Quotient of sum of exps

Percentage Accurate: 99.0% → 100.0%

Time: 8.2s

Alternatives: 17

Speedup: 2.9×

• could not determine a ground truth

  1. a = -2.00000000000000007e154
  2. b = -2.00000000000000007e154

Specification

Math

\[\begin{array}{l} \\ \frac{e^{a}}{e^{a} + e^{b}} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ (exp a) (+ (exp a) (exp b))))

C

double code(double a, double b) {
	return exp(a) / (exp(a) + exp(b));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = exp(a) / (exp(a) + exp(b))
end function

Java

public static double code(double a, double b) {
	return Math.exp(a) / (Math.exp(a) + Math.exp(b));
}

Python

def code(a, b):
	return math.exp(a) / (math.exp(a) + math.exp(b))

Julia

function code(a, b)
	return Float64(exp(a) / Float64(exp(a) + exp(b)))
end

MATLAB

function tmp = code(a, b)
	tmp = exp(a) / (exp(a) + exp(b));
end

Wolfram

code[a_, b_] := N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{a}}{e^{a} + e^{b}} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ (exp a) (+ (exp a) (exp b))))

C

double code(double a, double b) {
	return exp(a) / (exp(a) + exp(b));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = exp(a) / (exp(a) + exp(b))
end function

Java

public static double code(double a, double b) {
	return Math.exp(a) / (Math.exp(a) + Math.exp(b));
}

Python

def code(a, b):
	return math.exp(a) / (math.exp(a) + math.exp(b))

Julia

function code(a, b)
	return Float64(exp(a) / Float64(exp(a) + exp(b)))
end

MATLAB

function tmp = code(a, b)
	tmp = exp(a) / (exp(a) + exp(b));
end

Wolfram

code[a_, b_] := N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{-\mathsf{log1p}\left(e^{b - a}\right)} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (exp (- (log1p (exp (- b a))))))

C

double code(double a, double b) {
	return exp(-log1p(exp((b - a))));
}

Java

public static double code(double a, double b) {
	return Math.exp(-Math.log1p(Math.exp((b - a))));
}

Python

def code(a, b):
	return math.exp(-math.log1p(math.exp((b - a))))

Julia

function code(a, b)
	return exp(Float64(-log1p(exp(Float64(b - a)))))
end

Wolfram

code[a_, b_] := N[Exp[(-N[Log[1 + N[Exp[N[(b - a), $MachinePrecision]], $MachinePrecision]], $MachinePrecision])], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
2. Step-by-step derivation

   1. *-lft-identity 100.0%

      \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

   2. associate-*l/ 100.0%

      \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

   3. associate-/r/ 100.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

   4. remove-double-neg 100.0%

      \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

   5. unsub-neg 100.0%

      \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

   6. div-sub 71.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

   7. *-lft-identity 71.8%

      \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

   8. associate-*l/ 71.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

   9. lft-mult-inverse 100.0%

      \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

   10. sub-neg 100.0%

       \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

   11. distribute-frac-neg 100.0%

       \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

   12. remove-double-neg 100.0%

       \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

   13. div-exp 100.0%

       \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-exp-log 100.0%

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{1 + e^{b - a}}\right)}} \]

   2. log-rec 100.0%

      \[\leadsto e^{\color{blue}{-\log \left(1 + e^{b - a}\right)}} \]

   3. log1p-define 100.0%

      \[\leadsto e^{-\color{blue}{\mathsf{log1p}\left(e^{b - a}\right)}} \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{e^{-\mathsf{log1p}\left(e^{b - a}\right)}} \]
7. Add Preprocessing

Alternative 2: 94.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(a \cdot -0.16666666666666666\right) + -1\right)}\\ \mathbf{elif}\;a \leq -3100:\\ \;\;\;\;\frac{1}{1 + \frac{\frac{\frac{\frac{-1}{b} + -1}{b} + -1}{b} + -1}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1e+103)
   (/ 1.0 (+ 2.0 (* a (+ (* a (* a -0.16666666666666666)) -1.0))))
   (if (<= a -3100.0)
     (/ 1.0 (+ 1.0 (/ (+ (/ (+ (/ (+ (/ -1.0 b) -1.0) b) -1.0) b) -1.0) b)))
     (/ 1.0 (+ 1.0 (exp b))))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1e+103) {
		tmp = 1.0 / (2.0 + (a * ((a * (a * -0.16666666666666666)) + -1.0)));
	} else if (a <= -3100.0) {
		tmp = 1.0 / (1.0 + (((((((-1.0 / b) + -1.0) / b) + -1.0) / b) + -1.0) / b));
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-1d+103)) then
        tmp = 1.0d0 / (2.0d0 + (a * ((a * (a * (-0.16666666666666666d0))) + (-1.0d0))))
    else if (a <= (-3100.0d0)) then
        tmp = 1.0d0 / (1.0d0 + ((((((((-1.0d0) / b) + (-1.0d0)) / b) + (-1.0d0)) / b) + (-1.0d0)) / b))
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -1e+103) {
		tmp = 1.0 / (2.0 + (a * ((a * (a * -0.16666666666666666)) + -1.0)));
	} else if (a <= -3100.0) {
		tmp = 1.0 / (1.0 + (((((((-1.0 / b) + -1.0) / b) + -1.0) / b) + -1.0) / b));
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -1e+103:
		tmp = 1.0 / (2.0 + (a * ((a * (a * -0.16666666666666666)) + -1.0)))
	elif a <= -3100.0:
		tmp = 1.0 / (1.0 + (((((((-1.0 / b) + -1.0) / b) + -1.0) / b) + -1.0) / b))
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1e+103)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * Float64(a * -0.16666666666666666)) + -1.0))));
	elseif (a <= -3100.0)
		tmp = Float64(1.0 / Float64(1.0 + Float64(Float64(Float64(Float64(Float64(Float64(Float64(-1.0 / b) + -1.0) / b) + -1.0) / b) + -1.0) / b)));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1e+103)
		tmp = 1.0 / (2.0 + (a * ((a * (a * -0.16666666666666666)) + -1.0)));
	elseif (a <= -3100.0)
		tmp = 1.0 / (1.0 + (((((((-1.0 / b) + -1.0) / b) + -1.0) / b) + -1.0) / b));
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1e+103], N[(1.0 / N[(2.0 + N[(a * N[(N[(a * N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -3100.0], N[(1.0 / N[(1.0 + N[(N[(N[(N[(N[(N[(N[(-1.0 / b), $MachinePrecision] + -1.0), $MachinePrecision] / b), $MachinePrecision] + -1.0), $MachinePrecision] / b), $MachinePrecision] + -1.0), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1e103

