Quotient of sum of exps

Percentage Accurate: 99.0% → 100.0%

Time: 5.5s

Alternatives: 13

Speedup: 2.7×

• could not determine a ground truth

  1. a = 6.756660373856674e+47
  2. b = 2.686542662954431e+244

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, 2.7× 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]

Derivation

1. Initial program 99.2%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
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 65.0%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. remove-double-div N/A

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

   4. lift-exp.f64 N/A

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

   5. exp-neg N/A

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

   6. lift-neg.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. frac-2neg N/A

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

   9. metadata-eval N/A

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

   10. frac-times N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-neg.f64 65.0

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 65.0

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

7. Applied rewrites 65.0%

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

8. Taylor expanded in b around inf

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

   1. associate-*r* N/A

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

   2. distribute-lft-in N/A

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

   3. mul-1-neg N/A

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

   4. distribute-lft-neg-out N/A

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

   5. exp-neg N/A

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

   6. lft-mult-inverse N/A

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

   7. metadata-eval N/A

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

   8. mul-1-neg N/A

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

   9. distribute-lft-neg-in N/A

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

   10. unsub-neg N/A

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

   11. lower--.f64 N/A

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

   12. *-commutative N/A

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

   13. neg-mul-1 N/A

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

   14. prod-exp N/A

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

   15. lower-exp.f64 N/A

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

   16. neg-mul-1 N/A

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

   17. unsub-neg N/A

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

   18. lower--.f64 100.0

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

10. Applied rewrites 100.0%

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

Alternative 2: 98.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -55000000000000:\\ \;\;\;\;\frac{e^{a}}{1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -55000000000000.0) (/ (exp a) (+ 1.0 1.0)) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -55000000000000.0) {
		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 (a <= (-55000000000000.0d0)) 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 (a <= -55000000000000.0) {
		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 a <= -55000000000000.0:
		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 (a <= -55000000000000.0)
		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 (a <= -55000000000000.0)
		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[a, -55000000000000.0], 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 a < -5.5e13

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 -5.5e13 < a

   1. Initial program 98.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 98.2

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

   5. Applied rewrites 98.2%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -55000000000000:\\ \;\;\;\;\frac{e^{a}}{1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right), a, 1\right), a, 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -1e+103)
   (/ 1.0 (+ (fma (fma (fma 0.16666666666666666 a 0.5) a 1.0) a 1.0) 1.0))
   (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -1e+103) {
		tmp = 1.0 / (fma(fma(fma(0.16666666666666666, a, 0.5), a, 1.0), a, 1.0) + 1.0);
	} else {
		tmp = 1.0 / (exp(b) + 1.0);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -1e+103)
		tmp = Float64(1.0 / Float64(fma(fma(fma(0.16666666666666666, a, 0.5), a, 1.0), a, 1.0) + 1.0));
	else
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[a, -1e+103], N[(1.0 / N[(N[(N[(N[(0.16666666666666666 * a + 0.5), $MachinePrecision] * a + 1.0), $MachinePrecision] * a + 1.0), $MachinePrecision] + 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 a < -1e103

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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

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

      1. +-commutative N/A

         \[\leadsto \frac{e^{a}}{\color{blue}{\left(a \cdot \left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right)\right) + 1\right)} + 1} \]

      2. *-commutative N/A

         \[\leadsto \frac{e^{a}}{\left(\color{blue}{\left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right)\right) \cdot a} + 1\right) + 1} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{e^{a}}{\color{blue}{\mathsf{fma}\left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right), a, 1\right)} + 1} \]

      4. +-commutative N/A

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\color{blue}{a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right) + 1}, a, 1\right) + 1} \]

      5. *-commutative N/A

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot a\right) \cdot a} + 1, a, 1\right) + 1} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot a, a, 1\right)}, a, 1\right) + 1} \]

      7. +-commutative N/A

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot a + \frac{1}{2}}, a, 1\right), a, 1\right) + 1} \]

