rsin A (should all be same)

1. Initial program 76.3%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
2. Step-by-step derivation

   1. associate-*r/76.3%

      \[\leadsto \color{blue}{r \cdot \frac{\sin b}{\cos \left(a + b\right)}} \]

   2. *-commutative76.3%

      \[\leadsto \color{blue}{\frac{\sin b}{\cos \left(a + b\right)} \cdot r} \]

   3. +-commutative76.3%

      \[\leadsto \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \cdot r \]

3. Simplified76.3%

   \[\leadsto \color{blue}{\frac{\sin b}{\cos \left(b + a\right)} \cdot r} \]
4. Step-by-step derivation

   1. cos-sum99.5%

      \[\leadsto \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \cdot r \]

   2. *-un-lft-identity99.5%

      \[\leadsto \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{1 \cdot \left(\sin b \cdot \sin a\right)}} \cdot r \]

   3. *-un-lft-identity99.5%

      \[\leadsto \frac{\sin b}{\color{blue}{1 \cdot \left(\cos b \cdot \cos a\right)} - 1 \cdot \left(\sin b \cdot \sin a\right)} \cdot r \]

   4. prod-diff99.5%

      \[\leadsto \frac{\sin b}{\color{blue}{\mathsf{fma}\left(1, \cos b \cdot \cos a, -\left(\sin b \cdot \sin a\right) \cdot 1\right) + \mathsf{fma}\left(-\sin b \cdot \sin a, 1, \left(\sin b \cdot \sin a\right) \cdot 1\right)}} \cdot r \]

5. Applied egg-rr99.5%

   \[\leadsto \frac{\sin b}{\color{blue}{\mathsf{fma}\left(1, \cos b \cdot \cos a, -\left(\sin b \cdot \sin a\right) \cdot 1\right) + \mathsf{fma}\left(-\sin b \cdot \sin a, 1, \left(\sin b \cdot \sin a\right) \cdot 1\right)}} \cdot r \]
6. Step-by-step derivation

   1. *-rgt-identity99.5%

      \[\leadsto \frac{\sin b}{\mathsf{fma}\left(1, \cos b \cdot \cos a, -\color{blue}{\sin b \cdot \sin a}\right) + \mathsf{fma}\left(-\sin b \cdot \sin a, 1, \left(\sin b \cdot \sin a\right) \cdot 1\right)} \cdot r \]

   2. *-commutative99.5%

      \[\leadsto \frac{\sin b}{\mathsf{fma}\left(1, \color{blue}{\cos a \cdot \cos b}, -\sin b \cdot \sin a\right) + \mathsf{fma}\left(-\sin b \cdot \sin a, 1, \left(\sin b \cdot \sin a\right) \cdot 1\right)} \cdot r \]

   3. distribute-lft-neg-in99.5%

      \[\leadsto \frac{\sin b}{\mathsf{fma}\left(1, \cos a \cdot \cos b, \color{blue}{\left(-\sin b\right) \cdot \sin a}\right) + \mathsf{fma}\left(-\sin b \cdot \sin a, 1, \left(\sin b \cdot \sin a\right) \cdot 1\right)} \cdot r \]

   4. *-commutative99.5%

      \[\leadsto \frac{\sin b}{\mathsf{fma}\left(1, \cos a \cdot \cos b, \color{blue}{\sin a \cdot \left(-\sin b\right)}\right) + \mathsf{fma}\left(-\sin b \cdot \sin a, 1, \left(\sin b \cdot \sin a\right) \cdot 1\right)} \cdot r \]

   5. fma-udef99.5%

      \[\leadsto \frac{\sin b}{\mathsf{fma}\left(1, \cos a \cdot \cos b, \sin a \cdot \left(-\sin b\right)\right) + \color{blue}{\left(\left(-\sin b \cdot \sin a\right) \cdot 1 + \left(\sin b \cdot \sin a\right) \cdot 1\right)}} \cdot r \]

   6. *-rgt-identity99.5%

      \[\leadsto \frac{\sin b}{\mathsf{fma}\left(1, \cos a \cdot \cos b, \sin a \cdot \left(-\sin b\right)\right) + \left(\color{blue}{\left(-\sin b \cdot \sin a\right)} + \left(\sin b \cdot \sin a\right) \cdot 1\right)} \cdot r \]

   7. distribute-lft-neg-in99.5%

      \[\leadsto \frac{\sin b}{\mathsf{fma}\left(1, \cos a \cdot \cos b, \sin a \cdot \left(-\sin b\right)\right) + \left(\color{blue}{\left(-\sin b\right) \cdot \sin a} + \left(\sin b \cdot \sin a\right) \cdot 1\right)} \cdot r \]