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 100.0%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 0.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 0.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 0.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 100.0%

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

   6. Taylor expanded in a around 0 100.0%

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

   7. Taylor expanded in a around inf 100.0%

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

      1. *-commutative 100.0%

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

   9. Simplified 100.0%

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

   if -1e103 < a < -3100

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 100.0%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 0.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 0.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 0.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. exp-diff 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{e^{b}}{e^{a}}}} \]

      2. clear-num 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   7. Taylor expanded in a around 0 28.6%

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

      1. rec-exp 28.6%

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

   9. Simplified 28.6%

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

   10. Taylor expanded in b around 0 3.1%

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

       1. mul-1-neg 3.1%

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

       2. unsub-neg 3.1%

          \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 - b}}} \]

   12. Simplified 3.1%

       \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 - b}}} \]

   13. Taylor expanded in b around inf 54.5%

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

   15. Simplified 54.5%

       \[\leadsto \frac{1}{1 + \color{blue}{\frac{\frac{-1 + \frac{-1 + \frac{-1}{b}}{b}}{b} - 1}{b}}} \]

   if -3100 < a

   1. Initial program 99.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 99.9%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 99.9%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 99.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 99.9%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 99.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 99.4%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 99.4%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 99.4%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 99.9%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 99.9%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 99.9%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 99.9%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 99.9%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 99.5%

      \[\leadsto \frac{1}{\color{blue}{1 + e^{b}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(a \cdot -0.16666666666666666\right) + -1\right)}\\ \mathbf{elif}\;a \leq -3100:\\ \;\;\;\;\frac{1}{1 + \frac{\frac{\frac{\frac{-1}{b} + -1}{b} + -1}{b} + -1}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -8000:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -8000.0) (/ 1.0 (+ 1.0 (exp (- a)))) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -8000.0) {
		tmp = 1.0 / (1.0 + exp(-a));
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-8000.0d0)) then
        tmp = 1.0d0 / (1.0d0 + exp(-a))
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -8000.0) {
		tmp = 1.0 / (1.0 + Math.exp(-a));
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -8000.0:
		tmp = 1.0 / (1.0 + math.exp(-a))
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -8000.0)
		tmp = Float64(1.0 / Float64(1.0 + exp(Float64(-a))));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -8000.0)
		tmp = 1.0 / (1.0 + exp(-a));
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -8000.0], N[(1.0 / N[(1.0 + N[Exp[(-a)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -8e3

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 100.0%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 0.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 0.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 0.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 100.0%

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

   if -8e3 < a

   1. Initial program 99.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 99.9%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 99.9%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 99.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 99.9%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 99.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 99.4%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 99.4%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 99.4%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 99.9%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 99.9%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 99.9%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 99.9%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 99.9%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 99.5%

      \[\leadsto \frac{1}{\color{blue}{1 + e^{b}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 100.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{e^{b - a} + 1} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ 1.0 (+ (exp (- b a)) 1.0)))

C

double code(double a, double b) {
	return 1.0 / (exp((b - a)) + 1.0);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 / (exp((b - a)) + 1.0d0)
end function

Java

public static double code(double a, double b) {
	return 1.0 / (Math.exp((b - a)) + 1.0);
}

Python

def code(a, b):
	return 1.0 / (math.exp((b - a)) + 1.0)

Julia

function code(a, b)
	return Float64(1.0 / Float64(exp(Float64(b - a)) + 1.0))
end

MATLAB

function tmp = code(a, b)
	tmp = 1.0 / (exp((b - a)) + 1.0);
end

Wolfram

code[a_, b_] := N[(1.0 / N[(N[Exp[N[(b - a), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
2. Step-by-step derivation

