rsin B (should all be same)

1. Initial program 81.7%

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

   1. associate-*r/81.6%

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

   2. +-commutative81.6%

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

3. Simplified81.6%

   \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(b + a\right)}} \]
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-def99.6%

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

5. Applied egg-rr99.6%

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

   1. log1p-expm1-u99.6%

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

   2. distribute-lft-neg-out99.6%

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

7. Applied egg-rr99.6%

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

8. Final simplification99.6%

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

1. Initial program 81.7%

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

   1. associate-*r/81.6%

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

   2. +-commutative81.6%

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

3. Simplified81.6%

   \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(b + a\right)}} \]
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-def99.6%

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

5. Applied egg-rr99.6%

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

6. Final simplification99.6%

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

1. Initial program 81.7%

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

   1. +-commutative81.7%

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

3. Simplified81.7%

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

   1. cos-sum99.6%

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

5. Applied egg-rr99.6%

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

6. Final simplification99.6%

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

1. Initial program 81.7%

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

   1. associate-*r/81.6%

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

   2. +-commutative81.6%

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

3. Simplified81.6%

   \[\leadsto \color{blue}{\frac{r \cdot \sin b}{\cos \left(b + a\right)}} \]
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-def99.6%

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

5. Applied egg-rr99.6%

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

   1. log1p-expm1-u99.6%

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

   2. distribute-lft-neg-out99.6%

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

7. Applied egg-rr99.6%

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

8. Taylor expanded in r around 0 99.5%

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

   1. associate-/l*99.4%

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

   2. associate-/r/99.6%

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

   3. *-commutative99.6%

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

10. Simplified99.6%

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

    1. sin-mult82.6%

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

    2. div-sub82.6%

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

    3. cos-diff82.5%

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

    4. add-sqr-sqrt46.8%

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

    5. sqrt-unprod82.1%

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

    6. sqr-neg82.1%

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

    7. sqrt-unprod49.8%

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

    8. add-sqr-sqrt81.4%

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

    9. sub-neg81.4%

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

    10. cos-sum82.5%

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

12. Applied egg-rr82.5%

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

    1. +-inverses82.5%

       \[\leadsto \frac{\sin b}{\cos a \cdot \cos b - \color{blue}{0}} \cdot r \]

14. Simplified82.5%

    \[\leadsto \frac{\sin b}{\cos a \cdot \cos b - \color{blue}{0}} \cdot r \]

15. Final simplification82.5%

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

1. Split input into 2 regimes

2. if a < -6.5 or 5.7e14 < a

   1. Initial program 62.3%

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

      1. +-commutative62.3%

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

   3. Simplified62.3%

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

   4. Taylor expanded in b around 0 62.8%

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

   if -6.5 < a < 5.7e14

   1. Initial program 97.5%

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

      1. +-commutative97.5%

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

   3. Simplified97.5%

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

   4. Taylor expanded in a around 0 97.5%

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

4. Final simplification81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.5 \lor \neg \left(a \leq 5.7 \cdot 10^{+14}\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]

1. Split input into 2 regimes

2. if a < -6.5 or 5.7e14 < a

   1. Initial program 62.3%

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

      1. *-commutative62.3%

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

      2. associate-/r/62.3%

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

      3. +-commutative62.3%

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

   3. Simplified62.3%

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

      1. associate-/l*62.3%

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

      2. *-commutative62.3%

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

      3. clear-num62.3%

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

      4. associate-/r/62.2%

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

      5. *-commutative62.2%

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

   5. Applied egg-rr62.2%

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

   6. Taylor expanded in b around 0 62.7%

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

      1. associate-*l/62.8%

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

      2. *-un-lft-identity62.8%

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

      3. *-commutative62.8%

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

      4. associate-/l*62.8%

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

   8. Applied egg-rr62.8%

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

   if -6.5 < a < 5.7e14

   1. Initial program 97.5%

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

      1. +-commutative97.5%

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

   3. Simplified97.5%

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

   4. Taylor expanded in a around 0 97.5%

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

4. Final simplification81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.5 \lor \neg \left(a \leq 5.7 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \end{array} \]

