bug323 (missed optimization)

1. Initial program 6.0%

   \[\cos^{-1} \left(1 - x\right) \]
2. Step-by-step derivation

   1. acos-asin6.0%

      \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(1 - x\right)} \]

   2. flip--6.0%

      \[\leadsto \color{blue}{\frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)}} \]

   3. div-inv6.0%

      \[\leadsto \frac{\color{blue}{\left(\pi \cdot \frac{1}{2}\right)} \cdot \frac{\pi}{2} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

   4. metadata-eval6.0%

      \[\leadsto \frac{\left(\pi \cdot \color{blue}{0.5}\right) \cdot \frac{\pi}{2} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

   5. div-inv6.0%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \color{blue}{\left(\pi \cdot \frac{1}{2}\right)} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

   6. metadata-eval6.0%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot \color{blue}{0.5}\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

   7. div-inv6.0%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\color{blue}{\pi \cdot \frac{1}{2}} + \sin^{-1} \left(1 - x\right)} \]

   8. metadata-eval6.0%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot \color{blue}{0.5} + \sin^{-1} \left(1 - x\right)} \]

3. Applied egg-rr6.0%

   \[\leadsto \color{blue}{\frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)}} \]
4. Step-by-step derivation

   1. add-sqr-sqrt9.7%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \color{blue}{\left(\sqrt{\sin^{-1} \left(1 - x\right)} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}\right)} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

   2. pow29.7%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \color{blue}{{\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{2}} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

5. Applied egg-rr9.7%

   \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \color{blue}{{\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{2}} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]
6. Step-by-step derivation

   1. add-cbrt-cube9.7%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - {\color{blue}{\left(\sqrt[3]{\left(\sqrt{\sin^{-1} \left(1 - x\right)} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}\right) \cdot \sqrt{\sin^{-1} \left(1 - x\right)}}\right)}}^{2} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

   2. pow1/39.7%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - {\color{blue}{\left({\left(\left(\sqrt{\sin^{-1} \left(1 - x\right)} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}\right) \cdot \sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{0.3333333333333333}\right)}}^{2} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

   3. add-sqr-sqrt9.7%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - {\left({\left(\color{blue}{\sin^{-1} \left(1 - x\right)} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{0.3333333333333333}\right)}^{2} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

   4. pow19.7%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - {\left({\left(\color{blue}{{\sin^{-1} \left(1 - x\right)}^{1}} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{0.3333333333333333}\right)}^{2} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

   5. pow1/29.7%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - {\left({\left({\sin^{-1} \left(1 - x\right)}^{1} \cdot \color{blue}{{\sin^{-1} \left(1 - x\right)}^{0.5}}\right)}^{0.3333333333333333}\right)}^{2} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

   6. pow-prod-up9.7%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - {\left({\color{blue}{\left({\sin^{-1} \left(1 - x\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}\right)}^{2} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

   7. metadata-eval9.7%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - {\left({\left({\sin^{-1} \left(1 - x\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}\right)}^{2} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

7. Applied egg-rr9.7%

   \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - {\color{blue}{\left({\left({\sin^{-1} \left(1 - x\right)}^{1.5}\right)}^{0.3333333333333333}\right)}}^{2} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]
8. Step-by-step derivation

   1. unpow1/39.7%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - {\color{blue}{\left(\sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{1.5}}\right)}}^{2} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

9. Simplified9.7%

   \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - {\color{blue}{\left(\sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{1.5}}\right)}}^{2} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]
10. Step-by-step derivation

    1. add-cbrt-cube9.7%

       \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - {\left(\sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{1.5}}\right)}^{2} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \color{blue}{\sqrt[3]{\left(\sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)\right) \cdot \sin^{-1} \left(1 - x\right)}}} \]

    2. pow39.7%

       \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - {\left(\sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{1.5}}\right)}^{2} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sqrt[3]{\color{blue}{{\sin^{-1} \left(1 - x\right)}^{3}}}} \]

11. Applied egg-rr9.7%

    \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - {\left(\sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{1.5}}\right)}^{2} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \color{blue}{\sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{3}}}} \]

12. Final simplification9.7%

    \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot {\left(\sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{1.5}}\right)}^{2}}{\pi \cdot 0.5 + \sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{3}}} \]

1. Initial program 6.0%

   \[\cos^{-1} \left(1 - x\right) \]
2. Step-by-step derivation

   1. acos-asin6.0%

      \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(1 - x\right)} \]

   2. flip--6.0%

      \[\leadsto \color{blue}{\frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)}} \]

   3. div-inv6.0%

      \[\leadsto \frac{\color{blue}{\left(\pi \cdot \frac{1}{2}\right)} \cdot \frac{\pi}{2} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

   4. metadata-eval6.0%

      \[\leadsto \frac{\left(\pi \cdot \color{blue}{0.5}\right) \cdot \frac{\pi}{2} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

   5. div-inv6.0%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \color{blue}{\left(\pi \cdot \frac{1}{2}\right)} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

   6. metadata-eval6.0%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot \color{blue}{0.5}\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

   7. div-inv6.0%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\color{blue}{\pi \cdot \frac{1}{2}} + \sin^{-1} \left(1 - x\right)} \]

   8. metadata-eval6.0%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot \color{blue}{0.5} + \sin^{-1} \left(1 - x\right)} \]

3. Applied egg-rr6.0%

   \[\leadsto \color{blue}{\frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)}} \]
4. Step-by-step derivation

   1. add-sqr-sqrt9.7%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \color{blue}{\left(\sqrt{\sin^{-1} \left(1 - x\right)} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}\right)} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

