NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.6% → 98.8%

Time: 11.0s

Alternatives: 13

Speedup: 1.7×

• 37.8% 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: 73.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

3. Simplified 65.0%

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

   1. pow-neg 65.0%

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

   2. associate-*l/ 65.0%

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

   3. *-un-lft-identity 65.0%

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

   4. add-sqr-sqrt 35.8%

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

   5. sqrt-unprod 59.1%

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

   6. frac-times 59.2%

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

   7. metadata-eval 59.2%

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

   8. metadata-eval 59.2%

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

   9. frac-times 59.1%

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

   10. sqrt-unprod 28.9%

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

   11. add-sqr-sqrt 63.3%

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

   12. add-sqr-sqrt 40.3%

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

   13. sqrt-unprod 61.3%

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

   14. sqr-neg 61.3%

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

   15. sqrt-unprod 12.4%

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

   16. add-sqr-sqrt 24.7%

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

   17. pow-exp 36.3%

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

   18. *-commutative 36.3%

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

   19. +-commutative 36.3%

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

   20. distribute-rgt-in 36.3%

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

6. Applied egg-rr 75.7%

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

7. Taylor expanded in eps around inf 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 53.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq 3.7 \cdot 10^{-264}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps 3.7e-264)
   0.0
   (/ (+ (exp (* x (+ eps -1.0))) (exp (* eps (- x)))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= 3.7e-264) {
		tmp = 0.0;
	} else {
		tmp = (exp((x * (eps + -1.0))) + exp((eps * -x))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= 3.7d-264) then
        tmp = 0.0d0
    else
        tmp = (exp((x * (eps + (-1.0d0)))) + exp((eps * -x))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= 3.7e-264) {
		tmp = 0.0;
	} else {
		tmp = (Math.exp((x * (eps + -1.0))) + Math.exp((eps * -x))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= 3.7e-264:
		tmp = 0.0
	else:
		tmp = (math.exp((x * (eps + -1.0))) + math.exp((eps * -x))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= 3.7e-264)
		tmp = 0.0;
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps + -1.0))) + exp(Float64(eps * Float64(-x)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= 3.7e-264)
		tmp = 0.0;
	else
		tmp = (exp((x * (eps + -1.0))) + exp((eps * -x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, 3.7e-264], 0.0, N[(N[(N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(eps * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 3.69999999999999996e-264

   1. Initial program 78.2%

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

   3. Simplified 67.3%

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

   5. Taylor expanded in eps around 0 17.8%

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

   6. Taylor expanded in x around 0 18.1%

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

   if 3.69999999999999996e-264 < eps

   1. Initial program 77.1%

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

   3. Simplified 62.1%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 97.5%

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

      1. *-commutative 97.5%

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

      2. neg-mul-1 97.5%

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

      3. distribute-rgt-neg-in 97.5%

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

   8. Simplified 97.5%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 3.7 \cdot 10^{-264}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{\varepsilon \cdot \left(-x\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.65 \cdot 10^{+205}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot \left(-x\right)} + e^{x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 1.65e+205) (/ (+ (exp (* eps (- x))) (exp (* x eps))) 2.0) 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 1.65e+205) {
		tmp = (exp((eps * -x)) + exp((x * eps))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 1.65d+205) then
        tmp = (exp((eps * -x)) + exp((x * eps))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 1.65e+205) {
		tmp = (Math.exp((eps * -x)) + Math.exp((x * eps))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 1.65e+205:
		tmp = (math.exp((eps * -x)) + math.exp((x * eps))) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 1.65e+205)
		tmp = Float64(Float64(exp(Float64(eps * Float64(-x))) + exp(Float64(x * eps))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 1.65e+205)
		tmp = (exp((eps * -x)) + exp((x * eps))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 1.65e+205], N[(N[(N[Exp[N[(eps * (-x)), $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 < 1.6500000000000001e205

   1. Initial program 75.8%

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

   3. Simplified 62.1%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 93.4%

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

      1. *-commutative 93.4%

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

      2. neg-mul-1 93.4%

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

      3. distribute-rgt-neg-in 93.4%

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

   8. Simplified 93.4%

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

   9. Taylor expanded in eps around inf 91.7%

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

       1. *-commutative 91.7%

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

   11. Simplified 91.7%

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

   if 1.6500000000000001e205 < 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%

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

   5. Taylor expanded in eps around 0 70.5%

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

   6. Taylor expanded in x around 0 70.5%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.65 \cdot 10^{+205}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot \left(-x\right)} + e^{x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

Python

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

Julia

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

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

Wolfram

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 77.7%

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

3. Simplified 65.0%

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

5. Taylor expanded in eps around inf 100.0%

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

6. Final simplification 100.0%

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

Alternative 5: 74.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(\varepsilon + -1\right)\\ \mathbf{if}\;x \leq -4.7 \cdot 10^{+95}:\\ \;\;\;\;{0}^{-1}\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-245}:\\ \;\;\;\;\frac{\frac{1}{e^{x + x \cdot \varepsilon}} + \left(1 + t\_0\right)}{2}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+54}:\\ \;\;\;\;\frac{e^{t\_0} + \left(1 + x \cdot \left(-1 - \varepsilon\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+114}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+205}:\\ \;\;\;\;{0}^{-1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (+ eps -1.0))))
   (if (<= x -4.7e+95)
     (pow 0.0 -1.0)
     (if (<= x -5.8e-245)
       (/ (+ (/ 1.0 (exp (+ x (* x eps)))) (+ 1.0 t_0)) 2.0)
       (if (<= x 6.2e+54)
         (/ (+ (exp t_0) (+ 1.0 (* x (- -1.0 eps)))) 2.0)
         (if (<= x 1.25e+114) 0.0 (if (<= x 1.5e+205) (pow 0.0 -1.0) 0.0)))))))