   8. *-rgt-identity99.5%

      \[\leadsto \frac{\sin b}{\mathsf{fma}\left(1, \cos a \cdot \cos b, \sin a \cdot \left(-\sin b\right)\right) + \left(\left(-\sin b\right) \cdot \sin a + \color{blue}{\sin b \cdot \sin a}\right)} \cdot r \]

   9. fma-udef99.5%

      \[\leadsto \frac{\sin b}{\mathsf{fma}\left(1, \cos a \cdot \cos b, \sin a \cdot \left(-\sin b\right)\right) + \color{blue}{\mathsf{fma}\left(-\sin b, \sin a, \sin b \cdot \sin a\right)}} \cdot r \]

   10. *-commutative99.5%

       \[\leadsto \frac{\sin b}{\mathsf{fma}\left(1, \cos a \cdot \cos b, \sin a \cdot \left(-\sin b\right)\right) + \mathsf{fma}\left(-\sin b, \sin a, \color{blue}{\sin a \cdot \sin b}\right)} \cdot r \]

7. Simplified99.5%

   \[\leadsto \frac{\sin b}{\color{blue}{\mathsf{fma}\left(1, \cos a \cdot \cos b, \sin a \cdot \left(-\sin b\right)\right) + \mathsf{fma}\left(-\sin b, \sin a, \sin a \cdot \sin b\right)}} \cdot r \]

8. Taylor expanded in b around inf 99.5%

   \[\leadsto \color{blue}{\frac{\sin b}{-2 \cdot \left(\sin a \cdot \sin b\right) + \left(\cos a \cdot \cos b + \sin a \cdot \sin b\right)}} \cdot r \]
9. Step-by-step derivation

   1. fma-def99.5%

      \[\leadsto \frac{\sin b}{\color{blue}{\mathsf{fma}\left(-2, \sin a \cdot \sin b, \cos a \cdot \cos b + \sin a \cdot \sin b\right)}} \cdot r \]

   2. fma-def99.5%

      \[\leadsto \frac{\sin b}{\mathsf{fma}\left(-2, \sin a \cdot \sin b, \color{blue}{\mathsf{fma}\left(\cos a, \cos b, \sin a \cdot \sin b\right)}\right)} \cdot r \]

10. Simplified99.5%

    \[\leadsto \color{blue}{\frac{\sin b}{\mathsf{fma}\left(-2, \sin a \cdot \sin b, \mathsf{fma}\left(\cos a, \cos b, \sin a \cdot \sin b\right)\right)}} \cdot r \]

11. Final simplification99.5%

    \[\leadsto \frac{\sin b}{\mathsf{fma}\left(-2, \sin b \cdot \sin a, \mathsf{fma}\left(\cos a, \cos b, \sin b \cdot \sin a\right)\right)} \cdot r \]

1. Initial program 76.3%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
2. Step-by-step derivation

   1. +-commutative76.3%

      \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

3. Simplified76.3%

   \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(b + a\right)}} \]
4. Step-by-step derivation

   1. cos-sum99.5%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

   2. cancel-sign-sub-inv99.5%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a + \left(-\sin b\right) \cdot \sin a}} \]

   3. fma-def99.5%

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

5. Applied egg-rr99.5%

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

6. Final simplification99.5%

   \[\leadsto \frac{\sin b \cdot r}{\mathsf{fma}\left(\cos b, \cos a, \sin b \cdot \left(-\sin a\right)\right)} \]

1. Initial program 76.3%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
2. Step-by-step derivation

   1. associate-*r/76.3%

      \[\leadsto \color{blue}{r \cdot \frac{\sin b}{\cos \left(a + b\right)}} \]

   2. *-commutative76.3%

      \[\leadsto \color{blue}{\frac{\sin b}{\cos \left(a + b\right)} \cdot r} \]

   3. +-commutative76.3%

      \[\leadsto \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \cdot r \]

3. Simplified76.3%

   \[\leadsto \color{blue}{\frac{\sin b}{\cos \left(b + a\right)} \cdot r} \]
4. Step-by-step derivation

   1. cos-sum99.5%

      \[\leadsto \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \cdot r \]

5. Applied egg-rr99.5%

   \[\leadsto \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \cdot r \]

6. Final simplification99.5%

   \[\leadsto r \cdot \frac{\sin b}{\cos a \cdot \cos b - \sin b \cdot \sin a} \]

1. Initial program 76.3%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
2. Step-by-step derivation

   1. associate-*r/76.3%

      \[\leadsto \color{blue}{r \cdot \frac{\sin b}{\cos \left(a + b\right)}} \]

