NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.5% → 98.9%

Time: 14.7s

Alternatives: 11

Speedup: 1.1×

• 38.6% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.5% 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: 98.9% 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 67.0%

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

3. Simplified 61.4%

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

5. Taylor expanded in eps around inf 98.7%

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

6. Final simplification 98.7%

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

Alternative 2: 77.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-229}:\\ \;\;\;\;\frac{1 + e^{-x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{-x}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1e-229)
   (/ (+ 1.0 (exp (- (* x eps)))) 2.0)
   (/ (+ (exp (* x (+ eps -1.0))) (exp (- x))) 2.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1e-229) {
		tmp = (1.0 + exp(-(x * eps))) / 2.0;
	} else {
		tmp = (exp((x * (eps + -1.0))) + exp(-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 (x <= (-1d-229)) then
        tmp = (1.0d0 + exp(-(x * eps))) / 2.0d0
    else
        tmp = (exp((x * (eps + (-1.0d0)))) + exp(-x)) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1e-229) {
		tmp = (1.0 + Math.exp(-(x * eps))) / 2.0;
	} else {
		tmp = (Math.exp((x * (eps + -1.0))) + Math.exp(-x)) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1e-229:
		tmp = (1.0 + math.exp(-(x * eps))) / 2.0
	else:
		tmp = (math.exp((x * (eps + -1.0))) + math.exp(-x)) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1e-229)
		tmp = Float64(Float64(1.0 + exp(Float64(-Float64(x * eps)))) / 2.0);
	else
		tmp = Float64(Float64(exp(Float64(x * Float64(eps + -1.0))) + exp(Float64(-x))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1e-229)
		tmp = (1.0 + exp(-(x * eps))) / 2.0;
	else
		tmp = (exp((x * (eps + -1.0))) + exp(-x)) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1e-229], N[(N[(1.0 + N[Exp[(-N[(x * eps), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.00000000000000007e-229

   1. Initial program 66.0%

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

   3. Simplified 59.2%

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

   5. Taylor expanded in eps around inf 98.0%

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

   6. Taylor expanded in x around 0 73.7%

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

      1. sub-neg 73.7%

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

      2. metadata-eval 73.7%

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

      3. distribute-rgt-in 73.7%

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

      4. neg-mul-1 73.7%

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

      5. add-sqr-sqrt 73.7%

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

      6. sqrt-unprod 73.3%

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

      7. sqr-neg 73.3%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 73.7%

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

      10. +-commutative 73.7%

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

      11. *-commutative 73.7%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 70.8%

