Quotient of sum of exps

Percentage Accurate: 98.8% → 98.4%

Time: 6.4s

Alternatives: 12

Speedup: 1.0×

• could not determine a ground truth

  1. a = 1.784318441812281e+238
  2. b = -6.4782718527957345e+289

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: 98.8% 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: 98.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 96.4%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 99.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 99.0

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

   5. Applied rewrites 99.0%

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

4. Final simplification 99.2%

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

Alternative 2: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.4%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. clear-num N/A

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

   5. associate-/r/ N/A

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

   6. lower-*.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. rec-exp N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-neg.f64 98.4

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 98.4

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

4. Applied rewrites 98.4%

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

5. Final simplification 98.4%

   \[\leadsto \frac{1}{\left(e^{a} + e^{b}\right) \cdot e^{-a}} \]
6. Add Preprocessing

Alternative 3: 98.8% 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]

Derivation

1. Initial program 98.4%

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

Alternative 4: 94.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.1 \cdot 10^{+101}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, a, 0.5\right), a, -1\right), a, 2\right)}\\ \mathbf{elif}\;a \leq -54000:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot b\right) \cdot 0.020833333333333332\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -3.1e+101)
   (/ 1.0 (fma (fma (fma -0.16666666666666666 a 0.5) a -1.0) a 2.0))
   (if (<= a -54000.0)
     (* (* (* b b) b) 0.020833333333333332)
     (/ 1.0 (+ (exp b) 1.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -3.1e+101) {
		tmp = 1.0 / fma(fma(fma(-0.16666666666666666, a, 0.5), a, -1.0), a, 2.0);
	} else if (a <= -54000.0) {
		tmp = ((b * b) * b) * 0.020833333333333332;
	} else {
		tmp = 1.0 / (exp(b) + 1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -3.1e+101)
		tmp = Float64(1.0 / fma(fma(fma(-0.16666666666666666, a, 0.5), a, -1.0), a, 2.0));
	elseif (a <= -54000.0)
		tmp = Float64(Float64(Float64(b * b) * b) * 0.020833333333333332);
	else
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -3.1e+101], N[(1.0 / N[(N[(N[(-0.16666666666666666 * a + 0.5), $MachinePrecision] * a + -1.0), $MachinePrecision] * a + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -54000.0], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * 0.020833333333333332), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.09999999999999999e101

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. rec-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 100.0

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in b around 0

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. exp-neg N/A

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

      4. lft-mult-inverse N/A

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

      5. lower-+.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-exp.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 97.6%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, a, 0.5\right), a, -1\right), \color{blue}{a}, 2\right)} \]

   if -3.09999999999999999e101 < a < -54000

   1. Initial program 89.5%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 18.4

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

   5. Applied rewrites 18.4%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2} + \color{blue}{b \cdot \left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 2.9%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.020833333333333332, -0.25\right), \color{blue}{b}, 0.5\right) \]

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{48} \cdot {b}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 64.1%

       \[\leadsto \left(\left(b \cdot b\right) \cdot b\right) \cdot 0.020833333333333332 \]

   if -54000 < a

   1. Initial program 99.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 99.0

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

   5. Applied rewrites 99.0%

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

Alternative 5: 51.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot b\right) \cdot 0.020833333333333332\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.25, a, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0) (* (* (* b b) b) 0.020833333333333332) (fma 0.25 a 0.5)))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = ((b * b) * b) * 0.020833333333333332;
	} else {
		tmp = fma(0.25, a, 0.5);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = Float64(Float64(Float64(b * b) * b) * 0.020833333333333332);
	else
		tmp = fma(0.25, a, 0.5);
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * 0.020833333333333332), $MachinePrecision], N[(0.25 * a + 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 96.4%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 29.1

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

   5. Applied rewrites 29.1%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2} + \color{blue}{b \cdot \left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 2.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.020833333333333332, -0.25\right), \color{blue}{b}, 0.5\right) \]

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{48} \cdot {b}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 55.6%

       \[\leadsto \left(\left(b \cdot b\right) \cdot b\right) \cdot 0.020833333333333332 \]

   if 0.0 < (exp.f64 a)

   1. Initial program 99.0%

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

   3. Taylor expanded in b around 0

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

      1. associate-*r/ N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

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

      5. distribute-lft1-in N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in a around 0

      \[\leadsto \frac{1}{2} \cdot \left(1 + \frac{-1}{2} \cdot b\right) + \color{blue}{a \cdot \left(\frac{1}{2} \cdot \left(1 + \left(\frac{-1}{2} \cdot b + \frac{1}{4} \cdot b\right)\right) - \frac{1}{4} \cdot \left(1 + \frac{-1}{2} \cdot b\right)\right)} \]

   8. Applied rewrites 52.5%

      \[\leadsto \mathsf{fma}\left(0.25, \color{blue}{a}, \mathsf{fma}\left(-0.25, b, 0.5\right)\right) \]