   2. *-commutative76.3%

      \[\leadsto \color{blue}{\frac{\sin b}{\cos \left(a + b\right)} \cdot r} \]

   3. +-commutative76.3%

      \[\leadsto \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \cdot r \]

3. Simplified76.3%

   \[\leadsto \color{blue}{\frac{\sin b}{\cos \left(b + a\right)} \cdot r} \]
4. Step-by-step derivation

   1. cos-sum99.5%

      \[\leadsto \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \cdot r \]

   2. *-un-lft-identity99.5%

      \[\leadsto \frac{\sin b}{\cos b \cdot \cos a - \color{blue}{1 \cdot \left(\sin b \cdot \sin a\right)}} \cdot r \]

   3. *-un-lft-identity99.5%

      \[\leadsto \frac{\sin b}{\color{blue}{1 \cdot \left(\cos b \cdot \cos a\right)} - 1 \cdot \left(\sin b \cdot \sin a\right)} \cdot r \]

   4. prod-diff99.5%

      \[\leadsto \frac{\sin b}{\color{blue}{\mathsf{fma}\left(1, \cos b \cdot \cos a, -\left(\sin b \cdot \sin a\right) \cdot 1\right) + \mathsf{fma}\left(-\sin b \cdot \sin a, 1, \left(\sin b \cdot \sin a\right) \cdot 1\right)}} \cdot r \]

5. Applied egg-rr99.5%

   \[\leadsto \frac{\sin b}{\color{blue}{\mathsf{fma}\left(1, \cos b \cdot \cos a, -\left(\sin b \cdot \sin a\right) \cdot 1\right) + \mathsf{fma}\left(-\sin b \cdot \sin a, 1, \left(\sin b \cdot \sin a\right) \cdot 1\right)}} \cdot r \]
6. Step-by-step derivation

   1. *-rgt-identity99.5%

      \[\leadsto \frac{\sin b}{\mathsf{fma}\left(1, \cos b \cdot \cos a, -\color{blue}{\sin b \cdot \sin a}\right) + \mathsf{fma}\left(-\sin b \cdot \sin a, 1, \left(\sin b \cdot \sin a\right) \cdot 1\right)} \cdot r \]

   2. *-commutative99.5%

      \[\leadsto \frac{\sin b}{\mathsf{fma}\left(1, \color{blue}{\cos a \cdot \cos b}, -\sin b \cdot \sin a\right) + \mathsf{fma}\left(-\sin b \cdot \sin a, 1, \left(\sin b \cdot \sin a\right) \cdot 1\right)} \cdot r \]

   3. distribute-lft-neg-in99.5%

      \[\leadsto \frac{\sin b}{\mathsf{fma}\left(1, \cos a \cdot \cos b, \color{blue}{\left(-\sin b\right) \cdot \sin a}\right) + \mathsf{fma}\left(-\sin b \cdot \sin a, 1, \left(\sin b \cdot \sin a\right) \cdot 1\right)} \cdot r \]

   4. *-commutative99.5%

      \[\leadsto \frac{\sin b}{\mathsf{fma}\left(1, \cos a \cdot \cos b, \color{blue}{\sin a \cdot \left(-\sin b\right)}\right) + \mathsf{fma}\left(-\sin b \cdot \sin a, 1, \left(\sin b \cdot \sin a\right) \cdot 1\right)} \cdot r \]

   5. fma-udef99.5%

      \[\leadsto \frac{\sin b}{\mathsf{fma}\left(1, \cos a \cdot \cos b, \sin a \cdot \left(-\sin b\right)\right) + \color{blue}{\left(\left(-\sin b \cdot \sin a\right) \cdot 1 + \left(\sin b \cdot \sin a\right) \cdot 1\right)}} \cdot r \]

   6. *-rgt-identity99.5%

      \[\leadsto \frac{\sin b}{\mathsf{fma}\left(1, \cos a \cdot \cos b, \sin a \cdot \left(-\sin b\right)\right) + \left(\color{blue}{\left(-\sin b \cdot \sin a\right)} + \left(\sin b \cdot \sin a\right) \cdot 1\right)} \cdot r \]

   7. distribute-lft-neg-in99.5%

      \[\leadsto \frac{\sin b}{\mathsf{fma}\left(1, \cos a \cdot \cos b, \sin a \cdot \left(-\sin b\right)\right) + \left(\color{blue}{\left(-\sin b\right) \cdot \sin a} + \left(\sin b \cdot \sin a\right) \cdot 1\right)} \cdot r \]

   8. *-rgt-identity99.5%

      \[\leadsto \frac{\sin b}{\mathsf{fma}\left(1, \cos a \cdot \cos b, \sin a \cdot \left(-\sin b\right)\right) + \left(\left(-\sin b\right) \cdot \sin a + \color{blue}{\sin b \cdot \sin a}\right)} \cdot r \]