C

double code(double x, double eps) {
	double t_0 = x * (eps + -1.0);
	double tmp;
	if (x <= -4.7e+95) {
		tmp = pow(0.0, -1.0);
	} else if (x <= -5.8e-245) {
		tmp = ((1.0 / exp((x + (x * eps)))) + (1.0 + t_0)) / 2.0;
	} else if (x <= 6.2e+54) {
		tmp = (exp(t_0) + (1.0 + (x * (-1.0 - eps)))) / 2.0;
	} else if (x <= 1.25e+114) {
		tmp = 0.0;
	} else if (x <= 1.5e+205) {
		tmp = pow(0.0, -1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (eps + (-1.0d0))
    if (x <= (-4.7d+95)) then
        tmp = 0.0d0 ** (-1.0d0)
    else if (x <= (-5.8d-245)) then
        tmp = ((1.0d0 / exp((x + (x * eps)))) + (1.0d0 + t_0)) / 2.0d0
    else if (x <= 6.2d+54) then
        tmp = (exp(t_0) + (1.0d0 + (x * ((-1.0d0) - eps)))) / 2.0d0
    else if (x <= 1.25d+114) then
        tmp = 0.0d0
    else if (x <= 1.5d+205) then
        tmp = 0.0d0 ** (-1.0d0)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x * (eps + -1.0);
	double tmp;
	if (x <= -4.7e+95) {
		tmp = Math.pow(0.0, -1.0);
	} else if (x <= -5.8e-245) {
		tmp = ((1.0 / Math.exp((x + (x * eps)))) + (1.0 + t_0)) / 2.0;
	} else if (x <= 6.2e+54) {
		tmp = (Math.exp(t_0) + (1.0 + (x * (-1.0 - eps)))) / 2.0;
	} else if (x <= 1.25e+114) {
		tmp = 0.0;
	} else if (x <= 1.5e+205) {
		tmp = Math.pow(0.0, -1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x * (eps + -1.0)
	tmp = 0
	if x <= -4.7e+95:
		tmp = math.pow(0.0, -1.0)
	elif x <= -5.8e-245:
		tmp = ((1.0 / math.exp((x + (x * eps)))) + (1.0 + t_0)) / 2.0
	elif x <= 6.2e+54:
		tmp = (math.exp(t_0) + (1.0 + (x * (-1.0 - eps)))) / 2.0
	elif x <= 1.25e+114:
		tmp = 0.0
	elif x <= 1.5e+205:
		tmp = math.pow(0.0, -1.0)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(eps + -1.0))
	tmp = 0.0
	if (x <= -4.7e+95)
		tmp = 0.0 ^ -1.0;
	elseif (x <= -5.8e-245)
		tmp = Float64(Float64(Float64(1.0 / exp(Float64(x + Float64(x * eps)))) + Float64(1.0 + t_0)) / 2.0);
	elseif (x <= 6.2e+54)
		tmp = Float64(Float64(exp(t_0) + Float64(1.0 + Float64(x * Float64(-1.0 - eps)))) / 2.0);
	elseif (x <= 1.25e+114)
		tmp = 0.0;
	elseif (x <= 1.5e+205)
		tmp = 0.0 ^ -1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x * (eps + -1.0);
	tmp = 0.0;
	if (x <= -4.7e+95)
		tmp = 0.0 ^ -1.0;
	elseif (x <= -5.8e-245)
		tmp = ((1.0 / exp((x + (x * eps)))) + (1.0 + t_0)) / 2.0;
	elseif (x <= 6.2e+54)
		tmp = (exp(t_0) + (1.0 + (x * (-1.0 - eps)))) / 2.0;
	elseif (x <= 1.25e+114)
		tmp = 0.0;
	elseif (x <= 1.5e+205)
		tmp = 0.0 ^ -1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.7e+95], N[Power[0.0, -1.0], $MachinePrecision], If[LessEqual[x, -5.8e-245], N[(N[(N[(1.0 / N[Exp[N[(x + N[(x * eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 6.2e+54], N[(N[(N[Exp[t$95$0], $MachinePrecision] + N[(1.0 + N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.25e+114], 0.0, If[LessEqual[x, 1.5e+205], N[Power[0.0, -1.0], $MachinePrecision], 0.0]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.69999999999999972e95 or 1.25e114 < x < 1.5e205

   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%

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

   5. Taylor expanded in eps around 0 10.7%

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

   7. Applied egg-rr 45.9%

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

   8. Taylor expanded in eps around 0 90.0%

      \[\leadsto {\color{blue}{0}}^{-1} \]

   if -4.69999999999999972e95 < x < -5.7999999999999999e-245

   1. Initial program 63.1%

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

   3. Simplified 43.3%

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

      1. pow-neg 43.3%

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

      2. associate-*l/ 43.3%

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

      3. *-un-lft-identity 43.3%

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

      4. add-sqr-sqrt 25.2%

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

      5. sqrt-unprod 42.4%

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

      6. frac-times 42.5%

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

      7. metadata-eval 42.5%

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

      8. metadata-eval 42.5%

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

      9. frac-times 42.4%

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

      10. sqrt-unprod 17.5%

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

      11. add-sqr-sqrt 42.0%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 39.4%