   2. pow29.7%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \color{blue}{{\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{2}} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

5. Applied egg-rr9.7%

   \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \color{blue}{{\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{2}} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]
6. Step-by-step derivation

   1. add-cbrt-cube9.7%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - {\color{blue}{\left(\sqrt[3]{\left(\sqrt{\sin^{-1} \left(1 - x\right)} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}\right) \cdot \sqrt{\sin^{-1} \left(1 - x\right)}}\right)}}^{2} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

   2. pow1/39.7%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - {\color{blue}{\left({\left(\left(\sqrt{\sin^{-1} \left(1 - x\right)} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}\right) \cdot \sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{0.3333333333333333}\right)}}^{2} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

   3. add-sqr-sqrt9.7%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - {\left({\left(\color{blue}{\sin^{-1} \left(1 - x\right)} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{0.3333333333333333}\right)}^{2} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

   4. pow19.7%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - {\left({\left(\color{blue}{{\sin^{-1} \left(1 - x\right)}^{1}} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{0.3333333333333333}\right)}^{2} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

   5. pow1/29.7%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - {\left({\left({\sin^{-1} \left(1 - x\right)}^{1} \cdot \color{blue}{{\sin^{-1} \left(1 - x\right)}^{0.5}}\right)}^{0.3333333333333333}\right)}^{2} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

   6. pow-prod-up9.7%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - {\left({\color{blue}{\left({\sin^{-1} \left(1 - x\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}\right)}^{2} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

   7. metadata-eval9.7%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - {\left({\left({\sin^{-1} \left(1 - x\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}\right)}^{2} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

7. Applied egg-rr9.7%

   \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - {\color{blue}{\left({\left({\sin^{-1} \left(1 - x\right)}^{1.5}\right)}^{0.3333333333333333}\right)}}^{2} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]
8. Step-by-step derivation

   1. unpow1/39.7%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - {\color{blue}{\left(\sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{1.5}}\right)}}^{2} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

9. Simplified9.7%

   \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - {\color{blue}{\left(\sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{1.5}}\right)}}^{2} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

10. Final simplification9.7%

    \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot {\left(\sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{1.5}}\right)}^{2}}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

1. Initial program 6.0%

   \[\cos^{-1} \left(1 - x\right) \]
2. Step-by-step derivation

   1. acos-asin6.0%

      \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(1 - x\right)} \]

   2. flip--6.0%

      \[\leadsto \color{blue}{\frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)}} \]

   3. div-inv6.0%

      \[\leadsto \frac{\color{blue}{\left(\pi \cdot \frac{1}{2}\right)} \cdot \frac{\pi}{2} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

   4. metadata-eval6.0%

      \[\leadsto \frac{\left(\pi \cdot \color{blue}{0.5}\right) \cdot \frac{\pi}{2} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

   5. div-inv6.0%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \color{blue}{\left(\pi \cdot \frac{1}{2}\right)} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

   6. metadata-eval6.0%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot \color{blue}{0.5}\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

   7. div-inv6.0%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\color{blue}{\pi \cdot \frac{1}{2}} + \sin^{-1} \left(1 - x\right)} \]

   8. metadata-eval6.0%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot \color{blue}{0.5} + \sin^{-1} \left(1 - x\right)} \]

3. Applied egg-rr6.0%

   \[\leadsto \color{blue}{\frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)}} \]
4. Step-by-step derivation

   1. add-sqr-sqrt9.7%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \color{blue}{\left(\sqrt{\sin^{-1} \left(1 - x\right)} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}\right)} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

   2. pow29.7%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \color{blue}{{\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{2}} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

5. Applied egg-rr9.7%

   \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \color{blue}{{\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{2}} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

6. Final simplification9.7%

   \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot {\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{2}}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

1. Initial program 6.0%

   \[\cos^{-1} \left(1 - x\right) \]
2. Step-by-step derivation

   1. acos-asin6.0%

      \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(1 - x\right)} \]

   2. sub-neg6.0%

      \[\leadsto \color{blue}{\frac{\pi}{2} + \left(-\sin^{-1} \left(1 - x\right)\right)} \]

   3. div-inv6.0%

      \[\leadsto \color{blue}{\pi \cdot \frac{1}{2}} + \left(-\sin^{-1} \left(1 - x\right)\right) \]

   4. metadata-eval6.0%

      \[\leadsto \pi \cdot \color{blue}{0.5} + \left(-\sin^{-1} \left(1 - x\right)\right) \]

3. Applied egg-rr6.0%

   \[\leadsto \color{blue}{\pi \cdot 0.5 + \left(-\sin^{-1} \left(1 - x\right)\right)} \]
4. Step-by-step derivation

   1. sub-neg6.0%

      \[\leadsto \color{blue}{\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)} \]

5. Simplified6.0%

   \[\leadsto \color{blue}{\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)} \]
6. Step-by-step derivation

   1. add-sqr-sqrt9.7%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \color{blue}{\left(\sqrt{\sin^{-1} \left(1 - x\right)} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}\right)} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

   2. pow29.7%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \color{blue}{{\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{2}} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

7. Applied egg-rr9.7%

   \[\leadsto \pi \cdot 0.5 - \color{blue}{{\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{2}} \]
8. Step-by-step derivation

   1. add-cbrt-cube9.7%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - {\color{blue}{\left(\sqrt[3]{\left(\sqrt{\sin^{-1} \left(1 - x\right)} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}\right) \cdot \sqrt{\sin^{-1} \left(1 - x\right)}}\right)}}^{2} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

