Falkner and Boettcher, Equation (20:1,3)

1. Initial program 99.4%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
2. Step-by-step derivation

   1. *-commutative99.4%

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(1 - v \cdot v\right) \cdot \left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)}} \]

   2. associate-*l*99.5%

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(1 - v \cdot v\right) \cdot \color{blue}{\left(\pi \cdot \left(t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)\right)}} \]

   3. associate-*r*99.5%

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\left(1 - v \cdot v\right) \cdot \pi\right) \cdot \left(t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)}} \]

   4. associate-/r*99.6%

      \[\leadsto \color{blue}{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(1 - v \cdot v\right) \cdot \pi}}{t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}} \]

3. Simplified99.6%

   \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\pi \cdot \left(1 - v \cdot v\right)}}{t \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}}} \]

4. Final simplification99.6%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(v, v \cdot -5, 1\right)}{\pi \cdot \left(1 - v \cdot v\right)}}{t \cdot \sqrt{\mathsf{fma}\left(v \cdot v, -6, 2\right)}} \]

1. Initial program 99.4%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
2. Step-by-step derivation

   1. associate-*l*99.4%

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\pi \cdot t\right) \cdot \left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(1 - v \cdot v\right)\right)}} \]

   2. associate-/r*99.5%

      \[\leadsto \color{blue}{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi \cdot t}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(1 - v \cdot v\right)}} \]

   3. associate-/l/99.5%

      \[\leadsto \color{blue}{\frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}} \]

   4. sub-neg99.5%

      \[\leadsto \frac{\frac{\frac{\color{blue}{1 + \left(-5 \cdot \left(v \cdot v\right)\right)}}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \]

   5. +-commutative99.5%

      \[\leadsto \frac{\frac{\frac{\color{blue}{\left(-5 \cdot \left(v \cdot v\right)\right) + 1}}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \]

   6. *-commutative99.5%

      \[\leadsto \frac{\frac{\frac{\left(-\color{blue}{\left(v \cdot v\right) \cdot 5}\right) + 1}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \]

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

      \[\leadsto \frac{\frac{\frac{\color{blue}{\left(v \cdot v\right) \cdot \left(-5\right)} + 1}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \]

   8. fma-def99.5%

      \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left(v \cdot v, -5, 1\right)}}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \]

   9. metadata-eval99.5%

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(v \cdot v, \color{blue}{-5}, 1\right)}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \]

   10. sub-neg99.5%

       \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 \cdot \color{blue}{\left(1 + \left(-3 \cdot \left(v \cdot v\right)\right)\right)}}} \]

   11. distribute-rgt-in99.5%

       \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{\color{blue}{1 \cdot 2 + \left(-3 \cdot \left(v \cdot v\right)\right) \cdot 2}}} \]

3. Simplified99.5%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}} \]

4. Final simplification99.5%

   \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}} \]

1. Initial program 99.4%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
2. Step-by-step derivation

   1. expm1-log1p-u71.4%

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)\right)} \cdot \left(1 - v \cdot v\right)} \]

   2. expm1-udef25.3%

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(e^{\mathsf{log1p}\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)} - 1\right)} \cdot \left(1 - v \cdot v\right)} \]

   3. cancel-sign-sub-inv25.3%

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(e^{\mathsf{log1p}\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \color{blue}{\left(1 + \left(-3\right) \cdot \left(v \cdot v\right)\right)}}\right)} - 1\right) \cdot \left(1 - v \cdot v\right)} \]

   4. metadata-eval25.3%

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(e^{\mathsf{log1p}\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 + \color{blue}{-3} \cdot \left(v \cdot v\right)\right)}\right)} - 1\right) \cdot \left(1 - v \cdot v\right)} \]

3. Applied egg-rr25.3%

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(e^{\mathsf{log1p}\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 + -3 \cdot \left(v \cdot v\right)\right)}\right)} - 1\right)} \cdot \left(1 - v \cdot v\right)} \]
4. Step-by-step derivation

   1. expm1-def71.4%

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 + -3 \cdot \left(v \cdot v\right)\right)}\right)\right)} \cdot \left(1 - v \cdot v\right)} \]

