ENA, Section 1.4, Exercise 1

1. Initial program 94.5%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
2. Step-by-step derivation

   1. pow-exp95.2%

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{10}\right)}^{\left(x \cdot x\right)}} \]

   2. *-commutative95.2%

      \[\leadsto \color{blue}{{\left(e^{10}\right)}^{\left(x \cdot x\right)} \cdot \cos x} \]

   3. pow-exp94.5%

      \[\leadsto \color{blue}{e^{10 \cdot \left(x \cdot x\right)}} \cdot \cos x \]

   4. associate-*r*94.4%

      \[\leadsto e^{\color{blue}{\left(10 \cdot x\right) \cdot x}} \cdot \cos x \]

   5. add-log-exp94.4%

      \[\leadsto e^{\left(10 \cdot x\right) \cdot \color{blue}{\log \left(e^{x}\right)}} \cdot \cos x \]

   6. log-pow94.4%

      \[\leadsto e^{\color{blue}{\log \left({\left(e^{x}\right)}^{\left(10 \cdot x\right)}\right)}} \cdot \cos x \]

   7. pow-pow94.8%

      \[\leadsto e^{\log \color{blue}{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x}\right)}} \cdot \cos x \]

   8. add-exp-log96.8%

      \[\leadsto \color{blue}{{\left({\left(e^{x}\right)}^{10}\right)}^{x}} \cdot \cos x \]

   9. add-cbrt-cube96.8%

      \[\leadsto \color{blue}{\sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}}} \cdot \cos x \]

   10. add-cbrt-cube96.7%

       \[\leadsto \sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}} \cdot \color{blue}{\sqrt[3]{\left(\cos x \cdot \cos x\right) \cdot \cos x}} \]

   11. cbrt-unprod96.8%

       \[\leadsto \color{blue}{\sqrt[3]{\left(\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot \left(\left(\cos x \cdot \cos x\right) \cdot \cos x\right)}} \]

3. Applied egg-rr98.7%

   \[\leadsto \color{blue}{\sqrt[3]{{\left({\left(e^{30}\right)}^{x}\right)}^{x} \cdot {\cos x}^{3}}} \]
4. Step-by-step derivation

   1. add-sqr-sqrt98.7%

      \[\leadsto \sqrt[3]{{\color{blue}{\left(\sqrt{{\left(e^{30}\right)}^{x}} \cdot \sqrt{{\left(e^{30}\right)}^{x}}\right)}}^{x} \cdot {\cos x}^{3}} \]

   2. sqrt-unprod98.7%

      \[\leadsto \sqrt[3]{{\color{blue}{\left(\sqrt{{\left(e^{30}\right)}^{x} \cdot {\left(e^{30}\right)}^{x}}\right)}}^{x} \cdot {\cos x}^{3}} \]

   3. pow-prod-down98.8%

      \[\leadsto \sqrt[3]{{\left(\sqrt{\color{blue}{{\left(e^{30} \cdot e^{30}\right)}^{x}}}\right)}^{x} \cdot {\cos x}^{3}} \]

   4. prod-exp99.2%

      \[\leadsto \sqrt[3]{{\left(\sqrt{{\color{blue}{\left(e^{30 + 30}\right)}}^{x}}\right)}^{x} \cdot {\cos x}^{3}} \]

   5. metadata-eval99.2%

      \[\leadsto \sqrt[3]{{\left(\sqrt{{\left(e^{\color{blue}{60}}\right)}^{x}}\right)}^{x} \cdot {\cos x}^{3}} \]

5. Applied egg-rr99.2%

   \[\leadsto \sqrt[3]{{\color{blue}{\left(\sqrt{{\left(e^{60}\right)}^{x}}\right)}}^{x} \cdot {\cos x}^{3}} \]
6. Step-by-step derivation

   1. add-sqr-sqrt99.2%

      \[\leadsto \sqrt[3]{\color{blue}{\left(\sqrt{{\left(\sqrt{{\left(e^{60}\right)}^{x}}\right)}^{x}} \cdot \sqrt{{\left(\sqrt{{\left(e^{60}\right)}^{x}}\right)}^{x}}\right)} \cdot {\cos x}^{3}} \]

   2. sqrt-unprod99.2%

      \[\leadsto \sqrt[3]{\color{blue}{\sqrt{{\left(\sqrt{{\left(e^{60}\right)}^{x}}\right)}^{x} \cdot {\left(\sqrt{{\left(e^{60}\right)}^{x}}\right)}^{x}}} \cdot {\cos x}^{3}} \]

