = (1.0 + math.exp((eps_m * -x))) / 2.0
	elif x <= 75000000000000.0:
		tmp = (1.0 + math.exp((eps_m * x))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= -6.5e-257)
		tmp = Float64(Float64(1.0 + exp(Float64(eps_m * Float64(-x)))) / 2.0);
	elseif (x <= 75000000000000.0)
		tmp = Float64(Float64(1.0 + exp(Float64(eps_m * x))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= -6.5e-257)
		tmp = (1.0 + exp((eps_m * -x))) / 2.0;
	elseif (x <= 75000000000000.0)
		tmp = (1.0 + exp((eps_m * x))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, -6.5e-257], N[(N[(1.0 + N[Exp[N[(eps$95$m * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 75000000000000.0], N[(N[(1.0 + N[Exp[N[(eps$95$m * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -6.5000000000000002e-257

   1. Initial program 61.7%

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

   3. Simplified 61.7%

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

   5. Taylor expanded in x around 0 35.1%

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

   6. Taylor expanded in eps around inf 68.6%

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

      1. sub-neg 68.6%

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

      2. mul-1-neg 68.6%

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

      3. remove-double-neg 68.6%

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

      4. mul-1-neg 68.6%

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

      5. distribute-rgt-neg-in 68.6%

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

      6. distribute-neg-in 68.6%

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

      7. metadata-eval 68.6%

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

      8. unsub-neg 68.6%

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

   8. Simplified 68.6%

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

   9. Taylor expanded in eps around inf 69.3%

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

       1. associate-*r* 69.3%

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

       2. neg-mul-1 69.3%

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

   11. Simplified 69.3%

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

   if -6.5000000000000002e-257 < x < 7.5e13

   1. Initial program 52.8%

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

   3. Simplified 44.1%

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

   5. Taylor expanded in eps around inf 99.1%

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

   6. Taylor expanded in x around 0 91.8%

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

   7. Taylor expanded in eps around inf 92.1%

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

      1. *-commutative 99.2%

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

   9. Simplified 92.1%

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

   if 7.5e13 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 57.8%

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

      1. div-sub 57.8%

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

      2. mul-1-neg 57.8%

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

      3. rec-exp 57.8%

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

      4. +-inverses 57.8%

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

      5. metadata-eval 57.8%

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

   7. Simplified 57.8%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-257}:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \mathbf{elif}\;x \leq 75000000000000:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if eps < 1