   9. Taylor expanded in b around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \frac{1}{2}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 55.6%

       \[\leadsto \mathsf{fma}\left(0.25, a, 0.5\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 64.9% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, -1\right), a, 2\right)}\\ \mathbf{elif}\;a \leq -640:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot b\right) \cdot 0.020833333333333332\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1.9e+154)
   (/ 1.0 (fma (fma 0.5 a -1.0) a 2.0))
   (if (<= a -640.0)
     (* (* (* b b) b) 0.020833333333333332)
     (/ 1.0 (fma (fma 0.5 b 1.0) b 2.0)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1.9e+154) {
		tmp = 1.0 / fma(fma(0.5, a, -1.0), a, 2.0);
	} else if (a <= -640.0) {
		tmp = ((b * b) * b) * 0.020833333333333332;
	} else {
		tmp = 1.0 / fma(fma(0.5, b, 1.0), b, 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1.9e+154)
		tmp = Float64(1.0 / fma(fma(0.5, a, -1.0), a, 2.0));
	elseif (a <= -640.0)
		tmp = Float64(Float64(Float64(b * b) * b) * 0.020833333333333332);
	else
		tmp = Float64(1.0 / fma(fma(0.5, b, 1.0), b, 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1.9e+154], N[(1.0 / N[(N[(0.5 * a + -1.0), $MachinePrecision] * a + 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, -640.0], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * 0.020833333333333332), $MachinePrecision], N[(1.0 / N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.8999999999999999e154

   1. Initial program 100.0%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. rec-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 100.0

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in b around 0

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. exp-neg N/A

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

      4. lft-mult-inverse N/A

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

      5. lower-+.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-exp.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 100.0%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, -1\right), \color{blue}{a}, 2\right)} \]

   if -1.8999999999999999e154 < a < -640

   1. Initial program 92.9%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 27.3

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

   5. Applied rewrites 27.3%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2} + \color{blue}{b \cdot \left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 2.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.020833333333333332, -0.25\right), \color{blue}{b}, 0.5\right) \]

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{48} \cdot {b}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 56.2%

       \[\leadsto \left(\left(b \cdot b\right) \cdot b\right) \cdot 0.020833333333333332 \]

   if -640 < a

   1. Initial program 99.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 99.0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 62.8%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), \color{blue}{b}, 2\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 70.0% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{+63}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, a, 0.5\right), a, -1\right), a, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 8e+63)
   (/ 1.0 (fma (fma (fma -0.16666666666666666 a 0.5) a -1.0) a 2.0))
   (/ 1.0 (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 8e+63) {
		tmp = 1.0 / fma(fma(fma(-0.16666666666666666, a, 0.5), a, -1.0), a, 2.0);
	} else {
		tmp = 1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 8e+63)
		tmp = Float64(1.0 / fma(fma(fma(-0.16666666666666666, a, 0.5), a, -1.0), a, 2.0));
	else
		tmp = Float64(1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 8e+63], N[(1.0 / N[(N[(N[(-0.16666666666666666 * a + 0.5), $MachinePrecision] * a + -1.0), $MachinePrecision] * a + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.00000000000000046e63

   1. Initial program 98.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. rec-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 98.5

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in b around 0

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. exp-neg N/A

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

      4. lft-mult-inverse N/A

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

      5. lower-+.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-exp.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 74.6

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

   7. Applied rewrites 74.6%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 66.4%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, a, 0.5\right), a, -1\right), \color{blue}{a}, 2\right)} \]

   if 8.00000000000000046e63 < b

   1. Initial program 97.9%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 86.2%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), \color{blue}{b}, 2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 67.1% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.5 \cdot 10^{+63}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, -1\right), a, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.5e+63)
   (/ 1.0 (fma (fma 0.5 a -1.0) a 2.0))
   (/ 1.0 (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.5e+63) {
		tmp = 1.0 / fma(fma(0.5, a, -1.0), a, 2.0);
	} else {
		tmp = 1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.5e+63)
		tmp = Float64(1.0 / fma(fma(0.5, a, -1.0), a, 2.0));
	else
		tmp = Float64(1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.5e+63], N[(1.0 / N[(N[(0.5 * a + -1.0), $MachinePrecision] * a + 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.5e63

   1. Initial program 98.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lower-*.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. rec-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-neg.f64 98.5

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 98.5

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in b around 0

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. exp-neg N/A

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

      4. lft-mult-inverse N/A

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

      5. lower-+.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-exp.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 74.6