   2. pow1/39.7%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - {\color{blue}{\left({\left(\left(\sqrt{\sin^{-1} \left(1 - x\right)} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}\right) \cdot \sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{0.3333333333333333}\right)}}^{2} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

   3. add-sqr-sqrt9.7%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - {\left({\left(\color{blue}{\sin^{-1} \left(1 - x\right)} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{0.3333333333333333}\right)}^{2} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

   4. pow19.7%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - {\left({\left(\color{blue}{{\sin^{-1} \left(1 - x\right)}^{1}} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{0.3333333333333333}\right)}^{2} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

   5. pow1/29.7%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - {\left({\left({\sin^{-1} \left(1 - x\right)}^{1} \cdot \color{blue}{{\sin^{-1} \left(1 - x\right)}^{0.5}}\right)}^{0.3333333333333333}\right)}^{2} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

   6. pow-prod-up9.7%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - {\left({\color{blue}{\left({\sin^{-1} \left(1 - x\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}\right)}^{2} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

   7. metadata-eval9.7%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - {\left({\left({\sin^{-1} \left(1 - x\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}\right)}^{2} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

9. Applied egg-rr9.7%

   \[\leadsto \pi \cdot 0.5 - {\color{blue}{\left({\left({\sin^{-1} \left(1 - x\right)}^{1.5}\right)}^{0.3333333333333333}\right)}}^{2} \]
10. Step-by-step derivation

    1. unpow1/39.7%

       \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - {\color{blue}{\left(\sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{1.5}}\right)}}^{2} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

11. Simplified9.7%

    \[\leadsto \pi \cdot 0.5 - {\color{blue}{\left(\sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{1.5}}\right)}}^{2} \]

12. Final simplification9.7%

    \[\leadsto \pi \cdot 0.5 - {\left(\sqrt[3]{{\sin^{-1} \left(1 - x\right)}^{1.5}}\right)}^{2} \]

1. Initial program 6.0%

   \[\cos^{-1} \left(1 - x\right) \]
2. Step-by-step derivation

   1. acos-asin6.0%

      \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(1 - x\right)} \]

   2. sub-neg6.0%

      \[\leadsto \color{blue}{\frac{\pi}{2} + \left(-\sin^{-1} \left(1 - x\right)\right)} \]

   3. div-inv6.0%

      \[\leadsto \color{blue}{\pi \cdot \frac{1}{2}} + \left(-\sin^{-1} \left(1 - x\right)\right) \]

   4. metadata-eval6.0%

      \[\leadsto \pi \cdot \color{blue}{0.5} + \left(-\sin^{-1} \left(1 - x\right)\right) \]

3. Applied egg-rr6.0%

   \[\leadsto \color{blue}{\pi \cdot 0.5 + \left(-\sin^{-1} \left(1 - x\right)\right)} \]
4. Step-by-step derivation

   1. sub-neg6.0%

      \[\leadsto \color{blue}{\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)} \]

5. Simplified6.0%

   \[\leadsto \color{blue}{\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)} \]
6. Step-by-step derivation

   1. add-cube-cbrt9.6%

      \[\leadsto \pi \cdot 0.5 - \color{blue}{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)} \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}\right) \cdot \sqrt[3]{\sin^{-1} \left(1 - x\right)}} \]

   2. pow39.6%

      \[\leadsto \pi \cdot 0.5 - \color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}} \]

7. Applied egg-rr9.6%

   \[\leadsto \pi \cdot 0.5 - \color{blue}{{\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3}} \]

8. Final simplification9.6%

   \[\leadsto \pi \cdot 0.5 - {\left(\sqrt[3]{\sin^{-1} \left(1 - x\right)}\right)}^{3} \]

1. Initial program 6.0%

   \[\cos^{-1} \left(1 - x\right) \]
2. Step-by-step derivation

   1. acos-asin6.0%

      \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(1 - x\right)} \]

   2. sub-neg6.0%

      \[\leadsto \color{blue}{\frac{\pi}{2} + \left(-\sin^{-1} \left(1 - x\right)\right)} \]

   3. div-inv6.0%

      \[\leadsto \color{blue}{\pi \cdot \frac{1}{2}} + \left(-\sin^{-1} \left(1 - x\right)\right) \]

   4. metadata-eval6.0%

      \[\leadsto \pi \cdot \color{blue}{0.5} + \left(-\sin^{-1} \left(1 - x\right)\right) \]

3. Applied egg-rr6.0%

   \[\leadsto \color{blue}{\pi \cdot 0.5 + \left(-\sin^{-1} \left(1 - x\right)\right)} \]
4. Step-by-step derivation

   1. sub-neg6.0%

      \[\leadsto \color{blue}{\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)} \]

5. Simplified6.0%

   \[\leadsto \color{blue}{\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)} \]
6. Step-by-step derivation

   1. add-sqr-sqrt9.7%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \color{blue}{\left(\sqrt{\sin^{-1} \left(1 - x\right)} \cdot \sqrt{\sin^{-1} \left(1 - x\right)}\right)} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

   2. pow29.7%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \color{blue}{{\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{2}} \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)} \]

7. Applied egg-rr9.7%

   \[\leadsto \pi \cdot 0.5 - \color{blue}{{\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{2}} \]

8. Final simplification9.7%

   \[\leadsto \pi \cdot 0.5 - {\left(\sqrt{\sin^{-1} \left(1 - x\right)}\right)}^{2} \]

