symmetry log of sum of exp

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 9.6%

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

   2. Taylor expanded in b around 0 100.0%

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

      1. log1p-def100.0%

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

   4. Simplified100.0%

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

   5. Taylor expanded in a around 0 6.5%

      \[\leadsto \color{blue}{\log 2} + \frac{b}{1 + e^{a}} \]

   6. Taylor expanded in b around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 65.7%

      \[\log \left(e^{a} + e^{b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \]

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 9.6%

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

   2. Taylor expanded in b around 0 100.0%

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

      1. log1p-def100.0%

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

   4. Simplified100.0%

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

   5. Taylor expanded in a around 0 6.5%

      \[\leadsto \color{blue}{\log 2} + \frac{b}{1 + e^{a}} \]

   6. Taylor expanded in b around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 65.7%

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

   2. Taylor expanded in b around 0 61.6%

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

      1. log1p-def62.6%

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

   4. Simplified62.6%

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

   5. Taylor expanded in a around 0 62.6%

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

4. Final simplification72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right) + \frac{b}{2}\\ \end{array} \]

1. Initial program 51.7%

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

2. Taylor expanded in b around 0 71.2%

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

   1. log1p-def72.0%

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

4. Simplified72.0%

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

5. Final simplification72.0%

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

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 9.6%

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

   2. Taylor expanded in b around 0 100.0%

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

      1. log1p-def100.0%

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

   4. Simplified100.0%

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

   5. Taylor expanded in a around 0 6.5%

      \[\leadsto \color{blue}{\log 2} + \frac{b}{1 + e^{a}} \]

   6. Taylor expanded in b around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 65.7%

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

   2. Taylor expanded in b around 0 60.6%

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

      1. associate-+r+60.6%

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

      2. +-commutative60.6%

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

      3. associate-+l+60.6%

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

   4. Simplified60.6%

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

   5. Taylor expanded in a around inf 60.6%

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

      1. log1p-def61.6%

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

   7. Simplified61.6%

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

4. Final simplification71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a} + b\right)\\ \end{array} \]

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 9.6%

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

   2. Taylor expanded in b around 0 100.0%

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

      1. log1p-def100.0%

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

   4. Simplified100.0%

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

   5. Taylor expanded in a around 0 6.5%

      \[\leadsto \color{blue}{\log 2} + \frac{b}{1 + e^{a}} \]

   6. Taylor expanded in b around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 65.7%

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

   2. Taylor expanded in b around 0 61.0%

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

      1. log1p-def62.0%

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

   4. Simplified62.0%

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

4. Final simplification71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \end{array} \]

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 9.6%

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

   2. Taylor expanded in b around 0 100.0%

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

      1. log1p-def100.0%

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

   4. Simplified100.0%

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

   5. Taylor expanded in a around 0 6.5%

      \[\leadsto \color{blue}{\log 2} + \frac{b}{1 + e^{a}} \]

   6. Taylor expanded in b around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 65.7%

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

   2. Taylor expanded in a around 0 63.9%

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

      1. log1p-def63.9%

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

   4. Simplified63.9%

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

4. Final simplification72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]

1. Initial program 51.7%

   \[\log \left(e^{a} + e^{b}\right) \]
2. Step-by-step derivation

   1. add-sqr-sqrt50.6%

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

   2. log-prod51.0%

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

3. Applied egg-rr51.0%

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

   1. log-prod50.6%

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

   2. rem-square-sqrt51.7%

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

   3. log1p-expm151.7%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(e^{a} + e^{b}\right)\right)\right)} \]

   4. expm1-def51.7%

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

   5. rem-exp-log51.7%

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

   6. associate--l+52.4%

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

   7. expm1-def73.9%

      \[\leadsto \mathsf{log1p}\left(e^{a} + \color{blue}{\mathsf{expm1}\left(b\right)}\right) \]

5. Simplified73.9%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{a} + \mathsf{expm1}\left(b\right)\right)} \]

6. Final simplification73.9%

   \[\leadsto \mathsf{log1p}\left(e^{a} + \mathsf{expm1}\left(b\right)\right) \]

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 9.6%

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

   2. Taylor expanded in b around 0 100.0%

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

      1. log1p-def100.0%

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

   4. Simplified100.0%

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

   5. Taylor expanded in a around 0 6.5%

      \[\leadsto \color{blue}{\log 2} + \frac{b}{1 + e^{a}} \]

   6. Taylor expanded in b around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 65.7%

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

   2. Taylor expanded in a around 0 63.9%

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

   3. Taylor expanded in b around 0 61.3%

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

4. Final simplification71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \]

1. Split input into 2 regimes

2. if a < -125

   1. Initial program 9.6%

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

   2. Taylor expanded in a around 0 5.5%

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

   3. Taylor expanded in b around 0 5.1%

      \[\leadsto \color{blue}{0.5 \cdot b + \log 2} \]

   4. Taylor expanded in b around inf 18.8%

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

   if -125 < a

   1. Initial program 65.7%

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

   2. Taylor expanded in a around 0 63.9%

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

   3. Taylor expanded in b around 0 61.3%

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

4. Final simplification50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -125:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \]

1. Split input into 2 regimes

2. if a < -155

   1. Initial program 9.6%

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

   2. Taylor expanded in a around 0 5.5%

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

   3. Taylor expanded in b around 0 5.1%

      \[\leadsto \color{blue}{0.5 \cdot b + \log 2} \]

   4. Taylor expanded in b around inf 18.8%

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

   if -155 < a

   1. Initial program 65.7%

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

   2. Taylor expanded in a around 0 63.9%

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

   3. Taylor expanded in b around 0 60.3%

      \[\leadsto \log \color{blue}{\left(2 + b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -155:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \]

1. Split input into 2 regimes

2. if a < -96

   1. Initial program 9.6%

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

   2. Taylor expanded in a around 0 5.5%

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

   3. Taylor expanded in b around 0 5.1%

      \[\leadsto \color{blue}{0.5 \cdot b + \log 2} \]

   4. Taylor expanded in b around inf 18.8%

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

   if -96 < a

   1. Initial program 65.7%

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

   2. Taylor expanded in b around 0 60.6%

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

      1. associate-+r+60.6%

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

      2. +-commutative60.6%

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

      3. associate-+l+60.6%

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

   4. Simplified60.6%

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

   5. Taylor expanded in a around inf 60.6%

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

      1. log1p-def61.6%

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

   7. Simplified61.6%

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

   8. Taylor expanded in a around 0 60.3%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{1 + b}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -96:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(b + 1\right)\\ \end{array} \]

1. Split input into 2 regimes

2. if a < -105

   1. Initial program 9.6%

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

   2. Taylor expanded in a around 0 5.5%

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

   3. Taylor expanded in b around 0 5.1%

      \[\leadsto \color{blue}{0.5 \cdot b + \log 2} \]

   4. Taylor expanded in b around inf 18.8%

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

   if -105 < a

   1. Initial program 65.7%

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

   2. Taylor expanded in b around 0 61.0%

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

      1. log1p-def62.0%

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

   4. Simplified62.0%

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

   5. Taylor expanded in a around 0 60.7%

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

4. Final simplification50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -105:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]

1. Initial program 51.7%

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

2. Taylor expanded in a around 0 49.3%

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

3. Taylor expanded in b around 0 47.2%

   \[\leadsto \color{blue}{0.5 \cdot b + \log 2} \]

4. Taylor expanded in b around inf 7.4%

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

5. Final simplification7.4%

   \[\leadsto b \cdot 0.5 \]

herbie shell --seed 2023166 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))