Quotient of sum of exps

Percentage Accurate: 99.1% → 99.1%

Time: 5.9s

Alternatives: 12

Speedup: 1.0×

• could not determine a ground truth

  1. a = 3.4399616301689182e+78
  2. b = 5.233390180488412e+137

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: 99.1% 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: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Final simplification 99.2%

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

Alternative 2: 72.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{b} + e^{a}} \leq 0.508817835104878:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp b) (exp a))) 0.508817835104878)
   (/ 1.0 (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 2.0))
   1.0))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(b) + exp(a))) <= 0.508817835104878) {
		tmp = 1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(b) + exp(a))) <= 0.508817835104878)
		tmp = Float64(1.0 / fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 2.0));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[b], $MachinePrecision] + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.508817835104878], N[(1.0 / N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.50881783510487799

   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 74.7

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

   5. Applied rewrites 74.7%

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

   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 64.1%

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

   if 0.50881783510487799 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 95.0%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. pow-pow N/A

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

      8. sqr-pow N/A

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

      9. unpow-prod-down N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower-pow.f64 N/A

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

   4. Applied rewrites 95.0%

      \[\leadsto \frac{e^{a}}{\color{blue}{{\left({\left(e^{b} + e^{a}\right)}^{-0.5}\right)}^{-1} \cdot {\left({\left(e^{b} + e^{a}\right)}^{-0.5}\right)}^{-1}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. pow2 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-to-exp N/A

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

      7. pow-exp N/A

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

      8. div-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

   6. Applied rewrites 95.1%

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

      1. lift-exp.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. div-inv N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

   8. Applied rewrites 97.6%

      \[\leadsto \color{blue}{{\left(e^{a \cdot a - {\log \left(\sqrt{e^{b} + e^{a}}\right)}^{2} \cdot 4}\right)}^{\left({\left(\mathsf{fma}\left(2, \log \left(\sqrt{e^{b} + e^{a}}\right), a\right)\right)}^{-1}\right)}} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 97.6%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{b} + e^{a}} \leq 0.508817835104878:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{b} + e^{a}} \leq 0.508817835104878:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot b, b, 1\right), b, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp b) (exp a))) 0.508817835104878)
   (/ 1.0 (fma (fma (* 0.16666666666666666 b) b 1.0) b 2.0))
   1.0))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(b) + exp(a))) <= 0.508817835104878) {
		tmp = 1.0 / fma(fma((0.16666666666666666 * b), b, 1.0), b, 2.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(b) + exp(a))) <= 0.508817835104878)
		tmp = Float64(1.0 / fma(fma(Float64(0.16666666666666666 * b), b, 1.0), b, 2.0));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[b], $MachinePrecision] + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.508817835104878], N[(1.0 / N[(N[(N[(0.16666666666666666 * b), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.50881783510487799

   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 74.7

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

   5. Applied rewrites 74.7%

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

   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 64.1%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 63.6%

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

   if 0.50881783510487799 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 95.0%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. pow-pow N/A

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

      8. sqr-pow N/A

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

      9. unpow-prod-down N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower-pow.f64 N/A

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

   4. Applied rewrites 95.0%

      \[\leadsto \frac{e^{a}}{\color{blue}{{\left({\left(e^{b} + e^{a}\right)}^{-0.5}\right)}^{-1} \cdot {\left({\left(e^{b} + e^{a}\right)}^{-0.5}\right)}^{-1}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. pow2 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-to-exp N/A

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

      7. pow-exp N/A

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

      8. div-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

   6. Applied rewrites 95.1%

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

      1. lift-exp.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. div-inv N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