   9. fma-udef99.5%

      \[\leadsto \frac{\sin b}{\mathsf{fma}\left(1, \cos a \cdot \cos b, \sin a \cdot \left(-\sin b\right)\right) + \color{blue}{\mathsf{fma}\left(-\sin b, \sin a, \sin b \cdot \sin a\right)}} \cdot r \]

   10. *-commutative99.5%

       \[\leadsto \frac{\sin b}{\mathsf{fma}\left(1, \cos a \cdot \cos b, \sin a \cdot \left(-\sin b\right)\right) + \mathsf{fma}\left(-\sin b, \sin a, \color{blue}{\sin a \cdot \sin b}\right)} \cdot r \]

7. Simplified99.5%

   \[\leadsto \frac{\sin b}{\color{blue}{\mathsf{fma}\left(1, \cos a \cdot \cos b, \sin a \cdot \left(-\sin b\right)\right) + \mathsf{fma}\left(-\sin b, \sin a, \sin a \cdot \sin b\right)}} \cdot r \]

8. Taylor expanded in b around inf 99.5%

   \[\leadsto \color{blue}{\frac{\sin b}{-2 \cdot \left(\sin a \cdot \sin b\right) + \left(\cos a \cdot \cos b + \sin a \cdot \sin b\right)}} \cdot r \]
9. Step-by-step derivation

   1. fma-def99.5%

      \[\leadsto \frac{\sin b}{\color{blue}{\mathsf{fma}\left(-2, \sin a \cdot \sin b, \cos a \cdot \cos b + \sin a \cdot \sin b\right)}} \cdot r \]

   2. fma-def99.5%

      \[\leadsto \frac{\sin b}{\mathsf{fma}\left(-2, \sin a \cdot \sin b, \color{blue}{\mathsf{fma}\left(\cos a, \cos b, \sin a \cdot \sin b\right)}\right)} \cdot r \]

10. Simplified99.5%

    \[\leadsto \color{blue}{\frac{\sin b}{\mathsf{fma}\left(-2, \sin a \cdot \sin b, \mathsf{fma}\left(\cos a, \cos b, \sin a \cdot \sin b\right)\right)}} \cdot r \]
11. Step-by-step derivation

    1. fma-udef99.5%

       \[\leadsto \frac{\sin b}{\color{blue}{-2 \cdot \left(\sin a \cdot \sin b\right) + \mathsf{fma}\left(\cos a, \cos b, \sin a \cdot \sin b\right)}} \cdot r \]

    2. *-commutative99.5%

       \[\leadsto \frac{\sin b}{-2 \cdot \color{blue}{\left(\sin b \cdot \sin a\right)} + \mathsf{fma}\left(\cos a, \cos b, \sin a \cdot \sin b\right)} \cdot r \]

    3. fma-udef99.5%

       \[\leadsto \frac{\sin b}{-2 \cdot \left(\sin b \cdot \sin a\right) + \color{blue}{\left(\cos a \cdot \cos b + \sin a \cdot \sin b\right)}} \cdot r \]

    4. cos-diff77.1%

       \[\leadsto \frac{\sin b}{-2 \cdot \left(\sin b \cdot \sin a\right) + \color{blue}{\cos \left(a - b\right)}} \cdot r \]

12. Applied egg-rr77.1%

    \[\leadsto \frac{\sin b}{\color{blue}{-2 \cdot \left(\sin b \cdot \sin a\right) + \cos \left(a - b\right)}} \cdot r \]

13. Final simplification77.1%

    \[\leadsto r \cdot \frac{\sin b}{-2 \cdot \left(\sin b \cdot \sin a\right) + \cos \left(a - b\right)} \]

1. Split input into 2 regimes

2. if a < -1.6e-7 or 3.19999999999999981e-10 < a

   1. Initial program 53.4%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. associate-*r/53.5%

         \[\leadsto \color{blue}{r \cdot \frac{\sin b}{\cos \left(a + b\right)}} \]

      2. *-commutative53.5%

         \[\leadsto \color{blue}{\frac{\sin b}{\cos \left(a + b\right)} \cdot r} \]

      3. +-commutative53.5%

         \[\leadsto \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \cdot r \]

   3. Simplified53.5%

      \[\leadsto \color{blue}{\frac{\sin b}{\cos \left(b + a\right)} \cdot r} \]

   4. Taylor expanded in b around 0 52.8%

      \[\leadsto \frac{\sin b}{\color{blue}{\cos a}} \cdot r \]

   if -1.6e-7 < a < 3.19999999999999981e-10

   1. Initial program 99.6%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. associate-/l*99.5%

         \[\leadsto \color{blue}{\frac{r}{\frac{\cos \left(a + b\right)}{\sin b}}} \]