   1. *-lft-identity 100.0%

      \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

   2. associate-*l/ 100.0%

      \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

   3. associate-/r/ 100.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

   4. remove-double-neg 100.0%

      \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

   5. unsub-neg 100.0%

      \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

   6. div-sub 71.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

   7. *-lft-identity 71.8%

      \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

   8. associate-*l/ 71.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

   9. lft-mult-inverse 100.0%

      \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

   10. sub-neg 100.0%

       \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

   11. distribute-frac-neg 100.0%

       \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

   12. remove-double-neg 100.0%

       \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

   13. div-exp 100.0%

       \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
4. Add Preprocessing

5. Final simplification 100.0%

   \[\leadsto \frac{1}{e^{b - a} + 1} \]
6. Add Preprocessing

Alternative 5: 85.5% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{-16}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + b \cdot \left(b \cdot \left(0.5 + b \cdot -0.16666666666666666\right) + -1\right)}}\\ \mathbf{elif}\;b \leq 2.45 \cdot 10^{+67}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(a \cdot -0.16666666666666666 + 0.5\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(b + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -4.6e-16)
   (/
    1.0
    (+
     1.0
     (/ 1.0 (+ 1.0 (* b (+ (* b (+ 0.5 (* b -0.16666666666666666))) -1.0))))))
   (if (<= b 2.45e+67)
     (/ 1.0 (+ 2.0 (* a (+ (* a (+ (* a -0.16666666666666666) 0.5)) -1.0))))
     (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b (+ b 1.0)))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -4.6e-16) {
		tmp = 1.0 / (1.0 + (1.0 / (1.0 + (b * ((b * (0.5 + (b * -0.16666666666666666))) + -1.0)))));
	} else if (b <= 2.45e+67) {
		tmp = 1.0 / (2.0 + (a * ((a * ((a * -0.16666666666666666) + 0.5)) + -1.0)));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (b + 1.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-4.6d-16)) then
        tmp = 1.0d0 / (1.0d0 + (1.0d0 / (1.0d0 + (b * ((b * (0.5d0 + (b * (-0.16666666666666666d0)))) + (-1.0d0))))))
    else if (b <= 2.45d+67) then
        tmp = 1.0d0 / (2.0d0 + (a * ((a * ((a * (-0.16666666666666666d0)) + 0.5d0)) + (-1.0d0))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * (b + 1.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -4.6e-16) {
		tmp = 1.0 / (1.0 + (1.0 / (1.0 + (b * ((b * (0.5 + (b * -0.16666666666666666))) + -1.0)))));
	} else if (b <= 2.45e+67) {
		tmp = 1.0 / (2.0 + (a * ((a * ((a * -0.16666666666666666) + 0.5)) + -1.0)));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (b + 1.0)))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -4.6e-16:
		tmp = 1.0 / (1.0 + (1.0 / (1.0 + (b * ((b * (0.5 + (b * -0.16666666666666666))) + -1.0)))))
	elif b <= 2.45e+67:
		tmp = 1.0 / (2.0 + (a * ((a * ((a * -0.16666666666666666) + 0.5)) + -1.0)))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (b + 1.0)))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -4.6e-16)
		tmp = Float64(1.0 / Float64(1.0 + Float64(1.0 / Float64(1.0 + Float64(b * Float64(Float64(b * Float64(0.5 + Float64(b * -0.16666666666666666))) + -1.0))))));
	elseif (b <= 2.45e+67)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * Float64(Float64(a * -0.16666666666666666) + 0.5)) + -1.0))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * Float64(b + 1.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -4.6e-16)
		tmp = 1.0 / (1.0 + (1.0 / (1.0 + (b * ((b * (0.5 + (b * -0.16666666666666666))) + -1.0)))));
	elseif (b <= 2.45e+67)
		tmp = 1.0 / (2.0 + (a * ((a * ((a * -0.16666666666666666) + 0.5)) + -1.0)));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (b + 1.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -4.6e-16], N[(1.0 / N[(1.0 + N[(1.0 / N[(1.0 + N[(b * N[(N[(b * N[(0.5 + N[(b * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.45e+67], N[(1.0 / N[(2.0 + N[(a * N[(N[(a * N[(N[(a * -0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * N[(b + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.5999999999999998e-16

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 100.0%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 98.3%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 98.3%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 98.3%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. exp-diff 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{e^{b}}{e^{a}}}} \]

      2. clear-num 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   7. Taylor expanded in a around 0 98.4%

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

      1. rec-exp 98.4%

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

   9. Simplified 98.4%

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

   10. Taylor expanded in b around 0 96.8%

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

   if -4.5999999999999998e-16 < b < 2.44999999999999995e67

   1. Initial program 99.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 99.9%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 99.9%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 99.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 99.9%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 99.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 64.8%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 64.8%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 64.8%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 99.9%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 99.9%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 99.9%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 99.9%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 99.9%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 90.6%

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

   6. Taylor expanded in a around 0 80.3%

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

   if 2.44999999999999995e67 < b

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 100.0%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 60.4%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 60.4%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 60.4%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. exp-diff 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{e^{b}}{e^{a}}}} \]

      2. clear-num 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   7. Taylor expanded in a around 0 100.0%

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

      1. rec-exp 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in b around 0 3.1%

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

       1. mul-1-neg 3.1%

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

       2. unsub-neg 3.1%

          \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 - b}}} \]

   12. Simplified 3.1%

       \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 - b}}} \]

   13. Taylor expanded in b around 0 96.1%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{-16}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + b \cdot \left(b \cdot \left(0.5 + b \cdot -0.16666666666666666\right) + -1\right)}}\\ \mathbf{elif}\;b \leq 2.45 \cdot 10^{+67}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(a \cdot -0.16666666666666666 + 0.5\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(b + 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.4% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{-16}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + b \cdot \left(b \cdot 0.5 + -1\right)}}\\ \mathbf{elif}\;b \leq 2.45 \cdot 10^{+67}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(a \cdot -0.16666666666666666 + 0.5\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(b + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -4.6e-16)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (* b (+ (* b 0.5) -1.0))))))
   (if (<= b 2.45e+67)
     (/ 1.0 (+ 2.0 (* a (+ (* a (+ (* a -0.16666666666666666) 0.5)) -1.0))))
     (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b (+ b 1.0)))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -4.6e-16) {
		tmp = 1.0 / (1.0 + (1.0 / (1.0 + (b * ((b * 0.5) + -1.0)))));
	} else if (b <= 2.45e+67) {
		tmp = 1.0 / (2.0 + (a * ((a * ((a * -0.16666666666666666) + 0.5)) + -1.0)));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (b + 1.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-4.6d-16)) then
        tmp = 1.0d0 / (1.0d0 + (1.0d0 / (1.0d0 + (b * ((b * 0.5d0) + (-1.0d0))))))
    else if (b <= 2.45d+67) then
        tmp = 1.0d0 / (2.0d0 + (a * ((a * ((a * (-0.16666666666666666d0)) + 0.5d0)) + (-1.0d0))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * (b + 1.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -4.6e-16) {
		tmp = 1.0 / (1.0 + (1.0 / (1.0 + (b * ((b * 0.5) + -1.0)))));
	} else if (b <= 2.45e+67) {
		tmp = 1.0 / (2.0 + (a * ((a * ((a * -0.16666666666666666) + 0.5)) + -1.0)));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (b + 1.0)))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -4.6e-16:
		tmp = 1.0 / (1.0 + (1.0 / (1.0 + (b * ((b * 0.5) + -1.0)))))
	elif b <= 2.45e+67:
		tmp = 1.0 / (2.0 + (a * ((a * ((a * -0.16666666666666666) + 0.5)) + -1.0)))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (b + 1.0)))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -4.6e-16)
		tmp = Float64(1.0 / Float64(1.0 + Float64(1.0 / Float64(1.0 + Float64(b * Float64(Float64(b * 0.5) + -1.0))))));
	elseif (b <= 2.45e+67)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * Float64(Float64(a * -0.16666666666666666) + 0.5)) + -1.0))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * Float64(b + 1.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -4.6e-16)
		tmp = 1.0 / (1.0 + (1.0 / (1.0 + (b * ((b * 0.5) + -1.0)))));
	elseif (b <= 2.45e+67)
		tmp = 1.0 / (2.0 + (a * ((a * ((a * -0.16666666666666666) + 0.5)) + -1.0)));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (b + 1.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -4.6e-16], N[(1.0 / N[(1.0 + N[(1.0 / N[(1.0 + N[(b * N[(N[(b * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.45e+67], N[(1.0 / N[(2.0 + N[(a * N[(N[(a * N[(N[(a * -0.16666666666666666), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * N[(b + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.5999999999999998e-16