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

      14. sqr-neg 39.4%

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

      15. sqrt-unprod 26.4%

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

      16. add-sqr-sqrt 26.4%

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

      17. pow-exp 43.3%

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

      18. *-commutative 43.3%

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

      19. +-commutative 43.3%

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

      20. distribute-rgt-in 43.3%

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

   6. Applied egg-rr 61.0%

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

   7. Taylor expanded in eps around inf 100.0%

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

   8. Taylor expanded in x around 0 77.7%

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

   if -5.7999999999999999e-245 < x < 6.1999999999999999e54

   1. Initial program 67.4%

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

   3. Simplified 48.2%

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

      1. pow-neg 48.2%

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

      2. associate-*l/ 48.2%

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

      3. *-un-lft-identity 48.2%

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

      4. add-sqr-sqrt 27.8%

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

      5. sqrt-unprod 46.8%

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

      6. frac-times 46.9%

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

      7. metadata-eval 46.9%

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

      8. metadata-eval 46.9%

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

      9. frac-times 46.8%

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

      10. sqrt-unprod 20.2%

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

      11. add-sqr-sqrt 44.7%

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

      12. add-sqr-sqrt 44.3%

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

      13. sqrt-unprod 58.4%

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

      14. sqr-neg 58.4%

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

      15. sqrt-unprod 0.9%

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

      16. add-sqr-sqrt 19.8%

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

      17. pow-exp 38.1%

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

      18. *-commutative 38.1%

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

      19. +-commutative 38.1%

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

      20. distribute-rgt-in 38.1%

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

   6. Applied egg-rr 63.9%

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

   7. Taylor expanded in eps around inf 99.9%

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

   8. Taylor expanded in x around 0 74.0%

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

   10. Simplified 74.0%

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

   if 6.1999999999999999e54 < x < 1.25e114 or 1.5e205 < 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%

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

   5. Taylor expanded in eps around 0 77.0%

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

   6. Taylor expanded in x around 0 77.0%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+95}:\\ \;\;\;\;{0}^{-1}\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-245}:\\ \;\;\;\;\frac{\frac{1}{e^{x + x \cdot \varepsilon}} + \left(1 + x \cdot \left(\varepsilon + -1\right)\right)}{2}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+54}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \left(1 + x \cdot \left(-1 - \varepsilon\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+114}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+205}:\\ \;\;\;\;{0}^{-1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(\varepsilon + -1\right)\\ \mathbf{if}\;x \leq -4.7 \cdot 10^{+95}:\\ \;\;\;\;{0}^{-1}\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-245}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot \left(-x\right)} + \left(1 + t\_0\right)}{2}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+55}:\\ \;\;\;\;\frac{e^{t\_0} + \left(1 + x \cdot \left(-1 - \varepsilon\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+114}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+199}:\\ \;\;\;\;{0}^{-1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* x (+ eps -1.0))))
   (if (<= x -4.7e+95)
     (pow 0.0 -1.0)
     (if (<= x -5.8e-245)
       (/ (+ (exp (* eps (- x))) (+ 1.0 t_0)) 2.0)
       (if (<= x 2.5e+55)
         (/ (+ (exp t_0) (+ 1.0 (* x (- -1.0 eps)))) 2.0)
         (if (<= x 1.2e+114) 0.0 (if (<= x 5e+199) (pow 0.0 -1.0) 0.0)))))))

C

double code(double x, double eps) {
	double t_0 = x * (eps + -1.0);
	double tmp;
	if (x <= -4.7e+95) {
		tmp = pow(0.0, -1.0);
	} else if (x <= -5.8e-245) {
		tmp = (exp((eps * -x)) + (1.0 + t_0)) / 2.0;
	} else if (x <= 2.5e+55) {
		tmp = (exp(t_0) + (1.0 + (x * (-1.0 - eps)))) / 2.0;
	} else if (x <= 1.2e+114) {
		tmp = 0.0;
	} else if (x <= 5e+199) {
		tmp = pow(0.0, -1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (eps + (-1.0d0))
    if (x <= (-4.7d+95)) then
        tmp = 0.0d0 ** (-1.0d0)
    else if (x <= (-5.8d-245)) then
        tmp = (exp((eps * -x)) + (1.0d0 + t_0)) / 2.0d0
    else if (x <= 2.5d+55) then
        tmp = (exp(t_0) + (1.0d0 + (x * ((-1.0d0) - eps)))) / 2.0d0
    else if (x <= 1.2d+114) then
        tmp = 0.0d0
    else if (x <= 5d+199) then
        tmp = 0.0d0 ** (-1.0d0)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x * (eps + -1.0);
	double tmp;
	if (x <= -4.7e+95) {
		tmp = Math.pow(0.0, -1.0);
	} else if (x <= -5.8e-245) {
		tmp = (Math.exp((eps * -x)) + (1.0 + t_0)) / 2.0;
	} else if (x <= 2.5e+55) {
		tmp = (Math.exp(t_0) + (1.0 + (x * (-1.0 - eps)))) / 2.0;
	} else if (x <= 1.2e+114) {
		tmp = 0.0;
	} else if (x <= 5e+199) {
		tmp = Math.pow(0.0, -1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x * (eps + -1.0)
	tmp = 0
	if x <= -4.7e+95:
		tmp = math.pow(0.0, -1.0)
	elif x <= -5.8e-245:
		tmp = (math.exp((eps * -x)) + (1.0 + t_0)) / 2.0
	elif x <= 2.5e+55:
		tmp = (math.exp(t_0) + (1.0 + (x * (-1.0 - eps)))) / 2.0
	elif x <= 1.2e+114:
		tmp = 0.0
	elif x <= 5e+199:
		tmp = math.pow(0.0, -1.0)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x * Float64(eps + -1.0))
	tmp = 0.0
	if (x <= -4.7e+95)
		tmp = 0.0 ^ -1.0;
	elseif (x <= -5.8e-245)
		tmp = Float64(Float64(exp(Float64(eps * Float64(-x))) + Float64(1.0 + t_0)) / 2.0);
	elseif (x <= 2.5e+55)
		tmp = Float64(Float64(exp(t_0) + Float64(1.0 + Float64(x * Float64(-1.0 - eps)))) / 2.0);
	elseif (x <= 1.2e+114)
		tmp = 0.0;
	elseif (x <= 5e+199)
		tmp = 0.0 ^ -1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x * (eps + -1.0);
	tmp = 0.0;
	if (x <= -4.7e+95)
		tmp = 0.0 ^ -1.0;
	elseif (x <= -5.8e-245)
		tmp = (exp((eps * -x)) + (1.0 + t_0)) / 2.0;
	elseif (x <= 2.5e+55)
		tmp = (exp(t_0) + (1.0 + (x * (-1.0 - eps)))) / 2.0;
	elseif (x <= 1.2e+114)
		tmp = 0.0;
	elseif (x <= 5e+199)
		tmp = 0.0 ^ -1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.7e+95], N[Power[0.0, -1.0], $MachinePrecision], If[LessEqual[x, -5.8e-245], N[(N[(N[Exp[N[(eps * (-x)), $MachinePrecision]], $MachinePrecision] + N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 2.5e+55], N[(N[(N[Exp[t$95$0], $MachinePrecision] + N[(1.0 + N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.2e+114], 0.0, If[LessEqual[x, 5e+199], N[Power[0.0, -1.0], $MachinePrecision], 0.0]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.69999999999999972e95 or 1.2e114 < x < 4.9999999999999998e199