   8. Applied rewrites 97.6%

      \[\leadsto \color{blue}{{\left(e^{a \cdot a - {\log \left(\sqrt{e^{b} + e^{a}}\right)}^{2} \cdot 4}\right)}^{\left({\left(\mathsf{fma}\left(2, \log \left(\sqrt{e^{b} + e^{a}}\right), a\right)\right)}^{-1}\right)}} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 97.6%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{b} + e^{a}} \leq 0.508817835104878:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot b, b, 1\right), b, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{b} + e^{a}} \leq 0.508817835104878:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp b) (exp a))) 0.508817835104878)
   (/ 1.0 (fma (fma 0.5 b 1.0) b 2.0))
   1.0))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(b) + exp(a))) <= 0.508817835104878) {
		tmp = 1.0 / fma(fma(0.5, b, 1.0), b, 2.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(b) + exp(a))) <= 0.508817835104878)
		tmp = Float64(1.0 / fma(fma(0.5, b, 1.0), b, 2.0));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[b], $MachinePrecision] + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.508817835104878], N[(1.0 / N[(N[(0.5 * b + 1.0), $MachinePrecision] * b + 2.0), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.50881783510487799

   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 74.7

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

   5. Applied rewrites 74.7%

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

   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 60.0%

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

   if 0.50881783510487799 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 95.0%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. pow-pow N/A

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

      8. sqr-pow N/A

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

      9. unpow-prod-down N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower-pow.f64 N/A

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

   4. Applied rewrites 95.0%

      \[\leadsto \frac{e^{a}}{\color{blue}{{\left({\left(e^{b} + e^{a}\right)}^{-0.5}\right)}^{-1} \cdot {\left({\left(e^{b} + e^{a}\right)}^{-0.5}\right)}^{-1}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. pow2 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-to-exp N/A

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

      7. pow-exp N/A

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

      8. div-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

   6. Applied rewrites 95.1%

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

      1. lift-exp.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. div-inv N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

   8. Applied rewrites 97.6%

      \[\leadsto \color{blue}{{\left(e^{a \cdot a - {\log \left(\sqrt{e^{b} + e^{a}}\right)}^{2} \cdot 4}\right)}^{\left({\left(\mathsf{fma}\left(2, \log \left(\sqrt{e^{b} + e^{a}}\right), a\right)\right)}^{-1}\right)}} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 97.6%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{b} + e^{a}} \leq 0.508817835104878:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, b, 1\right), b, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{b} + e^{a}} \leq 0.508817835104878:\\ \;\;\;\;\frac{1}{2 + b}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp b) (exp a))) 0.508817835104878)
   (/ 1.0 (+ 2.0 b))
   1.0))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(b) + exp(a))) <= 0.508817835104878) {
		tmp = 1.0 / (2.0 + b);
	} else {
		tmp = 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) / (exp(b) + exp(a))) <= 0.508817835104878d0) then
        tmp = 1.0d0 / (2.0d0 + b)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((Math.exp(a) / (Math.exp(b) + Math.exp(a))) <= 0.508817835104878) {
		tmp = 1.0 / (2.0 + b);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (math.exp(a) / (math.exp(b) + math.exp(a))) <= 0.508817835104878:
		tmp = 1.0 / (2.0 + b)
	else:
		tmp = 1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(b) + exp(a))) <= 0.508817835104878)
		tmp = Float64(1.0 / Float64(2.0 + b));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((exp(a) / (exp(b) + exp(a))) <= 0.508817835104878)
		tmp = 1.0 / (2.0 + b);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[b], $MachinePrecision] + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.508817835104878], N[(1.0 / N[(2.0 + b), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.50881783510487799

   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 74.7

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

   5. Applied rewrites 74.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 43.9%

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

   if 0.50881783510487799 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 95.0%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. pow-pow N/A

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

      8. sqr-pow N/A

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

      9. unpow-prod-down N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower-pow.f64 N/A

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

   4. Applied rewrites 95.0%

      \[\leadsto \frac{e^{a}}{\color{blue}{{\left({\left(e^{b} + e^{a}\right)}^{-0.5}\right)}^{-1} \cdot {\left({\left(e^{b} + e^{a}\right)}^{-0.5}\right)}^{-1}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. pow2 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-to-exp N/A