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 100.0%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 98.3%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 98.3%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 98.3%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. exp-diff 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{e^{b}}{e^{a}}}} \]

      2. clear-num 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   7. Taylor expanded in a around 0 98.4%

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

      1. rec-exp 98.4%

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

   9. Simplified 98.4%

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

   10. Taylor expanded in b around 0 96.5%

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

   if -4.5999999999999998e-16 < b < 2.44999999999999995e67

   1. Initial program 99.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 99.9%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 99.9%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 99.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 99.9%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 99.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 64.8%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 64.8%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 64.8%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 99.9%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 99.9%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 99.9%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 99.9%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 99.9%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 90.6%

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

   6. Taylor expanded in a around 0 80.3%

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

   if 2.44999999999999995e67 < b

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 100.0%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 60.4%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 60.4%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 60.4%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. exp-diff 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{e^{b}}{e^{a}}}} \]

      2. clear-num 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   7. Taylor expanded in a around 0 100.0%

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

      1. rec-exp 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in b around 0 3.1%

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

       1. mul-1-neg 3.1%

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

       2. unsub-neg 3.1%

          \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 - b}}} \]

   12. Simplified 3.1%

       \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 - b}}} \]

   13. Taylor expanded in b around 0 96.1%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{-16}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + b \cdot \left(b \cdot 0.5 + -1\right)}}\\ \mathbf{elif}\;b \leq 2.45 \cdot 10^{+67}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(a \cdot -0.16666666666666666 + 0.5\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(b + 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.2% accurate, 13.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{-16}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + b \cdot \left(b \cdot 0.5 + -1\right)}}\\ \mathbf{elif}\;b \leq 2.45 \cdot 10^{+67}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(a \cdot -0.16666666666666666\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(b + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -4.6e-16)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (* b (+ (* b 0.5) -1.0))))))
   (if (<= b 2.45e+67)
     (/ 1.0 (+ 2.0 (* a (+ (* a (* a -0.16666666666666666)) -1.0))))
     (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b (+ b 1.0)))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -4.6e-16) {
		tmp = 1.0 / (1.0 + (1.0 / (1.0 + (b * ((b * 0.5) + -1.0)))));
	} else if (b <= 2.45e+67) {
		tmp = 1.0 / (2.0 + (a * ((a * (a * -0.16666666666666666)) + -1.0)));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (b + 1.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-4.6d-16)) then
        tmp = 1.0d0 / (1.0d0 + (1.0d0 / (1.0d0 + (b * ((b * 0.5d0) + (-1.0d0))))))
    else if (b <= 2.45d+67) then
        tmp = 1.0d0 / (2.0d0 + (a * ((a * (a * (-0.16666666666666666d0))) + (-1.0d0))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * (b + 1.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -4.6e-16) {
		tmp = 1.0 / (1.0 + (1.0 / (1.0 + (b * ((b * 0.5) + -1.0)))));
	} else if (b <= 2.45e+67) {
		tmp = 1.0 / (2.0 + (a * ((a * (a * -0.16666666666666666)) + -1.0)));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (b + 1.0)))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -4.6e-16:
		tmp = 1.0 / (1.0 + (1.0 / (1.0 + (b * ((b * 0.5) + -1.0)))))
	elif b <= 2.45e+67:
		tmp = 1.0 / (2.0 + (a * ((a * (a * -0.16666666666666666)) + -1.0)))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (b + 1.0)))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -4.6e-16)
		tmp = Float64(1.0 / Float64(1.0 + Float64(1.0 / Float64(1.0 + Float64(b * Float64(Float64(b * 0.5) + -1.0))))));
	elseif (b <= 2.45e+67)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * Float64(a * -0.16666666666666666)) + -1.0))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * Float64(b + 1.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -4.6e-16)
		tmp = 1.0 / (1.0 + (1.0 / (1.0 + (b * ((b * 0.5) + -1.0)))));
	elseif (b <= 2.45e+67)
		tmp = 1.0 / (2.0 + (a * ((a * (a * -0.16666666666666666)) + -1.0)));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (b + 1.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -4.6e-16], N[(1.0 / N[(1.0 + N[(1.0 / N[(1.0 + N[(b * N[(N[(b * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.45e+67], N[(1.0 / N[(2.0 + N[(a * N[(N[(a * N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * N[(b + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.5999999999999998e-16

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 100.0%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 98.3%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 98.3%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 98.3%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. exp-diff 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{e^{b}}{e^{a}}}} \]

      2. clear-num 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   7. Taylor expanded in a around 0 98.4%

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

      1. rec-exp 98.4%

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

   9. Simplified 98.4%

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

   10. Taylor expanded in b around 0 96.5%

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

   if -4.5999999999999998e-16 < b < 2.44999999999999995e67

   1. Initial program 99.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 99.9%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 99.9%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 99.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 99.9%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 99.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 64.8%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 64.8%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 64.8%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 99.9%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 99.9%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 99.9%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 99.9%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 99.9%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 90.6%

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

   6. Taylor expanded in a around 0 80.3%

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

   7. Taylor expanded in a around inf 80.1%

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

      1. *-commutative 80.1%

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

   9. Simplified 80.1%

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

   if 2.44999999999999995e67 < b

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 100.0%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 60.4%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 60.4%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 60.4%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. exp-diff 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{e^{b}}{e^{a}}}} \]

      2. clear-num 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   7. Taylor expanded in a around 0 100.0%

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

      1. rec-exp 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in b around 0 3.1%

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

       1. mul-1-neg 3.1%

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

       2. unsub-neg 3.1%

          \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 - b}}} \]

   12. Simplified 3.1%

       \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 - b}}} \]

