Quotient of sum of exps

Percentage Accurate: 99.0% → 100.0%

Time: 5.2s

Alternatives: 8

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, 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]

Derivation

1. Initial program 99.2%

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

   1. clear-num 99.2%

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

   2. inv-pow 99.2%

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

4. Applied egg-rr 99.2%

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

5. Taylor expanded in a around inf 99.2%

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

   1. remove-double-div 99.2%

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

   2. rec-exp 99.2%

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

   3. +-commutative 99.2%

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

   4. associate-/r* 99.2%

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

   5. +-commutative 99.2%

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

   6. distribute-rgt-in 71.9%

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

   7. rec-exp 71.9%

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

   8. rgt-mult-inverse 100.0%

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

   9. rec-exp 100.0%

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

   10. associate-*r/ 100.0%

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

   11. *-rgt-identity 100.0%

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

   12. div-exp 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 98.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-88}:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 5e-88) (/ 1.0 (+ 1.0 (exp (- a)))) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 5e-88) {
		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 (exp(a) <= 5d-88) 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 (Math.exp(a) <= 5e-88) {
		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 math.exp(a) <= 5e-88:
		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 (exp(a) <= 5e-88)
		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 (exp(a) <= 5e-88)
		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[N[Exp[a], $MachinePrecision], 5e-88], 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 (exp.f64 a) < 5.00000000000000009e-88

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in a around inf 100.0%

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

      1. remove-double-div 100.0%

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

      2. rec-exp 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-/r* 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-rgt-in 2.8%

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

      7. rec-exp 2.8%

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

      8. rgt-mult-inverse 100.0%

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

      9. rec-exp 100.0%

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

      10. associate-*r/ 100.0%

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

      11. *-rgt-identity 100.0%

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

      12. div-exp 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in b around 0 98.8%

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

   if 5.00000000000000009e-88 < (exp.f64 a)

   1. Initial program 98.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 98.5%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-88}:\\ \;\;\;\;\frac{1}{1 + e^{-a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{e^{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0) (/ (exp a) b) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = exp(a) / 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 (exp(a) <= 0.0d0) then
        tmp = exp(a) / 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 (Math.exp(a) <= 0.0) {
		tmp = Math.exp(a) / b;
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = math.exp(a) / b
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = Float64(exp(a) / 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 (exp(a) <= 0.0)
		tmp = exp(a) / b;
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(N[Exp[a], $MachinePrecision] / b), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0 100.0%

      \[\leadsto \frac{e^{a}}{\color{blue}{1 + \left(b + e^{a}\right)}} \]
   4. Step-by-step derivation

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 98.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 98.0%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{e^{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -205:\\ \;\;\;\;\frac{e^{a}}{b}\\ \mathbf{elif}\;a \leq -1.55 \cdot 10^{-172}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{elif}\;a \leq -2.3 \cdot 10^{-223}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -205.0)
   (/ (exp a) b)
   (if (<= a -1.55e-172) (/ 1.0 (- 2.0 a)) (if (<= a -2.3e-223) 1.0 0.5))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -205.0) {
		tmp = exp(a) / b;
	} else if (a <= -1.55e-172) {
		tmp = 1.0 / (2.0 - a);
	} else if (a <= -2.3e-223) {
		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 (a <= (-205.0d0)) then
        tmp = exp(a) / b
    else if (a <= (-1.55d-172)) then
        tmp = 1.0d0 / (2.0d0 - a)
    else if (a <= (-2.3d-223)) 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 (a <= -205.0) {
		tmp = Math.exp(a) / b;
	} else if (a <= -1.55e-172) {
		tmp = 1.0 / (2.0 - a);
	} else if (a <= -2.3e-223) {
		tmp = 1.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -205.0:
		tmp = math.exp(a) / b
	elif a <= -1.55e-172:
		tmp = 1.0 / (2.0 - a)
	elif a <= -2.3e-223:
		tmp = 1.0
	else:
		tmp = 0.5
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -205.0)
		tmp = Float64(exp(a) / b);
	elseif (a <= -1.55e-172)
		tmp = Float64(1.0 / Float64(2.0 - a));
	elseif (a <= -2.3e-223)
		tmp = 1.0;
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -205.0)
		tmp = exp(a) / b;
	elseif (a <= -1.55e-172)
		tmp = 1.0 / (2.0 - a);
	elseif (a <= -2.3e-223)
		tmp = 1.0;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -205.0], N[(N[Exp[a], $MachinePrecision] / b), $MachinePrecision], If[LessEqual[a, -1.55e-172], N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -2.3e-223], 1.0, 0.5]]]

Derivation

1. Split input into 4 regimes

2. if a < -205

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0 100.0%

      \[\leadsto \frac{e^{a}}{\color{blue}{1 + \left(b + e^{a}\right)}} \]
   4. Step-by-step derivation

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 100.0%

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

   if -205 < a < -1.5500000000000001e-172

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in a around inf 100.0%

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

      1. remove-double-div 100.0%

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

      2. rec-exp 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-/r* 99.9%

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

      5. +-commutative 99.9%

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

      6. distribute-rgt-in 99.9%

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

      7. rec-exp 100.0%

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

      8. rgt-mult-inverse 100.0%

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

      9. rec-exp 100.0%

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

      10. associate-*r/ 99.9%

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

      11. *-rgt-identity 99.9%

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

      12. div-exp 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in b around 0 58.7%