      8. lower-fma.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 100.0%

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

   if -1e103 < 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 94.9

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

   5. Applied rewrites 94.9%

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

Alternative 4: 70.2% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{+79}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, a, 0.5\right), a, 1\right), a, 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3e+79)
   (/ 1.0 (+ (fma (fma (fma 0.16666666666666666 a 0.5) a 1.0) a 1.0) 1.0))
   (/ 1.0 (* (* (fma 0.16666666666666666 b 0.5) b) b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3e+79) {
		tmp = 1.0 / (fma(fma(fma(0.16666666666666666, a, 0.5), a, 1.0), a, 1.0) + 1.0);
	} else {
		tmp = 1.0 / ((fma(0.16666666666666666, b, 0.5) * b) * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3e+79)
		tmp = Float64(1.0 / Float64(fma(fma(fma(0.16666666666666666, a, 0.5), a, 1.0), a, 1.0) + 1.0));
	else
		tmp = Float64(1.0 / Float64(Float64(fma(0.16666666666666666, b, 0.5) * b) * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3e+79], N[(1.0 / N[(N[(N[(N[(0.16666666666666666 * a + 0.5), $MachinePrecision] * a + 1.0), $MachinePrecision] * a + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.99999999999999974e79

   1. Initial program 99.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 71.0%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{e^{a}}{\color{blue}{\left(a \cdot \left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right)\right) + 1\right)} + 1} \]

      2. *-commutative N/A

         \[\leadsto \frac{e^{a}}{\left(\color{blue}{\left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right)\right) \cdot a} + 1\right) + 1} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{e^{a}}{\color{blue}{\mathsf{fma}\left(1 + a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right), a, 1\right)} + 1} \]

      4. +-commutative N/A

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\color{blue}{a \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot a\right) + 1}, a, 1\right) + 1} \]

      5. *-commutative N/A

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot a\right) \cdot a} + 1, a, 1\right) + 1} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot a, a, 1\right)}, a, 1\right) + 1} \]

      7. +-commutative N/A

         \[\leadsto \frac{e^{a}}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot a + \frac{1}{2}}, a, 1\right), a, 1\right) + 1} \]

      8. lower-fma.f64 70.7

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

   8. Applied rewrites 70.7%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 66.0%

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

   if 2.99999999999999974e79 < b

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 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 90.9%

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

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{{b}^{3} \cdot \left(\frac{1}{6} + \color{blue}{\frac{1}{2} \cdot \frac{1}{b}}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 90.9%

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

Alternative 5: 66.9% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{+79}:\\ \;\;\;\;\frac{1 + a}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 3e+79)
   (/ (+ 1.0 a) (+ (fma (fma 0.5 a 1.0) a 1.0) 1.0))
   (/ 1.0 (* (* (fma 0.16666666666666666 b 0.5) b) b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3e+79) {
		tmp = (1.0 + a) / (fma(fma(0.5, a, 1.0), a, 1.0) + 1.0);
	} else {
		tmp = 1.0 / ((fma(0.16666666666666666, b, 0.5) * b) * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3e+79)
		tmp = Float64(Float64(1.0 + a) / Float64(fma(fma(0.5, a, 1.0), a, 1.0) + 1.0));
	else
		tmp = Float64(1.0 / Float64(Float64(fma(0.16666666666666666, b, 0.5) * b) * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3e+79], N[(N[(1.0 + a), $MachinePrecision] / N[(N[(N[(0.5 * a + 1.0), $MachinePrecision] * a + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.99999999999999974e79

   1. Initial program 99.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 71.0%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 70.3

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

   8. Applied rewrites 70.3%

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

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 47.4

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

   11. Applied rewrites 47.4%

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

   12. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 60.0

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

   14. Applied rewrites 60.0%

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

   if 2.99999999999999974e79 < b

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 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 90.9%