      2. +-commutative99.5%

         \[\leadsto \frac{r}{\frac{\cos \color{blue}{\left(b + a\right)}}{\sin b}} \]

   3. Simplified99.5%

      \[\leadsto \color{blue}{\frac{r}{\frac{\cos \left(b + a\right)}{\sin b}}} \]

   4. Taylor expanded in a around 0 99.5%

      \[\leadsto \frac{r}{\color{blue}{\frac{\cos b}{\sin b} + -1 \cdot a}} \]
   5. Step-by-step derivation

      1. mul-1-neg99.5%

         \[\leadsto \frac{r}{\frac{\cos b}{\sin b} + \color{blue}{\left(-a\right)}} \]

      2. unsub-neg99.5%

         \[\leadsto \frac{r}{\color{blue}{\frac{\cos b}{\sin b} - a}} \]

   6. Simplified99.5%

      \[\leadsto \frac{r}{\color{blue}{\frac{\cos b}{\sin b} - a}} \]
   7. Step-by-step derivation

      1. div-inv99.6%

         \[\leadsto \color{blue}{r \cdot \frac{1}{\frac{\cos b}{\sin b} - a}} \]

      2. clear-num99.6%

         \[\leadsto r \cdot \frac{1}{\color{blue}{\frac{1}{\frac{\sin b}{\cos b}}} - a} \]

      3. quot-tan99.7%

         \[\leadsto r \cdot \frac{1}{\frac{1}{\color{blue}{\tan b}} - a} \]

   8. Applied egg-rr99.7%

      \[\leadsto \color{blue}{r \cdot \frac{1}{\frac{1}{\tan b} - a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.6 \cdot 10^{-7} \lor \neg \left(a \leq 3.2 \cdot 10^{-10}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{1}{\frac{1}{\tan b} - a}\\ \end{array} \]

1. Split input into 2 regimes

2. if b < -7.50000000000000019e-6 or 2.35000000000000007e-20 < b

   1. Initial program 57.4%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. associate-*r/57.4%

         \[\leadsto \color{blue}{r \cdot \frac{\sin b}{\cos \left(a + b\right)}} \]

      2. *-commutative57.4%

         \[\leadsto \color{blue}{\frac{\sin b}{\cos \left(a + b\right)} \cdot r} \]

      3. +-commutative57.4%

         \[\leadsto \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \cdot r \]

   3. Simplified57.4%

      \[\leadsto \color{blue}{\frac{\sin b}{\cos \left(b + a\right)} \cdot r} \]

   4. Taylor expanded in a around 0 57.7%

      \[\leadsto \frac{\sin b}{\color{blue}{\cos b}} \cdot r \]

   if -7.50000000000000019e-6 < b < 2.35000000000000007e-20

   1. Initial program 99.2%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. +-commutative99.2%

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

   3. Simplified99.2%

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(b + a\right)}} \]
   4. Step-by-step derivation

      1. cos-sum99.7%

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      2. cancel-sign-sub-inv99.7%

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a + \left(-\sin b\right) \cdot \sin a}} \]

      3. fma-def99.7%

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

   5. Applied egg-rr99.7%

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

   6. Taylor expanded in b around 0 99.2%

      \[\leadsto \color{blue}{\frac{b \cdot r}{\cos a}} \]
   7. Step-by-step derivation

      1. *-commutative99.2%

         \[\leadsto \frac{\color{blue}{r \cdot b}}{\cos a} \]

      2. associate-/l*99.0%

         \[\leadsto \color{blue}{\frac{r}{\frac{\cos a}{b}}} \]

      3. associate-/r/99.3%

         \[\leadsto \color{blue}{\frac{r}{\cos a} \cdot b} \]

   8. Simplified99.3%

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

4. Final simplification76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.5 \cdot 10^{-6} \lor \neg \left(b \leq 2.35 \cdot 10^{-20}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]

1. Split input into 3 regimes

2. if b < -3.60000000000000009e-5

   1. Initial program 47.1%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. associate-*r/47.1%

         \[\leadsto \color{blue}{r \cdot \frac{\sin b}{\cos \left(a + b\right)}} \]

      2. *-commutative47.1%

         \[\leadsto \color{blue}{\frac{\sin b}{\cos \left(a + b\right)} \cdot r} \]

      3. +-commutative47.1%

         \[\leadsto \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \cdot r \]

   3. Simplified47.1%

      \[\leadsto \color{blue}{\frac{\sin b}{\cos \left(b + a\right)} \cdot r} \]