1. Split input into 2 regimes

2. if (-.f64 1 x) < 1

   1. Initial program 6.0%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Step-by-step derivation

      1. add-log-exp6.0%

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

   3. Applied egg-rr6.0%

      \[\leadsto \color{blue}{\log \left(e^{\cos^{-1} \left(1 - x\right)}\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u6.0%

         \[\leadsto \log \left(e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)\right)}}\right) \]

      2. expm1-udef6.0%

         \[\leadsto \log \left(e^{\color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)} - 1}}\right) \]

      3. log1p-udef6.0%

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

      4. add-exp-log6.0%

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

   5. Applied egg-rr6.0%

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

   if 1 < (-.f64 1 x)

   1. Initial program 6.0%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Step-by-step derivation

      1. acos-asin6.0%

         \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(1 - x\right)} \]

      2. flip--6.0%

         \[\leadsto \color{blue}{\frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)}} \]

      3. div-inv6.0%

         \[\leadsto \frac{\color{blue}{\left(\pi \cdot \frac{1}{2}\right)} \cdot \frac{\pi}{2} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

      4. metadata-eval6.0%

         \[\leadsto \frac{\left(\pi \cdot \color{blue}{0.5}\right) \cdot \frac{\pi}{2} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

      5. div-inv6.0%

         \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \color{blue}{\left(\pi \cdot \frac{1}{2}\right)} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

      6. metadata-eval6.0%

         \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot \color{blue}{0.5}\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

      7. div-inv6.0%

         \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\color{blue}{\pi \cdot \frac{1}{2}} + \sin^{-1} \left(1 - x\right)} \]

      8. metadata-eval6.0%

         \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot \color{blue}{0.5} + \sin^{-1} \left(1 - x\right)} \]

   3. Applied egg-rr6.0%

      \[\leadsto \color{blue}{\frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)}} \]
   4. Step-by-step derivation

      1. flip--6.0%

         \[\leadsto \color{blue}{\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)} \]

      2. metadata-eval6.0%

         \[\leadsto \pi \cdot \color{blue}{\frac{1}{2}} - \sin^{-1} \left(1 - x\right) \]

      3. div-inv6.0%

         \[\leadsto \color{blue}{\frac{\pi}{2}} - \sin^{-1} \left(1 - x\right) \]

      4. acos-asin6.0%

         \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right)} \]

      5. expm1-log1p-u6.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)\right)} \]

      6. expm1-udef6.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)} - 1} \]

      7. log1p-udef6.0%

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

      8. add-exp-log6.0%

         \[\leadsto \color{blue}{\left(1 + \cos^{-1} \left(1 - x\right)\right)} - 1 \]

      9. associate--l+6.0%

         \[\leadsto \color{blue}{1 + \left(\cos^{-1} \left(1 - x\right) - 1\right)} \]

      10. +-commutative6.0%

          \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) - 1\right) + 1} \]

      11. sub-neg6.0%

          \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) + \left(-1\right)\right)} + 1 \]

      12. metadata-eval6.0%

          \[\leadsto \left(\cos^{-1} \left(1 - x\right) + \color{blue}{-1}\right) + 1 \]

   5. Applied egg-rr6.0%

      \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) + -1\right) + 1} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt0.0%

         \[\leadsto \color{blue}{\sqrt{\cos^{-1} \left(1 - x\right) + -1} \cdot \sqrt{\cos^{-1} \left(1 - x\right) + -1}} + 1 \]

      2. sqrt-unprod6.9%

         \[\leadsto \color{blue}{\sqrt{\left(\cos^{-1} \left(1 - x\right) + -1\right) \cdot \left(\cos^{-1} \left(1 - x\right) + -1\right)}} + 1 \]

      3. pow26.9%

         \[\leadsto \sqrt{\color{blue}{{\left(\cos^{-1} \left(1 - x\right) + -1\right)}^{2}}} + 1 \]

   7. Applied egg-rr6.9%

      \[\leadsto \color{blue}{\sqrt{{\left(\cos^{-1} \left(1 - x\right) + -1\right)}^{2}}} + 1 \]
   8. Step-by-step derivation

      1. unpow26.9%

         \[\leadsto \sqrt{\color{blue}{\left(\cos^{-1} \left(1 - x\right) + -1\right) \cdot \left(\cos^{-1} \left(1 - x\right) + -1\right)}} + 1 \]

      2. rem-sqrt-square6.9%

         \[\leadsto \color{blue}{\left|\cos^{-1} \left(1 - x\right) + -1\right|} + 1 \]

      3. +-commutative6.9%

         \[\leadsto \left|\color{blue}{-1 + \cos^{-1} \left(1 - x\right)}\right| + 1 \]

   9. Simplified6.9%

      \[\leadsto \color{blue}{\left|-1 + \cos^{-1} \left(1 - x\right)\right|} + 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification6.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;\log \left(e^{-1 + \left(1 + \cos^{-1} \left(1 - x\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left|\cos^{-1} \left(1 - x\right) + -1\right|\\ \end{array} \]

1. Split input into 2 regimes

2. if (-.f64 1 x) < 1

   1. Initial program 6.0%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Step-by-step derivation

      1. acos-asin6.0%

         \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(1 - x\right)} \]

      2. flip--6.0%

         \[\leadsto \color{blue}{\frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)}} \]

      3. div-inv6.0%

         \[\leadsto \frac{\color{blue}{\left(\pi \cdot \frac{1}{2}\right)} \cdot \frac{\pi}{2} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

      4. metadata-eval6.0%

         \[\leadsto \frac{\left(\pi \cdot \color{blue}{0.5}\right) \cdot \frac{\pi}{2} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

      5. div-inv6.0%

         \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \color{blue}{\left(\pi \cdot \frac{1}{2}\right)} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