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

      14. sqr-neg 70.8%

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

      15. sqrt-unprod 72.7%

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

      16. add-sqr-sqrt 72.7%

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

      17. cancel-sign-sub-inv 72.7%

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

      18. *-commutative 72.7%

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

      19. sub-neg 72.7%

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

      20. *-commutative 72.7%

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

   8. Applied egg-rr 72.7%

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

      1. sub-neg 72.7%

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

   10. Simplified 72.7%

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

   11. Taylor expanded in eps around inf 72.8%

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

       1. associate-*r* 72.8%

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

       2. neg-mul-1 72.8%

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

       3. *-commutative 72.8%

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

   13. Simplified 72.8%

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

   if -1.00000000000000007e-229 < x

   1. Initial program 67.7%

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

   3. Simplified 62.8%

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

   5. Taylor expanded in eps around inf 99.1%

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

   6. Taylor expanded in eps around 0 84.2%

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

      1. rec-exp 84.2%

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

   8. Simplified 84.2%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-229}:\\ \;\;\;\;\frac{1 + e^{-x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \left(\varepsilon + -1\right)} + e^{-x}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-248}:\\ \;\;\;\;\frac{1 + e^{-x \cdot \varepsilon}}{2}\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{+85}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{2}{e^{x}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -6e-248)
   (/ (+ 1.0 (exp (- (* x eps)))) 2.0)
   (if (<= x 9.2e+85)
     (/ (+ 1.0 (exp (* x eps))) 2.0)
     (/ (* x (/ 2.0 (exp x))) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -6e-248) {
		tmp = (1.0 + exp(-(x * eps))) / 2.0;
	} else if (x <= 9.2e+85) {
		tmp = (1.0 + exp((x * eps))) / 2.0;
	} else {
		tmp = (x * (2.0 / exp(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 (x <= (-6d-248)) then
        tmp = (1.0d0 + exp(-(x * eps))) / 2.0d0
    else if (x <= 9.2d+85) then
        tmp = (1.0d0 + exp((x * eps))) / 2.0d0
    else
        tmp = (x * (2.0d0 / exp(x))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -6e-248) {
		tmp = (1.0 + Math.exp(-(x * eps))) / 2.0;
	} else if (x <= 9.2e+85) {
		tmp = (1.0 + Math.exp((x * eps))) / 2.0;
	} else {
		tmp = (x * (2.0 / Math.exp(x))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -6e-248:
		tmp = (1.0 + math.exp(-(x * eps))) / 2.0
	elif x <= 9.2e+85:
		tmp = (1.0 + math.exp((x * eps))) / 2.0
	else:
		tmp = (x * (2.0 / math.exp(x))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -6e-248)
		tmp = Float64(Float64(1.0 + exp(Float64(-Float64(x * eps)))) / 2.0);
	elseif (x <= 9.2e+85)
		tmp = Float64(Float64(1.0 + exp(Float64(x * eps))) / 2.0);
	else
		tmp = Float64(Float64(x * Float64(2.0 / exp(x))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -6e-248)
		tmp = (1.0 + exp(-(x * eps))) / 2.0;
	elseif (x <= 9.2e+85)
		tmp = (1.0 + exp((x * eps))) / 2.0;
	else
		tmp = (x * (2.0 / exp(x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -6e-248], N[(N[(1.0 + N[Exp[(-N[(x * eps), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 9.2e+85], N[(N[(1.0 + N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.00000000000000027e-248

   1. Initial program 66.0%

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

   3. Simplified 59.2%

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

   5. Taylor expanded in eps around inf 98.0%

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

   6. Taylor expanded in x around 0 73.7%

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

      1. sub-neg 73.7%

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

      2. metadata-eval 73.7%

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

      3. distribute-rgt-in 73.7%

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

      4. neg-mul-1 73.7%

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

      5. add-sqr-sqrt 73.7%

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

      6. sqrt-unprod 73.3%

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

      7. sqr-neg 73.3%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 73.7%

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

      10. +-commutative 73.7%

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

      11. *-commutative 73.7%

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

      12. add-sqr-sqrt 0.0%

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

      13. sqrt-unprod 70.8%

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

      14. sqr-neg 70.8%

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

      15. sqrt-unprod 72.7%

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

      16. add-sqr-sqrt 72.7%

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

      17. cancel-sign-sub-inv 72.7%

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

      18. *-commutative 72.7%

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

      19. sub-neg 72.7%

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

      20. *-commutative 72.7%

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

   8. Applied egg-rr 72.7%

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

      1. sub-neg 72.7%

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

   10. Simplified 72.7%

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

   11. Taylor expanded in eps around inf 72.8%

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

       1. associate-*r* 72.8%

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

       2. neg-mul-1 72.8%

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

       3. *-commutative 72.8%

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

   13. Simplified 72.8%

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

   if -6.00000000000000027e-248 < x < 9.1999999999999996e85

   1. Initial program 57.6%

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

   3. Simplified 51.2%

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

   5. Taylor expanded in eps around inf 98.8%

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

   6. Taylor expanded in x around 0 80.2%

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

   7. Taylor expanded in eps around inf 80.4%

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

      1. *-commutative 80.4%

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

   9. Simplified 80.4%

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

   if 9.1999999999999996e85 < 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 70.7%