   3. pow-prod-down99.1%

      \[\leadsto \sqrt[3]{\sqrt{\color{blue}{{\left(\sqrt{{\left(e^{60}\right)}^{x}} \cdot \sqrt{{\left(e^{60}\right)}^{x}}\right)}^{x}}} \cdot {\cos x}^{3}} \]

   4. add-sqr-sqrt99.2%

      \[\leadsto \sqrt[3]{\sqrt{{\color{blue}{\left({\left(e^{60}\right)}^{x}\right)}}^{x}} \cdot {\cos x}^{3}} \]

7. Applied egg-rr99.2%

   \[\leadsto \sqrt[3]{\color{blue}{\sqrt{{\left({\left(e^{60}\right)}^{x}\right)}^{x}}} \cdot {\cos x}^{3}} \]

8. Final simplification99.2%

   \[\leadsto \sqrt[3]{\sqrt{{\left({\left(e^{60}\right)}^{x}\right)}^{x}} \cdot {\cos x}^{3}} \]

1. Initial program 94.5%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
2. Step-by-step derivation

   1. pow-exp95.2%

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{10}\right)}^{\left(x \cdot x\right)}} \]

   2. *-commutative95.2%

      \[\leadsto \color{blue}{{\left(e^{10}\right)}^{\left(x \cdot x\right)} \cdot \cos x} \]

   3. pow-exp94.5%

      \[\leadsto \color{blue}{e^{10 \cdot \left(x \cdot x\right)}} \cdot \cos x \]

   4. associate-*r*94.4%

      \[\leadsto e^{\color{blue}{\left(10 \cdot x\right) \cdot x}} \cdot \cos x \]

   5. add-log-exp94.4%

      \[\leadsto e^{\left(10 \cdot x\right) \cdot \color{blue}{\log \left(e^{x}\right)}} \cdot \cos x \]

   6. log-pow94.4%

      \[\leadsto e^{\color{blue}{\log \left({\left(e^{x}\right)}^{\left(10 \cdot x\right)}\right)}} \cdot \cos x \]

   7. pow-pow94.8%

      \[\leadsto e^{\log \color{blue}{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x}\right)}} \cdot \cos x \]

   8. add-exp-log96.8%

      \[\leadsto \color{blue}{{\left({\left(e^{x}\right)}^{10}\right)}^{x}} \cdot \cos x \]

   9. add-cbrt-cube96.8%

      \[\leadsto \color{blue}{\sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}}} \cdot \cos x \]

   10. add-cbrt-cube96.7%

       \[\leadsto \sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}} \cdot \color{blue}{\sqrt[3]{\left(\cos x \cdot \cos x\right) \cdot \cos x}} \]

   11. cbrt-unprod96.8%

       \[\leadsto \color{blue}{\sqrt[3]{\left(\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot \left(\left(\cos x \cdot \cos x\right) \cdot \cos x\right)}} \]

3. Applied egg-rr98.7%

   \[\leadsto \color{blue}{\sqrt[3]{{\left({\left(e^{30}\right)}^{x}\right)}^{x} \cdot {\cos x}^{3}}} \]
4. Step-by-step derivation

   1. add-sqr-sqrt98.7%

      \[\leadsto \sqrt[3]{{\color{blue}{\left(\sqrt{{\left(e^{30}\right)}^{x}} \cdot \sqrt{{\left(e^{30}\right)}^{x}}\right)}}^{x} \cdot {\cos x}^{3}} \]

   2. sqrt-unprod98.7%

      \[\leadsto \sqrt[3]{{\color{blue}{\left(\sqrt{{\left(e^{30}\right)}^{x} \cdot {\left(e^{30}\right)}^{x}}\right)}}^{x} \cdot {\cos x}^{3}} \]

   3. pow-prod-down98.8%

      \[\leadsto \sqrt[3]{{\left(\sqrt{\color{blue}{{\left(e^{30} \cdot e^{30}\right)}^{x}}}\right)}^{x} \cdot {\cos x}^{3}} \]

   4. prod-exp99.2%

      \[\leadsto \sqrt[3]{{\left(\sqrt{{\color{blue}{\left(e^{30 + 30}\right)}}^{x}}\right)}^{x} \cdot {\cos x}^{3}} \]

   5. metadata-eval99.2%

      \[\leadsto \sqrt[3]{{\left(\sqrt{{\left(e^{\color{blue}{60}}\right)}^{x}}\right)}^{x} \cdot {\cos x}^{3}} \]

5. Applied egg-rr99.2%

   \[\leadsto \sqrt[3]{{\color{blue}{\left(\sqrt{{\left(e^{60}\right)}^{x}}\right)}}^{x} \cdot {\cos x}^{3}} \]