      6. metadata-eval6.0%

         \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot \color{blue}{0.5}\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

      7. div-inv6.0%

         \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\color{blue}{\pi \cdot \frac{1}{2}} + \sin^{-1} \left(1 - x\right)} \]

      8. metadata-eval6.0%

         \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot \color{blue}{0.5} + \sin^{-1} \left(1 - x\right)} \]

   3. Applied egg-rr6.0%

      \[\leadsto \color{blue}{\frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)}} \]
   4. Step-by-step derivation

      1. flip--6.0%

         \[\leadsto \color{blue}{\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)} \]

      2. metadata-eval6.0%

         \[\leadsto \pi \cdot \color{blue}{\frac{1}{2}} - \sin^{-1} \left(1 - x\right) \]

      3. div-inv6.0%

         \[\leadsto \color{blue}{\frac{\pi}{2}} - \sin^{-1} \left(1 - x\right) \]

      4. acos-asin6.0%

         \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right)} \]

      5. expm1-log1p-u6.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)\right)} \]

      6. expm1-udef6.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)} - 1} \]

      7. log1p-udef6.0%

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

      8. add-exp-log6.0%

         \[\leadsto \color{blue}{\left(1 + \cos^{-1} \left(1 - x\right)\right)} - 1 \]

      9. associate--l+6.0%

         \[\leadsto \color{blue}{1 + \left(\cos^{-1} \left(1 - x\right) - 1\right)} \]

      10. +-commutative6.0%

          \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) - 1\right) + 1} \]

      11. sub-neg6.0%

          \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) + \left(-1\right)\right)} + 1 \]

      12. metadata-eval6.0%

          \[\leadsto \left(\cos^{-1} \left(1 - x\right) + \color{blue}{-1}\right) + 1 \]

   5. Applied egg-rr6.0%

      \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) + -1\right) + 1} \]
   6. Step-by-step derivation

      1. add-log-exp6.0%

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

   7. Applied egg-rr6.0%

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

   if 1 < (-.f64 1 x)

   1. Initial program 6.0%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Step-by-step derivation

      1. acos-asin6.0%

         \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(1 - x\right)} \]

      2. flip--6.0%

         \[\leadsto \color{blue}{\frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)}} \]

      3. div-inv6.0%

         \[\leadsto \frac{\color{blue}{\left(\pi \cdot \frac{1}{2}\right)} \cdot \frac{\pi}{2} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

      4. metadata-eval6.0%

         \[\leadsto \frac{\left(\pi \cdot \color{blue}{0.5}\right) \cdot \frac{\pi}{2} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

      5. div-inv6.0%

         \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \color{blue}{\left(\pi \cdot \frac{1}{2}\right)} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

      6. metadata-eval6.0%

         \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot \color{blue}{0.5}\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

      7. div-inv6.0%

         \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\color{blue}{\pi \cdot \frac{1}{2}} + \sin^{-1} \left(1 - x\right)} \]

      8. metadata-eval6.0%

         \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot \color{blue}{0.5} + \sin^{-1} \left(1 - x\right)} \]

   3. Applied egg-rr6.0%

      \[\leadsto \color{blue}{\frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)}} \]
   4. Step-by-step derivation

      1. flip--6.0%

         \[\leadsto \color{blue}{\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)} \]

      2. metadata-eval6.0%

         \[\leadsto \pi \cdot \color{blue}{\frac{1}{2}} - \sin^{-1} \left(1 - x\right) \]

      3. div-inv6.0%

         \[\leadsto \color{blue}{\frac{\pi}{2}} - \sin^{-1} \left(1 - x\right) \]

      4. acos-asin6.0%

         \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right)} \]

      5. expm1-log1p-u6.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)\right)} \]

      6. expm1-udef6.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)} - 1} \]

      7. log1p-udef6.0%

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

      8. add-exp-log6.0%

         \[\leadsto \color{blue}{\left(1 + \cos^{-1} \left(1 - x\right)\right)} - 1 \]

      9. associate--l+6.0%

         \[\leadsto \color{blue}{1 + \left(\cos^{-1} \left(1 - x\right) - 1\right)} \]

      10. +-commutative6.0%

          \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) - 1\right) + 1} \]

      11. sub-neg6.0%

          \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) + \left(-1\right)\right)} + 1 \]

      12. metadata-eval6.0%

          \[\leadsto \left(\cos^{-1} \left(1 - x\right) + \color{blue}{-1}\right) + 1 \]

   5. Applied egg-rr6.0%

      \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) + -1\right) + 1} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt0.0%

         \[\leadsto \color{blue}{\sqrt{\cos^{-1} \left(1 - x\right) + -1} \cdot \sqrt{\cos^{-1} \left(1 - x\right) + -1}} + 1 \]

      2. sqrt-unprod6.9%

         \[\leadsto \color{blue}{\sqrt{\left(\cos^{-1} \left(1 - x\right) + -1\right) \cdot \left(\cos^{-1} \left(1 - x\right) + -1\right)}} + 1 \]

      3. pow26.9%

         \[\leadsto \sqrt{\color{blue}{{\left(\cos^{-1} \left(1 - x\right) + -1\right)}^{2}}} + 1 \]

   7. Applied egg-rr6.9%

      \[\leadsto \color{blue}{\sqrt{{\left(\cos^{-1} \left(1 - x\right) + -1\right)}^{2}}} + 1 \]
   8. Step-by-step derivation

      1. unpow26.9%

         \[\leadsto \sqrt{\color{blue}{\left(\cos^{-1} \left(1 - x\right) + -1\right) \cdot \left(\cos^{-1} \left(1 - x\right) + -1\right)}} + 1 \]