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

   7. Simplified 70.7%

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

   8. Taylor expanded in eps around 0 70.7%

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

   9. Taylor expanded in x around inf 70.7%

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

       1. neg-mul-1 70.7%

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

       2. *-commutative 70.7%

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

       3. neg-mul-1 70.7%

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

       4. rec-exp 70.7%

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

       5. associate-*r* 70.7%

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

       6. *-commutative 70.7%

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

       7. associate-*r/ 70.7%

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

       8. metadata-eval 70.7%

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

   11. Simplified 70.7%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{-248}:\\ \;\;\;\;\frac{1 + e^{-x \cdot \varepsilon}}{2}\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{+85}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{2}{e^{x}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-243}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+85}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{2}{e^{x}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 8e-243)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (<= x 7e+85)
     (/ (+ 1.0 (exp (* x eps))) 2.0)
     (/ (* x (/ 2.0 (exp x))) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 8e-243) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 7e+85) {
		tmp = (1.0 + exp((x * eps))) / 2.0;
	} else {
		tmp = (x * (2.0 / exp(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 (x <= 8d-243) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if (x <= 7d+85) then
        tmp = (1.0d0 + exp((x * eps))) / 2.0d0
    else
        tmp = (x * (2.0d0 / exp(x))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 8e-243) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 7e+85) {
		tmp = (1.0 + Math.exp((x * eps))) / 2.0;
	} else {
		tmp = (x * (2.0 / Math.exp(x))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 8e-243:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 7e+85:
		tmp = (1.0 + math.exp((x * eps))) / 2.0
	else:
		tmp = (x * (2.0 / math.exp(x))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 8e-243)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 7e+85)
		tmp = Float64(Float64(1.0 + exp(Float64(x * eps))) / 2.0);
	else
		tmp = Float64(Float64(x * Float64(2.0 / exp(x))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 8e-243)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif (x <= 7e+85)
		tmp = (1.0 + exp((x * eps))) / 2.0;
	else
		tmp = (x * (2.0 / exp(x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 8e-243], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 7e+85], N[(N[(1.0 + N[Exp[N[(x * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 7.99999999999999996e-243

   1. Initial program 61.9%

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

   3. Simplified 56.7%

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

   5. Taylor expanded in eps around inf 98.5%

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

   6. Taylor expanded in x around 0 80.4%

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

   7. Taylor expanded in eps around 0 87.8%

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

      1. neg-mul-1 87.8%

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

   9. Simplified 87.8%

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

   if 7.99999999999999996e-243 < x < 7.0000000000000001e85

   1. Initial program 60.8%

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

   3. Simplified 51.9%

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

   5. Taylor expanded in eps around inf 98.3%

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

   6. Taylor expanded in x around 0 72.2%

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

   7. Taylor expanded in eps around inf 72.5%

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

      1. *-commutative 72.5%

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

   9. Simplified 72.5%

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

   if 7.0000000000000001e85 < 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 70.7%