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

      7. pow-exp N/A

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

      8. div-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

   6. Applied rewrites 95.1%

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

      1. lift-exp.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. div-inv N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

   8. Applied rewrites 97.6%

      \[\leadsto \color{blue}{{\left(e^{a \cdot a - {\log \left(\sqrt{e^{b} + e^{a}}\right)}^{2} \cdot 4}\right)}^{\left({\left(\mathsf{fma}\left(2, \log \left(\sqrt{e^{b} + e^{a}}\right), a\right)\right)}^{-1}\right)}} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 97.6%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{b} + e^{a}} \leq 0.508817835104878:\\ \;\;\;\;\frac{1}{2 + b}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{b} + e^{a}} \leq 0.508817835104878:\\ \;\;\;\;\mathsf{fma}\left(0.25, a, \mathsf{fma}\left(-0.25, b, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp b) (exp a))) 0.508817835104878)
   (fma 0.25 a (fma -0.25 b 0.5))
   1.0))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(b) + exp(a))) <= 0.508817835104878) {
		tmp = fma(0.25, a, fma(-0.25, b, 0.5));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(b) + exp(a))) <= 0.508817835104878)
		tmp = fma(0.25, a, fma(-0.25, b, 0.5));
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[b], $MachinePrecision] + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.508817835104878], N[(0.25 * a + N[(-0.25 * b + 0.5), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.50881783510487799

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 79.8%

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

   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 43.4%

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

   if 0.50881783510487799 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 95.0%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. pow-pow N/A

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

      8. sqr-pow N/A

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

      9. unpow-prod-down N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower-pow.f64 N/A

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

   4. Applied rewrites 95.0%

      \[\leadsto \frac{e^{a}}{\color{blue}{{\left({\left(e^{b} + e^{a}\right)}^{-0.5}\right)}^{-1} \cdot {\left({\left(e^{b} + e^{a}\right)}^{-0.5}\right)}^{-1}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. pow2 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-to-exp N/A

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

      7. pow-exp N/A

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

      8. div-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

   6. Applied rewrites 95.1%

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

      1. lift-exp.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. div-inv N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

   8. Applied rewrites 97.6%

      \[\leadsto \color{blue}{{\left(e^{a \cdot a - {\log \left(\sqrt{e^{b} + e^{a}}\right)}^{2} \cdot 4}\right)}^{\left({\left(\mathsf{fma}\left(2, \log \left(\sqrt{e^{b} + e^{a}}\right), a\right)\right)}^{-1}\right)}} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 97.6%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{b} + e^{a}} \leq 0.508817835104878:\\ \;\;\;\;\mathsf{fma}\left(0.25, a, \mathsf{fma}\left(-0.25, b, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{b} + e^{a}} \leq 0.508817835104878:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp b) (exp a))) 0.508817835104878)
   (fma -0.25 b 0.5)
   1.0))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(b) + exp(a))) <= 0.508817835104878) {
		tmp = fma(-0.25, b, 0.5);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(b) + exp(a))) <= 0.508817835104878)
		tmp = fma(-0.25, b, 0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[b], $MachinePrecision] + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.508817835104878], N[(-0.25 * b + 0.5), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.50881783510487799

   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 74.7

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

   5. Applied rewrites 74.7%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2} + \color{blue}{\frac{-1}{4} \cdot b} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.3%

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

   if 0.50881783510487799 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 95.0%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. pow-pow N/A

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

      8. sqr-pow N/A

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

      9. unpow-prod-down N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower-pow.f64 N/A

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

   4. Applied rewrites 95.0%

      \[\leadsto \frac{e^{a}}{\color{blue}{{\left({\left(e^{b} + e^{a}\right)}^{-0.5}\right)}^{-1} \cdot {\left({\left(e^{b} + e^{a}\right)}^{-0.5}\right)}^{-1}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. pow2 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-to-exp N/A