   2. expm1-log1p99.4%

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 + -3 \cdot \left(v \cdot v\right)\right)}\right)} \cdot \left(1 - v \cdot v\right)} \]

   3. associate-*l*99.5%

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\pi \cdot \left(t \cdot \sqrt{2 \cdot \left(1 + -3 \cdot \left(v \cdot v\right)\right)}\right)\right)} \cdot \left(1 - v \cdot v\right)} \]

   4. unpow299.5%

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\pi \cdot \left(t \cdot \sqrt{2 \cdot \left(1 + -3 \cdot \color{blue}{{v}^{2}}\right)}\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   5. *-commutative99.5%

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\pi \cdot \left(t \cdot \sqrt{2 \cdot \left(1 + \color{blue}{{v}^{2} \cdot -3}\right)}\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   6. unpow299.5%

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\pi \cdot \left(t \cdot \sqrt{2 \cdot \left(1 + \color{blue}{\left(v \cdot v\right)} \cdot -3\right)}\right)\right) \cdot \left(1 - v \cdot v\right)} \]

5. Simplified99.5%

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\pi \cdot \left(t \cdot \sqrt{2 \cdot \left(1 + \left(v \cdot v\right) \cdot -3\right)}\right)\right)} \cdot \left(1 - v \cdot v\right)} \]

6. Final simplification99.5%

   \[\leadsto \frac{1 - \left(v \cdot v\right) \cdot 5}{\left(1 - v \cdot v\right) \cdot \left(\pi \cdot \left(t \cdot \sqrt{2 \cdot \left(1 + \left(v \cdot v\right) \cdot -3\right)}\right)\right)} \]

1. Initial program 99.4%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
2. Step-by-step derivation

   1. expm1-log1p-u71.4%

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)\right)} \cdot \left(1 - v \cdot v\right)} \]

   2. expm1-udef25.3%

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(e^{\mathsf{log1p}\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)} - 1\right)} \cdot \left(1 - v \cdot v\right)} \]

   3. cancel-sign-sub-inv25.3%

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(e^{\mathsf{log1p}\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \color{blue}{\left(1 + \left(-3\right) \cdot \left(v \cdot v\right)\right)}}\right)} - 1\right) \cdot \left(1 - v \cdot v\right)} \]

   4. metadata-eval25.3%

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(e^{\mathsf{log1p}\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 + \color{blue}{-3} \cdot \left(v \cdot v\right)\right)}\right)} - 1\right) \cdot \left(1 - v \cdot v\right)} \]

3. Applied egg-rr25.3%

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(e^{\mathsf{log1p}\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 + -3 \cdot \left(v \cdot v\right)\right)}\right)} - 1\right)} \cdot \left(1 - v \cdot v\right)} \]
4. Step-by-step derivation

   1. expm1-def71.4%

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 + -3 \cdot \left(v \cdot v\right)\right)}\right)\right)} \cdot \left(1 - v \cdot v\right)} \]

   2. expm1-log1p99.4%

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 + -3 \cdot \left(v \cdot v\right)\right)}\right)} \cdot \left(1 - v \cdot v\right)} \]

   3. associate-*l*99.5%

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\pi \cdot \left(t \cdot \sqrt{2 \cdot \left(1 + -3 \cdot \left(v \cdot v\right)\right)}\right)\right)} \cdot \left(1 - v \cdot v\right)} \]

   4. unpow299.5%

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\pi \cdot \left(t \cdot \sqrt{2 \cdot \left(1 + -3 \cdot \color{blue}{{v}^{2}}\right)}\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   5. *-commutative99.5%

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\pi \cdot \left(t \cdot \sqrt{2 \cdot \left(1 + \color{blue}{{v}^{2} \cdot -3}\right)}\right)\right) \cdot \left(1 - v \cdot v\right)} \]

   6. unpow299.5%

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\pi \cdot \left(t \cdot \sqrt{2 \cdot \left(1 + \color{blue}{\left(v \cdot v\right)} \cdot -3\right)}\right)\right) \cdot \left(1 - v \cdot v\right)} \]