   13. Taylor expanded in b around 0 96.1%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{-16}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + b \cdot \left(b \cdot 0.5 + -1\right)}}\\ \mathbf{elif}\;b \leq 2.45 \cdot 10^{+67}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(a \cdot -0.16666666666666666\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(b + 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.5% accurate, 13.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -11:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.45 \cdot 10^{+67}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(a \cdot -0.16666666666666666\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(b + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -11.0)
   1.0
   (if (<= b 2.45e+67)
     (/ 1.0 (+ 2.0 (* a (+ (* a (* a -0.16666666666666666)) -1.0))))
     (/ 1.0 (+ 2.0 (* b (+ 1.0 (* b (+ b 1.0)))))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -11.0) {
		tmp = 1.0;
	} else if (b <= 2.45e+67) {
		tmp = 1.0 / (2.0 + (a * ((a * (a * -0.16666666666666666)) + -1.0)));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (b + 1.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-11.0d0)) then
        tmp = 1.0d0
    else if (b <= 2.45d+67) then
        tmp = 1.0d0 / (2.0d0 + (a * ((a * (a * (-0.16666666666666666d0))) + (-1.0d0))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (1.0d0 + (b * (b + 1.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -11.0) {
		tmp = 1.0;
	} else if (b <= 2.45e+67) {
		tmp = 1.0 / (2.0 + (a * ((a * (a * -0.16666666666666666)) + -1.0)));
	} else {
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (b + 1.0)))));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -11.0:
		tmp = 1.0
	elif b <= 2.45e+67:
		tmp = 1.0 / (2.0 + (a * ((a * (a * -0.16666666666666666)) + -1.0)))
	else:
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (b + 1.0)))))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -11.0)
		tmp = 1.0;
	elseif (b <= 2.45e+67)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * Float64(a * -0.16666666666666666)) + -1.0))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(1.0 + Float64(b * Float64(b + 1.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -11.0)
		tmp = 1.0;
	elseif (b <= 2.45e+67)
		tmp = 1.0 / (2.0 + (a * ((a * (a * -0.16666666666666666)) + -1.0)));
	else
		tmp = 1.0 / (2.0 + (b * (1.0 + (b * (b + 1.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -11.0], 1.0, If[LessEqual[b, 2.45e+67], N[(1.0 / N[(2.0 + N[(a * N[(N[(a * N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(1.0 + N[(b * N[(b + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -11

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 100.0%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 100.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 100.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. exp-diff 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{e^{b}}{e^{a}}}} \]

      2. clear-num 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   7. Taylor expanded in a around 0 100.0%

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

      1. rec-exp 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in b around 0 99.0%

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

       1. mul-1-neg 99.0%

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

       2. unsub-neg 99.0%

          \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 - b}}} \]

   12. Simplified 99.0%

       \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 - b}}} \]

   13. Taylor expanded in b around inf 100.0%

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

   if -11 < b < 2.44999999999999995e67

   1. Initial program 99.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 99.9%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 99.9%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 99.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 99.9%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 99.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 65.7%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 65.7%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 65.7%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 99.9%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 99.9%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 99.9%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 99.9%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 99.9%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 89.4%

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

   6. Taylor expanded in a around 0 79.6%

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

   7. Taylor expanded in a around inf 79.4%

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

      1. *-commutative 79.4%

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

   9. Simplified 79.4%

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

   if 2.44999999999999995e67 < b

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 100.0%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 60.4%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 60.4%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 60.4%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. exp-diff 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{e^{b}}{e^{a}}}} \]

      2. clear-num 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   7. Taylor expanded in a around 0 100.0%

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

      1. rec-exp 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in b around 0 3.1%

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

       1. mul-1-neg 3.1%

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

       2. unsub-neg 3.1%

          \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 - b}}} \]

   12. Simplified 3.1%

       \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 - b}}} \]

   13. Taylor expanded in b around 0 96.1%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -11:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 2.45 \cdot 10^{+67}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(a \cdot -0.16666666666666666\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(1 + b \cdot \left(b + 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.8% accurate, 13.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -17.5:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{+149}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(a \cdot -0.16666666666666666\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(b + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -17.5)
   1.0
   (if (<= b 1.45e+149)
     (/ 1.0 (+ 2.0 (* a (+ (* a (* a -0.16666666666666666)) -1.0))))
     (/ 1.0 (+ 2.0 (* b (+ b 1.0)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -17.5) {
		tmp = 1.0;
	} else if (b <= 1.45e+149) {
		tmp = 1.0 / (2.0 + (a * ((a * (a * -0.16666666666666666)) + -1.0)));
	} else {
		tmp = 1.0 / (2.0 + (b * (b + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-17.5d0)) then
        tmp = 1.0d0
    else if (b <= 1.45d+149) then
        tmp = 1.0d0 / (2.0d0 + (a * ((a * (a * (-0.16666666666666666d0))) + (-1.0d0))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (b + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -17.5) {
		tmp = 1.0;
	} else if (b <= 1.45e+149) {
		tmp = 1.0 / (2.0 + (a * ((a * (a * -0.16666666666666666)) + -1.0)));
	} else {
		tmp = 1.0 / (2.0 + (b * (b + 1.0)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -17.5:
		tmp = 1.0
	elif b <= 1.45e+149:
		tmp = 1.0 / (2.0 + (a * ((a * (a * -0.16666666666666666)) + -1.0)))
	else:
		tmp = 1.0 / (2.0 + (b * (b + 1.0)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -17.5)
		tmp = 1.0;
	elseif (b <= 1.45e+149)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * Float64(a * -0.16666666666666666)) + -1.0))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(b + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -17.5)
		tmp = 1.0;
	elseif (b <= 1.45e+149)
		tmp = 1.0 / (2.0 + (a * ((a * (a * -0.16666666666666666)) + -1.0)));
	else
		tmp = 1.0 / (2.0 + (b * (b + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -17.5], 1.0, If[LessEqual[b, 1.45e+149], N[(1.0 / N[(2.0 + N[(a * N[(N[(a * N[(a * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(b + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -17.5

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 100.0%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 100.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 100.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. exp-diff 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{e^{b}}{e^{a}}}} \]

      2. clear-num 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   7. Taylor expanded in a around 0 100.0%

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

      1. rec-exp 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in b around 0 99.0%

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

       1. mul-1-neg 99.0%

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

       2. unsub-neg 99.0%

          \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 - b}}} \]

   12. Simplified 99.0%

       \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 - b}}} \]

   13. Taylor expanded in b around inf 100.0%

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

   if -17.5 < b < 1.4500000000000001e149

   1. Initial program 99.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 99.9%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 99.9%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 99.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 99.9%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 99.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 65.4%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 65.4%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 65.4%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 99.9%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 99.9%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 99.9%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 99.9%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 99.9%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 86.5%