   1. Initial program 58.3%

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

   3. Simplified 47.3%

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

   5. Taylor expanded in eps around 0 34.6%

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

      1. associate-+r+ 75.7%

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

      2. mul-1-neg 75.7%

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

      3. sub-neg 75.7%

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

      4. +-inverses 75.7%

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

      5. distribute-lft-out 75.7%

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

      6. distribute-rgt1-in 76.2%

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

      7. mul-1-neg 76.2%

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

   7. Simplified 76.2%

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

   8. Taylor expanded in eps around 0 76.2%

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

      1. +-commutative 76.2%

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

   10. Simplified 76.2%

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

   if 1 < eps

   1. Initial program 100.0%

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

   3. Simplified 76.9%

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

   5. Taylor expanded in eps around 0 27.7%

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

      1. associate-+r+ 27.7%

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

      2. mul-1-neg 27.7%

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

      3. sub-neg 27.7%

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

      4. +-inverses 27.7%

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

      5. distribute-lft-out 27.7%

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

      6. distribute-rgt1-in 27.7%

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

      7. mul-1-neg 27.7%

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

   7. Simplified 27.7%

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

   8. Taylor expanded in x around 0 27.3%

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

      1. neg-mul-1 27.3%

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

      2. sub-neg 27.3%

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

   10. Simplified 27.3%

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

       1. associate-*r* 27.3%

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

       2. sub-neg 27.3%

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

       3. distribute-rgt-in 27.3%

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

       4. *-un-lft-identity 27.3%

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

       5. +-commutative 27.3%

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

       6. distribute-rgt-in 27.3%

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

       7. metadata-eval 27.3%

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

       8. add-sqr-sqrt 10.7%

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

       9. sqrt-unprod 45.5%

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

       10. sqr-neg 45.5%

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

       11. sqrt-unprod 34.9%

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

       12. add-sqr-sqrt 58.9%

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

       13. +-commutative 58.9%

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

       14. distribute-rgt-in 58.9%

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

       15. metadata-eval 58.9%

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

   12. Applied egg-rr 58.9%

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

       1. distribute-rgt1-in 58.9%

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

       2. +-commutative 58.9%

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

   14. Simplified 58.9%

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

4. Final simplification 71.6%

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

Alternative 5: 78.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;eps\_m \leq 1.6 \cdot 10^{+56}:\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{eps\_m \cdot \left(\left(1 + x\right) \cdot \left(2 + x \cdot 2\right)\right)}{eps\_m}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= eps_m 1.6e+56)
   (exp (- x))
   (/ (/ (* eps_m (* (+ 1.0 x) (+ 2.0 (* x 2.0)))) eps_m) 2.0)))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.6e+56) {
		tmp = exp(-x);
	} else {
		tmp = ((eps_m * ((1.0 + x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (eps_m <= 1.6d+56) then
        tmp = exp(-x)
    else
        tmp = ((eps_m * ((1.0d0 + x) * (2.0d0 + (x * 2.0d0)))) / eps_m) / 2.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (eps_m <= 1.6e+56) {
		tmp = Math.exp(-x);
	} else {
		tmp = ((eps_m * ((1.0 + x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if eps_m <= 1.6e+56:
		tmp = math.exp(-x)
	else:
		tmp = ((eps_m * ((1.0 + x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (eps_m <= 1.6e+56)
		tmp = exp(Float64(-x));
	else
		tmp = Float64(Float64(Float64(eps_m * Float64(Float64(1.0 + x) * Float64(2.0 + Float64(x * 2.0)))) / eps_m) / 2.0);
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (eps_m <= 1.6e+56)
		tmp = exp(-x);
	else
		tmp = ((eps_m * ((1.0 + x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[eps$95$m, 1.6e+56], N[Exp[(-x)], $MachinePrecision], N[(N[(N[(eps$95$m * N[(N[(1.0 + x), $MachinePrecision] * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.60000000000000002e56