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

      7. pow-exp N/A

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

      8. div-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

   6. Applied rewrites 95.1%

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

      1. lift-exp.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. div-inv N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

   8. Applied rewrites 97.6%

      \[\leadsto \color{blue}{{\left(e^{a \cdot a - {\log \left(\sqrt{e^{b} + e^{a}}\right)}^{2} \cdot 4}\right)}^{\left({\left(\mathsf{fma}\left(2, \log \left(\sqrt{e^{b} + e^{a}}\right), a\right)\right)}^{-1}\right)}} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 97.6%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{b} + e^{a}} \leq 0.508817835104878:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-13}:\\ \;\;\;\;\frac{e^{a}}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 1e-13) (/ (exp a) (+ 1.0 (exp a))) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 1e-13) {
		tmp = exp(a) / (1.0 + exp(a));
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 1d-13) then
        tmp = exp(a) / (1.0d0 + exp(a))
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 1e-13) {
		tmp = Math.exp(a) / (1.0 + Math.exp(a));
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 1e-13:
		tmp = math.exp(a) / (1.0 + math.exp(a))
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 1e-13)
		tmp = Float64(exp(a) / Float64(1.0 + exp(a)));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 1e-13)
		tmp = exp(a) / (1.0 + exp(a));
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 1e-13], N[(N[Exp[a], $MachinePrecision] / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 1e-13

   1. Initial program 98.8%

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

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

   if 1e-13 < (exp.f64 a)

   1. Initial program 99.4%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 98.8

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

   5. Applied rewrites 98.8%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-13}:\\ \;\;\;\;\frac{e^{a}}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{b} + e^{a}} \leq 0.75:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (/ (exp a) (+ (exp b) (exp a))) 0.75) 0.5 1.0))

C

double code(double a, double b) {
	double tmp;
	if ((exp(a) / (exp(b) + exp(a))) <= 0.75) {
		tmp = 0.5;
	} else {
		tmp = 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) / (exp(b) + exp(a))) <= 0.75d0) then
        tmp = 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if ((Math.exp(a) / (Math.exp(b) + Math.exp(a))) <= 0.75) {
		tmp = 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if (math.exp(a) / (math.exp(b) + math.exp(a))) <= 0.75:
		tmp = 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (Float64(exp(a) / Float64(exp(b) + exp(a))) <= 0.75)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if ((exp(a) / (exp(b) + exp(a))) <= 0.75)
		tmp = 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[b], $MachinePrecision] + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.75], 0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b))) < 0.75

   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 74.7

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

   5. Applied rewrites 74.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 42.5%

      \[\leadsto 0.5 \]

   if 0.75 < (/.f64 (exp.f64 a) (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 95.0%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. pow-flip N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. pow-pow N/A

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

      8. sqr-pow N/A

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

      9. unpow-prod-down N/A

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

      10. lower-*.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower-pow.f64 N/A

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

   4. Applied rewrites 95.0%

      \[\leadsto \frac{e^{a}}{\color{blue}{{\left({\left(e^{b} + e^{a}\right)}^{-0.5}\right)}^{-1} \cdot {\left({\left(e^{b} + e^{a}\right)}^{-0.5}\right)}^{-1}}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. pow2 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-to-exp N/A

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

      7. pow-exp N/A

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

      8. div-exp N/A

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

      9. lower-exp.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

   6. Applied rewrites 95.1%

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

      1. lift-exp.f64 N/A

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. div-inv N/A

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

      5. exp-prod N/A

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

      6. lower-pow.f64 N/A

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

   8. Applied rewrites 97.6%

      \[\leadsto \color{blue}{{\left(e^{a \cdot a - {\log \left(\sqrt{e^{b} + e^{a}}\right)}^{2} \cdot 4}\right)}^{\left({\left(\mathsf{fma}\left(2, \log \left(\sqrt{e^{b} + e^{a}}\right), a\right)\right)}^{-1}\right)}} \]