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

   6. Taylor expanded in a around 0 76.2%

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

   7. Taylor expanded in a around inf 76.0%

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

      1. *-commutative 76.0%

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

   9. Simplified 76.0%

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

   if 1.4500000000000001e149 < b

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 100.0%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 60.5%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 60.5%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 60.5%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. exp-diff 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{e^{b}}{e^{a}}}} \]

      2. clear-num 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   7. Taylor expanded in a around 0 100.0%

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

      1. rec-exp 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in b around 0 3.1%

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

       1. mul-1-neg 3.1%

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

       2. unsub-neg 3.1%

          \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 - b}}} \]

   12. Simplified 3.1%

       \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 - b}}} \]

   13. Taylor expanded in b around 0 93.1%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -17.5:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{+149}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot \left(a \cdot -0.16666666666666666\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(b + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.4% accurate, 14.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -12:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+148}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot 0.5 + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(b + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -12.0)
   1.0
   (if (<= b 6e+148)
     (/ 1.0 (+ 2.0 (* a (+ (* a 0.5) -1.0))))
     (/ 1.0 (+ 2.0 (* b (+ b 1.0)))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -12.0) {
		tmp = 1.0;
	} else if (b <= 6e+148) {
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	} else {
		tmp = 1.0 / (2.0 + (b * (b + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-12.0d0)) then
        tmp = 1.0d0
    else if (b <= 6d+148) then
        tmp = 1.0d0 / (2.0d0 + (a * ((a * 0.5d0) + (-1.0d0))))
    else
        tmp = 1.0d0 / (2.0d0 + (b * (b + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -12.0) {
		tmp = 1.0;
	} else if (b <= 6e+148) {
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	} else {
		tmp = 1.0 / (2.0 + (b * (b + 1.0)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -12.0:
		tmp = 1.0
	elif b <= 6e+148:
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)))
	else:
		tmp = 1.0 / (2.0 + (b * (b + 1.0)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -12.0)
		tmp = 1.0;
	elseif (b <= 6e+148)
		tmp = Float64(1.0 / Float64(2.0 + Float64(a * Float64(Float64(a * 0.5) + -1.0))));
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(b + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -12.0)
		tmp = 1.0;
	elseif (b <= 6e+148)
		tmp = 1.0 / (2.0 + (a * ((a * 0.5) + -1.0)));
	else
		tmp = 1.0 / (2.0 + (b * (b + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -12.0], 1.0, If[LessEqual[b, 6e+148], N[(1.0 / N[(2.0 + N[(a * N[(N[(a * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 + N[(b * N[(b + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -12

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 100.0%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 100.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 100.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. exp-diff 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{e^{b}}{e^{a}}}} \]

      2. clear-num 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   7. Taylor expanded in a around 0 100.0%

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

      1. rec-exp 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in b around 0 99.0%

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

       1. mul-1-neg 99.0%

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

       2. unsub-neg 99.0%

          \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 - b}}} \]

   12. Simplified 99.0%

       \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 - b}}} \]

   13. Taylor expanded in b around inf 100.0%

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

   if -12 < b < 6.00000000000000029e148

   1. Initial program 99.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 99.9%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 99.9%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 99.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 99.9%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 99.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 65.4%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 65.4%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 65.4%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 99.9%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 99.9%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 99.9%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 99.9%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 99.9%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 86.5%

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

   6. Taylor expanded in a around 0 70.9%

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

   if 6.00000000000000029e148 < b

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 100.0%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 60.5%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 60.5%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 60.5%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. exp-diff 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{e^{b}}{e^{a}}}} \]

      2. clear-num 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   7. Taylor expanded in a around 0 100.0%

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

      1. rec-exp 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in b around 0 3.1%

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

       1. mul-1-neg 3.1%

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

       2. unsub-neg 3.1%

          \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 - b}}} \]

   12. Simplified 3.1%

       \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 - b}}} \]

   13. Taylor expanded in b around 0 93.1%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -12:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 6 \cdot 10^{+148}:\\ \;\;\;\;\frac{1}{2 + a \cdot \left(a \cdot 0.5 + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(b + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.1% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.05:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(b + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -1.05) 1.0 (/ 1.0 (+ 2.0 (* b (+ b 1.0))))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -1.05) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / (2.0 + (b * (b + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-1.05d0)) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 / (2.0d0 + (b * (b + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -1.05) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / (2.0 + (b * (b + 1.0)));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -1.05:
		tmp = 1.0
	else:
		tmp = 1.0 / (2.0 + (b * (b + 1.0)))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -1.05)
		tmp = 1.0;
	else
		tmp = Float64(1.0 / Float64(2.0 + Float64(b * Float64(b + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -1.05)
		tmp = 1.0;
	else
		tmp = 1.0 / (2.0 + (b * (b + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -1.05], 1.0, N[(1.0 / N[(2.0 + N[(b * N[(b + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.05000000000000004

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 100.0%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 100.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 100.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. exp-diff 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{e^{b}}{e^{a}}}} \]

      2. clear-num 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   7. Taylor expanded in a around 0 100.0%

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

      1. rec-exp 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in b around 0 99.0%

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

       1. mul-1-neg 99.0%

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

       2. unsub-neg 99.0%

          \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 - b}}} \]

   12. Simplified 99.0%

       \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 - b}}} \]

   13. Taylor expanded in b around inf 100.0%

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

   if -1.05000000000000004 < b

   1. Initial program 99.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 99.9%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 99.9%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 99.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 99.9%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 99.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 64.5%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 64.5%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 64.5%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. exp-diff 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{e^{b}}{e^{a}}}} \]

      2. clear-num 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   7. Taylor expanded in a around 0 77.6%

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

      1. rec-exp 77.6%

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

   9. Simplified 77.6%

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

   10. Taylor expanded in b around 0 44.6%

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

       1. mul-1-neg 44.6%

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

       2. unsub-neg 44.6%

          \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 - b}}} \]

   12. Simplified 44.6%

       \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 - b}}} \]