5. Simplified99.5%

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\pi \cdot \left(t \cdot \sqrt{2 \cdot \left(1 + \left(v \cdot v\right) \cdot -3\right)}\right)\right)} \cdot \left(1 - v \cdot v\right)} \]

6. Taylor expanded in v around 0 98.3%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{2} \cdot \left(t \cdot \pi\right)}} \]
7. Step-by-step derivation

   1. *-commutative98.3%

      \[\leadsto \frac{1}{\color{blue}{\left(t \cdot \pi\right) \cdot \sqrt{2}}} \]

   2. associate-/r*98.4%

      \[\leadsto \color{blue}{\frac{\frac{1}{t \cdot \pi}}{\sqrt{2}}} \]

8. Simplified98.4%

   \[\leadsto \color{blue}{\frac{\frac{1}{t \cdot \pi}}{\sqrt{2}}} \]
9. Step-by-step derivation

   1. frac-2neg98.4%

      \[\leadsto \color{blue}{\frac{-\frac{1}{t \cdot \pi}}{-\sqrt{2}}} \]

   2. div-inv98.2%

      \[\leadsto \color{blue}{\left(-\frac{1}{t \cdot \pi}\right) \cdot \frac{1}{-\sqrt{2}}} \]

   3. clear-num98.2%

      \[\leadsto \left(-\color{blue}{\frac{1}{\frac{t \cdot \pi}{1}}}\right) \cdot \frac{1}{-\sqrt{2}} \]

   4. distribute-neg-frac98.2%

      \[\leadsto \color{blue}{\frac{-1}{\frac{t \cdot \pi}{1}}} \cdot \frac{1}{-\sqrt{2}} \]

   5. metadata-eval98.2%

      \[\leadsto \frac{\color{blue}{-1}}{\frac{t \cdot \pi}{1}} \cdot \frac{1}{-\sqrt{2}} \]

   6. /-rgt-identity98.2%

      \[\leadsto \frac{-1}{\color{blue}{t \cdot \pi}} \cdot \frac{1}{-\sqrt{2}} \]

10. Applied egg-rr98.2%

    \[\leadsto \color{blue}{\frac{-1}{t \cdot \pi} \cdot \frac{1}{-\sqrt{2}}} \]
11. Step-by-step derivation

    1. associate-*l/98.4%

       \[\leadsto \color{blue}{\frac{-1 \cdot \frac{1}{-\sqrt{2}}}{t \cdot \pi}} \]

    2. *-commutative98.4%

       \[\leadsto \frac{-1 \cdot \frac{1}{-\sqrt{2}}}{\color{blue}{\pi \cdot t}} \]

    3. associate-/r*98.7%

       \[\leadsto \color{blue}{\frac{\frac{-1 \cdot \frac{1}{-\sqrt{2}}}{\pi}}{t}} \]

    4. mul-1-neg98.7%

       \[\leadsto \frac{\frac{\color{blue}{-\frac{1}{-\sqrt{2}}}}{\pi}}{t} \]

    5. neg-mul-198.7%

       \[\leadsto \frac{\frac{-\frac{1}{\color{blue}{-1 \cdot \sqrt{2}}}}{\pi}}{t} \]

    6. associate-/r*98.7%

       \[\leadsto \frac{\frac{-\color{blue}{\frac{\frac{1}{-1}}{\sqrt{2}}}}{\pi}}{t} \]

    7. metadata-eval98.7%

       \[\leadsto \frac{\frac{-\frac{\color{blue}{-1}}{\sqrt{2}}}{\pi}}{t} \]

12. Simplified98.7%

    \[\leadsto \color{blue}{\frac{\frac{-\frac{-1}{\sqrt{2}}}{\pi}}{t}} \]

13. Final simplification98.7%

    \[\leadsto \frac{\frac{\frac{--1}{\sqrt{2}}}{\pi}}{t} \]

1. Initial program 99.4%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
2. Step-by-step derivation

   1. associate-*l*99.4%

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\pi \cdot t\right) \cdot \left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(1 - v \cdot v\right)\right)}} \]

   2. associate-/r*99.5%

      \[\leadsto \color{blue}{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi \cdot t}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(1 - v \cdot v\right)}} \]

