1. Initial program 99.2%

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

   1. add-exp-log99.2%

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

   2. div-exp99.2%

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

3. Applied egg-rr99.2%

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

4. Taylor expanded in a around 0 99.3%

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

   1. +-commutative99.3%

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

   2. log1p-def99.3%

      \[\leadsto e^{a - \left(\color{blue}{\mathsf{log1p}\left(e^{b}\right)} + \frac{a}{1 + e^{b}}\right)} \]

6. Simplified99.3%

   \[\leadsto e^{a - \color{blue}{\left(\mathsf{log1p}\left(e^{b}\right) + \frac{a}{1 + e^{b}}\right)}} \]

7. Final simplification99.3%

   \[\leadsto e^{a - \left(\mathsf{log1p}\left(e^{b}\right) + \frac{a}{e^{b} + 1}\right)} \]

1. Initial program 99.2%

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

   1. add-exp-log99.2%

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

   2. div-exp99.2%

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

3. Applied egg-rr99.2%

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

4. Final simplification99.2%

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

1. Split input into 2 regimes

2. if (exp.f64 a) < 4.9999999999999996e-41

   1. Initial program 98.6%

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

      1. add-exp-log98.6%

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

      2. div-exp98.7%

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

   3. Applied egg-rr98.7%

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

   4. Taylor expanded in a around inf 98.7%

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

   if 4.9999999999999996e-41 < (exp.f64 a)

   1. Initial program 99.4%

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

      1. add-exp-log99.4%

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

      2. div-exp99.4%

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

   3. Applied egg-rr99.4%

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

   4. Taylor expanded in a around 0 99.4%

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

      1. neg-mul-199.4%

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

      2. log1p-def99.4%

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

   6. Simplified99.4%

      \[\leadsto e^{\color{blue}{-\mathsf{log1p}\left(e^{b}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-41}:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;e^{-\mathsf{log1p}\left(e^{b}\right)}\\ \end{array} \]

1. Initial program 99.2%

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

2. Final simplification99.2%

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

1. Split input into 2 regimes

2. if (exp.f64 a) < 4.9999999999999996e-41

   1. Initial program 98.6%

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

      1. add-exp-log98.6%

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

      2. div-exp98.7%

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

   3. Applied egg-rr98.7%

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

   4. Taylor expanded in a around inf 98.7%

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

   if 4.9999999999999996e-41 < (exp.f64 a)

   1. Initial program 99.4%

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

   2. Taylor expanded in a around 0 99.4%

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

4. Final simplification99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-41}:\\ \;\;\;\;e^{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]

1. Split input into 2 regimes

2. if b < -12

   1. Initial program 97.9%

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

      1. add-exp-log97.9%

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

      2. div-exp97.9%

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

   3. Applied egg-rr97.9%

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

   4. Taylor expanded in a around 0 99.6%

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

      1. neg-mul-199.6%

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

      2. log1p-def99.6%

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

   6. Simplified99.6%

      \[\leadsto e^{\color{blue}{-\mathsf{log1p}\left(e^{b}\right)}} \]
   7. Step-by-step derivation

      1. exp-neg99.6%

         \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(e^{b}\right)}}} \]

      2. log1p-udef99.6%

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

      3. add-exp-log99.6%

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

      4. add-sqr-sqrt99.6%

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

      5. associate-/r*99.6%

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

   8. Applied egg-rr98.9%

      \[\leadsto \color{blue}{\frac{\mathsf{hypot}\left(1, \sqrt{e^{b}}\right)}{\mathsf{hypot}\left(1, \sqrt{e^{b}}\right)}} \]
   9. Step-by-step derivation

      1. *-inverses98.9%

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

   10. Simplified98.9%

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

   if -12 < b

   1. Initial program 99.5%

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

   2. Taylor expanded in b around 0 73.1%

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

   3. Taylor expanded in a around 0 72.7%

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

4. Final simplification77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -12:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{a}}{2}\\ \end{array} \]