   1. Initial program 60.0%

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

   3. Simplified 57.2%

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

   5. Taylor expanded in eps around inf 97.6%

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

   6. Taylor expanded in x around -inf 97.6%

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

      1. rec-exp 97.6%

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

      2. *-commutative 97.6%

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

      3. fma-neg 97.6%

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

      4. neg-mul-1 97.6%

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

      5. remove-double-neg 97.6%

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

      6. neg-mul-1 97.6%

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

      7. exp-prod 97.6%

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

      8. fma-define 97.6%

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

      9. +-commutative 97.6%

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

      10. *-rgt-identity 97.6%

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

      11. distribute-lft-in 97.6%

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

      12. exp-prod 97.6%

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

      13. mul-1-neg 97.6%

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

      14. distribute-rgt-neg-in 97.6%

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

      15. distribute-neg-in 97.6%

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

      16. metadata-eval 97.6%

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

      17. unsub-neg 97.6%

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

   8. Simplified 97.6%

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

   9. Taylor expanded in eps around 0 84.6%

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

       1. neg-mul-1 84.6%

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

   11. Simplified 84.6%

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

   if 1.60000000000000002e56 < eps

   1. Initial program 100.0%

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

   3. Simplified 77.0%

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

   5. Taylor expanded in eps around 0 27.9%

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

      1. associate-+r+ 27.9%

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

      2. mul-1-neg 27.9%

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

      3. sub-neg 27.9%

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

      4. +-inverses 27.9%

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

      5. distribute-lft-out 27.9%

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

      6. distribute-rgt1-in 27.9%

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

      7. mul-1-neg 27.9%

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

   7. Simplified 27.9%

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

   8. Taylor expanded in x around 0 27.5%

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

      1. neg-mul-1 27.5%

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

      2. sub-neg 27.5%

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

   10. Simplified 27.5%

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

       1. associate-*r* 27.5%

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

       2. sub-neg 27.5%

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

       3. distribute-rgt-in 27.5%

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

       4. *-un-lft-identity 27.5%

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

       5. +-commutative 27.5%

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

       6. distribute-rgt-in 27.5%

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

       7. metadata-eval 27.5%

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

       8. add-sqr-sqrt 8.8%

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

       9. sqrt-unprod 44.7%

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

       10. sqr-neg 44.7%

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

       11. sqrt-unprod 35.9%

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

       12. add-sqr-sqrt 58.2%

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

       13. +-commutative 58.2%

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

       14. distribute-rgt-in 58.2%

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

       15. metadata-eval 58.2%

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

   12. Applied egg-rr 58.2%

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

       1. distribute-rgt1-in 58.2%

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

       2. +-commutative 58.2%

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

   14. Simplified 58.2%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.6 \cdot 10^{+56}:\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(\left(1 + x\right) \cdot \left(2 + x \cdot 2\right)\right)}{\varepsilon}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.3% accurate, 11.3× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 6400000:\\ \;\;\;\;\frac{\frac{eps\_m \cdot \left(\left(1 + x\right) \cdot \left(2 + x \cdot 2\right)\right)}{eps\_m}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 6400000.0)
   (/ (/ (* eps_m (* (+ 1.0 x) (+ 2.0 (* x 2.0)))) eps_m) 2.0)
   0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 6400000.0) {
		tmp = ((eps_m * ((1.0 + x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 6400000.0d0) then
        tmp = ((eps_m * ((1.0d0 + x) * (2.0d0 + (x * 2.0d0)))) / eps_m) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 6400000.0) {
		tmp = ((eps_m * ((1.0 + x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 6400000.0:
		tmp = ((eps_m * ((1.0 + x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 6400000.0)
		tmp = Float64(Float64(Float64(eps_m * Float64(Float64(1.0 + x) * Float64(2.0 + Float64(x * 2.0)))) / eps_m) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 6400000.0)
		tmp = ((eps_m * ((1.0 + x) * (2.0 + (x * 2.0)))) / eps_m) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 6400000.0], N[(N[(N[(eps$95$m * N[(N[(1.0 + x), $MachinePrecision] * N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eps$95$m), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 6.4e6

   1. Initial program 57.8%

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

   3. Simplified 38.3%

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

   5. Taylor expanded in eps around 0 23.3%

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

      1. associate-+r+ 64.8%

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

      2. mul-1-neg 64.8%

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

      3. sub-neg 64.8%

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

      4. +-inverses 64.8%

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

      5. distribute-lft-out 64.8%

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

      6. distribute-rgt1-in 65.4%

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

      7. mul-1-neg 65.4%

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

   7. Simplified 65.4%

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

   8. Taylor expanded in x around 0 63.5%

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

      1. neg-mul-1 63.5%

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

      2. sub-neg 63.5%

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

   10. Simplified 63.5%

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

       1. associate-*r* 63.5%

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

       2. sub-neg 63.5%

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

       3. distribute-rgt-in 63.5%

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

       4. *-un-lft-identity 63.5%

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

       5. +-commutative 63.5%

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

       6. distribute-rgt-in 63.5%

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

       7. metadata-eval 63.5%

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

       8. add-sqr-sqrt 32.9%

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

       9. sqrt-unprod 63.4%

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

       10. sqr-neg 63.4%

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

       11. sqrt-unprod 30.6%

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

       12. add-sqr-sqrt 76.6%

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

       13. +-commutative 76.6%

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

       14. distribute-rgt-in 76.6%

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

       15. metadata-eval 76.6%

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

   12. Applied egg-rr 76.6%

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

       1. distribute-rgt1-in 76.6%

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

       2. +-commutative 76.6%

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

   14. Simplified 76.6%

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

   if 6.4e6 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 57.8%

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

      1. div-sub 57.8%

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

      2. mul-1-neg 57.8%

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

      3. rec-exp 57.8%

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

      4. +-inverses 57.8%

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

      5. metadata-eval 57.8%

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

   7. Simplified 57.8%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6400000:\\ \;\;\;\;\frac{\frac{\varepsilon \cdot \left(\left(1 + x\right) \cdot \left(2 + x \cdot 2\right)\right)}{\varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.0% accurate, 16.2× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;1 + \left(x \cdot -0.5\right) \cdot \left(eps\_m + 1\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (if (<= x 2.0) (+ 1.0 (* (* x -0.5) (+ eps_m 1.0))) 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.0) {
		tmp = 1.0 + ((x * -0.5) * (eps_m + 1.0));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 2.0d0) then
        tmp = 1.0d0 + ((x * (-0.5d0)) * (eps_m + 1.0d0))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 2.0) {
		tmp = 1.0 + ((x * -0.5) * (eps_m + 1.0));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 2.0:
		tmp = 1.0 + ((x * -0.5) * (eps_m + 1.0))
	else:
		tmp = 0.0
	return tmp