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

   7. Simplified 70.7%

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

   8. Taylor expanded in eps around 0 70.7%

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

   9. Taylor expanded in x around inf 70.7%

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

       1. neg-mul-1 70.7%

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

       2. *-commutative 70.7%

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

       3. neg-mul-1 70.7%

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

       4. rec-exp 70.7%

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

       5. associate-*r* 70.7%

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

       6. *-commutative 70.7%

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

       7. associate-*r/ 70.7%

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

       8. metadata-eval 70.7%

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

   11. Simplified 70.7%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-243}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+85}:\\ \;\;\;\;\frac{1 + e^{x \cdot \varepsilon}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{2}{e^{x}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-49}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+84}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{2}{e^{x}}}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 5e-49)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (<= x 3e+84) (/ (+ 1.0 (exp x)) 2.0) (/ (* x (/ 2.0 (exp x))) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 5e-49) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 3e+84) {
		tmp = (1.0 + exp(x)) / 2.0;
	} else {
		tmp = (x * (2.0 / exp(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 (x <= 5d-49) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if (x <= 3d+84) then
        tmp = (1.0d0 + exp(x)) / 2.0d0
    else
        tmp = (x * (2.0d0 / exp(x))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 5e-49) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 3e+84) {
		tmp = (1.0 + Math.exp(x)) / 2.0;
	} else {
		tmp = (x * (2.0 / Math.exp(x))) / 2.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 5e-49:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 3e+84:
		tmp = (1.0 + math.exp(x)) / 2.0
	else:
		tmp = (x * (2.0 / math.exp(x))) / 2.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 5e-49)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 3e+84)
		tmp = Float64(Float64(1.0 + exp(x)) / 2.0);
	else
		tmp = Float64(Float64(x * Float64(2.0 / exp(x))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 5e-49)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif (x <= 3e+84)
		tmp = (1.0 + exp(x)) / 2.0;
	else
		tmp = (x * (2.0 / exp(x))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 5e-49], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 3e+84], N[(N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(x * N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 4.9999999999999999e-49

   1. Initial program 57.9%

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

   3. Simplified 51.5%

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

   5. Taylor expanded in eps around inf 98.9%

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

   6. Taylor expanded in x around 0 81.8%

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

   7. Taylor expanded in eps around 0 84.6%

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

      1. neg-mul-1 84.6%

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

   9. Simplified 84.6%

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

   if 4.9999999999999999e-49 < x < 2.99999999999999996e84

   1. Initial program 83.9%

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

   3. Simplified 76.1%

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

   5. Taylor expanded in eps around inf 95.3%

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

   6. Taylor expanded in x around 0 48.9%

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

      1. sub-neg 48.9%

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

      2. metadata-eval 48.9%

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

      3. distribute-rgt-in 48.9%

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

      4. neg-mul-1 48.9%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 48.7%

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

      7. sqr-neg 48.7%

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

      8. sqrt-unprod 48.7%

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

      9. add-sqr-sqrt 48.7%

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

      10. +-commutative 48.7%

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

      11. *-commutative 48.7%

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

      12. add-sqr-sqrt 48.7%

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

      13. sqrt-unprod 48.7%

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

      14. sqr-neg 48.7%

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

      15. sqrt-unprod 0.0%

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

      16. add-sqr-sqrt 45.5%

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

      17. cancel-sign-sub-inv 45.5%

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

      18. *-commutative 45.5%

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

      19. sub-neg 45.5%

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

      20. *-commutative 45.5%

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

   8. Applied egg-rr 45.5%

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

      1. sub-neg 45.5%

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

   10. Simplified 45.5%

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

   11. Taylor expanded in eps around 0 56.5%

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

   if 2.99999999999999996e84 < 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 70.7%

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

   7. Simplified 70.7%

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

   8. Taylor expanded in eps around 0 70.7%

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

   9. Taylor expanded in x around inf 70.7%

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

       1. neg-mul-1 70.7%

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

       2. *-commutative 70.7%

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

       3. neg-mul-1 70.7%

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

       4. rec-exp 70.7%

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

       5. associate-*r* 70.7%

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

       6. *-commutative 70.7%

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

       7. associate-*r/ 70.7%

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

       8. metadata-eval 70.7%

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

   11. Simplified 70.7%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-49}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+84}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{2}{e^{x}}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-49}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+85}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 5e-49)
   (/ (+ 1.0 (exp (- x))) 2.0)
   (if (<= x 1.5e+85) (/ (+ 1.0 (exp x)) 2.0) 0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 5e-49) {
		tmp = (1.0 + exp(-x)) / 2.0;
	} else if (x <= 1.5e+85) {
		tmp = (1.0 + exp(x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 5d-49) then
        tmp = (1.0d0 + exp(-x)) / 2.0d0
    else if (x <= 1.5d+85) then
        tmp = (1.0d0 + exp(x)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 5e-49) {
		tmp = (1.0 + Math.exp(-x)) / 2.0;
	} else if (x <= 1.5e+85) {
		tmp = (1.0 + Math.exp(x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 5e-49:
		tmp = (1.0 + math.exp(-x)) / 2.0
	elif x <= 1.5e+85:
		tmp = (1.0 + math.exp(x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 5e-49)
		tmp = Float64(Float64(1.0 + exp(Float64(-x))) / 2.0);
	elseif (x <= 1.5e+85)
		tmp = Float64(Float64(1.0 + exp(x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 5e-49)
		tmp = (1.0 + exp(-x)) / 2.0;
	elseif (x <= 1.5e+85)
		tmp = (1.0 + exp(x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 5e-49], N[(N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 1.5e+85], N[(N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < 4.9999999999999999e-49