1. Split input into 3 regimes

2. if a < -1.34999999999999992e-45

   1. Initial program 98.8%

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

      1. add-exp-log98.8%

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

      2. div-exp98.8%

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

   3. Applied egg-rr98.8%

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

   4. Taylor expanded in a around inf 90.6%

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

   if -1.34999999999999992e-45 < a < 4.40000000000000024e-44

   1. Initial program 100.0%

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

   2. Taylor expanded in b around 0 53.0%

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

   3. Taylor expanded in a around 0 53.0%

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

      1. *-commutative53.0%

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

   5. Simplified53.0%

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

   if 4.40000000000000024e-44 < a

   1. Initial program 95.3%

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

      1. add-exp-log95.3%

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

      2. div-exp95.4%

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

   3. Applied egg-rr95.4%

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

   4. Taylor expanded in a around 0 95.3%

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

      1. neg-mul-195.3%

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

      2. log1p-def95.3%

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

   6. Simplified95.3%

      \[\leadsto e^{\color{blue}{-\mathsf{log1p}\left(e^{b}\right)}} \]
   7. Step-by-step derivation

      1. exp-neg95.3%

         \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(e^{b}\right)}}} \]

      2. log1p-udef95.3%

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

      3. add-exp-log95.3%

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

      4. add-sqr-sqrt94.7%

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

      5. associate-/r*94.9%

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

   8. Applied egg-rr56.7%

      \[\leadsto \color{blue}{\frac{\mathsf{hypot}\left(1, \sqrt{e^{b}}\right)}{\mathsf{hypot}\left(1, \sqrt{e^{b}}\right)}} \]
   9. Step-by-step derivation

      1. *-inverses57.2%

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

   10. Simplified57.2%

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

4. Final simplification65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{-45}:\\ \;\;\;\;e^{a}\\ \mathbf{elif}\;a \leq 4.4 \cdot 10^{-44}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

1. Split input into 2 regimes

2. if b < -1.6000000000000001

   1. Initial program 97.9%

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

      1. add-exp-log97.9%

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

      2. div-exp97.9%

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

   3. Applied egg-rr97.9%

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

   4. Taylor expanded in a around 0 99.6%

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

      1. neg-mul-199.6%

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

      2. log1p-def99.6%

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

   6. Simplified99.6%

      \[\leadsto e^{\color{blue}{-\mathsf{log1p}\left(e^{b}\right)}} \]
   7. Step-by-step derivation

      1. exp-neg99.6%

         \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(e^{b}\right)}}} \]

      2. log1p-udef99.6%

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

      3. add-exp-log99.6%

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

      4. add-sqr-sqrt99.6%

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

      5. associate-/r*99.6%

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

   8. Applied egg-rr98.9%

      \[\leadsto \color{blue}{\frac{\mathsf{hypot}\left(1, \sqrt{e^{b}}\right)}{\mathsf{hypot}\left(1, \sqrt{e^{b}}\right)}} \]
   9. Step-by-step derivation

      1. *-inverses98.9%

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

   10. Simplified98.9%

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

   if -1.6000000000000001 < b

   1. Initial program 99.5%

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

   2. Taylor expanded in a around 0 82.8%

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

   3. Taylor expanded in b around 0 40.4%

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

      1. *-commutative40.4%

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

   5. Simplified40.4%

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

4. Final simplification51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 + b \cdot -0.25\\ \end{array} \]

1. Split input into 2 regimes

2. if b < -1.1000000000000001

   1. Initial program 97.9%

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

      1. add-exp-log97.9%

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

      2. div-exp97.9%

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

   3. Applied egg-rr97.9%

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

   4. Taylor expanded in a around 0 99.6%

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

      1. neg-mul-199.6%

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

      2. log1p-def99.6%

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

   6. Simplified99.6%

      \[\leadsto e^{\color{blue}{-\mathsf{log1p}\left(e^{b}\right)}} \]
   7. Step-by-step derivation

      1. exp-neg99.6%

         \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(e^{b}\right)}}} \]

      2. log1p-udef99.6%

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

      3. add-exp-log99.6%

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

      4. add-sqr-sqrt99.6%

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

      5. associate-/r*99.6%

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

   8. Applied egg-rr98.9%

      \[\leadsto \color{blue}{\frac{\mathsf{hypot}\left(1, \sqrt{e^{b}}\right)}{\mathsf{hypot}\left(1, \sqrt{e^{b}}\right)}} \]
   9. Step-by-step derivation

      1. *-inverses98.9%

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

   10. Simplified98.9%

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

   if -1.1000000000000001 < b

   1. Initial program 99.5%

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

   2. Taylor expanded in a around 0 82.8%

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

   3. Taylor expanded in b around 0 40.2%

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

4. Final simplification51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

1. Initial program 99.2%

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

2. Taylor expanded in a around 0 86.0%

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

3. Taylor expanded in b around 0 36.1%

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

4. Final simplification36.1%

   \[\leadsto 0.5 \]

herbie shell --seed 2023166 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

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

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