   4. Taylor expanded in a around 0 47.8%

      \[\leadsto \frac{\sin b}{\color{blue}{\cos b}} \cdot r \]

   if -3.60000000000000009e-5 < b < 4.10000000000000036e-15

   1. Initial program 99.2%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. +-commutative99.2%

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

   3. Simplified99.2%

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(b + a\right)}} \]
   4. Step-by-step derivation

      1. cos-sum99.7%

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      2. cancel-sign-sub-inv99.7%

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a + \left(-\sin b\right) \cdot \sin a}} \]

      3. fma-def99.7%

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

   5. Applied egg-rr99.7%

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

   6. Taylor expanded in b around 0 99.2%

      \[\leadsto \color{blue}{\frac{b \cdot r}{\cos a}} \]
   7. Step-by-step derivation

      1. *-commutative99.2%

         \[\leadsto \frac{\color{blue}{r \cdot b}}{\cos a} \]

      2. associate-/l*99.1%

         \[\leadsto \color{blue}{\frac{r}{\frac{\cos a}{b}}} \]

      3. associate-/r/99.3%

         \[\leadsto \color{blue}{\frac{r}{\cos a} \cdot b} \]

   8. Simplified99.3%

      \[\leadsto \color{blue}{\frac{r}{\cos a} \cdot b} \]

   if 4.10000000000000036e-15 < b

   1. Initial program 64.7%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. associate-/l*64.8%

         \[\leadsto \color{blue}{\frac{r}{\frac{\cos \left(a + b\right)}{\sin b}}} \]

      2. +-commutative64.8%

         \[\leadsto \frac{r}{\frac{\cos \color{blue}{\left(b + a\right)}}{\sin b}} \]

   3. Simplified64.8%

      \[\leadsto \color{blue}{\frac{r}{\frac{\cos \left(b + a\right)}{\sin b}}} \]

   4. Taylor expanded in a around 0 64.7%

      \[\leadsto \frac{r}{\frac{\color{blue}{\cos b}}{\sin b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.6 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{elif}\;b \leq 4.1 \cdot 10^{-15}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \end{array} \]

1. Initial program 76.3%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
2. Step-by-step derivation

   1. associate-/l*76.3%

      \[\leadsto \color{blue}{\frac{r}{\frac{\cos \left(a + b\right)}{\sin b}}} \]

   2. +-commutative76.3%

      \[\leadsto \frac{r}{\frac{\cos \color{blue}{\left(b + a\right)}}{\sin b}} \]

3. Simplified76.3%

   \[\leadsto \color{blue}{\frac{r}{\frac{\cos \left(b + a\right)}{\sin b}}} \]
4. Step-by-step derivation

   1. associate-/r/76.4%

      \[\leadsto \color{blue}{\frac{r}{\cos \left(b + a\right)} \cdot \sin b} \]

5. Applied egg-rr76.4%

   \[\leadsto \color{blue}{\frac{r}{\cos \left(b + a\right)} \cdot \sin b} \]

6. Final simplification76.4%

   \[\leadsto \sin b \cdot \frac{r}{\cos \left(b + a\right)} \]

1. Split input into 2 regimes

2. if b < -1.9000000000000001e-4 or 2.35000000000000007e-20 < b

   1. Initial program 57.4%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. associate-/l*57.4%

         \[\leadsto \color{blue}{\frac{r}{\frac{\cos \left(a + b\right)}{\sin b}}} \]

      2. +-commutative57.4%

         \[\leadsto \frac{r}{\frac{\cos \color{blue}{\left(b + a\right)}}{\sin b}} \]

   3. Simplified57.4%

      \[\leadsto \color{blue}{\frac{r}{\frac{\cos \left(b + a\right)}{\sin b}}} \]

   4. Taylor expanded in a around 0 53.5%

      \[\leadsto \frac{r}{\color{blue}{\frac{\cos b}{\sin b} + -1 \cdot a}} \]
   5. Step-by-step derivation

      1. mul-1-neg53.5%

         \[\leadsto \frac{r}{\frac{\cos b}{\sin b} + \color{blue}{\left(-a\right)}} \]

      2. unsub-neg53.5%

         \[\leadsto \frac{r}{\color{blue}{\frac{\cos b}{\sin b} - a}} \]

   6. Simplified53.5%

      \[\leadsto \frac{r}{\color{blue}{\frac{\cos b}{\sin b} - a}} \]
   7. Step-by-step derivation

      1. div-inv53.4%

         \[\leadsto \color{blue}{r \cdot \frac{1}{\frac{\cos b}{\sin b} - a}} \]