6. Taylor expanded in x around inf 94.2%

   \[\leadsto \color{blue}{\cos x \cdot {\left(1 \cdot e^{\log \left(\sqrt{{\left(e^{60}\right)}^{x}}\right) \cdot x}\right)}^{0.3333333333333333}} \]
7. Step-by-step derivation

   1. unpow1/395.3%

      \[\leadsto \cos x \cdot \color{blue}{\sqrt[3]{1 \cdot e^{\log \left(\sqrt{{\left(e^{60}\right)}^{x}}\right) \cdot x}}} \]

   2. *-lft-identity95.3%

      \[\leadsto \cos x \cdot \sqrt[3]{\color{blue}{e^{\log \left(\sqrt{{\left(e^{60}\right)}^{x}}\right) \cdot x}}} \]

   3. exp-to-pow99.0%

      \[\leadsto \cos x \cdot \sqrt[3]{\color{blue}{{\left(\sqrt{{\left(e^{60}\right)}^{x}}\right)}^{x}}} \]

8. Simplified99.0%

   \[\leadsto \color{blue}{\cos x \cdot \sqrt[3]{{\left(\sqrt{{\left(e^{60}\right)}^{x}}\right)}^{x}}} \]

9. Final simplification99.0%

   \[\leadsto \cos x \cdot \sqrt[3]{{\left(\sqrt{{\left(e^{60}\right)}^{x}}\right)}^{x}} \]

1. Initial program 94.5%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
2. Step-by-step derivation

   1. associate-*r*94.4%

      \[\leadsto \cos x \cdot e^{\color{blue}{\left(10 \cdot x\right) \cdot x}} \]

   2. exp-prod95.0%

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{10 \cdot x}\right)}^{x}} \]

   3. sqr-pow95.0%

      \[\leadsto \cos x \cdot \color{blue}{\left({\left(e^{10 \cdot x}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(e^{10 \cdot x}\right)}^{\left(\frac{x}{2}\right)}\right)} \]

   4. sqr-pow95.0%

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{10 \cdot x}\right)}^{x}} \]

   5. exp-prod98.0%

      \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{10}\right)}^{x}\right)}}^{x} \]

3. Simplified98.0%

   \[\leadsto \color{blue}{\cos x \cdot {\left({\left(e^{10}\right)}^{x}\right)}^{x}} \]
4. Step-by-step derivation

   1. add-cbrt-cube98.1%

      \[\leadsto \cos x \cdot {\color{blue}{\left(\sqrt[3]{\left({\left(e^{10}\right)}^{x} \cdot {\left(e^{10}\right)}^{x}\right) \cdot {\left(e^{10}\right)}^{x}}\right)}}^{x} \]

   2. pow-prod-down98.0%

      \[\leadsto \cos x \cdot {\left(\sqrt[3]{\color{blue}{{\left(e^{10} \cdot e^{10}\right)}^{x}} \cdot {\left(e^{10}\right)}^{x}}\right)}^{x} \]

   3. pow-prod-down97.7%

      \[\leadsto \cos x \cdot {\left(\sqrt[3]{\color{blue}{{\left(\left(e^{10} \cdot e^{10}\right) \cdot e^{10}\right)}^{x}}}\right)}^{x} \]

   4. pow397.7%

      \[\leadsto \cos x \cdot {\left(\sqrt[3]{{\color{blue}{\left({\left(e^{10}\right)}^{3}\right)}}^{x}}\right)}^{x} \]

   5. metadata-eval97.7%

      \[\leadsto \cos x \cdot {\left(\sqrt[3]{{\left({\left(e^{10}\right)}^{\color{blue}{\left(2 + 1\right)}}\right)}^{x}}\right)}^{x} \]

   6. pow-exp98.6%

      \[\leadsto \cos x \cdot {\left(\sqrt[3]{{\color{blue}{\left(e^{10 \cdot \left(2 + 1\right)}\right)}}^{x}}\right)}^{x} \]

   7. metadata-eval98.6%

      \[\leadsto \cos x \cdot {\left(\sqrt[3]{{\left(e^{10 \cdot \color{blue}{3}}\right)}^{x}}\right)}^{x} \]

   8. metadata-eval98.6%

      \[\leadsto \cos x \cdot {\left(\sqrt[3]{{\left(e^{\color{blue}{30}}\right)}^{x}}\right)}^{x} \]

5. Applied egg-rr98.6%

   \[\leadsto \cos x \cdot {\color{blue}{\left(\sqrt[3]{{\left(e^{30}\right)}^{x}}\right)}}^{x} \]
6. Step-by-step derivation

   1. add-sqr-sqrt98.2%

      \[\leadsto \cos x \cdot {\color{blue}{\left(\sqrt{\sqrt[3]{{\left(e^{30}\right)}^{x}}} \cdot \sqrt{\sqrt[3]{{\left(e^{30}\right)}^{x}}}\right)}}^{x} \]