   1. Initial program 57.9%

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

   3. Simplified 51.5%

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

   5. Taylor expanded in eps around inf 98.9%

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

   6. Taylor expanded in x around 0 81.8%

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

   7. Taylor expanded in eps around 0 84.6%

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

      1. neg-mul-1 84.6%

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

   9. Simplified 84.6%

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

   if 4.9999999999999999e-49 < x < 1.5e85

   1. Initial program 83.9%

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

   3. Simplified 76.1%

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

   5. Taylor expanded in eps around inf 95.3%

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

   6. Taylor expanded in x around 0 48.9%

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

      1. sub-neg 48.9%

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

      2. metadata-eval 48.9%

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

      3. distribute-rgt-in 48.9%

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

      4. neg-mul-1 48.9%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 48.7%

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

      7. sqr-neg 48.7%

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

      8. sqrt-unprod 48.7%

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

      9. add-sqr-sqrt 48.7%

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

      10. +-commutative 48.7%

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

      11. *-commutative 48.7%

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

      12. add-sqr-sqrt 48.7%

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

      13. sqrt-unprod 48.7%

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

      14. sqr-neg 48.7%

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

      15. sqrt-unprod 0.0%

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

      16. add-sqr-sqrt 45.5%

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

      17. cancel-sign-sub-inv 45.5%

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

      18. *-commutative 45.5%

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

      19. sub-neg 45.5%

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

      20. *-commutative 45.5%

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

   8. Applied egg-rr 45.5%

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

      1. sub-neg 45.5%

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

   10. Simplified 45.5%

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

   11. Taylor expanded in eps around 0 56.5%

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

   if 1.5e85 < 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 70.7%

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

      1. mul-1-neg 70.7%

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

      2. mul-1-neg 70.7%

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

      3. rec-exp 70.7%

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

      4. sub-neg 70.7%

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

      5. div-sub 70.7%

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

      6. mul-1-neg 70.7%

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

      7. rec-exp 70.7%

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

      8. +-inverses 70.7%

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

   7. Simplified 70.7%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-49}:\\ \;\;\;\;\frac{1 + e^{-x}}{2}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+85}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-237}:\\ \;\;\;\;\frac{2 - x \cdot \left(\varepsilon + 2\right)}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+84}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.4e-237)
   (/ (- 2.0 (* x (+ eps 2.0))) 2.0)
   (if (<= x 5e+84) (/ (+ 1.0 (exp x)) 2.0) 0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.4e-237) {
		tmp = (2.0 - (x * (eps + 2.0))) / 2.0;
	} else if (x <= 5e+84) {
		tmp = (1.0 + exp(x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-3.4d-237)) then
        tmp = (2.0d0 - (x * (eps + 2.0d0))) / 2.0d0
    else if (x <= 5d+84) then
        tmp = (1.0d0 + exp(x)) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -3.4e-237) {
		tmp = (2.0 - (x * (eps + 2.0))) / 2.0;
	} else if (x <= 5e+84) {
		tmp = (1.0 + Math.exp(x)) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -3.4e-237:
		tmp = (2.0 - (x * (eps + 2.0))) / 2.0
	elif x <= 5e+84:
		tmp = (1.0 + math.exp(x)) / 2.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.4e-237)
		tmp = Float64(Float64(2.0 - Float64(x * Float64(eps + 2.0))) / 2.0);
	elseif (x <= 5e+84)
		tmp = Float64(Float64(1.0 + exp(x)) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -3.4e-237)
		tmp = (2.0 - (x * (eps + 2.0))) / 2.0;
	elseif (x <= 5e+84)
		tmp = (1.0 + exp(x)) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.4e-237], N[(N[(2.0 - N[(x * N[(eps + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], If[LessEqual[x, 5e+84], N[(N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -3.4000000000000002e-237