   3. associate-/l/99.5%

      \[\leadsto \color{blue}{\frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}} \]

   4. sub-neg99.5%

      \[\leadsto \frac{\frac{\frac{\color{blue}{1 + \left(-5 \cdot \left(v \cdot v\right)\right)}}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \]

   5. +-commutative99.5%

      \[\leadsto \frac{\frac{\frac{\color{blue}{\left(-5 \cdot \left(v \cdot v\right)\right) + 1}}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \]

   6. *-commutative99.5%

      \[\leadsto \frac{\frac{\frac{\left(-\color{blue}{\left(v \cdot v\right) \cdot 5}\right) + 1}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \]

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

      \[\leadsto \frac{\frac{\frac{\color{blue}{\left(v \cdot v\right) \cdot \left(-5\right)} + 1}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \]

   8. fma-def99.5%

      \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left(v \cdot v, -5, 1\right)}}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \]

   9. metadata-eval99.5%

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(v \cdot v, \color{blue}{-5}, 1\right)}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \]

   10. sub-neg99.5%

       \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 \cdot \color{blue}{\left(1 + \left(-3 \cdot \left(v \cdot v\right)\right)\right)}}} \]

   11. distribute-rgt-in99.5%

       \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{\color{blue}{1 \cdot 2 + \left(-3 \cdot \left(v \cdot v\right)\right) \cdot 2}}} \]

3. Simplified99.5%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}} \]

4. Taylor expanded in v around 0 97.9%

   \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{t \cdot \pi}} \]
5. Step-by-step derivation

   1. div-inv97.9%

      \[\leadsto \color{blue}{\sqrt{0.5} \cdot \frac{1}{t \cdot \pi}} \]

6. Applied egg-rr97.9%

   \[\leadsto \color{blue}{\sqrt{0.5} \cdot \frac{1}{t \cdot \pi}} \]

7. Final simplification97.9%

   \[\leadsto \sqrt{0.5} \cdot \frac{1}{\pi \cdot t} \]

1. Initial program 99.4%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]

2. Taylor expanded in v around 0 98.3%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{2} \cdot \left(t \cdot \pi\right)}} \]
3. Step-by-step derivation

   1. associate-*r*98.4%

      \[\leadsto \frac{1}{\color{blue}{\left(\sqrt{2} \cdot t\right) \cdot \pi}} \]

   2. *-commutative98.4%

      \[\leadsto \frac{1}{\color{blue}{\left(t \cdot \sqrt{2}\right)} \cdot \pi} \]

   3. associate-*l*98.4%

      \[\leadsto \frac{1}{\color{blue}{t \cdot \left(\sqrt{2} \cdot \pi\right)}} \]

4. Simplified98.4%

   \[\leadsto \color{blue}{\frac{1}{t \cdot \left(\sqrt{2} \cdot \pi\right)}} \]

5. Final simplification98.4%

   \[\leadsto \frac{1}{t \cdot \left(\pi \cdot \sqrt{2}\right)} \]

1. Initial program 99.4%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
2. Step-by-step derivation

   1. associate-*l*99.4%

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\pi \cdot t\right) \cdot \left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(1 - v \cdot v\right)\right)}} \]

   2. associate-/r*99.5%

      \[\leadsto \color{blue}{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi \cdot t}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(1 - v \cdot v\right)}} \]

   3. associate-/l/99.5%

      \[\leadsto \color{blue}{\frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}} \]

   4. sub-neg99.5%

      \[\leadsto \frac{\frac{\frac{\color{blue}{1 + \left(-5 \cdot \left(v \cdot v\right)\right)}}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \]

   5. +-commutative99.5%

      \[\leadsto \frac{\frac{\frac{\color{blue}{\left(-5 \cdot \left(v \cdot v\right)\right) + 1}}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \]

   6. *-commutative99.5%

      \[\leadsto \frac{\frac{\frac{\left(-\color{blue}{\left(v \cdot v\right) \cdot 5}\right) + 1}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \]

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

      \[\leadsto \frac{\frac{\frac{\color{blue}{\left(v \cdot v\right) \cdot \left(-5\right)} + 1}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \]