   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%

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

   5. Taylor expanded in eps around 0 10.7%

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

   7. Applied egg-rr 45.9%

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

   8. Taylor expanded in eps around 0 90.0%

      \[\leadsto {\color{blue}{0}}^{-1} \]

   if -4.69999999999999972e95 < x < -5.7999999999999999e-245

   1. Initial program 63.1%

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

   3. Simplified 43.3%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. neg-mul-1 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 77.7%

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

   if -5.7999999999999999e-245 < x < 2.50000000000000023e55

   1. Initial program 67.4%

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

   3. Simplified 48.2%

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

      1. pow-neg 48.2%

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

      2. associate-*l/ 48.2%

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

      3. *-un-lft-identity 48.2%

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

      4. add-sqr-sqrt 27.8%

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

      5. sqrt-unprod 46.8%

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

      6. frac-times 46.9%

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

      7. metadata-eval 46.9%

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

      8. metadata-eval 46.9%

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

      9. frac-times 46.8%

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

      10. sqrt-unprod 20.2%

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

      11. add-sqr-sqrt 44.7%

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

      12. add-sqr-sqrt 44.3%

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

      13. sqrt-unprod 58.4%

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

      14. sqr-neg 58.4%

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

      15. sqrt-unprod 0.9%

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

      16. add-sqr-sqrt 19.8%

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

      17. pow-exp 38.1%

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

      18. *-commutative 38.1%

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

      19. +-commutative 38.1%

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

      20. distribute-rgt-in 38.1%

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

   6. Applied egg-rr 63.9%

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

   7. Taylor expanded in eps around inf 99.9%

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

   8. Taylor expanded in x around 0 74.0%

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

   10. Simplified 74.0%

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

   if 2.50000000000000023e55 < x < 1.2e114 or 4.9999999999999998e199 < 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%

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

   5. Taylor expanded in eps around 0 77.0%

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

   6. Taylor expanded in x around 0 77.0%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+95}:\\ \;\;\;\;{0}^{-1}\\ \mathbf{elif}\;x \leq -5.8 \cdot 10^{-245}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot \left(-x\right)} + \left(1 + x \cdot \left(\varepsilon + -1\right)\right)}{2}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+55}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + \left(1 + x \cdot \left(-1 - \varepsilon\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+114}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+199}:\\ \;\;\;\;{0}^{-1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+95}:\\ \;\;\;\;{0}^{-1}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-75}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot \left(-x\right)} + \left(1 + x \cdot \left(\varepsilon + -1\right)\right)}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+199}:\\ \;\;\;\;{0}^{-1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -4.7e+95)
   (pow 0.0 -1.0)
   (if (<= x 3.6e-75)
     (/ (+ (exp (* eps (- x))) (+ 1.0 (* x (+ eps -1.0)))) 2.0)
     (if (<= x 5e+199) (pow 0.0 -1.0) 0.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -4.7e+95) {
		tmp = pow(0.0, -1.0);
	} else if (x <= 3.6e-75) {
		tmp = (exp((eps * -x)) + (1.0 + (x * (eps + -1.0)))) / 2.0;
	} else if (x <= 5e+199) {
		tmp = pow(0.0, -1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-4.7d+95)) then
        tmp = 0.0d0 ** (-1.0d0)
    else if (x <= 3.6d-75) then
        tmp = (exp((eps * -x)) + (1.0d0 + (x * (eps + (-1.0d0))))) / 2.0d0
    else if (x <= 5d+199) then
        tmp = 0.0d0 ** (-1.0d0)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -4.7e+95) {
		tmp = Math.pow(0.0, -1.0);
	} else if (x <= 3.6e-75) {
		tmp = (Math.exp((eps * -x)) + (1.0 + (x * (eps + -1.0)))) / 2.0;
	} else if (x <= 5e+199) {
		tmp = Math.pow(0.0, -1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -4.7e+95:
		tmp = math.pow(0.0, -1.0)
	elif x <= 3.6e-75:
		tmp = (math.exp((eps * -x)) + (1.0 + (x * (eps + -1.0)))) / 2.0
	elif x <= 5e+199:
		tmp = math.pow(0.0, -1.0)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -4.7e+95)
		tmp = 0.0 ^ -1.0;
	elseif (x <= 3.6e-75)
		tmp = Float64(Float64(exp(Float64(eps * Float64(-x))) + Float64(1.0 + Float64(x * Float64(eps + -1.0)))) / 2.0);
	elseif (x <= 5e+199)
		tmp = 0.0 ^ -1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -4.7e+95)
		tmp = 0.0 ^ -1.0;
	elseif (x <= 3.6e-75)
		tmp = (exp((eps * -x)) + (1.0 + (x * (eps + -1.0)))) / 2.0;
	elseif (x <= 5e+199)
		tmp = 0.0 ^ -1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -4.7e+95], N[Power[0.0, -1.0], $MachinePrecision], If[LessEqual[x, 3.6e-75], N[(N[(N[Exp[N[(eps * (-x)), $MachinePrecision]], $MachinePrecision] + N[(1.0 + N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5e+199], N[Power[0.0, -1.0], $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.69999999999999972e95 or 3.6e-75 < x < 4.9999999999999998e199