      2. rem-sqrt-square6.9%

         \[\leadsto \color{blue}{\left|\cos^{-1} \left(1 - x\right) + -1\right|} + 1 \]

      3. +-commutative6.9%

         \[\leadsto \left|\color{blue}{-1 + \cos^{-1} \left(1 - x\right)}\right| + 1 \]

   9. Simplified6.9%

      \[\leadsto \color{blue}{\left|-1 + \cos^{-1} \left(1 - x\right)\right|} + 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification6.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;1 + \log \left(e^{\cos^{-1} \left(1 - x\right) + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left|\cos^{-1} \left(1 - x\right) + -1\right|\\ \end{array} \]

1. Split input into 2 regimes

2. if (-.f64 1 x) < 1

   1. Initial program 6.0%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Step-by-step derivation

      1. add-cube-cbrt6.0%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\cos^{-1} \left(1 - x\right)} \cdot \sqrt[3]{\cos^{-1} \left(1 - x\right)}\right) \cdot \sqrt[3]{\cos^{-1} \left(1 - x\right)}} \]

      2. pow36.0%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos^{-1} \left(1 - x\right)}\right)}^{3}} \]

   3. Applied egg-rr6.0%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos^{-1} \left(1 - x\right)}\right)}^{3}} \]

   if 1 < (-.f64 1 x)

   1. Initial program 6.0%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Step-by-step derivation

      1. acos-asin6.0%

         \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(1 - x\right)} \]

      2. flip--6.0%

         \[\leadsto \color{blue}{\frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)}} \]

      3. div-inv6.0%

         \[\leadsto \frac{\color{blue}{\left(\pi \cdot \frac{1}{2}\right)} \cdot \frac{\pi}{2} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

      4. metadata-eval6.0%

         \[\leadsto \frac{\left(\pi \cdot \color{blue}{0.5}\right) \cdot \frac{\pi}{2} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

      5. div-inv6.0%

         \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \color{blue}{\left(\pi \cdot \frac{1}{2}\right)} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

      6. metadata-eval6.0%

         \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot \color{blue}{0.5}\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

      7. div-inv6.0%

         \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\color{blue}{\pi \cdot \frac{1}{2}} + \sin^{-1} \left(1 - x\right)} \]

      8. metadata-eval6.0%

         \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot \color{blue}{0.5} + \sin^{-1} \left(1 - x\right)} \]

   3. Applied egg-rr6.0%

      \[\leadsto \color{blue}{\frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)}} \]
   4. Step-by-step derivation

      1. flip--6.0%

         \[\leadsto \color{blue}{\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)} \]

      2. metadata-eval6.0%

         \[\leadsto \pi \cdot \color{blue}{\frac{1}{2}} - \sin^{-1} \left(1 - x\right) \]

      3. div-inv6.0%

         \[\leadsto \color{blue}{\frac{\pi}{2}} - \sin^{-1} \left(1 - x\right) \]

      4. acos-asin6.0%

         \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right)} \]

      5. expm1-log1p-u6.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)\right)} \]

      6. expm1-udef6.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)} - 1} \]

      7. log1p-udef6.0%

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

      8. add-exp-log6.0%

         \[\leadsto \color{blue}{\left(1 + \cos^{-1} \left(1 - x\right)\right)} - 1 \]

      9. associate--l+6.0%

         \[\leadsto \color{blue}{1 + \left(\cos^{-1} \left(1 - x\right) - 1\right)} \]

      10. +-commutative6.0%

          \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) - 1\right) + 1} \]

      11. sub-neg6.0%

          \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) + \left(-1\right)\right)} + 1 \]

      12. metadata-eval6.0%

          \[\leadsto \left(\cos^{-1} \left(1 - x\right) + \color{blue}{-1}\right) + 1 \]

   5. Applied egg-rr6.0%

      \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) + -1\right) + 1} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt0.0%

         \[\leadsto \color{blue}{\sqrt{\cos^{-1} \left(1 - x\right) + -1} \cdot \sqrt{\cos^{-1} \left(1 - x\right) + -1}} + 1 \]

      2. sqrt-unprod6.9%

         \[\leadsto \color{blue}{\sqrt{\left(\cos^{-1} \left(1 - x\right) + -1\right) \cdot \left(\cos^{-1} \left(1 - x\right) + -1\right)}} + 1 \]

      3. pow26.9%

         \[\leadsto \sqrt{\color{blue}{{\left(\cos^{-1} \left(1 - x\right) + -1\right)}^{2}}} + 1 \]

   7. Applied egg-rr6.9%

      \[\leadsto \color{blue}{\sqrt{{\left(\cos^{-1} \left(1 - x\right) + -1\right)}^{2}}} + 1 \]
   8. Step-by-step derivation

      1. unpow26.9%

         \[\leadsto \sqrt{\color{blue}{\left(\cos^{-1} \left(1 - x\right) + -1\right) \cdot \left(\cos^{-1} \left(1 - x\right) + -1\right)}} + 1 \]

      2. rem-sqrt-square6.9%

         \[\leadsto \color{blue}{\left|\cos^{-1} \left(1 - x\right) + -1\right|} + 1 \]

      3. +-commutative6.9%

         \[\leadsto \left|\color{blue}{-1 + \cos^{-1} \left(1 - x\right)}\right| + 1 \]