   2. sqrt-unprod98.6%

      \[\leadsto \cos x \cdot {\color{blue}{\left(\sqrt{\sqrt[3]{{\left(e^{30}\right)}^{x}} \cdot \sqrt[3]{{\left(e^{30}\right)}^{x}}}\right)}}^{x} \]

   3. cbrt-unprod98.6%

      \[\leadsto \cos x \cdot {\left(\sqrt{\color{blue}{\sqrt[3]{{\left(e^{30}\right)}^{x} \cdot {\left(e^{30}\right)}^{x}}}}\right)}^{x} \]

   4. pow-prod-down98.8%

      \[\leadsto \cos x \cdot {\left(\sqrt{\sqrt[3]{\color{blue}{{\left(e^{30} \cdot e^{30}\right)}^{x}}}}\right)}^{x} \]

   5. prod-exp99.0%

      \[\leadsto \cos x \cdot {\left(\sqrt{\sqrt[3]{{\color{blue}{\left(e^{30 + 30}\right)}}^{x}}}\right)}^{x} \]

   6. metadata-eval99.0%

      \[\leadsto \cos x \cdot {\left(\sqrt{\sqrt[3]{{\left(e^{\color{blue}{60}}\right)}^{x}}}\right)}^{x} \]

   7. add-sqr-sqrt99.0%

      \[\leadsto \cos x \cdot {\left(\sqrt{\sqrt[3]{\color{blue}{\sqrt{{\left(e^{60}\right)}^{x}} \cdot \sqrt{{\left(e^{60}\right)}^{x}}}}}\right)}^{x} \]

   8. cbrt-prod98.6%

      \[\leadsto \cos x \cdot {\left(\sqrt{\color{blue}{\sqrt[3]{\sqrt{{\left(e^{60}\right)}^{x}}} \cdot \sqrt[3]{\sqrt{{\left(e^{60}\right)}^{x}}}}}\right)}^{x} \]

   9. cbrt-prod99.0%

      \[\leadsto \cos x \cdot {\left(\sqrt{\color{blue}{\sqrt[3]{\sqrt{{\left(e^{60}\right)}^{x}} \cdot \sqrt{{\left(e^{60}\right)}^{x}}}}}\right)}^{x} \]

   10. add-sqr-sqrt99.0%

       \[\leadsto \cos x \cdot {\left(\sqrt{\sqrt[3]{\color{blue}{{\left(e^{60}\right)}^{x}}}}\right)}^{x} \]

7. Applied egg-rr99.0%

   \[\leadsto \cos x \cdot {\color{blue}{\left(\sqrt{\sqrt[3]{{\left(e^{60}\right)}^{x}}}\right)}}^{x} \]

8. Final simplification99.0%

   \[\leadsto \cos x \cdot {\left(\sqrt{\sqrt[3]{{\left(e^{60}\right)}^{x}}}\right)}^{x} \]

1. Initial program 94.5%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
2. Step-by-step derivation

   1. associate-*r*94.4%

      \[\leadsto \cos x \cdot e^{\color{blue}{\left(10 \cdot x\right) \cdot x}} \]

   2. add-log-exp94.4%

      \[\leadsto \cos x \cdot e^{\left(10 \cdot x\right) \cdot \color{blue}{\log \left(e^{x}\right)}} \]

   3. log-pow94.4%

      \[\leadsto \cos x \cdot e^{\color{blue}{\log \left({\left(e^{x}\right)}^{\left(10 \cdot x\right)}\right)}} \]

   4. pow-pow94.8%

      \[\leadsto \cos x \cdot e^{\log \color{blue}{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x}\right)}} \]

   5. add-exp-log96.8%

      \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{x}\right)}^{10}\right)}^{x}} \]

   6. add-cbrt-cube96.8%

      \[\leadsto \cos x \cdot \color{blue}{\sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}}} \]

   7. add-exp-log96.4%

      \[\leadsto \cos x \cdot \sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot \color{blue}{e^{\log \left({\left({\left(e^{x}\right)}^{10}\right)}^{x}\right)}}} \]

   8. pow-pow96.1%

      \[\leadsto \cos x \cdot \sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot e^{\log \color{blue}{\left({\left(e^{x}\right)}^{\left(10 \cdot x\right)}\right)}}} \]

   9. log-pow96.3%

      \[\leadsto \cos x \cdot \sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot e^{\color{blue}{\left(10 \cdot x\right) \cdot \log \left(e^{x}\right)}}} \]

   10. add-log-exp96.3%

       \[\leadsto \cos x \cdot \sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot e^{\left(10 \cdot x\right) \cdot \color{blue}{x}}} \]