   8. fma-def99.5%

      \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left(v \cdot v, -5, 1\right)}}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \]

   9. metadata-eval99.5%

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(v \cdot v, \color{blue}{-5}, 1\right)}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \]

   10. sub-neg99.5%

       \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 \cdot \color{blue}{\left(1 + \left(-3 \cdot \left(v \cdot v\right)\right)\right)}}} \]

   11. distribute-rgt-in99.5%

       \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{\color{blue}{1 \cdot 2 + \left(-3 \cdot \left(v \cdot v\right)\right) \cdot 2}}} \]

3. Simplified99.5%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}} \]

4. Taylor expanded in v around 0 97.9%

   \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{t \cdot \pi}} \]

5. Final simplification97.9%

   \[\leadsto \frac{\sqrt{0.5}}{\pi \cdot t} \]

1. Initial program 99.4%

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)} \]
2. Step-by-step derivation

   1. associate-*l*99.4%

      \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\pi \cdot t\right) \cdot \left(\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(1 - v \cdot v\right)\right)}} \]

   2. associate-/r*99.5%

      \[\leadsto \color{blue}{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi \cdot t}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)} \cdot \left(1 - v \cdot v\right)}} \]

   3. associate-/l/99.5%

      \[\leadsto \color{blue}{\frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}}} \]

   4. sub-neg99.5%

      \[\leadsto \frac{\frac{\frac{\color{blue}{1 + \left(-5 \cdot \left(v \cdot v\right)\right)}}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \]

   5. +-commutative99.5%

      \[\leadsto \frac{\frac{\frac{\color{blue}{\left(-5 \cdot \left(v \cdot v\right)\right) + 1}}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \]

   6. *-commutative99.5%

      \[\leadsto \frac{\frac{\frac{\left(-\color{blue}{\left(v \cdot v\right) \cdot 5}\right) + 1}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \]

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

      \[\leadsto \frac{\frac{\frac{\color{blue}{\left(v \cdot v\right) \cdot \left(-5\right)} + 1}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \]

   8. fma-def99.5%

      \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left(v \cdot v, -5, 1\right)}}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \]

   9. metadata-eval99.5%

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(v \cdot v, \color{blue}{-5}, 1\right)}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}} \]

   10. sub-neg99.5%

       \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 \cdot \color{blue}{\left(1 + \left(-3 \cdot \left(v \cdot v\right)\right)\right)}}} \]

   11. distribute-rgt-in99.5%

       \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{\color{blue}{1 \cdot 2 + \left(-3 \cdot \left(v \cdot v\right)\right) \cdot 2}}} \]

3. Simplified99.5%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\mathsf{fma}\left(v \cdot v, -5, 1\right)}{\pi \cdot t}}{1 - v \cdot v}}{\sqrt{2 + \left(v \cdot v\right) \cdot -6}}} \]

4. Taylor expanded in v around 0 97.9%

   \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{t \cdot \pi}} \]
5. Step-by-step derivation

   1. div-inv97.9%

      \[\leadsto \color{blue}{\sqrt{0.5} \cdot \frac{1}{t \cdot \pi}} \]

6. Applied egg-rr97.9%

   \[\leadsto \color{blue}{\sqrt{0.5} \cdot \frac{1}{t \cdot \pi}} \]

7. Taylor expanded in t around 0 97.9%

   \[\leadsto \color{blue}{\frac{\sqrt{0.5}}{t \cdot \pi}} \]
8. Step-by-step derivation

   1. associate-/r*97.9%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{0.5}}{t}}{\pi}} \]

9. Simplified97.9%

   \[\leadsto \color{blue}{\frac{\frac{\sqrt{0.5}}{t}}{\pi}} \]

10. Final simplification97.9%

    \[\leadsto \frac{\frac{\sqrt{0.5}}{t}}{\pi} \]

herbie shell --seed 2023166 
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  :precision binary64
  (/ (- 1.0 (* 5.0 (* v v))) (* (* (* PI t) (sqrt (* 2.0 (- 1.0 (* 3.0 (* v v)))))) (- 1.0 (* v v)))))