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{1} \]
   10. Step-by-step derivation

   11. Applied rewrites 97.6%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{a}}{e^{b} + e^{a}} \leq 0.75:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-13}:\\ \;\;\;\;\frac{e^{a}}{1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 1e-13) (/ (exp a) (+ 1.0 1.0)) (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 1e-13) {
		tmp = exp(a) / (1.0 + 1.0);
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 1d-13) then
        tmp = exp(a) / (1.0d0 + 1.0d0)
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 1e-13) {
		tmp = Math.exp(a) / (1.0 + 1.0);
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 1e-13:
		tmp = math.exp(a) / (1.0 + 1.0)
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 1e-13)
		tmp = Float64(exp(a) / Float64(1.0 + 1.0));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 1e-13)
		tmp = exp(a) / (1.0 + 1.0);
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 1e-13], N[(N[Exp[a], $MachinePrecision] / N[(1.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 1e-13

   1. Initial program 98.8%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 99.0%

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

   if 1e-13 < (exp.f64 a)

   1. Initial program 99.4%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 98.8

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

   5. Applied rewrites 98.8%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-13}:\\ \;\;\;\;\frac{e^{a}}{1 + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 86.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7.8 \cdot 10^{+23}:\\ \;\;\;\;-0.0020833333333333333 \cdot {b}^{5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -7.8e+23)
   (* -0.0020833333333333333 (pow b 5.0))
   (/ 1.0 (+ 1.0 (exp b)))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -7.8e+23) {
		tmp = -0.0020833333333333333 * pow(b, 5.0);
	} else {
		tmp = 1.0 / (1.0 + exp(b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-7.8d+23)) then
        tmp = (-0.0020833333333333333d0) * (b ** 5.0d0)
    else
        tmp = 1.0d0 / (1.0d0 + exp(b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -7.8e+23) {
		tmp = -0.0020833333333333333 * Math.pow(b, 5.0);
	} else {
		tmp = 1.0 / (1.0 + Math.exp(b));
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -7.8e+23:
		tmp = -0.0020833333333333333 * math.pow(b, 5.0)
	else:
		tmp = 1.0 / (1.0 + math.exp(b))
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -7.8e+23)
		tmp = Float64(-0.0020833333333333333 * (b ^ 5.0));
	else
		tmp = Float64(1.0 / Float64(1.0 + exp(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -7.8e+23)
		tmp = -0.0020833333333333333 * (b ^ 5.0);
	else
		tmp = 1.0 / (1.0 + exp(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -7.8e+23], N[(-0.0020833333333333333 * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -7.8000000000000001e23

   1. Initial program 98.7%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 32.8

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

   5. Applied rewrites 32.8%

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

   6. Taylor expanded in b around 0

      \[\leadsto \frac{1}{2} + \color{blue}{b \cdot \left({b}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {b}^{2}\right) - \frac{1}{4}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 2.7%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0020833333333333333, b \cdot b, 0.020833333333333332\right), b \cdot b, -0.25\right), \color{blue}{b}, 0.5\right) \]

   9. Taylor expanded in b around inf

      \[\leadsto \frac{-1}{480} \cdot {b}^{\color{blue}{5}} \]
   10. Step-by-step derivation

   11. Applied rewrites 55.9%

       \[\leadsto {b}^{5} \cdot -0.0020833333333333333 \]

   if -7.8000000000000001e23 < a

   1. Initial program 99.4%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   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 96.7

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

   5. Applied rewrites 96.7%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.8 \cdot 10^{+23}:\\ \;\;\;\;-0.0020833333333333333 \cdot {b}^{5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + e^{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 39.4% 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 78.0

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

5. Applied rewrites 78.0%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 38.8%

   \[\leadsto 0.5 \]
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 2024284 
(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))))