   1. Initial program 66.0%

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

   3. Simplified 59.2%

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

   5. Taylor expanded in eps around inf 98.0%

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

   6. Taylor expanded in eps around 0 90.2%

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

      1. neg-mul-1 83.5%

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

   8. Simplified 90.2%

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

   9. Taylor expanded in x around 0 61.5%

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

       1. mul-1-neg 61.5%

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

       2. unsub-neg 61.5%

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

   11. Simplified 61.5%

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

   if -3.4000000000000002e-237 < x < 5.0000000000000001e84

   1. Initial program 57.6%

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

   3. Simplified 51.2%

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

   5. Taylor expanded in eps around inf 98.8%

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

   6. Taylor expanded in x around 0 80.2%

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

      1. sub-neg 80.2%

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

      2. metadata-eval 80.2%

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

      3. distribute-rgt-in 80.2%

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

      4. neg-mul-1 80.2%

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

      5. add-sqr-sqrt 20.2%

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

      6. sqrt-unprod 80.2%

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

      7. sqr-neg 80.2%

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

      8. sqrt-unprod 60.0%

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

      9. add-sqr-sqrt 80.2%

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

      10. +-commutative 80.2%

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

      11. *-commutative 80.2%

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

      12. add-sqr-sqrt 60.0%

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

      13. sqrt-unprod 79.7%

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

      14. sqr-neg 79.7%

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

      15. sqrt-unprod 20.2%

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

      16. add-sqr-sqrt 81.8%

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

      17. cancel-sign-sub-inv 81.8%

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

      18. *-commutative 81.8%

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

      19. sub-neg 81.8%

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

      20. *-commutative 81.8%

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

   8. Applied egg-rr 81.8%

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

      1. sub-neg 81.8%

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

   10. Simplified 81.8%

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

   11. Taylor expanded in eps around 0 78.5%

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

   if 5.0000000000000001e84 < 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 70.7%

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

      1. mul-1-neg 70.7%

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

      2. mul-1-neg 70.7%

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

      3. rec-exp 70.7%

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

      4. sub-neg 70.7%

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

      5. div-sub 70.7%

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

      6. mul-1-neg 70.7%

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

      7. rec-exp 70.7%

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

      8. +-inverses 70.7%

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

   7. Simplified 70.7%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-237}:\\ \;\;\;\;\frac{2 - x \cdot \left(\varepsilon + 2\right)}{2}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+84}:\\ \;\;\;\;\frac{1 + e^{x}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.9% accurate, 16.2× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	double tmp;
	if (x <= 1.0) {
		tmp = (2.0 - (x * (eps + 2.0))) / 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 <= 1.0d0) then
        tmp = (2.0d0 - (x * (eps + 2.0d0))) / 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 <= 1.0) {
		tmp = (2.0 - (x * (eps + 2.0))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 58.2%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in eps around inf 98.3%