Julia

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

MATLAB

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 2.0], N[(1.0 + N[(N[(x * -0.5), $MachinePrecision] * N[(eps$95$m + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 57.1%

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

   3. Simplified 57.1%

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

   5. Taylor expanded in x around 0 38.1%

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

   6. Taylor expanded in eps around inf 77.9%

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

      1. sub-neg 77.9%

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

      2. mul-1-neg 77.9%

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

      3. remove-double-neg 77.9%

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

      4. mul-1-neg 77.9%

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

      5. distribute-rgt-neg-in 77.9%

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

      6. distribute-neg-in 77.9%

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

      7. metadata-eval 77.9%

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

      8. unsub-neg 77.9%

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

   8. Simplified 77.9%

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

   9. Taylor expanded in x around 0 67.5%

      \[\leadsto \color{blue}{1 + -0.5 \cdot \left(x \cdot \left(1 + \varepsilon\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* 67.5%

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

   11. Simplified 67.5%

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

   if 2 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 55.5%

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

      1. div-sub 55.5%

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

      2. mul-1-neg 55.5%

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

      3. rec-exp 55.5%

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

      4. +-inverses 55.5%

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

      5. metadata-eval 55.5%

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

   7. Simplified 55.5%

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

4. Final simplification 64.1%

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

Alternative 8: 57.4% accurate, 37.7× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 6400000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 (if (<= x 6400000.0) 1.0 0.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double tmp;
	if (x <= 6400000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

eps_m = abs(eps)
real(8) function code(x, eps_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps_m
    real(8) :: tmp
    if (x <= 6400000.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double tmp;
	if (x <= 6400000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	tmp = 0
	if x <= 6400000.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	tmp = 0.0
	if (x <= 6400000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

eps_m = abs(eps);
function tmp_2 = code(x, eps_m)
	tmp = 0.0;
	if (x <= 6400000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := If[LessEqual[x, 6400000.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 6.4e6

   1. Initial program 57.8%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in eps around inf 97.4%

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

   6. Taylor expanded in x around -inf 97.4%

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

      1. rec-exp 97.4%

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

      2. *-commutative 97.4%

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

      3. fma-neg 97.4%

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

      4. neg-mul-1 97.4%

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

      5. remove-double-neg 97.4%

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

      6. neg-mul-1 97.4%

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

      7. exp-prod 97.4%

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

      8. fma-define 97.4%

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

      9. +-commutative 97.4%

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

      10. *-rgt-identity 97.4%

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

      11. distribute-lft-in 97.4%

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

      12. exp-prod 97.4%

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

      13. mul-1-neg 97.4%

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

      14. distribute-rgt-neg-in 97.4%

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

      15. distribute-neg-in 97.4%

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

      16. metadata-eval 97.4%

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

      17. unsub-neg 97.4%

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

   8. Simplified 97.4%

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

   9. Taylor expanded in x around 0 64.1%

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

   if 6.4e6 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in eps around 0 57.8%

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

      1. div-sub 57.8%

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

      2. mul-1-neg 57.8%

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

      3. rec-exp 57.8%

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

      4. +-inverses 57.8%

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

      5. metadata-eval 57.8%

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

   7. Simplified 57.8%

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

Alternative 9: 16.2% accurate, 227.0× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ 0 \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	return 0.0

Julia

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

MATLAB

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

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := 0.0

Derivation

1. Initial program 69.3%

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

3. Simplified 64.4%

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

5. Taylor expanded in eps around 0 17.4%

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

   1. div-sub 17.4%

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

   2. mul-1-neg 17.4%

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

   3. rec-exp 17.4%

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

   4. +-inverses 17.7%

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

   5. metadata-eval 17.7%

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

7. Simplified 17.7%

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

Reproduce

herbie shell --seed 2024146 
(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))