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

   9. Taylor expanded in a around 0 55.4%

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

       1. mul-1-neg 55.4%

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

       2. unsub-neg 55.4%

          \[\leadsto \frac{1}{\color{blue}{2 - a}} \]

   11. Simplified 55.4%

       \[\leadsto \frac{1}{\color{blue}{2 - a}} \]

   if -1.5500000000000001e-172 < a < -2.3e-223

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in a around inf 100.0%

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

      1. remove-double-div 100.0%

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

      2. rec-exp 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-/r* 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-rgt-in 100.0%

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

      7. rec-exp 100.0%

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

      8. rgt-mult-inverse 100.0%

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

      9. rec-exp 100.0%

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

      10. associate-*r/ 100.0%

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

      11. *-rgt-identity 100.0%

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

      12. div-exp 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in b around 0 23.2%

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

   10. Applied egg-rr 66.0%

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

   if -2.3e-223 < a

   1. Initial program 98.3%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0 52.4%

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

   4. Taylor expanded in a around 0 52.7%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -205:\\ \;\;\;\;\frac{e^{a}}{b}\\ \mathbf{elif}\;a \leq -1.55 \cdot 10^{-172}:\\ \;\;\;\;\frac{1}{2 - a}\\ \mathbf{elif}\;a \leq -2.3 \cdot 10^{-223}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.0% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.7:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -1.7) 1.0 (+ 0.5 (* b -0.25))))

C

double code(double a, double b) {
	double tmp;
	if (b <= -1.7) {
		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 <= (-1.7d0)) 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 <= -1.7) {
		tmp = 1.0;
	} else {
		tmp = 0.5 + (b * -0.25);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -1.7:
		tmp = 1.0
	else:
		tmp = 0.5 + (b * -0.25)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -1.7)
		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 <= -1.7)
		tmp = 1.0;
	else
		tmp = 0.5 + (b * -0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -1.7], 1.0, N[(0.5 + N[(b * -0.25), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.69999999999999996

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in a around inf 100.0%

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

      1. remove-double-div 100.0%

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

      2. rec-exp 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-/r* 99.9%

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

      5. +-commutative 99.9%

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

      6. distribute-rgt-in 99.9%

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

      7. rec-exp 100.0%

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

      8. rgt-mult-inverse 100.0%

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

      9. rec-exp 100.0%

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

      10. associate-*r/ 100.0%

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

      11. *-rgt-identity 100.0%

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

      12. div-exp 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in b around 0 18.5%

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

   10. Applied egg-rr 100.0%

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

   if -1.69999999999999996 < b

   1. Initial program 99.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 77.7%

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

   4. Taylor expanded in b around 0 42.9%

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

   5. Taylor expanded in b around 0 41.5%

      \[\leadsto \color{blue}{0.5 + -0.25 \cdot b} \]
   6. Step-by-step derivation

      1. *-commutative 41.5%

         \[\leadsto 0.5 + \color{blue}{b \cdot -0.25} \]

   7. Simplified 41.5%

      \[\leadsto \color{blue}{0.5 + b \cdot -0.25} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.7:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.8% 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. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in a around inf 100.0%

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

      1. remove-double-div 100.0%

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

      2. rec-exp 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-/r* 99.9%

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

      5. +-commutative 99.9%

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

      6. distribute-rgt-in 99.9%

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

      7. rec-exp 100.0%

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

      8. rgt-mult-inverse 100.0%

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

      9. rec-exp 100.0%

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

      10. associate-*r/ 100.0%

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

      11. *-rgt-identity 100.0%

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

      12. div-exp 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in b around 0 18.5%

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

   10. Applied egg-rr 100.0%

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

   if -1 < b

   1. Initial program 99.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 77.7%

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

   4. Taylor expanded in b around 0 42.9%

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

4. Final simplification 53.6%

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

Alternative 7: 53.7% accurate, 50.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -0.96:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b -0.96) 1.0 0.5))

C

double code(double a, double b) {
	double tmp;
	if (b <= -0.96) {
		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 <= (-0.96d0)) 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 <= -0.96) {
		tmp = 1.0;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -0.96:
		tmp = 1.0
	else:
		tmp = 0.5
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -0.96)
		tmp = 1.0;
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -0.96)
		tmp = 1.0;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -0.96], 1.0, 0.5]

Derivation

1. Split input into 2 regimes

2. if b < -0.95999999999999996

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in a around inf 100.0%

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

      1. remove-double-div 100.0%

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

      2. rec-exp 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-/r* 99.9%

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

      5. +-commutative 99.9%

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

      6. distribute-rgt-in 99.9%

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

      7. rec-exp 100.0%

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

      8. rgt-mult-inverse 100.0%

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

      9. rec-exp 100.0%

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

      10. associate-*r/ 100.0%

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

      11. *-rgt-identity 100.0%

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

      12. div-exp 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in b around 0 18.5%

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

   10. Applied egg-rr 100.0%

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

   if -0.95999999999999996 < b

   1. Initial program 99.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0 74.9%

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

   4. Taylor expanded in a around 0 41.5%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.96:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 38.8% 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 99.2%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
2. Add Preprocessing

3. Taylor expanded in b around 0 64.3%

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

4. Taylor expanded in a around 0 37.2%

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

5. Final simplification 37.2%

   \[\leadsto 0.5 \]
6. Add Preprocessing

Developer target: 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 2024027 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ 1.0 (exp (- b a))))

  (/ (exp a) (+ (exp a) (exp b))))