      2. clear-num53.4%

         \[\leadsto r \cdot \frac{1}{\color{blue}{\frac{1}{\frac{\sin b}{\cos b}}} - a} \]

      3. quot-tan53.5%

         \[\leadsto r \cdot \frac{1}{\frac{1}{\color{blue}{\tan b}} - a} \]

   8. Applied egg-rr53.5%

      \[\leadsto \color{blue}{r \cdot \frac{1}{\frac{1}{\tan b} - a}} \]

   if -1.9000000000000001e-4 < b < 2.35000000000000007e-20

   1. Initial program 99.2%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. +-commutative99.2%

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

   3. Simplified99.2%

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(b + a\right)}} \]
   4. Step-by-step derivation

      1. cos-sum99.7%

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      2. cancel-sign-sub-inv99.7%

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a + \left(-\sin b\right) \cdot \sin a}} \]

      3. fma-def99.7%

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

   5. Applied egg-rr99.7%

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

   6. Taylor expanded in b around 0 99.2%

      \[\leadsto \color{blue}{\frac{b \cdot r}{\cos a}} \]
   7. Step-by-step derivation

      1. *-commutative99.2%

         \[\leadsto \frac{\color{blue}{r \cdot b}}{\cos a} \]

      2. associate-/l*99.0%

         \[\leadsto \color{blue}{\frac{r}{\frac{\cos a}{b}}} \]

      3. associate-/r/99.3%

         \[\leadsto \color{blue}{\frac{r}{\cos a} \cdot b} \]

   8. Simplified99.3%

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

4. Final simplification74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.00019 \lor \neg \left(b \leq 2.35 \cdot 10^{-20}\right):\\ \;\;\;\;r \cdot \frac{1}{\frac{1}{\tan b} - a}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]

1. Split input into 2 regimes

2. if b < -1.05e-4 or 4.10000000000000036e-15 < b

   1. Initial program 56.8%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. associate-/l*56.8%

         \[\leadsto \color{blue}{\frac{r}{\frac{\cos \left(a + b\right)}{\sin b}}} \]

      2. +-commutative56.8%

         \[\leadsto \frac{r}{\frac{\cos \color{blue}{\left(b + a\right)}}{\sin b}} \]

   3. Simplified56.8%

      \[\leadsto \color{blue}{\frac{r}{\frac{\cos \left(b + a\right)}{\sin b}}} \]

   4. Taylor expanded in a around 0 52.8%

      \[\leadsto \frac{r}{\color{blue}{\frac{\cos b}{\sin b} + -1 \cdot a}} \]
   5. Step-by-step derivation

      1. mul-1-neg52.8%

         \[\leadsto \frac{r}{\frac{\cos b}{\sin b} + \color{blue}{\left(-a\right)}} \]

      2. unsub-neg52.8%

         \[\leadsto \frac{r}{\color{blue}{\frac{\cos b}{\sin b} - a}} \]

   6. Simplified52.8%

      \[\leadsto \frac{r}{\color{blue}{\frac{\cos b}{\sin b} - a}} \]
   7. Step-by-step derivation

      1. expm1-log1p-u40.6%

         \[\leadsto \frac{r}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\cos b}{\sin b}\right)\right)} - a} \]

      2. expm1-udef40.2%

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

      3. clear-num40.3%

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

      4. quot-tan40.3%

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

   8. Applied egg-rr40.3%

      \[\leadsto \frac{r}{\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\tan b}\right)} - 1\right)} - a} \]
   9. Step-by-step derivation

      1. expm1-def40.6%

         \[\leadsto \frac{r}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\tan b}\right)\right)} - a} \]

      2. expm1-log1p52.8%

         \[\leadsto \frac{r}{\color{blue}{\frac{1}{\tan b}} - a} \]

   10. Simplified52.8%

       \[\leadsto \frac{r}{\color{blue}{\frac{1}{\tan b}} - a} \]

   if -1.05e-4 < b < 4.10000000000000036e-15

   1. Initial program 99.2%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. +-commutative99.2%

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

   3. Simplified99.2%

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(b + a\right)}} \]
   4. Step-by-step derivation

      1. cos-sum99.7%

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      2. cancel-sign-sub-inv99.7%

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a + \left(-\sin b\right) \cdot \sin a}} \]

      3. fma-def99.7%

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

   5. Applied egg-rr99.7%

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

   6. Taylor expanded in b around 0 99.2%

      \[\leadsto \color{blue}{\frac{b \cdot r}{\cos a}} \]
   7. Step-by-step derivation

      1. *-commutative99.2%

         \[\leadsto \frac{\color{blue}{r \cdot b}}{\cos a} \]

      2. associate-/l*99.1%

         \[\leadsto \color{blue}{\frac{r}{\frac{\cos a}{b}}} \]