   11. associate-*r*96.4%

       \[\leadsto \cos x \cdot \sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot e^{\color{blue}{10 \cdot \left(x \cdot x\right)}}} \]

   12. pow-exp96.9%

       \[\leadsto \cos x \cdot \sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot \color{blue}{{\left(e^{10}\right)}^{\left(x \cdot x\right)}}} \]

   13. *-commutative96.9%

       \[\leadsto \cos x \cdot \sqrt[3]{\color{blue}{{\left(e^{10}\right)}^{\left(x \cdot x\right)} \cdot \left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right)}} \]

3. Applied egg-rr98.7%

   \[\leadsto \cos x \cdot \color{blue}{\sqrt[3]{{\left({\left(e^{30}\right)}^{x}\right)}^{x}}} \]

4. Final simplification98.7%

   \[\leadsto \cos x \cdot \sqrt[3]{{\left({\left(e^{30}\right)}^{x}\right)}^{x}} \]

1. Initial program 94.5%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
2. Step-by-step derivation

   1. associate-*r*94.4%

      \[\leadsto \cos x \cdot e^{\color{blue}{\left(10 \cdot x\right) \cdot x}} \]

   2. exp-prod95.0%

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{10 \cdot x}\right)}^{x}} \]

   3. sqr-pow95.0%

      \[\leadsto \cos x \cdot \color{blue}{\left({\left(e^{10 \cdot x}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(e^{10 \cdot x}\right)}^{\left(\frac{x}{2}\right)}\right)} \]

   4. sqr-pow95.0%

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{10 \cdot x}\right)}^{x}} \]

   5. exp-prod98.0%

      \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{10}\right)}^{x}\right)}}^{x} \]

3. Simplified98.0%

   \[\leadsto \color{blue}{\cos x \cdot {\left({\left(e^{10}\right)}^{x}\right)}^{x}} \]

4. Final simplification98.0%

   \[\leadsto \cos x \cdot {\left({\left(e^{10}\right)}^{x}\right)}^{x} \]

1. Initial program 94.5%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
2. Step-by-step derivation

   1. associate-*r*94.4%

      \[\leadsto \cos x \cdot e^{\color{blue}{\left(10 \cdot x\right) \cdot x}} \]

   2. exp-prod95.0%

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{10 \cdot x}\right)}^{x}} \]

   3. sqr-pow95.0%

      \[\leadsto \cos x \cdot \color{blue}{\left({\left(e^{10 \cdot x}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(e^{10 \cdot x}\right)}^{\left(\frac{x}{2}\right)}\right)} \]

   4. sqr-pow95.0%

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{10 \cdot x}\right)}^{x}} \]

   5. exp-prod98.0%

      \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{10}\right)}^{x}\right)}}^{x} \]

3. Simplified98.0%

   \[\leadsto \color{blue}{\cos x \cdot {\left({\left(e^{10}\right)}^{x}\right)}^{x}} \]
4. Step-by-step derivation

   1. add-cbrt-cube98.1%

      \[\leadsto \cos x \cdot {\color{blue}{\left(\sqrt[3]{\left({\left(e^{10}\right)}^{x} \cdot {\left(e^{10}\right)}^{x}\right) \cdot {\left(e^{10}\right)}^{x}}\right)}}^{x} \]

   2. pow-prod-down98.0%

      \[\leadsto \cos x \cdot {\left(\sqrt[3]{\color{blue}{{\left(e^{10} \cdot e^{10}\right)}^{x}} \cdot {\left(e^{10}\right)}^{x}}\right)}^{x} \]

   3. pow-prod-down97.7%

      \[\leadsto \cos x \cdot {\left(\sqrt[3]{\color{blue}{{\left(\left(e^{10} \cdot e^{10}\right) \cdot e^{10}\right)}^{x}}}\right)}^{x} \]

   4. pow397.7%

      \[\leadsto \cos x \cdot {\left(\sqrt[3]{{\color{blue}{\left({\left(e^{10}\right)}^{3}\right)}}^{x}}\right)}^{x} \]

   5. metadata-eval97.7%

      \[\leadsto \cos x \cdot {\left(\sqrt[3]{{\left({\left(e^{10}\right)}^{\color{blue}{\left(2 + 1\right)}}\right)}^{x}}\right)}^{x} \]

   6. pow-exp98.6%

      \[\leadsto \cos x \cdot {\left(\sqrt[3]{{\color{blue}{\left(e^{10 \cdot \left(2 + 1\right)}\right)}}^{x}}\right)}^{x} \]