   13. Taylor expanded in b around 0 61.7%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.05:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + b \cdot \left(b + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 55.1% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -1.0) 1.0 (/ 1.0 (+ b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -1.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / (b + 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-1.0d0)) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 / (b + 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -1.0) {
		tmp = 1.0;
	} else {
		tmp = 1.0 / (b + 2.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -1.0:
		tmp = 1.0
	else:
		tmp = 1.0 / (b + 2.0)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -1.0)
		tmp = 1.0;
	else
		tmp = 1.0 / (b + 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -1.0], 1.0, N[(1.0 / N[(b + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 100.0%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 100.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 100.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. exp-diff 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{e^{b}}{e^{a}}}} \]

      2. clear-num 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   7. Taylor expanded in a around 0 100.0%

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

      1. rec-exp 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in b around 0 99.0%

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

       1. mul-1-neg 99.0%

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

       2. unsub-neg 99.0%

          \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 - b}}} \]

   12. Simplified 99.0%

       \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 - b}}} \]

   13. Taylor expanded in b around inf 100.0%

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

   if -1 < b

   1. Initial program 99.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 99.9%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 99.9%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 99.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 99.9%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 99.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 64.5%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 64.5%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 64.5%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 77.6%

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

   6. Taylor expanded in b around 0 45.4%

      \[\leadsto \frac{1}{\color{blue}{2 + b}} \]
   7. Step-by-step derivation

      1. +-commutative 45.4%

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

   8. Simplified 45.4%

      \[\leadsto \frac{1}{\color{blue}{b + 2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 54.4% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -2.0) 1.0 (+ 0.5 (* b -0.25))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -2.0) {
		tmp = 1.0;
	} else {
		tmp = 0.5 + (b * -0.25);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-2.0d0)) then
        tmp = 1.0d0
    else
        tmp = 0.5d0 + (b * (-0.25d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -2.0) {
		tmp = 1.0;
	} else {
		tmp = 0.5 + (b * -0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -2.0:
		tmp = 1.0
	else:
		tmp = 0.5 + (b * -0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -2.0)
		tmp = 1.0;
	else
		tmp = Float64(0.5 + Float64(b * -0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -2.0)
		tmp = 1.0;
	else
		tmp = 0.5 + (b * -0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -2.0], 1.0, N[(0.5 + N[(b * -0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 100.0%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 100.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 100.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. exp-diff 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{e^{b}}{e^{a}}}} \]

      2. clear-num 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   7. Taylor expanded in a around 0 100.0%

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

      1. rec-exp 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in b around 0 99.0%

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

       1. mul-1-neg 99.0%

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

       2. unsub-neg 99.0%

          \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 - b}}} \]

   12. Simplified 99.0%

       \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 - b}}} \]

   13. Taylor expanded in b around inf 100.0%

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

   if -2 < b

   1. Initial program 99.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 99.9%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 99.9%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 99.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 99.9%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 99.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 64.5%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 64.5%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 64.5%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 77.6%

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

   6. Taylor expanded in b around 0 44.9%

      \[\leadsto \color{blue}{0.5 + -0.25 \cdot b} \]
   7. Step-by-step derivation

      1. *-commutative 44.9%

         \[\leadsto 0.5 + \color{blue}{b \cdot -0.25} \]

   8. Simplified 44.9%

      \[\leadsto \color{blue}{0.5 + b \cdot -0.25} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 54.4% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.98:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -0.98) 1.0 (+ 0.5 (* a 0.25))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -0.98) {
		tmp = 1.0;
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-0.98d0)) then
        tmp = 1.0d0
    else
        tmp = 0.5d0 + (a * 0.25d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -0.98) {
		tmp = 1.0;
	} else {
		tmp = 0.5 + (a * 0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -0.98:
		tmp = 1.0
	else:
		tmp = 0.5 + (a * 0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -0.98)
		tmp = 1.0;
	else
		tmp = Float64(0.5 + Float64(a * 0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -0.98)
		tmp = 1.0;
	else
		tmp = 0.5 + (a * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -0.98], 1.0, N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -0.97999999999999998

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 100.0%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 100.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 100.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. exp-diff 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{e^{b}}{e^{a}}}} \]

      2. clear-num 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   7. Taylor expanded in a around 0 100.0%

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

      1. rec-exp 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in b around 0 99.0%

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

       1. mul-1-neg 99.0%

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

       2. unsub-neg 99.0%

          \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 - b}}} \]

   12. Simplified 99.0%

       \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 - b}}} \]

   13. Taylor expanded in b around inf 100.0%

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

   if -0.97999999999999998 < b

   1. Initial program 99.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 99.9%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 99.9%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 99.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 99.9%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 99.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 64.5%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 64.5%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 64.5%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 78.0%

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

   6. Taylor expanded in a around 0 43.4%

      \[\leadsto \color{blue}{0.5 + 0.25 \cdot a} \]
   7. Step-by-step derivation

      1. *-commutative 43.4%

         \[\leadsto 0.5 + \color{blue}{a \cdot 0.25} \]

   8. Simplified 43.4%

      \[\leadsto \color{blue}{0.5 + a \cdot 0.25} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.98:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 54.1% accurate, 33.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \frac{1}{1 - b}} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ 1.0 (+ 1.0 (/ 1.0 (- 1.0 b)))))

C

double code(double a, double b) {
	return 1.0 / (1.0 + (1.0 / (1.0 - b)));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 / (1.0d0 + (1.0d0 / (1.0d0 - b)))
end function

Java

public static double code(double a, double b) {
	return 1.0 / (1.0 + (1.0 / (1.0 - b)));
}

Python

def code(a, b):
	return 1.0 / (1.0 + (1.0 / (1.0 - b)))

Julia

function code(a, b)
	return Float64(1.0 / Float64(1.0 + Float64(1.0 / Float64(1.0 - b))))
end

MATLAB

function tmp = code(a, b)
	tmp = 1.0 / (1.0 + (1.0 / (1.0 - b)));
end

Wolfram

code[a_, b_] := N[(1.0 / N[(1.0 + N[(1.0 / N[(1.0 - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
2. Step-by-step derivation

   1. *-lft-identity 100.0%

      \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

   2. associate-*l/ 100.0%

      \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

   3. associate-/r/ 100.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

   4. remove-double-neg 100.0%

      \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

   5. unsub-neg 100.0%

      \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

   6. div-sub 71.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

   7. *-lft-identity 71.8%

      \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

   8. associate-*l/ 71.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

   9. lft-mult-inverse 100.0%

      \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

   10. sub-neg 100.0%

       \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

   11. distribute-frac-neg 100.0%

       \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

   12. remove-double-neg 100.0%

       \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

   13. div-exp 100.0%

       \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. exp-diff 100.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\frac{e^{b}}{e^{a}}}} \]

