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-exp95.3%

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

   2. *-commutative95.3%

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

   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-pow94.3%

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

   7. pow-pow94.9%

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

   8. add-exp-log96.9%

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

   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-cube96.8%

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

   11. cbrt-unprod96.9%

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

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-sqrt98.6%

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

   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. Final simplification99.2%

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

1. Initial program 94.5%

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

   1. pow-exp95.3%

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

   2. *-commutative95.3%

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

   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-pow94.3%

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

   7. pow-pow94.9%

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

   8. add-exp-log96.9%

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

   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-cube96.8%

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

   11. cbrt-unprod96.9%

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

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-sqrt98.6%

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

   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-sqrt99.1%

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

   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-sqrt99.1%

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

7. Applied egg-rr99.1%

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

8. Final simplification99.1%

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

1. Initial program 94.5%

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

   1. pow-exp95.3%

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

   2. *-commutative95.3%

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

   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-pow94.3%

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

   7. pow-pow94.9%

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

   8. add-exp-log96.9%

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

   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-cube96.8%

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

   11. cbrt-unprod96.9%

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

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-sqrt98.6%

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

   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. exp-to-pow93.9%

      \[\leadsto \cos x \cdot {\left(1 \cdot \color{blue}{{\left(\sqrt{{\left(e^{60}\right)}^{x}}\right)}^{x}}\right)}^{0.3333333333333333} \]

   2. *-lft-identity93.9%

      \[\leadsto \cos x \cdot {\color{blue}{\left({\left(\sqrt{{\left(e^{60}\right)}^{x}}\right)}^{x}\right)}}^{0.3333333333333333} \]

   3. unpow1/399.1%

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

8. Simplified99.1%

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

9. Final simplification99.1%

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

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. add-log-exp94.4%

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

   3. log-pow94.3%

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

   4. pow-pow94.9%

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

   5. add-exp-log96.9%

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

   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-log96.5%

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

   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-pow96.2%

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

   10. add-log-exp96.2%

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

   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-exp96.8%

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

   13. *-commutative96.8%

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

3. Applied egg-rr98.6%

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

4. Final simplification98.6%

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

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-prod94.9%

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

   3. sqr-pow94.8%

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

   4. sqr-pow94.9%

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

   5. exp-prod97.9%

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

3. Simplified97.9%

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

4. Final simplification97.9%

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

1. Initial program 94.5%

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

   1. *-un-lft-identity94.5%

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

   2. exp-prod94.0%

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

   3. pow-unpow94.0%

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

3. Applied egg-rr94.0%

   \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{1}\right)}^{10}\right)}^{\left(x \cdot x\right)}} \]
4. Step-by-step derivation

   1. exp-1-e94.0%

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

5. Simplified94.0%

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

   1. e-exp-194.0%

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

   2. pow-exp95.3%

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

   3. metadata-eval95.3%

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

   4. add-sqr-sqrt95.2%

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

   5. unpow-prod-down95.3%

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

   6. metadata-eval95.3%

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

   7. pow-exp94.9%

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

   8. e-exp-194.9%

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

   9. sqrt-pow194.9%

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

   10. e-exp-194.9%

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

   11. pow-exp95.3%

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

   12. metadata-eval95.3%

       \[\leadsto \cos x \cdot \left({\left(e^{1 \cdot \color{blue}{5}}\right)}^{\left(x \cdot x\right)} \cdot {\left(\sqrt{e^{10}}\right)}^{\left(x \cdot x\right)}\right) \]

   13. metadata-eval95.3%

       \[\leadsto \cos x \cdot \left({\left(e^{\color{blue}{5}}\right)}^{\left(x \cdot x\right)} \cdot {\left(\sqrt{e^{10}}\right)}^{\left(x \cdot x\right)}\right) \]

   14. metadata-eval95.3%

       \[\leadsto \cos x \cdot \left({\left(e^{5}\right)}^{\left(x \cdot x\right)} \cdot {\left(\sqrt{e^{\color{blue}{1 \cdot 10}}}\right)}^{\left(x \cdot x\right)}\right) \]

   15. pow-exp94.9%

       \[\leadsto \cos x \cdot \left({\left(e^{5}\right)}^{\left(x \cdot x\right)} \cdot {\left(\sqrt{\color{blue}{{\left(e^{1}\right)}^{10}}}\right)}^{\left(x \cdot x\right)}\right) \]