   7. metadata-eval98.6%

      \[\leadsto \cos x \cdot {\left(\sqrt[3]{{\left(e^{10 \cdot \color{blue}{3}}\right)}^{x}}\right)}^{x} \]

   8. metadata-eval98.6%

      \[\leadsto \cos x \cdot {\left(\sqrt[3]{{\left(e^{\color{blue}{30}}\right)}^{x}}\right)}^{x} \]

5. Applied egg-rr98.6%

   \[\leadsto \cos x \cdot {\color{blue}{\left(\sqrt[3]{{\left(e^{30}\right)}^{x}}\right)}}^{x} \]
6. Step-by-step derivation

   1. add-cbrt-cube98.4%

      \[\leadsto \cos x \cdot \color{blue}{\sqrt[3]{\left({\left(\sqrt[3]{{\left(e^{30}\right)}^{x}}\right)}^{x} \cdot {\left(\sqrt[3]{{\left(e^{30}\right)}^{x}}\right)}^{x}\right) \cdot {\left(\sqrt[3]{{\left(e^{30}\right)}^{x}}\right)}^{x}}} \]

   2. pow-prod-down98.4%

      \[\leadsto \cos x \cdot \sqrt[3]{\color{blue}{{\left(\sqrt[3]{{\left(e^{30}\right)}^{x}} \cdot \sqrt[3]{{\left(e^{30}\right)}^{x}}\right)}^{x}} \cdot {\left(\sqrt[3]{{\left(e^{30}\right)}^{x}}\right)}^{x}} \]

   3. cbrt-unprod98.5%

      \[\leadsto \cos x \cdot \sqrt[3]{{\color{blue}{\left(\sqrt[3]{{\left(e^{30}\right)}^{x} \cdot {\left(e^{30}\right)}^{x}}\right)}}^{x} \cdot {\left(\sqrt[3]{{\left(e^{30}\right)}^{x}}\right)}^{x}} \]

   4. pow-prod-down98.5%

      \[\leadsto \cos x \cdot \sqrt[3]{{\left(\sqrt[3]{\color{blue}{{\left(e^{30} \cdot e^{30}\right)}^{x}}}\right)}^{x} \cdot {\left(\sqrt[3]{{\left(e^{30}\right)}^{x}}\right)}^{x}} \]

   5. prod-exp98.7%

      \[\leadsto \cos x \cdot \sqrt[3]{{\left(\sqrt[3]{{\color{blue}{\left(e^{30 + 30}\right)}}^{x}}\right)}^{x} \cdot {\left(\sqrt[3]{{\left(e^{30}\right)}^{x}}\right)}^{x}} \]

   6. metadata-eval98.7%

      \[\leadsto \cos x \cdot \sqrt[3]{{\left(\sqrt[3]{{\left(e^{\color{blue}{60}}\right)}^{x}}\right)}^{x} \cdot {\left(\sqrt[3]{{\left(e^{30}\right)}^{x}}\right)}^{x}} \]

   7. add-sqr-sqrt98.8%

      \[\leadsto \cos x \cdot \sqrt[3]{{\left(\sqrt[3]{\color{blue}{\sqrt{{\left(e^{60}\right)}^{x}} \cdot \sqrt{{\left(e^{60}\right)}^{x}}}}\right)}^{x} \cdot {\left(\sqrt[3]{{\left(e^{30}\right)}^{x}}\right)}^{x}} \]

   8. cbrt-prod98.6%

      \[\leadsto \cos x \cdot \sqrt[3]{{\color{blue}{\left(\sqrt[3]{\sqrt{{\left(e^{60}\right)}^{x}}} \cdot \sqrt[3]{\sqrt{{\left(e^{60}\right)}^{x}}}\right)}}^{x} \cdot {\left(\sqrt[3]{{\left(e^{30}\right)}^{x}}\right)}^{x}} \]

7. Applied egg-rr98.7%

   \[\leadsto \cos x \cdot \color{blue}{\sqrt[3]{{\left({\left(e^{30}\right)}^{x}\right)}^{x}}} \]
8. Step-by-step derivation

   1. exp-prod95.6%

      \[\leadsto \cos x \cdot \sqrt[3]{{\color{blue}{\left(e^{30 \cdot x}\right)}}^{x}} \]

   2. *-commutative95.6%

      \[\leadsto \cos x \cdot \sqrt[3]{{\left(e^{\color{blue}{x \cdot 30}}\right)}^{x}} \]

   3. exp-prod95.3%

      \[\leadsto \cos x \cdot \sqrt[3]{\color{blue}{e^{\left(x \cdot 30\right) \cdot x}}} \]

   4. *-commutative95.3%

      \[\leadsto \cos x \cdot \sqrt[3]{e^{\color{blue}{x \cdot \left(x \cdot 30\right)}}} \]