1. Split input into 3 regimes

2. if a < -6.5

   1. Initial program 67.2%

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

      1. *-commutative67.2%

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

      2. associate-/r/67.3%

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

      3. +-commutative67.3%

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

   3. Simplified67.3%

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

   4. Taylor expanded in b around 0 67.4%

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

   if -6.5 < a < 5.7e14

   1. Initial program 97.5%

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

      1. +-commutative97.5%

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

   3. Simplified97.5%

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

   4. Taylor expanded in a around 0 97.5%

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

   if 5.7e14 < a

   1. Initial program 55.6%

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

      1. *-commutative55.6%

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

      2. associate-/r/55.5%

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

      3. +-commutative55.5%

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

   3. Simplified55.5%

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

      1. associate-/l*55.6%

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

      2. *-commutative55.6%

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

      3. clear-num55.7%

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

      4. associate-/r/55.5%

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

      5. *-commutative55.5%

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

   5. Applied egg-rr55.5%

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

   6. Taylor expanded in b around 0 56.5%

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

      1. associate-*l/56.6%

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

      2. *-un-lft-identity56.6%

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

      3. *-commutative56.6%

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

      4. associate-/l*56.6%

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

   8. Applied egg-rr56.6%

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

4. Final simplification81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.5:\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \mathbf{elif}\;a \leq 5.7 \cdot 10^{+14}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \end{array} \]

1. Initial program 81.7%

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

2. Final simplification81.7%

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

1. Initial program 81.7%

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

   1. +-commutative81.7%

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

3. Simplified81.7%

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

4. Taylor expanded in b around 0 59.8%

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

5. Final simplification59.8%

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

1. Split input into 2 regimes

2. if b < -3.2e23

   1. Initial program 67.9%

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

      1. *-commutative67.9%

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

      2. associate-/r/67.9%

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

      3. +-commutative67.9%

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

   3. Simplified67.9%

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

   4. Taylor expanded in b around 0 9.2%

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

      1. mul-1-neg9.2%

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

      2. unsub-neg9.2%

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

      3. associate-/l*8.7%

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

   6. Simplified8.7%

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

   7. Taylor expanded in a around 0 12.2%

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

   if -3.2e23 < b

   1. Initial program 85.9%

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

      1. associate-*r/85.9%

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

      2. +-commutative85.9%

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

   3. Simplified85.9%

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

   4. Taylor expanded in b around 0 72.6%

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

4. Final simplification58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{+23}:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos \left(b + a\right)}\\ \end{array} \]

1. Split input into 2 regimes

2. if b < -1.65999999999999992

   1. Initial program 67.6%

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

      1. *-commutative67.6%

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

      2. associate-/r/67.5%

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

      3. +-commutative67.5%

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

   3. Simplified67.5%

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

   4. Taylor expanded in b around 0 9.0%

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

      1. mul-1-neg9.0%

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

      2. unsub-neg9.0%

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

      3. associate-/l*8.5%

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

   6. Simplified8.5%

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

   7. Taylor expanded in a around 0 12.1%

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

   if -1.65999999999999992 < b

   1. Initial program 86.2%

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

      1. +-commutative86.2%

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

   3. Simplified86.2%

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

   4. Taylor expanded in b around 0 72.9%

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

4. Final simplification58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.66:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

1. Split input into 2 regimes

2. if b < -4.70000000000000018

   1. Initial program 67.6%

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

      1. *-commutative67.6%

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

      2. associate-/r/67.5%

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

      3. +-commutative67.5%

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

   3. Simplified67.5%

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

   4. Taylor expanded in b around 0 9.0%

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

      1. mul-1-neg9.0%

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

      2. unsub-neg9.0%

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

      3. associate-/l*8.5%

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

   6. Simplified8.5%

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

   7. Taylor expanded in a around 0 12.1%

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

   if -4.70000000000000018 < b

   1. Initial program 86.2%

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

      1. +-commutative86.2%

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

   3. Simplified86.2%

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

   4. Taylor expanded in b around 0 72.9%

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

4. Final simplification58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.7:\\ \;\;\;\;r \cdot \sin b\\ \mathbf{else}:\\ \;\;\;\;\frac{r \cdot b}{\cos a}\\ \end{array} \]

1. Initial program 81.7%

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

   1. *-commutative81.7%

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

   2. associate-/r/81.5%

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

   3. +-commutative81.5%

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

3. Simplified81.5%

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

4. Taylor expanded in b around 0 58.1%

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

   1. mul-1-neg58.1%

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

   2. unsub-neg58.1%

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

   3. associate-/l*58.0%

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

6. Simplified58.0%

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

7. Taylor expanded in a around 0 41.0%

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

8. Final simplification41.0%

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

1. Initial program 81.7%

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

   1. +-commutative81.7%

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

3. Simplified81.7%

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

4. Taylor expanded in b around 0 56.2%

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

5. Taylor expanded in a around 0 37.7%

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

6. Final simplification37.7%

   \[\leadsto r \cdot b \]

herbie shell --seed 2023167 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))