      3. associate-/r/99.3%

         \[\leadsto \color{blue}{\frac{r}{\cos a} \cdot b} \]

   8. Simplified99.3%

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

4. Final simplification74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -0.000105 \lor \neg \left(b \leq 4.1 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{r}{\frac{1}{\tan b} - a}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]

1. Split input into 2 regimes

2. if b < -54000 or 8.8e10 < b

   1. Initial program 55.1%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. +-commutative55.1%

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

   3. Simplified55.1%

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(b + a\right)}} \]

   4. Taylor expanded in b around 0 12.6%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos a}} \]

   5. Taylor expanded in a around 0 12.6%

      \[\leadsto \color{blue}{\sin b \cdot r} \]

   if -54000 < b < 8.8e10

   1. Initial program 98.2%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. associate-*r/98.2%

         \[\leadsto \color{blue}{r \cdot \frac{\sin b}{\cos \left(a + b\right)}} \]

      2. *-commutative98.2%

         \[\leadsto \color{blue}{\frac{\sin b}{\cos \left(a + b\right)} \cdot r} \]

      3. +-commutative98.2%

         \[\leadsto \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \cdot r \]

   3. Simplified98.2%

      \[\leadsto \color{blue}{\frac{\sin b}{\cos \left(b + a\right)} \cdot r} \]

   4. Taylor expanded in b around 0 95.6%

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

4. Final simplification53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -54000 \lor \neg \left(b \leq 88000000000\right):\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

1. Split input into 2 regimes

2. if b < -54000 or 8.8e10 < b

   1. Initial program 55.1%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. +-commutative55.1%

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

   3. Simplified55.1%

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(b + a\right)}} \]

   4. Taylor expanded in b around 0 12.6%

      \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos a}} \]

   5. Taylor expanded in a around 0 12.6%

      \[\leadsto \color{blue}{\sin b \cdot r} \]

   if -54000 < b < 8.8e10

   1. Initial program 98.2%

      \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   2. Step-by-step derivation

      1. +-commutative98.2%

         \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

   3. Simplified98.2%

      \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(b + a\right)}} \]
   4. Step-by-step derivation

      1. cos-sum99.7%

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \]

      2. cancel-sign-sub-inv99.7%

         \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos b \cdot \cos a + \left(-\sin b\right) \cdot \sin a}} \]

      3. fma-def99.7%

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

   5. Applied egg-rr99.7%

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

   6. Taylor expanded in b around 0 95.5%

      \[\leadsto \color{blue}{\frac{b \cdot r}{\cos a}} \]
   7. Step-by-step derivation

      1. *-commutative95.5%

         \[\leadsto \frac{\color{blue}{r \cdot b}}{\cos a} \]

      2. associate-/l*95.4%

         \[\leadsto \color{blue}{\frac{r}{\frac{\cos a}{b}}} \]

      3. associate-/r/95.7%

         \[\leadsto \color{blue}{\frac{r}{\cos a} \cdot b} \]

   8. Simplified95.7%

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

4. Final simplification53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -54000 \lor \neg \left(b \leq 88000000000\right):\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]

1. Initial program 76.3%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
2. Step-by-step derivation

   1. +-commutative76.3%

      \[\leadsto \frac{r \cdot \sin b}{\cos \color{blue}{\left(b + a\right)}} \]

3. Simplified76.3%

   \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(b + a\right)}} \]

4. Taylor expanded in b around 0 53.2%

   \[\leadsto \frac{r \cdot \sin b}{\color{blue}{\cos a}} \]

5. Taylor expanded in a around 0 37.4%

   \[\leadsto \color{blue}{\sin b \cdot r} \]

6. Final simplification37.4%

   \[\leadsto \sin b \cdot r \]

1. Initial program 76.3%

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
2. Step-by-step derivation

   1. associate-*r/76.3%

      \[\leadsto \color{blue}{r \cdot \frac{\sin b}{\cos \left(a + b\right)}} \]

   2. *-commutative76.3%

      \[\leadsto \color{blue}{\frac{\sin b}{\cos \left(a + b\right)} \cdot r} \]

   3. +-commutative76.3%

      \[\leadsto \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \cdot r \]

3. Simplified76.3%

   \[\leadsto \color{blue}{\frac{\sin b}{\cos \left(b + a\right)} \cdot r} \]

4. Taylor expanded in b around 0 48.9%

   \[\leadsto \color{blue}{\frac{b}{\cos a}} \cdot r \]

5. Taylor expanded in a around 0 33.0%

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

6. Final simplification33.0%

   \[\leadsto b \cdot r \]

herbie shell --seed 2023166 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))