   16. e-exp-194.9%

       \[\leadsto \cos x \cdot \left({\left(e^{5}\right)}^{\left(x \cdot x\right)} \cdot {\left(\sqrt{{\color{blue}{e}}^{10}}\right)}^{\left(x \cdot x\right)}\right) \]

   17. sqrt-pow194.9%

       \[\leadsto \cos x \cdot \left({\left(e^{5}\right)}^{\left(x \cdot x\right)} \cdot {\color{blue}{\left({e}^{\left(\frac{10}{2}\right)}\right)}}^{\left(x \cdot x\right)}\right) \]

   18. e-exp-194.9%

       \[\leadsto \cos x \cdot \left({\left(e^{5}\right)}^{\left(x \cdot x\right)} \cdot {\left({\color{blue}{\left(e^{1}\right)}}^{\left(\frac{10}{2}\right)}\right)}^{\left(x \cdot x\right)}\right) \]

   19. pow-exp95.3%

       \[\leadsto \cos x \cdot \left({\left(e^{5}\right)}^{\left(x \cdot x\right)} \cdot {\color{blue}{\left(e^{1 \cdot \frac{10}{2}}\right)}}^{\left(x \cdot x\right)}\right) \]

   20. metadata-eval95.3%

       \[\leadsto \cos x \cdot \left({\left(e^{5}\right)}^{\left(x \cdot x\right)} \cdot {\left(e^{1 \cdot \color{blue}{5}}\right)}^{\left(x \cdot x\right)}\right) \]

   21. metadata-eval95.3%

       \[\leadsto \cos x \cdot \left({\left(e^{5}\right)}^{\left(x \cdot x\right)} \cdot {\left(e^{\color{blue}{5}}\right)}^{\left(x \cdot x\right)}\right) \]

7. Applied egg-rr95.3%

   \[\leadsto \cos x \cdot \color{blue}{\left({\left(e^{5}\right)}^{\left(x \cdot x\right)} \cdot {\left(e^{5}\right)}^{\left(x \cdot x\right)}\right)} \]
8. Step-by-step derivation

   1. unpow295.3%

      \[\leadsto \cos x \cdot \left({\left(e^{5}\right)}^{\color{blue}{\left({x}^{2}\right)}} \cdot {\left(e^{5}\right)}^{\left(x \cdot x\right)}\right) \]

   2. unpow295.3%

      \[\leadsto \cos x \cdot \left({\left(e^{5}\right)}^{\left({x}^{2}\right)} \cdot {\left(e^{5}\right)}^{\color{blue}{\left({x}^{2}\right)}}\right) \]

   3. pow-sqr95.3%

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{5}\right)}^{\left(2 \cdot {x}^{2}\right)}} \]

   4. unpow295.3%

      \[\leadsto \cos x \cdot {\left(e^{5}\right)}^{\left(2 \cdot \color{blue}{\left(x \cdot x\right)}\right)} \]

9. Simplified95.3%

   \[\leadsto \cos x \cdot \color{blue}{{\left(e^{5}\right)}^{\left(2 \cdot \left(x \cdot x\right)\right)}} \]

10. Final simplification95.3%

    \[\leadsto \cos x \cdot {\left(e^{5}\right)}^{\left(2 \cdot \left(x \cdot x\right)\right)} \]

1. Initial program 94.5%

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

   1. exp-prod95.3%

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

3. Simplified95.3%

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

4. Final simplification95.3%

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

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. Taylor expanded in x around 0 18.2%

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

   1. unpow218.2%

      \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

4. Simplified18.2%

   \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(x \cdot x\right)\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]

5. Final simplification18.2%

   \[\leadsto e^{10 \cdot \left(x \cdot x\right)} \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right) \]

1. Initial program 94.5%

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

   1. 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-pow94.3%

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

   4. pow-pow94.9%

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

   5. add-exp-log96.9%

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

   6. pow-pow94.8%

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

   7. add-sqr-sqrt93.0%

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

   8. pow-unpow93.2%

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

3. Applied egg-rr93.2%

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

   1. metadata-eval93.2%

      \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{\left(\sqrt{\color{blue}{\left(2 \cdot 5\right)} \cdot x}\right)}\right)}^{\left(\sqrt{10 \cdot x}\right)} \]

   2. associate-*r*93.2%

      \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{\left(\sqrt{\color{blue}{2 \cdot \left(5 \cdot x\right)}}\right)}\right)}^{\left(\sqrt{10 \cdot x}\right)} \]

   3. metadata-eval93.2%

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

   4. associate-/r/93.5%

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

   5. associate-*r/93.5%

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

   6. metadata-eval93.5%

      \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{\left(\sqrt{\frac{\color{blue}{20}}{\frac{2}{x}}}\right)}\right)}^{\left(\sqrt{10 \cdot x}\right)} \]