   1. Initial program 94.0%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in eps around 0 18.7%

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

   7. Applied egg-rr 36.7%

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

   8. Taylor expanded in eps around 0 74.9%

      \[\leadsto {\color{blue}{0}}^{-1} \]

   if -4.69999999999999972e95 < x < 3.6e-75

   1. Initial program 59.0%

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

   3. Simplified 33.5%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in eps around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. neg-mul-1 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 87.0%

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

   if 4.9999999999999998e199 < 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%

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

   5. Taylor expanded in eps around 0 70.5%

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

   6. Taylor expanded in x around 0 70.5%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+95}:\\ \;\;\;\;{0}^{-1}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-75}:\\ \;\;\;\;\frac{e^{\varepsilon \cdot \left(-x\right)} + \left(1 + x \cdot \left(\varepsilon + -1\right)\right)}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+199}:\\ \;\;\;\;{0}^{-1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-5}:\\ \;\;\;\;{0}^{-1}\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-239}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + \left(\varepsilon + \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon + -1}\right)\right)}{2}\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-82}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+203}:\\ \;\;\;\;{0}^{-1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.9e-5)
   (pow 0.0 -1.0)
   (if (<= x -3.2e-239)
     (/
      (+ 2.0 (* x (+ -1.0 (+ eps (/ (- 1.0 (* eps eps)) (+ eps -1.0))))))
      2.0)
     (if (<= x 4.9e-82) 1.0 (if (<= x 5.8e+203) (pow 0.0 -1.0) 0.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.9e-5) {
		tmp = pow(0.0, -1.0);
	} else if (x <= -3.2e-239) {
		tmp = (2.0 + (x * (-1.0 + (eps + ((1.0 - (eps * eps)) / (eps + -1.0)))))) / 2.0;
	} else if (x <= 4.9e-82) {
		tmp = 1.0;
	} else if (x <= 5.8e+203) {
		tmp = pow(0.0, -1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-1.9d-5)) then
        tmp = 0.0d0 ** (-1.0d0)
    else if (x <= (-3.2d-239)) then
        tmp = (2.0d0 + (x * ((-1.0d0) + (eps + ((1.0d0 - (eps * eps)) / (eps + (-1.0d0))))))) / 2.0d0
    else if (x <= 4.9d-82) then
        tmp = 1.0d0
    else if (x <= 5.8d+203) then
        tmp = 0.0d0 ** (-1.0d0)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.9e-5) {
		tmp = Math.pow(0.0, -1.0);
	} else if (x <= -3.2e-239) {
		tmp = (2.0 + (x * (-1.0 + (eps + ((1.0 - (eps * eps)) / (eps + -1.0)))))) / 2.0;
	} else if (x <= 4.9e-82) {
		tmp = 1.0;
	} else if (x <= 5.8e+203) {
		tmp = Math.pow(0.0, -1.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.9e-5:
		tmp = math.pow(0.0, -1.0)
	elif x <= -3.2e-239:
		tmp = (2.0 + (x * (-1.0 + (eps + ((1.0 - (eps * eps)) / (eps + -1.0)))))) / 2.0
	elif x <= 4.9e-82:
		tmp = 1.0
	elif x <= 5.8e+203:
		tmp = math.pow(0.0, -1.0)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.9e-5)
		tmp = 0.0 ^ -1.0;
	elseif (x <= -3.2e-239)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(eps + Float64(Float64(1.0 - Float64(eps * eps)) / Float64(eps + -1.0)))))) / 2.0);
	elseif (x <= 4.9e-82)
		tmp = 1.0;
	elseif (x <= 5.8e+203)
		tmp = 0.0 ^ -1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.9e-5)
		tmp = 0.0 ^ -1.0;
	elseif (x <= -3.2e-239)
		tmp = (2.0 + (x * (-1.0 + (eps + ((1.0 - (eps * eps)) / (eps + -1.0)))))) / 2.0;
	elseif (x <= 4.9e-82)
		tmp = 1.0;
	elseif (x <= 5.8e+203)
		tmp = 0.0 ^ -1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.9e-5], N[Power[0.0, -1.0], $MachinePrecision], If[LessEqual[x, -3.2e-239], N[(N[(2.0 + N[(x * N[(-1.0 + N[(eps + N[(N[(1.0 - N[(eps * eps), $MachinePrecision]), $MachinePrecision] / N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 4.9e-82], 1.0, If[LessEqual[x, 5.8e+203], N[Power[0.0, -1.0], $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.9000000000000001e-5 or 4.9000000000000003e-82 < x < 5.80000000000000021e203