   2. clear-num 100.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

7. Taylor expanded in a around 0 82.2%

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

   1. rec-exp 82.2%

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

9. Simplified 82.2%

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

10. Taylor expanded in b around 0 55.9%

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

    1. mul-1-neg 55.9%

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

    2. unsub-neg 55.9%

       \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 - b}}} \]

12. Simplified 55.9%

    \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 - b}}} \]
13. Add Preprocessing

Alternative 16: 54.1% accurate, 50.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -1.1) 1.0 0.5))

C

double code(double a, double b) {
	double tmp;
	if (b <= -1.1) {
		tmp = 1.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-1.1d0)) then
        tmp = 1.0d0
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -1.1) {
		tmp = 1.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -1.1:
		tmp = 1.0
	else:
		tmp = 0.5
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -1.1)
		tmp = 1.0;
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -1.1)
		tmp = 1.0;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -1.1], 1.0, 0.5]

Derivation

1. Split input into 2 regimes

2. if b < -1.1000000000000001

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 100.0%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 100.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 100.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 100.0%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. exp-diff 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{e^{b}}{e^{a}}}} \]

      2. clear-num 100.0%

         \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{\frac{e^{a}}{e^{b}}}}} \]

   7. Taylor expanded in a around 0 100.0%

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

      1. rec-exp 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in b around 0 99.0%

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

       1. mul-1-neg 99.0%

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

       2. unsub-neg 99.0%

          \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 - b}}} \]

   12. Simplified 99.0%

       \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 - b}}} \]

   13. Taylor expanded in b around inf 100.0%

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

   if -1.1000000000000001 < b

   1. Initial program 99.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Step-by-step derivation

      1. *-lft-identity 99.9%

         \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

      2. associate-*l/ 99.9%

         \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

      3. associate-/r/ 99.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

      4. remove-double-neg 99.9%

         \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

      5. unsub-neg 99.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

      6. div-sub 64.5%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

      7. *-lft-identity 64.5%

         \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      8. associate-*l/ 64.5%

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

      9. lft-mult-inverse 100.0%

         \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

      10. sub-neg 100.0%

          \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

      11. distribute-frac-neg 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

      12. remove-double-neg 100.0%

          \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

      13. div-exp 100.0%

          \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 77.6%

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

   6. Taylor expanded in b around 0 43.3%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 38.9% accurate, 305.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 0.5)

C

double code(double a, double b) {
	return 0.5;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 0.5d0
end function

Java

public static double code(double a, double b) {
	return 0.5;
}

Python

def code(a, b):
	return 0.5

Julia

function code(a, b)
	return 0.5
end

MATLAB

function tmp = code(a, b)
	tmp = 0.5;
end

Wolfram

code[a_, b_] := 0.5

Derivation

1. Initial program 100.0%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
2. Step-by-step derivation

   1. *-lft-identity 100.0%

      \[\leadsto \frac{\color{blue}{1 \cdot e^{a}}}{e^{a} + e^{b}} \]

   2. associate-*l/ 100.0%

      \[\leadsto \color{blue}{\frac{1}{e^{a} + e^{b}} \cdot e^{a}} \]

   3. associate-/r/ 100.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{a} + e^{b}}{e^{a}}}} \]

   4. remove-double-neg 100.0%

      \[\leadsto \frac{1}{\frac{e^{a} + \color{blue}{\left(-\left(-e^{b}\right)\right)}}{e^{a}}} \]

   5. unsub-neg 100.0%

      \[\leadsto \frac{1}{\frac{\color{blue}{e^{a} - \left(-e^{b}\right)}}{e^{a}}} \]

   6. div-sub 71.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{a}}{e^{a}} - \frac{-e^{b}}{e^{a}}}} \]

   7. *-lft-identity 71.8%

      \[\leadsto \frac{1}{\frac{\color{blue}{1 \cdot e^{a}}}{e^{a}} - \frac{-e^{b}}{e^{a}}} \]

   8. associate-*l/ 71.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{e^{a}} \cdot e^{a}} - \frac{-e^{b}}{e^{a}}} \]

   9. lft-mult-inverse 100.0%

      \[\leadsto \frac{1}{\color{blue}{1} - \frac{-e^{b}}{e^{a}}} \]

   10. sub-neg 100.0%

       \[\leadsto \frac{1}{\color{blue}{1 + \left(-\frac{-e^{b}}{e^{a}}\right)}} \]

   11. distribute-frac-neg 100.0%

       \[\leadsto \frac{1}{1 + \color{blue}{\frac{-\left(-e^{b}\right)}{e^{a}}}} \]

   12. remove-double-neg 100.0%

       \[\leadsto \frac{1}{1 + \frac{\color{blue}{e^{b}}}{e^{a}}} \]

   13. div-exp 100.0%

       \[\leadsto \frac{1}{1 + \color{blue}{e^{b - a}}} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{1}{1 + e^{b - a}}} \]
4. Add Preprocessing

5. Taylor expanded in a around 0 82.2%

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

6. Taylor expanded in b around 0 38.2%

   \[\leadsto \color{blue}{0.5} \]
7. Add Preprocessing

Developer Target 1: 100.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + e^{b - a}} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ 1.0 (+ 1.0 (exp (- b a)))))

C

double code(double a, double b) {
	return 1.0 / (1.0 + exp((b - a)));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 / (1.0d0 + exp((b - a)))
end function

Java

public static double code(double a, double b) {
	return 1.0 / (1.0 + Math.exp((b - a)));
}

Python

def code(a, b):
	return 1.0 / (1.0 + math.exp((b - a)))

Julia

function code(a, b)
	return Float64(1.0 / Float64(1.0 + exp(Float64(b - a))))
end

MATLAB

function tmp = code(a, b)
	tmp = 1.0 / (1.0 + exp((b - a)));
end

Wolfram

code[a_, b_] := N[(1.0 / N[(1.0 + N[Exp[N[(b - a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024131 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

  :alt
  (! :herbie-platform default (/ 1 (+ 1 (exp (- b a)))))

  (/ (exp a) (+ (exp a) (exp b))))