   7. associate-/l*93.2%

      \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{\left(\sqrt{\color{blue}{\frac{20 \cdot x}{2}}}\right)}\right)}^{\left(\sqrt{10 \cdot x}\right)} \]

   8. *-commutative93.2%

      \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{\left(\sqrt{\frac{\color{blue}{x \cdot 20}}{2}}\right)}\right)}^{\left(\sqrt{10 \cdot x}\right)} \]

   9. associate-/l*93.7%

      \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{\left(\sqrt{\color{blue}{\frac{x}{\frac{2}{20}}}}\right)}\right)}^{\left(\sqrt{10 \cdot x}\right)} \]

   10. metadata-eval93.7%

       \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{\left(\sqrt{\frac{x}{\color{blue}{0.1}}}\right)}\right)}^{\left(\sqrt{10 \cdot x}\right)} \]

   11. metadata-eval93.7%

       \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{\left(\sqrt{\frac{x}{0.1}}\right)}\right)}^{\left(\sqrt{\color{blue}{\left(2 \cdot 5\right)} \cdot x}\right)} \]

   12. associate-*r*93.7%

       \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{\left(\sqrt{\frac{x}{0.1}}\right)}\right)}^{\left(\sqrt{\color{blue}{2 \cdot \left(5 \cdot x\right)}}\right)} \]

   13. metadata-eval93.7%

       \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{\left(\sqrt{\frac{x}{0.1}}\right)}\right)}^{\left(\sqrt{2 \cdot \left(\color{blue}{\frac{10}{2}} \cdot x\right)}\right)} \]

   14. associate-/r/93.7%

       \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{\left(\sqrt{\frac{x}{0.1}}\right)}\right)}^{\left(\sqrt{2 \cdot \color{blue}{\frac{10}{\frac{2}{x}}}}\right)} \]

   15. associate-*r/93.7%

       \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{\left(\sqrt{\frac{x}{0.1}}\right)}\right)}^{\left(\sqrt{\color{blue}{\frac{2 \cdot 10}{\frac{2}{x}}}}\right)} \]

   16. metadata-eval93.7%

       \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{\left(\sqrt{\frac{x}{0.1}}\right)}\right)}^{\left(\sqrt{\frac{\color{blue}{20}}{\frac{2}{x}}}\right)} \]

   17. associate-/l*93.7%

       \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{\left(\sqrt{\frac{x}{0.1}}\right)}\right)}^{\left(\sqrt{\color{blue}{\frac{20 \cdot x}{2}}}\right)} \]

   18. *-commutative93.7%

       \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{\left(\sqrt{\frac{x}{0.1}}\right)}\right)}^{\left(\sqrt{\frac{\color{blue}{x \cdot 20}}{2}}\right)} \]

   19. associate-/l*93.1%

       \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{\left(\sqrt{\frac{x}{0.1}}\right)}\right)}^{\left(\sqrt{\color{blue}{\frac{x}{\frac{2}{20}}}}\right)} \]

   20. metadata-eval93.1%

       \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{\left(\sqrt{\frac{x}{0.1}}\right)}\right)}^{\left(\sqrt{\frac{x}{\color{blue}{0.1}}}\right)} \]

5. Simplified93.1%

   \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{x}\right)}^{\left(\sqrt{\frac{x}{0.1}}\right)}\right)}^{\left(\sqrt{\frac{x}{0.1}}\right)}} \]

6. Taylor expanded in x around 0 9.7%

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

   1. *-commutative9.7%

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

   2. unpow29.7%

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

8. Simplified9.7%

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

9. Taylor expanded in x around inf 9.7%

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

    1. *-commutative9.7%

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

    2. unpow29.7%

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

11. Simplified9.7%

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

12. Final simplification9.7%

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

1. Initial program 94.5%

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

   1. pow-exp95.3%

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

   2. *-commutative95.3%

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

   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-pow94.3%

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

   7. pow-pow94.9%

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

   8. add-exp-log96.9%

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

   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-cube96.8%

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

   11. cbrt-unprod96.9%

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

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-sqrt98.6%

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

   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-sqrt99.1%

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

   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-sqrt99.1%

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

7. Applied egg-rr99.1%

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

8. Taylor expanded in x around 0 1.5%

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

9. Final simplification1.5%

   \[\leadsto 1 \]

herbie shell --seed 2023166 
(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)))))