   1. Initial program 93.9%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in eps around 0 16.5%

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

   7. Applied egg-rr 35.5%

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

   8. Taylor expanded in eps around 0 77.1%

      \[\leadsto {\color{blue}{0}}^{-1} \]

   if -1.9000000000000001e-5 < x < -3.1999999999999999e-239

   1. Initial program 53.6%

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

   3. Simplified 29.3%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 76.7%

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

      1. mul-1-neg 76.7%

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

      2. distribute-neg-in 76.7%

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

      3. metadata-eval 76.7%

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

      4. flip-+ 82.3%

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

      5. metadata-eval 82.3%

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

   8. Applied egg-rr 82.3%

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

   if -3.1999999999999999e-239 < x < 4.9000000000000003e-82

   1. Initial program 54.8%

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

   3. Simplified 21.9%

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

      1. pow-neg 21.9%

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

      2. associate-*l/ 21.9%

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

      3. *-un-lft-identity 21.9%

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

      4. add-sqr-sqrt 13.4%

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

      5. sqrt-unprod 20.7%

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

      6. frac-times 20.9%

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

      7. metadata-eval 20.9%

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

      8. metadata-eval 20.9%

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

      9. frac-times 20.7%

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

      10. sqrt-unprod 7.6%

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

      11. add-sqr-sqrt 17.1%

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

      12. add-sqr-sqrt 16.0%

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

      13. sqrt-unprod 41.2%

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

      14. sqr-neg 41.2%

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

      15. sqrt-unprod 1.6%

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

      16. add-sqr-sqrt 9.4%

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

      17. pow-exp 40.5%

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

      18. *-commutative 40.5%

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

      19. +-commutative 40.5%

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

      20. distribute-rgt-in 40.5%

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

   6. Applied egg-rr 50.0%

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

   7. Taylor expanded in x around 0 38.2%

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

      1. associate-*r/ 38.2%

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

      2. metadata-eval 38.2%

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

   9. Simplified 38.2%

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

   10. Taylor expanded in eps around 0 38.2%

       \[\leadsto \color{blue}{\frac{1 + \varepsilon}{\varepsilon}} \]

   11. Taylor expanded in eps around inf 88.2%

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

   if 5.80000000000000021e203 < 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%

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

   5. Taylor expanded in eps around 0 70.5%

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

   6. Taylor expanded in x around 0 70.5%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-5}:\\ \;\;\;\;{0}^{-1}\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-239}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + \left(\varepsilon + \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon + -1}\right)\right)}{2}\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-82}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+203}:\\ \;\;\;\;{0}^{-1}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.1% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-242}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + \left(\varepsilon + \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon + -1}\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+45}:\\ \;\;\;\;1 + x \cdot \varepsilon\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+121}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+203}:\\ \;\;\;\;x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -6.8e-242)
   (/ (+ 2.0 (* x (+ -1.0 (+ eps (/ (- 1.0 (* eps eps)) (+ eps -1.0)))))) 2.0)
   (if (<= x 1.1e+45)
     (+ 1.0 (* x eps))
     (if (<= x 1.55e+121) 0.0 (if (<= x 5.6e+203) (* x eps) 0.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -6.8e-242) {
		tmp = (2.0 + (x * (-1.0 + (eps + ((1.0 - (eps * eps)) / (eps + -1.0)))))) / 2.0;
	} else if (x <= 1.1e+45) {
		tmp = 1.0 + (x * eps);
	} else if (x <= 1.55e+121) {
		tmp = 0.0;
	} else if (x <= 5.6e+203) {
		tmp = x * eps;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-6.8d-242)) then
        tmp = (2.0d0 + (x * ((-1.0d0) + (eps + ((1.0d0 - (eps * eps)) / (eps + (-1.0d0))))))) / 2.0d0
    else if (x <= 1.1d+45) then
        tmp = 1.0d0 + (x * eps)
    else if (x <= 1.55d+121) then
        tmp = 0.0d0
    else if (x <= 5.6d+203) then
        tmp = x * eps
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -6.8e-242) {
		tmp = (2.0 + (x * (-1.0 + (eps + ((1.0 - (eps * eps)) / (eps + -1.0)))))) / 2.0;
	} else if (x <= 1.1e+45) {
		tmp = 1.0 + (x * eps);
	} else if (x <= 1.55e+121) {
		tmp = 0.0;
	} else if (x <= 5.6e+203) {
		tmp = x * eps;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -6.8e-242:
		tmp = (2.0 + (x * (-1.0 + (eps + ((1.0 - (eps * eps)) / (eps + -1.0)))))) / 2.0
	elif x <= 1.1e+45:
		tmp = 1.0 + (x * eps)
	elif x <= 1.55e+121:
		tmp = 0.0
	elif x <= 5.6e+203:
		tmp = x * eps
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -6.8e-242)
		tmp = Float64(Float64(2.0 + Float64(x * Float64(-1.0 + Float64(eps + Float64(Float64(1.0 - Float64(eps * eps)) / Float64(eps + -1.0)))))) / 2.0);
	elseif (x <= 1.1e+45)
		tmp = Float64(1.0 + Float64(x * eps));
	elseif (x <= 1.55e+121)
		tmp = 0.0;
	elseif (x <= 5.6e+203)
		tmp = Float64(x * eps);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -6.8e-242)
		tmp = (2.0 + (x * (-1.0 + (eps + ((1.0 - (eps * eps)) / (eps + -1.0)))))) / 2.0;
	elseif (x <= 1.1e+45)
		tmp = 1.0 + (x * eps);
	elseif (x <= 1.55e+121)
		tmp = 0.0;
	elseif (x <= 5.6e+203)
		tmp = x * eps;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -6.8e-242], N[(N[(2.0 + N[(x * N[(-1.0 + N[(eps + N[(N[(1.0 - N[(eps * eps), $MachinePrecision]), $MachinePrecision] / N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.1e+45], N[(1.0 + N[(x * eps), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.55e+121], 0.0, If[LessEqual[x, 5.6e+203], N[(x * eps), $MachinePrecision], 0.0]]]]