   9. Simplified6.9%

      \[\leadsto \color{blue}{\left|-1 + \cos^{-1} \left(1 - x\right)\right|} + 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification6.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;{\left(\sqrt[3]{\cos^{-1} \left(1 - x\right)}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;1 + \left|\cos^{-1} \left(1 - x\right) + -1\right|\\ \end{array} \]

1. Split input into 2 regimes

2. if (-.f64 1 x) < 1

   1. Initial program 6.0%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Step-by-step derivation

      1. acos-asin6.0%

         \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(1 - x\right)} \]

      2. flip--6.0%

         \[\leadsto \color{blue}{\frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)}} \]

      3. div-inv6.0%

         \[\leadsto \frac{\color{blue}{\left(\pi \cdot \frac{1}{2}\right)} \cdot \frac{\pi}{2} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

      4. metadata-eval6.0%

         \[\leadsto \frac{\left(\pi \cdot \color{blue}{0.5}\right) \cdot \frac{\pi}{2} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

      5. div-inv6.0%

         \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \color{blue}{\left(\pi \cdot \frac{1}{2}\right)} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

      6. metadata-eval6.0%

         \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot \color{blue}{0.5}\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

      7. div-inv6.0%

         \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\color{blue}{\pi \cdot \frac{1}{2}} + \sin^{-1} \left(1 - x\right)} \]

      8. metadata-eval6.0%

         \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot \color{blue}{0.5} + \sin^{-1} \left(1 - x\right)} \]

   3. Applied egg-rr6.0%

      \[\leadsto \color{blue}{\frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)}} \]
   4. Step-by-step derivation

      1. flip--6.0%

         \[\leadsto \color{blue}{\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)} \]

      2. metadata-eval6.0%

         \[\leadsto \pi \cdot \color{blue}{\frac{1}{2}} - \sin^{-1} \left(1 - x\right) \]

      3. div-inv6.0%

         \[\leadsto \color{blue}{\frac{\pi}{2}} - \sin^{-1} \left(1 - x\right) \]

      4. acos-asin6.0%

         \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right)} \]

      5. expm1-log1p-u6.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)\right)} \]

      6. expm1-udef6.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)} - 1} \]

      7. log1p-udef6.0%

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

      8. add-exp-log6.0%

         \[\leadsto \color{blue}{\left(1 + \cos^{-1} \left(1 - x\right)\right)} - 1 \]

      9. associate--l+6.0%

         \[\leadsto \color{blue}{1 + \left(\cos^{-1} \left(1 - x\right) - 1\right)} \]

      10. +-commutative6.0%

          \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) - 1\right) + 1} \]

      11. sub-neg6.0%

          \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) + \left(-1\right)\right)} + 1 \]

      12. metadata-eval6.0%

          \[\leadsto \left(\cos^{-1} \left(1 - x\right) + \color{blue}{-1}\right) + 1 \]

   5. Applied egg-rr6.0%

      \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) + -1\right) + 1} \]

   if 1 < (-.f64 1 x)

   1. Initial program 6.0%

      \[\cos^{-1} \left(1 - x\right) \]
   2. Step-by-step derivation

      1. acos-asin6.0%

         \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(1 - x\right)} \]

      2. flip--6.0%

         \[\leadsto \color{blue}{\frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)}} \]

      3. div-inv6.0%

         \[\leadsto \frac{\color{blue}{\left(\pi \cdot \frac{1}{2}\right)} \cdot \frac{\pi}{2} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

      4. metadata-eval6.0%

         \[\leadsto \frac{\left(\pi \cdot \color{blue}{0.5}\right) \cdot \frac{\pi}{2} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

      5. div-inv6.0%

         \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \color{blue}{\left(\pi \cdot \frac{1}{2}\right)} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

      6. metadata-eval6.0%

         \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot \color{blue}{0.5}\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

      7. div-inv6.0%

         \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\color{blue}{\pi \cdot \frac{1}{2}} + \sin^{-1} \left(1 - x\right)} \]

      8. metadata-eval6.0%

         \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot \color{blue}{0.5} + \sin^{-1} \left(1 - x\right)} \]

   3. Applied egg-rr6.0%

      \[\leadsto \color{blue}{\frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)}} \]
   4. Step-by-step derivation

      1. flip--6.0%

         \[\leadsto \color{blue}{\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)} \]

      2. metadata-eval6.0%

         \[\leadsto \pi \cdot \color{blue}{\frac{1}{2}} - \sin^{-1} \left(1 - x\right) \]

      3. div-inv6.0%

         \[\leadsto \color{blue}{\frac{\pi}{2}} - \sin^{-1} \left(1 - x\right) \]

      4. acos-asin6.0%

         \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right)} \]

      5. expm1-log1p-u6.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)\right)} \]

      6. expm1-udef6.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)} - 1} \]

      7. log1p-udef6.0%

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

      8. add-exp-log6.0%

         \[\leadsto \color{blue}{\left(1 + \cos^{-1} \left(1 - x\right)\right)} - 1 \]

      9. associate--l+6.0%

         \[\leadsto \color{blue}{1 + \left(\cos^{-1} \left(1 - x\right) - 1\right)} \]

      10. +-commutative6.0%

          \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) - 1\right) + 1} \]

      11. sub-neg6.0%

          \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) + \left(-1\right)\right)} + 1 \]

      12. metadata-eval6.0%

          \[\leadsto \left(\cos^{-1} \left(1 - x\right) + \color{blue}{-1}\right) + 1 \]

   5. Applied egg-rr6.0%

      \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) + -1\right) + 1} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt0.0%

         \[\leadsto \color{blue}{\sqrt{\cos^{-1} \left(1 - x\right) + -1} \cdot \sqrt{\cos^{-1} \left(1 - x\right) + -1}} + 1 \]