   5. exp-prod95.4%

      \[\leadsto \cos x \cdot \sqrt[3]{\color{blue}{{\left(e^{x}\right)}^{\left(x \cdot 30\right)}}} \]

9. Simplified95.4%

   \[\leadsto \cos x \cdot \color{blue}{\sqrt[3]{{\left(e^{x}\right)}^{\left(x \cdot 30\right)}}} \]
10. Step-by-step derivation

    1. pow-exp95.3%

       \[\leadsto \cos x \cdot \sqrt[3]{\color{blue}{e^{x \cdot \left(x \cdot 30\right)}}} \]

11. Applied egg-rr95.3%

    \[\leadsto \cos x \cdot \sqrt[3]{\color{blue}{e^{x \cdot \left(x \cdot 30\right)}}} \]

12. Final simplification95.3%

    \[\leadsto \cos x \cdot \sqrt[3]{e^{x \cdot \left(x \cdot 30\right)}} \]

1. Initial program 94.5%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
2. Step-by-step derivation

   1. exp-prod95.2%

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{10}\right)}^{\left(x \cdot x\right)}} \]

3. Simplified95.2%

   \[\leadsto \color{blue}{\cos x \cdot {\left(e^{10}\right)}^{\left(x \cdot x\right)}} \]

4. Final simplification95.2%

   \[\leadsto \cos x \cdot {\left(e^{10}\right)}^{\left(x \cdot x\right)} \]

1. Initial program 94.5%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]

2. Final simplification94.5%

   \[\leadsto \cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]

1. Initial program 94.5%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
2. Step-by-step derivation

   1. *-commutative94.5%

      \[\leadsto \cos x \cdot e^{\color{blue}{\left(x \cdot x\right) \cdot 10}} \]

   2. exp-prod95.2%

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{x \cdot x}\right)}^{10}} \]

   3. exp-prod96.6%

      \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{x}\right)}^{x}\right)}}^{10} \]

   4. pow-unpow94.8%

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{x}\right)}^{\left(x \cdot 10\right)}} \]

   5. add-sqr-sqrt93.5%

      \[\leadsto \cos x \cdot {\left(e^{x}\right)}^{\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot 10\right)} \]

   6. associate-*l*93.5%

      \[\leadsto \cos x \cdot {\left(e^{x}\right)}^{\color{blue}{\left(\sqrt{x} \cdot \left(\sqrt{x} \cdot 10\right)\right)}} \]

   7. pow-unpow93.5%

      \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{x}\right)}^{\left(\sqrt{x}\right)}\right)}^{\left(\sqrt{x} \cdot 10\right)}} \]

3. Applied egg-rr93.5%

   \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{x}\right)}^{\left(\sqrt{x}\right)}\right)}^{\left(\sqrt{x} \cdot 10\right)}} \]

4. Taylor expanded in x around 0 9.7%

   \[\leadsto \color{blue}{1 + -0.5 \cdot {x}^{2}} \]
5. Step-by-step derivation

   1. *-commutative9.7%

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot -0.5} \]

   2. unpow29.7%

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

6. Simplified9.7%

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

7. Taylor expanded in x around inf 9.7%

   \[\leadsto \color{blue}{-0.5 \cdot {x}^{2}} \]
8. Step-by-step derivation

   1. *-commutative9.7%

      \[\leadsto \color{blue}{{x}^{2} \cdot -0.5} \]

   2. unpow29.7%

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

   3. associate-*r*9.7%

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

9. Simplified9.7%

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

10. Final simplification9.7%

    \[\leadsto x \cdot \left(x \cdot -0.5\right) \]

1. Initial program 94.5%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
2. Step-by-step derivation

   1. pow-exp95.2%

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{10}\right)}^{\left(x \cdot x\right)}} \]

   2. *-commutative95.2%

      \[\leadsto \color{blue}{{\left(e^{10}\right)}^{\left(x \cdot x\right)} \cdot \cos x} \]

   3. pow-exp94.5%

      \[\leadsto \color{blue}{e^{10 \cdot \left(x \cdot x\right)}} \cdot \cos x \]

   4. associate-*r*94.4%

      \[\leadsto e^{\color{blue}{\left(10 \cdot x\right) \cdot x}} \cdot \cos x \]

   5. add-log-exp94.4%

      \[\leadsto e^{\left(10 \cdot x\right) \cdot \color{blue}{\log \left(e^{x}\right)}} \cdot \cos x \]

   6. log-pow94.4%

      \[\leadsto e^{\color{blue}{\log \left({\left(e^{x}\right)}^{\left(10 \cdot x\right)}\right)}} \cdot \cos x \]