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

   6. Taylor expanded in eps around 0 89.7%

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

      1. neg-mul-1 81.5%

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

   8. Simplified 89.7%

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

   9. Taylor expanded in x around 0 70.2%

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

       1. mul-1-neg 70.2%

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

       2. unsub-neg 70.2%

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

   11. Simplified 70.2%

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

   if 1 < 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 58.1%

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

      1. mul-1-neg 58.1%

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

      2. mul-1-neg 58.1%

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

      3. rec-exp 58.1%

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

      4. sub-neg 58.1%

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

      5. div-sub 58.1%

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

      6. mul-1-neg 58.1%

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

      7. rec-exp 58.1%

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

      8. +-inverses 58.1%

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

   7. Simplified 58.1%

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

4. Final simplification 67.6%

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

Alternative 9: 59.9% accurate, 20.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{-x \cdot \varepsilon}{2}\\ \mathbf{elif}\;x \leq 105000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.8e-15) (/ (- (* x eps)) 2.0) (if (<= x 105000000.0) 1.0 0.0)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.8e-15) {
		tmp = -(x * eps) / 2.0;
	} else if (x <= 105000000.0) {
		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 <= (-2.8d-15)) then
        tmp = -(x * eps) / 2.0d0
    else if (x <= 105000000.0d0) 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 <= -2.8e-15) {
		tmp = -(x * eps) / 2.0;
	} else if (x <= 105000000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.8e-15:
		tmp = -(x * eps) / 2.0
	elif x <= 105000000.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.8e-15)
		tmp = Float64(Float64(-Float64(x * eps)) / 2.0);
	elseif (x <= 105000000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.8e-15)
		tmp = -(x * eps) / 2.0;
	elseif (x <= 105000000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.8e-15], N[((-N[(x * eps), $MachinePrecision]) / 2.0), $MachinePrecision], If[LessEqual[x, 105000000.0], 1.0, 0.0]]

Derivation

1. Split input into 3 regimes

2. if x < -2.80000000000000014e-15

   1. Initial program 95.1%

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

   3. Simplified 95.1%

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

   5. Taylor expanded in eps around inf 95.1%

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

   6. Taylor expanded in x around 0 59.3%

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

      1. associate-*r* 59.3%

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

      2. neg-mul-1 59.3%

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

   8. Simplified 59.3%

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

   9. Taylor expanded in eps around inf 35.1%

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

       1. mul-1-neg 35.1%

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

       2. distribute-lft-neg-out 35.1%

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

       3. *-commutative 35.1%

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

   11. Simplified 35.1%

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

   if -2.80000000000000014e-15 < x < 1.05e8

   1. Initial program 49.8%

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

   3. Simplified 49.8%

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

   5. Taylor expanded in x around 0 78.6%

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

   if 1.05e8 < 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 61.4%

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

      1. mul-1-neg 61.4%

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

      2. mul-1-neg 61.4%

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

      3. rec-exp 61.4%

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

      4. sub-neg 61.4%

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

      5. div-sub 61.4%

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

      6. mul-1-neg 61.4%

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

      7. rec-exp 61.4%

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

      8. +-inverses 61.4%

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

   7. Simplified 61.4%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{-x \cdot \varepsilon}{2}\\ \mathbf{elif}\;x \leq 105000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.6% accurate, 37.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 105000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (if (<= x 105000000.0) 1.0 0.0))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 105000000.0) {
		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 <= 105000000.0d0) 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 <= 105000000.0) {
		tmp = 1.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 105000000.0:
		tmp = 1.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 105000000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 105000000.0)
		tmp = 1.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 105000000.0], 1.0, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.05e8

   1. Initial program 58.8%

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

   3. Simplified 58.8%

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

   5. Taylor expanded in x around 0 63.5%

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

   if 1.05e8 < 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 61.4%

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

      1. mul-1-neg 61.4%

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

      2. mul-1-neg 61.4%

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

      3. rec-exp 61.4%

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

      4. sub-neg 61.4%

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

      5. div-sub 61.4%

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

      6. mul-1-neg 61.4%

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

      7. rec-exp 61.4%

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

      8. +-inverses 61.4%

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

   7. Simplified 61.4%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 105000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 16.0% 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 67.0%

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

3. Simplified 52.2%

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

5. Taylor expanded in eps around 0 14.0%

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

   1. mul-1-neg 14.0%

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

   2. mul-1-neg 14.0%

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

   3. rec-exp 14.0%

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

   4. sub-neg 14.0%

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

   5. div-sub 14.0%

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

   6. mul-1-neg 14.0%

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

   7. rec-exp 14.0%

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

   8. +-inverses 14.3%

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

7. Simplified 14.3%

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

8. Final simplification 14.3%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024100 
(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))