Derivation

1. Split input into 4 regimes

2. if x < -6.8000000000000001e-242

   1. Initial program 76.0%

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

   3. Simplified 63.5%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 42.2%

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

      1. mul-1-neg 42.2%

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

      2. distribute-neg-in 42.2%

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

      3. metadata-eval 42.2%

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

      4. flip-+ 53.7%

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

      5. metadata-eval 53.7%

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

   8. Applied egg-rr 53.7%

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

   if -6.8000000000000001e-242 < x < 1.1e45

   1. Initial program 65.5%

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

   3. Simplified 44.9%

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

   5. Taylor expanded in eps around inf 99.9%

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

   6. Taylor expanded in x around 0 62.9%

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

      1. associate--l+ 63.1%

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

      2. add-sqr-sqrt 12.1%

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

      3. sqrt-unprod 77.9%

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

      4. mul-1-neg 77.9%

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

      5. mul-1-neg 77.9%

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

      6. sqr-neg 77.9%

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

      7. sqrt-unprod 54.0%

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

      8. add-sqr-sqrt 64.4%

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

      9. add-exp-log 54.0%

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

      10. expm1-undefine 54.0%

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

      11. log1p-define 54.0%

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

      12. expm1-log1p-u 64.4%

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

   8. Applied egg-rr 64.4%

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

   9. Taylor expanded in x around 0 64.4%

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

       1. +-commutative 64.4%

          \[\leadsto \color{blue}{\varepsilon \cdot x + 1} \]

   11. Simplified 64.4%

       \[\leadsto \color{blue}{\varepsilon \cdot x + 1} \]

   if 1.1e45 < x < 1.55000000000000004e121 or 5.5999999999999998e203 < 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%

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

   5. Taylor expanded in eps around 0 71.9%

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

   6. Taylor expanded in x around 0 71.9%

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

   if 1.55000000000000004e121 < x < 5.5999999999999998e203

   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%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 0.7%

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

      1. associate--l+ 2.9%

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

      2. add-sqr-sqrt 4.8%

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

      3. sqrt-unprod 53.8%

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

      4. mul-1-neg 53.8%

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

      5. mul-1-neg 53.8%

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

      6. sqr-neg 53.8%

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

      7. sqrt-unprod 20.1%

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

      8. add-sqr-sqrt 20.1%

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

      9. add-exp-log 20.1%

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

      10. expm1-undefine 20.1%

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

      11. log1p-define 20.1%

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

      12. expm1-log1p-u 20.1%

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

   8. Applied egg-rr 20.1%

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

   9. Taylor expanded in x around inf 20.2%

      \[\leadsto \color{blue}{\varepsilon \cdot x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-242}:\\ \;\;\;\;\frac{2 + x \cdot \left(-1 + \left(\varepsilon + \frac{1 - \varepsilon \cdot \varepsilon}{\varepsilon + -1}\right)\right)}{2}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+45}:\\ \;\;\;\;1 + x \cdot \varepsilon\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+121}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+203}:\\ \;\;\;\;x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.0% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;1 - x\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+119}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+203}:\\ \;\;\;\;x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 1.0)
   (- 1.0 x)
   (if (<= x 1.65e+119) 0.0 (if (<= x 9.5e+203) (* x eps) 0.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 - x;
	} else if (x <= 1.65e+119) {
		tmp = 0.0;
	} else if (x <= 9.5e+203) {
		tmp = x * eps;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = 1.0d0 - x
    else if (x <= 1.65d+119) then
        tmp = 0.0d0
    else if (x <= 9.5d+203) then
        tmp = x * eps
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 - x;
	} else if (x <= 1.65e+119) {
		tmp = 0.0;
	} else if (x <= 9.5e+203) {
		tmp = x * eps;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 1.0:
		tmp = 1.0 - x
	elif x <= 1.65e+119:
		tmp = 0.0
	elif x <= 9.5e+203:
		tmp = x * eps
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(1.0 - x);
	elseif (x <= 1.65e+119)
		tmp = 0.0;
	elseif (x <= 9.5e+203)
		tmp = Float64(x * eps);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = 1.0 - x;
	elseif (x <= 1.65e+119)
		tmp = 0.0;
	elseif (x <= 9.5e+203)
		tmp = x * eps;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 1.0], N[(1.0 - x), $MachinePrecision], If[LessEqual[x, 1.65e+119], 0.0, If[LessEqual[x, 9.5e+203], N[(x * eps), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 1

   1. Initial program 68.4%

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

   3. Simplified 50.5%

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

   5. Taylor expanded in eps around inf 99.9%

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

   6. Taylor expanded in x around 0 56.7%

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

   7. Taylor expanded in x around inf 56.2%

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

   9. Simplified 56.6%

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

   10. Taylor expanded in x around 0 56.7%

       \[\leadsto \color{blue}{1 + -1 \cdot x} \]
   11. Step-by-step derivation

       1. neg-mul-1 56.7%

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

       2. unsub-neg 56.7%

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

   12. Simplified 56.7%

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

   if 1 < x < 1.6500000000000001e119 or 9.4999999999999995e203 < 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%

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

   5. Taylor expanded in eps around 0 59.0%

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

   6. Taylor expanded in x around 0 59.0%

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

   if 1.6500000000000001e119 < x < 9.4999999999999995e203

   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%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 0.7%