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

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{{b}^{3} \cdot \left(\frac{1}{6} + \color{blue}{\frac{1}{2} \cdot \frac{1}{b}}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 90.9%

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

Alternative 6: 58.3% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.8 \cdot 10^{-13}:\\ \;\;\;\;\frac{1 + a}{\left(1 + a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 8.8e-13)
   (/ (+ 1.0 a) (+ (+ 1.0 a) 1.0))
   (/ 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 <= 8.8e-13) {
		tmp = (1.0 + a) / ((1.0 + a) + 1.0);
	} 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 <= 8.8e-13)
		tmp = Float64(Float64(1.0 + a) / Float64(Float64(1.0 + a) + 1.0));
	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, 8.8e-13], N[(N[(1.0 + a), $MachinePrecision] / N[(N[(1.0 + a), $MachinePrecision] + 1.0), $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 < 8.79999999999999986e-13

   1. Initial program 98.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 74.1%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 73.4

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

   8. Applied rewrites 73.4%

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

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 51.4

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

   11. Applied rewrites 51.4%

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

   if 8.79999999999999986e-13 < b

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 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 73.1%

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

Alternative 7: 58.1% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.15:\\ \;\;\;\;\frac{1 + a}{\left(1 + a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b, b, b\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.15)
   (/ (+ 1.0 a) (+ (+ 1.0 a) 1.0))
   (/ 1.0 (fma (* (fma 0.16666666666666666 b 0.5) b) b b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.15) {
		tmp = (1.0 + a) / ((1.0 + a) + 1.0);
	} else {
		tmp = 1.0 / fma((fma(0.16666666666666666, b, 0.5) * b), b, b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.15)
		tmp = Float64(Float64(1.0 + a) / Float64(Float64(1.0 + a) + 1.0));
	else
		tmp = Float64(1.0 / fma(Float64(fma(0.16666666666666666, b, 0.5) * b), b, b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.15], N[(N[(1.0 + a), $MachinePrecision] / N[(N[(1.0 + a), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b), $MachinePrecision] * b + b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.1499999999999999

   1. Initial program 98.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 73.8%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 73.1

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

   8. Applied rewrites 73.1%

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

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 51.4

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

   11. Applied rewrites 51.4%

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

   if 1.1499999999999999 < b

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 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 73.0%

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

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{{b}^{3} \cdot \left(\frac{1}{6} + \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{b} + \frac{1}{{b}^{2}}\right)}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 73.0%

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

Alternative 8: 58.1% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.65:\\ \;\;\;\;\frac{1 + a}{\left(1 + a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right) \cdot b\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.65)
   (/ (+ 1.0 a) (+ (+ 1.0 a) 1.0))
   (/ 1.0 (* (* (fma 0.16666666666666666 b 0.5) b) b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.65) {
		tmp = (1.0 + a) / ((1.0 + a) + 1.0);
	} else {
		tmp = 1.0 / ((fma(0.16666666666666666, b, 0.5) * b) * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.65)
		tmp = Float64(Float64(1.0 + a) / Float64(Float64(1.0 + a) + 1.0));
	else
		tmp = Float64(1.0 / Float64(Float64(fma(0.16666666666666666, b, 0.5) * b) * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.65], N[(N[(1.0 + a), $MachinePrecision] / N[(N[(1.0 + a), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.6499999999999999

   1. Initial program 98.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 73.8%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 73.1

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

   8. Applied rewrites 73.1%

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

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 51.4

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

   11. Applied rewrites 51.4%

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

   if 1.6499999999999999 < b

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 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 73.0%

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

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{{b}^{3} \cdot \left(\frac{1}{6} + \color{blue}{\frac{1}{2} \cdot \frac{1}{b}}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 73.0%