   7. pow-pow94.8%

      \[\leadsto e^{\log \color{blue}{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x}\right)}} \cdot \cos x \]

   8. add-exp-log96.8%

      \[\leadsto \color{blue}{{\left({\left(e^{x}\right)}^{10}\right)}^{x}} \cdot \cos x \]

   9. add-cbrt-cube96.8%

      \[\leadsto \color{blue}{\sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}}} \cdot \cos x \]

   10. add-cbrt-cube96.7%

       \[\leadsto \sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}} \cdot \color{blue}{\sqrt[3]{\left(\cos x \cdot \cos x\right) \cdot \cos x}} \]

   11. cbrt-unprod96.8%

       \[\leadsto \color{blue}{\sqrt[3]{\left(\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot \left(\left(\cos x \cdot \cos x\right) \cdot \cos x\right)}} \]

3. Applied egg-rr98.7%

   \[\leadsto \color{blue}{\sqrt[3]{{\left({\left(e^{30}\right)}^{x}\right)}^{x} \cdot {\cos x}^{3}}} \]
4. Step-by-step derivation

   1. add-sqr-sqrt98.7%

      \[\leadsto \sqrt[3]{{\color{blue}{\left(\sqrt{{\left(e^{30}\right)}^{x}} \cdot \sqrt{{\left(e^{30}\right)}^{x}}\right)}}^{x} \cdot {\cos x}^{3}} \]

   2. sqrt-unprod98.7%

      \[\leadsto \sqrt[3]{{\color{blue}{\left(\sqrt{{\left(e^{30}\right)}^{x} \cdot {\left(e^{30}\right)}^{x}}\right)}}^{x} \cdot {\cos x}^{3}} \]

   3. pow-prod-down98.8%

      \[\leadsto \sqrt[3]{{\left(\sqrt{\color{blue}{{\left(e^{30} \cdot e^{30}\right)}^{x}}}\right)}^{x} \cdot {\cos x}^{3}} \]

   4. prod-exp99.2%

      \[\leadsto \sqrt[3]{{\left(\sqrt{{\color{blue}{\left(e^{30 + 30}\right)}}^{x}}\right)}^{x} \cdot {\cos x}^{3}} \]

   5. metadata-eval99.2%

      \[\leadsto \sqrt[3]{{\left(\sqrt{{\left(e^{\color{blue}{60}}\right)}^{x}}\right)}^{x} \cdot {\cos x}^{3}} \]

5. Applied egg-rr99.2%

   \[\leadsto \sqrt[3]{{\color{blue}{\left(\sqrt{{\left(e^{60}\right)}^{x}}\right)}}^{x} \cdot {\cos x}^{3}} \]
6. Step-by-step derivation

   1. add-sqr-sqrt99.2%

      \[\leadsto \sqrt[3]{\color{blue}{\left(\sqrt{{\left(\sqrt{{\left(e^{60}\right)}^{x}}\right)}^{x}} \cdot \sqrt{{\left(\sqrt{{\left(e^{60}\right)}^{x}}\right)}^{x}}\right)} \cdot {\cos x}^{3}} \]

   2. sqrt-unprod99.2%

      \[\leadsto \sqrt[3]{\color{blue}{\sqrt{{\left(\sqrt{{\left(e^{60}\right)}^{x}}\right)}^{x} \cdot {\left(\sqrt{{\left(e^{60}\right)}^{x}}\right)}^{x}}} \cdot {\cos x}^{3}} \]

   3. pow-prod-down99.1%

      \[\leadsto \sqrt[3]{\sqrt{\color{blue}{{\left(\sqrt{{\left(e^{60}\right)}^{x}} \cdot \sqrt{{\left(e^{60}\right)}^{x}}\right)}^{x}}} \cdot {\cos x}^{3}} \]

   4. add-sqr-sqrt99.2%

      \[\leadsto \sqrt[3]{\sqrt{{\color{blue}{\left({\left(e^{60}\right)}^{x}\right)}}^{x}} \cdot {\cos x}^{3}} \]

7. Applied egg-rr99.2%

   \[\leadsto \sqrt[3]{\color{blue}{\sqrt{{\left({\left(e^{60}\right)}^{x}\right)}^{x}}} \cdot {\cos x}^{3}} \]

8. Taylor expanded in x around 0 1.5%

   \[\leadsto \color{blue}{1} \]

9. Final simplification1.5%

   \[\leadsto 1 \]

herbie shell --seed 2023167 
(FPCore (x)
  :name "ENA, Section 1.4, Exercise 1"
  :precision binary64
  :pre (and (<= 1.99 x) (<= x 2.01))
  (* (cos x) (exp (* 10.0 (* x x)))))