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

      1. associate--l+ 2.9%

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

      2. add-sqr-sqrt 4.8%

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

      3. sqrt-unprod 53.8%

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

      4. mul-1-neg 53.8%

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

      5. mul-1-neg 53.8%

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

      6. sqr-neg 53.8%

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

      7. sqrt-unprod 20.1%

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

      8. add-sqr-sqrt 20.1%

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

      9. add-exp-log 20.1%

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

      10. expm1-undefine 20.1%

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

      11. log1p-define 20.1%

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

      12. expm1-log1p-u 20.1%

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

   8. Applied egg-rr 20.1%

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

   9. Taylor expanded in x around inf 20.2%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;1 - x\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+119}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+203}:\\ \;\;\;\;x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 54.9% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 105000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{+119}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{+204}:\\ \;\;\;\;x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 105000.0)
   1.0
   (if (<= x 9.6e+119) 0.0 (if (<= x 5.3e+204) (* x eps) 0.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 105000.0) {
		tmp = 1.0;
	} else if (x <= 9.6e+119) {
		tmp = 0.0;
	} else if (x <= 5.3e+204) {
		tmp = x * eps;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 105000.0d0) then
        tmp = 1.0d0
    else if (x <= 9.6d+119) then
        tmp = 0.0d0
    else if (x <= 5.3d+204) then
        tmp = x * eps
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 105000.0) {
		tmp = 1.0;
	} else if (x <= 9.6e+119) {
		tmp = 0.0;
	} else if (x <= 5.3e+204) {
		tmp = x * eps;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 105000.0:
		tmp = 1.0
	elif x <= 9.6e+119:
		tmp = 0.0
	elif x <= 5.3e+204:
		tmp = x * eps
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 105000.0)
		tmp = 1.0;
	elseif (x <= 9.6e+119)
		tmp = 0.0;
	elseif (x <= 5.3e+204)
		tmp = Float64(x * eps);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 105000.0)
		tmp = 1.0;
	elseif (x <= 9.6e+119)
		tmp = 0.0;
	elseif (x <= 5.3e+204)
		tmp = x * eps;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 105000.0], 1.0, If[LessEqual[x, 9.6e+119], 0.0, If[LessEqual[x, 5.3e+204], N[(x * eps), $MachinePrecision], 0.0]]]

Derivation

1. Split input into 3 regimes

2. if x < 105000

   1. Initial program 68.8%

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

   3. Simplified 51.1%

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

      1. pow-neg 51.1%

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

      2. associate-*l/ 51.1%

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

      3. *-un-lft-identity 51.1%

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

      4. add-sqr-sqrt 27.2%

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

      5. sqrt-unprod 50.5%

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

      6. frac-times 50.6%

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

      7. metadata-eval 50.6%

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

      8. metadata-eval 50.6%

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

      9. frac-times 50.5%

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

      10. sqrt-unprod 23.5%

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

      11. add-sqr-sqrt 48.6%

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

      12. add-sqr-sqrt 16.4%

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

      13. sqrt-unprod 45.8%

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

      14. sqr-neg 45.8%

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

      15. sqrt-unprod 17.4%

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

      16. add-sqr-sqrt 25.3%

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

      17. pow-exp 41.5%

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

      18. *-commutative 41.5%

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

      19. +-commutative 41.5%

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

      20. distribute-rgt-in 41.5%

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

   6. Applied egg-rr 66.0%

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

   7. Taylor expanded in x around 0 21.6%

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

      1. associate-*r/ 21.6%

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

      2. metadata-eval 21.6%

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

   9. Simplified 21.6%

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

   10. Taylor expanded in eps around 0 21.6%

       \[\leadsto \color{blue}{\frac{1 + \varepsilon}{\varepsilon}} \]

   11. Taylor expanded in eps around inf 55.6%

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

   if 105000 < x < 9.6e119 or 5.3e204 < 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%

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

   5. Taylor expanded in eps around 0 61.5%

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

   6. Taylor expanded in x around 0 61.5%

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

   if 9.6e119 < x < 5.3e204

   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%

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

   5. Taylor expanded in eps around inf 100.0%

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

   6. Taylor expanded in x around 0 0.7%

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

      1. associate--l+ 2.9%

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

      2. add-sqr-sqrt 4.8%

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

      3. sqrt-unprod 53.8%

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

      4. mul-1-neg 53.8%

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

      5. mul-1-neg 53.8%

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

      6. sqr-neg 53.8%

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

      7. sqrt-unprod 20.1%

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

      8. add-sqr-sqrt 20.1%

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

      9. add-exp-log 20.1%

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

      10. expm1-undefine 20.1%

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

      11. log1p-define 20.1%

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

      12. expm1-log1p-u 20.1%

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

   8. Applied egg-rr 20.1%

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

   9. Taylor expanded in x around inf 20.2%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 105000:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{+119}:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{+204}:\\ \;\;\;\;x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.6% accurate, 37.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 105000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (if (<= x 105000.0) 1.0 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 105000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 105000.0d0) then
        tmp = 1.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 105000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 105000.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 105000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 105000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 105000.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 105000

   1. Initial program 68.8%

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

   3. Simplified 51.1%

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

      1. pow-neg 51.1%

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

      2. associate-*l/ 51.1%

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

      3. *-un-lft-identity 51.1%

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

      4. add-sqr-sqrt 27.2%

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

      5. sqrt-unprod 50.5%

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

      6. frac-times 50.6%

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

      7. metadata-eval 50.6%

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

      8. metadata-eval 50.6%

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

      9. frac-times 50.5%

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

      10. sqrt-unprod 23.5%

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

      11. add-sqr-sqrt 48.6%

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

      12. add-sqr-sqrt 16.4%

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

      13. sqrt-unprod 45.8%

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

      14. sqr-neg 45.8%

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

      15. sqrt-unprod 17.4%

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

      16. add-sqr-sqrt 25.3%

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

      17. pow-exp 41.5%

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

      18. *-commutative 41.5%

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

      19. +-commutative 41.5%

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

      20. distribute-rgt-in 41.5%

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

   6. Applied egg-rr 66.0%

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

   7. Taylor expanded in x around 0 21.6%

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

      1. associate-*r/ 21.6%

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

      2. metadata-eval 21.6%

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

   9. Simplified 21.6%

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

   10. Taylor expanded in eps around 0 21.6%

       \[\leadsto \color{blue}{\frac{1 + \varepsilon}{\varepsilon}} \]

   11. Taylor expanded in eps around inf 55.6%

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

   if 105000 < 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%

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

   5. Taylor expanded in eps around 0 47.4%

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

   6. Taylor expanded in x around 0 47.4%

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

Alternative 13: 16.1% accurate, 227.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 77.7%

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

3. Simplified 65.0%

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

5. Taylor expanded in eps around 0 15.0%

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

6. Taylor expanded in x around 0 15.2%

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

Reproduce

herbie shell --seed 2024137 
(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))