      2. sqrt-unprod6.9%

         \[\leadsto \color{blue}{\sqrt{\left(\cos^{-1} \left(1 - x\right) + -1\right) \cdot \left(\cos^{-1} \left(1 - x\right) + -1\right)}} + 1 \]

      3. pow26.9%

         \[\leadsto \sqrt{\color{blue}{{\left(\cos^{-1} \left(1 - x\right) + -1\right)}^{2}}} + 1 \]

   7. Applied egg-rr6.9%

      \[\leadsto \color{blue}{\sqrt{{\left(\cos^{-1} \left(1 - x\right) + -1\right)}^{2}}} + 1 \]
   8. Step-by-step derivation

      1. unpow26.9%

         \[\leadsto \sqrt{\color{blue}{\left(\cos^{-1} \left(1 - x\right) + -1\right) \cdot \left(\cos^{-1} \left(1 - x\right) + -1\right)}} + 1 \]

      2. rem-sqrt-square6.9%

         \[\leadsto \color{blue}{\left|\cos^{-1} \left(1 - x\right) + -1\right|} + 1 \]

      3. +-commutative6.9%

         \[\leadsto \left|\color{blue}{-1 + \cos^{-1} \left(1 - x\right)}\right| + 1 \]

   9. Simplified6.9%

      \[\leadsto \color{blue}{\left|-1 + \cos^{-1} \left(1 - x\right)\right|} + 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification6.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - x \leq 1:\\ \;\;\;\;1 + \left(\cos^{-1} \left(1 - x\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left|\cos^{-1} \left(1 - x\right) + -1\right|\\ \end{array} \]

1. Initial program 6.0%

   \[\cos^{-1} \left(1 - x\right) \]
2. Step-by-step derivation

   1. acos-asin6.0%

      \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(1 - x\right)} \]

   2. flip--6.0%

      \[\leadsto \color{blue}{\frac{\frac{\pi}{2} \cdot \frac{\pi}{2} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)}} \]

   3. div-inv6.0%

      \[\leadsto \frac{\color{blue}{\left(\pi \cdot \frac{1}{2}\right)} \cdot \frac{\pi}{2} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

   4. metadata-eval6.0%

      \[\leadsto \frac{\left(\pi \cdot \color{blue}{0.5}\right) \cdot \frac{\pi}{2} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

   5. div-inv6.0%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \color{blue}{\left(\pi \cdot \frac{1}{2}\right)} - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

   6. metadata-eval6.0%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot \color{blue}{0.5}\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\frac{\pi}{2} + \sin^{-1} \left(1 - x\right)} \]

   7. div-inv6.0%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\color{blue}{\pi \cdot \frac{1}{2}} + \sin^{-1} \left(1 - x\right)} \]

   8. metadata-eval6.0%

      \[\leadsto \frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot \color{blue}{0.5} + \sin^{-1} \left(1 - x\right)} \]

3. Applied egg-rr6.0%

   \[\leadsto \color{blue}{\frac{\left(\pi \cdot 0.5\right) \cdot \left(\pi \cdot 0.5\right) - \sin^{-1} \left(1 - x\right) \cdot \sin^{-1} \left(1 - x\right)}{\pi \cdot 0.5 + \sin^{-1} \left(1 - x\right)}} \]
4. Step-by-step derivation

   1. flip--6.0%

      \[\leadsto \color{blue}{\pi \cdot 0.5 - \sin^{-1} \left(1 - x\right)} \]

   2. metadata-eval6.0%

      \[\leadsto \pi \cdot \color{blue}{\frac{1}{2}} - \sin^{-1} \left(1 - x\right) \]

   3. div-inv6.0%

      \[\leadsto \color{blue}{\frac{\pi}{2}} - \sin^{-1} \left(1 - x\right) \]

   4. acos-asin6.0%

      \[\leadsto \color{blue}{\cos^{-1} \left(1 - x\right)} \]

   5. expm1-log1p-u6.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)\right)} \]

   6. expm1-udef6.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos^{-1} \left(1 - x\right)\right)} - 1} \]

   7. log1p-udef6.0%

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

   8. add-exp-log6.0%

      \[\leadsto \color{blue}{\left(1 + \cos^{-1} \left(1 - x\right)\right)} - 1 \]

   9. associate--l+6.0%

      \[\leadsto \color{blue}{1 + \left(\cos^{-1} \left(1 - x\right) - 1\right)} \]

   10. +-commutative6.0%

       \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) - 1\right) + 1} \]

   11. sub-neg6.0%

       \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) + \left(-1\right)\right)} + 1 \]

   12. metadata-eval6.0%

       \[\leadsto \left(\cos^{-1} \left(1 - x\right) + \color{blue}{-1}\right) + 1 \]

5. Applied egg-rr6.0%

   \[\leadsto \color{blue}{\left(\cos^{-1} \left(1 - x\right) + -1\right) + 1} \]

6. Final simplification6.0%

   \[\leadsto 1 + \left(\cos^{-1} \left(1 - x\right) + -1\right) \]

1. Initial program 6.0%

   \[\cos^{-1} \left(1 - x\right) \]

2. Final simplification6.0%

   \[\leadsto \cos^{-1} \left(1 - x\right) \]

herbie shell --seed 2023167 
(FPCore (x)
  :name "bug323 (missed optimization)"
  :precision binary64
  :pre (and (<= 0.0 x) (<= x 0.5))

  :herbie-target
  (* 2.0 (asin (sqrt (/ x 2.0))))

  (acos (- 1.0 x)))