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

Alternative 9: 53.4% accurate, 10.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.8 \cdot 10^{-13}:\\ \;\;\;\;\frac{1 + a}{\left(1 + a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 8.8e-13)
   (/ (+ 1.0 a) (+ (+ 1.0 a) 1.0))
   (/ 1.0 (fma (fma 0.5 b 1.0) b 2.0))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 8.8e-13) {
		tmp = (1.0 + a) / ((1.0 + a) + 1.0);
	} 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 (b <= 8.8e-13)
		tmp = Float64(Float64(1.0 + a) / Float64(Float64(1.0 + a) + 1.0));
	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[b, 8.8e-13], N[(N[(1.0 + a), $MachinePrecision] / N[(N[(1.0 + a), $MachinePrecision] + 1.0), $MachinePrecision]), $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 b < 8.79999999999999986e-13

   1. Initial program 98.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 74.1%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 73.4

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

   8. Applied rewrites 73.4%

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

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 51.4

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

   11. Applied rewrites 51.4%

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

   if 8.79999999999999986e-13 < b

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 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

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

   8. Applied rewrites 57.4%

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

Alternative 10: 53.1% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.22:\\ \;\;\;\;\frac{1 + a}{\left(1 + a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(b, 0.5, 1\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b 1.22)
   (/ (+ 1.0 a) (+ (+ 1.0 a) 1.0))
   (/ 1.0 (* (fma b 0.5 1.0) b))))

C

double code(double a, double b) {
	double tmp;
	if (b <= 1.22) {
		tmp = (1.0 + a) / ((1.0 + a) + 1.0);
	} else {
		tmp = 1.0 / (fma(b, 0.5, 1.0) * b);
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 1.22)
		tmp = Float64(Float64(1.0 + a) / Float64(Float64(1.0 + a) + 1.0));
	else
		tmp = Float64(1.0 / Float64(fma(b, 0.5, 1.0) * b));
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[b, 1.22], N[(N[(1.0 + a), $MachinePrecision] / N[(N[(1.0 + a), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(b * 0.5 + 1.0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.21999999999999997

   1. Initial program 98.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 73.8%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 73.1

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

   8. Applied rewrites 73.1%

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

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 51.4

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

   11. Applied rewrites 51.4%

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

   if 1.21999999999999997 < b

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 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

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

   8. Applied rewrites 57.0%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 57.0%

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

Alternative 11: 53.1% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2:\\ \;\;\;\;\frac{1 + a}{\left(1 + a\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(0.5 \cdot b\right) \cdot b}\\ \end{array} \end{array} \]

FPCore

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

C

double code(double a, double b) {
	double tmp;
	if (b <= 2.0) {
		tmp = (1.0 + a) / ((1.0 + a) + 1.0);
	} else {
		tmp = 1.0 / ((0.5 * b) * b);
	}
	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 + a) / ((1.0d0 + a) + 1.0d0)
    else
        tmp = 1.0d0 / ((0.5d0 * b) * b)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < 2

   1. Initial program 98.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 73.8%

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

   6. Taylor expanded in a around 0

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

      1. lower-+.f64 73.1

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

   8. Applied rewrites 73.1%

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

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 51.4

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

   11. Applied rewrites 51.4%

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

   if 2 < b

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 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

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

   8. Applied rewrites 57.0%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 57.0%

       \[\leadsto \frac{1}{\left(b \cdot 0.5\right) \cdot b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.1%

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

Alternative 12: 38.7% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(1 + a\right) + 1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(a, b):
	return 1.0 / ((1.0 + a) + 1.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
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 65.0%

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

6. Taylor expanded in a around 0

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

   1. lower-+.f64 64.5

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

8. Applied rewrites 64.5%

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

9. Taylor expanded in a around 0

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

11. Applied rewrites 36.9%

    \[\leadsto \frac{\color{blue}{1}}{\left(1 + a\right) + 1} \]
12. Add Preprocessing

Alternative 13: 38.3% 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 99.2%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
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 83.2

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

5. Applied rewrites 83.2%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 36.3%

   \[\leadsto 0.5 \]
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 2024254 
(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))))