Difference of squares

1. Split input into 2 regimes

2. if a < -2.2000000000000002e174

   1. Initial program 71.8%

      \[a \cdot a - b \cdot b \]

   2. Taylor expanded in a around inf 94.9%

      \[\leadsto \color{blue}{{a}^{2}} \]
   3. Step-by-step derivation

      1. unpow294.9%

         \[\leadsto \color{blue}{a \cdot a} \]

   4. Simplified94.9%

      \[\leadsto \color{blue}{a \cdot a} \]

   if -2.2000000000000002e174 < a

   1. Initial program 95.4%

      \[a \cdot a - b \cdot b \]
   2. Step-by-step derivation

      1. fma-neg98.2%

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

      2. distribute-rgt-neg-in98.2%

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

   3. Simplified98.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.2 \cdot 10^{+174}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\\ \end{array} \]

1. Split input into 2 regimes

2. if (*.f64 a a) < 0.0259999999999999988 or 3.5000000000000001e77 < (*.f64 a a) < 3.2000000000000002e129

   1. Initial program 100.0%

      \[a \cdot a - b \cdot b \]

   2. Taylor expanded in a around 0 76.1%

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

      1. unpow276.1%

         \[\leadsto -1 \cdot \color{blue}{\left(b \cdot b\right)} \]

      2. mul-1-neg76.1%

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

      3. distribute-rgt-neg-in76.1%

         \[\leadsto \color{blue}{b \cdot \left(-b\right)} \]

   4. Simplified76.1%

      \[\leadsto \color{blue}{b \cdot \left(-b\right)} \]

   if 0.0259999999999999988 < (*.f64 a a) < 3.5000000000000001e77 or 3.2000000000000002e129 < (*.f64 a a)

   1. Initial program 83.8%

      \[a \cdot a - b \cdot b \]

   2. Taylor expanded in a around inf 85.2%

      \[\leadsto \color{blue}{{a}^{2}} \]
   3. Step-by-step derivation

      1. unpow285.2%

         \[\leadsto \color{blue}{a \cdot a} \]

   4. Simplified85.2%

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

4. Final simplification80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 0.026 \lor \neg \left(a \cdot a \leq 3.5 \cdot 10^{+77}\right) \land a \cdot a \leq 3.2 \cdot 10^{+129}:\\ \;\;\;\;b \cdot \left(-b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]

1. Split input into 2 regimes

2. if (*.f64 a a) < 3.49999999999999976e287

   1. Initial program 100.0%

      \[a \cdot a - b \cdot b \]

   if 3.49999999999999976e287 < (*.f64 a a)

   1. Initial program 75.6%

      \[a \cdot a - b \cdot b \]

   2. Taylor expanded in a around inf 90.7%

      \[\leadsto \color{blue}{{a}^{2}} \]
   3. Step-by-step derivation

      1. unpow290.7%

         \[\leadsto \color{blue}{a \cdot a} \]

   4. Simplified90.7%

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

4. Final simplification96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot a \leq 3.5 \cdot 10^{+287}:\\ \;\;\;\;a \cdot a - b \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]

1. Initial program 91.8%

   \[a \cdot a - b \cdot b \]

2. Taylor expanded in a around inf 62.4%

   \[\leadsto \color{blue}{{a}^{2}} \]
3. Step-by-step derivation

   1. unpow262.4%

      \[\leadsto \color{blue}{a \cdot a} \]

4. Simplified62.4%

   \[\leadsto \color{blue}{a \cdot a} \]

5. Final simplification62.4%

   \[\leadsto a \cdot a \]

herbie shell --seed 2023167 
(FPCore (a b)
  :name "Difference of squares"
  :precision binary64

  :herbie-target
  (* (+ a b) (- a b))

  (- (* a a) (* b b)))