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

   7. Applied rewrites 74.6%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 64.2%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, -1\right), \color{blue}{a}, 2\right)} \]

   if 1.5e63 < b

   1. Initial program 97.9%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 86.2%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), \color{blue}{b}, 2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 57.6% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -640:\\ \;\;\;\;\left(\left(b \cdot b\right) \cdot b\right) \cdot 0.020833333333333332\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -640.0)
   (* (* (* b b) b) 0.020833333333333332)
   (/ 1.0 (fma (fma 0.5 b 1.0) b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -640.0) {
		tmp = ((b * b) * b) * 0.020833333333333332;
	} else {
		tmp = 1.0 / fma(fma(0.5, b, 1.0), b, 2.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -640.0)
		tmp = Float64(Float64(Float64(b * b) * b) * 0.020833333333333332);
	else
		tmp = Float64(1.0 / fma(fma(0.5, b, 1.0), b, 2.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -640.0], N[(N[(N[(b * b), $MachinePrecision] * b), $MachinePrecision] * 0.020833333333333332), $MachinePrecision], N[(1.0 / N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -640

   1. Initial program 96.4%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 29.1

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

   5. Applied rewrites 29.1%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2} + \color{blue}{b \cdot \left(\frac{1}{48} \cdot {b}^{2} - \frac{1}{4}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 2.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, 0.020833333333333332, -0.25\right), \color{blue}{b}, 0.5\right) \]

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{48} \cdot {b}^{\color{blue}{3}} \]
   10. Step-by-step derivation

   11. Applied rewrites 55.6%

       \[\leadsto \left(\left(b \cdot b\right) \cdot b\right) \cdot 0.020833333333333332 \]

   if -640 < a

   1. Initial program 99.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 99.0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 62.8%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), \color{blue}{b}, 2\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 39.9% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2 - a} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ 1.0 (- 2.0 a)))

C

double code(double a, double b) {
	return 1.0 / (2.0 - a);
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 / (2.0d0 - a)
end function

Java

public static double code(double a, double b) {
	return 1.0 / (2.0 - a);
}

Python

def code(a, b):
	return 1.0 / (2.0 - a)

Julia

function code(a, b)
	return Float64(1.0 / Float64(2.0 - a))
end

MATLAB

function tmp = code(a, b)
	tmp = 1.0 / (2.0 - a);
end

Wolfram

code[a_, b_] := N[(1.0 / N[(2.0 - a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.4%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. clear-num N/A

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

   5. associate-/r/ N/A

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

   6. lower-*.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. rec-exp N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-neg.f64 98.4

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

   11. lift-+.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 98.4

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

4. Applied rewrites 98.4%

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

5. Taylor expanded in b around 0

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

   1. distribute-lft-in N/A

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

   2. *-rgt-identity N/A

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

   3. exp-neg N/A

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

   4. lft-mult-inverse N/A

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

   5. lower-+.f64 N/A

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

   6. neg-mul-1 N/A

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

   7. lower-exp.f64 N/A

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

   8. neg-mul-1 N/A

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

   9. lower-neg.f64 65.7

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

7. Applied rewrites 65.7%

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

8. Taylor expanded in a around 0

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

10. Applied rewrites 44.4%

    \[\leadsto \frac{1}{2 - \color{blue}{a}} \]
11. Add Preprocessing

Alternative 11: 39.2% accurate, 45.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.25, a, 0.5\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (fma 0.25 a 0.5))

C

double code(double a, double b) {
	return fma(0.25, a, 0.5);
}

Julia

function code(a, b)
	return fma(0.25, a, 0.5)
end

Wolfram

code[a_, b_] := N[(0.25 * a + 0.5), $MachinePrecision]

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in b around 0

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

   1. associate-*r/ N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. times-frac N/A

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

   5. distribute-lft1-in N/A

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

   6. lower-*.f64 N/A

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

5. Applied rewrites 62.8%

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

6. Taylor expanded in a around 0

   \[\leadsto \frac{1}{2} \cdot \left(1 + \frac{-1}{2} \cdot b\right) + \color{blue}{a \cdot \left(\frac{1}{2} \cdot \left(1 + \left(\frac{-1}{2} \cdot b + \frac{1}{4} \cdot b\right)\right) - \frac{1}{4} \cdot \left(1 + \frac{-1}{2} \cdot b\right)\right)} \]

8. Applied rewrites 41.5%

   \[\leadsto \mathsf{fma}\left(0.25, \color{blue}{a}, \mathsf{fma}\left(-0.25, b, 0.5\right)\right) \]

9. Taylor expanded in b around 0

   \[\leadsto \mathsf{fma}\left(\frac{1}{4}, a, \frac{1}{2}\right) \]
10. Step-by-step derivation

11. Applied rewrites 43.9%

    \[\leadsto \mathsf{fma}\left(0.25, a, 0.5\right) \]
12. Add Preprocessing

Alternative 12: 39.0% accurate, 315.0× 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 98.4%

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

3. Taylor expanded in a around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. lower-exp.f64 83.7

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

5. Applied rewrites 83.7%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 43.7%

   \[\leadsto 0.5 \]
9. Add Preprocessing

Developer Target 1: 100.0% accurate, 2.7× 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 2024240 
(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))))