quad2p (problem 3.2.1, positive)

1. Split input into 4 regimes

2. if b_2 < -4.4000000000000001e101

   1. Initial program 57.6%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative57.6%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg57.6%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified57.6%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around -inf 91.5%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]

   if -4.4000000000000001e101 < b_2 < 1.95000000000000005e-129

   1. Initial program 79.9%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative79.9%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg79.9%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified79.9%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   if 1.95000000000000005e-129 < b_2 < 1.46e-72 or 1.1e-21 < b_2

   1. Initial program 13.8%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative13.8%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg13.8%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified13.8%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around inf 94.4%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]

   if 1.46e-72 < b_2 < 1.1e-21

   1. Initial program 70.5%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative70.5%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg70.5%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified70.5%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
   4. Step-by-step derivation

      1. add-cbrt-cube62.1%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}} - b_2}{a} \]

      2. pow362.3%

         \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}^{3}}} - b_2}{a} \]

      3. pow1/357.5%

         \[\leadsto \frac{\color{blue}{{\left({\left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}^{3}\right)}^{0.3333333333333333}} - b_2}{a} \]

      4. sqrt-pow257.5%

         \[\leadsto \frac{{\color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(\frac{3}{2}\right)}\right)}}^{0.3333333333333333} - b_2}{a} \]

      5. fma-neg57.5%

         \[\leadsto \frac{{\left({\color{blue}{\left(\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)\right)}}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

      6. *-commutative57.5%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{c \cdot a}\right)\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

      7. distribute-rgt-neg-in57.5%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{c \cdot \left(-a\right)}\right)\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

      8. metadata-eval57.5%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} - b_2}{a} \]

   5. Applied egg-rr57.5%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b_2}{a} \]
   6. Step-by-step derivation

      1. unpow1/362.3%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{1.5}}} - b_2}{a} \]

      2. distribute-rgt-neg-out62.3%

         \[\leadsto \frac{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{-c \cdot a}\right)\right)}^{1.5}} - b_2}{a} \]

      3. *-commutative62.3%

         \[\leadsto \frac{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{a \cdot c}\right)\right)}^{1.5}} - b_2}{a} \]

      4. fma-neg62.1%

         \[\leadsto \frac{\sqrt[3]{{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right)}}^{1.5}} - b_2}{a} \]

      5. *-commutative62.1%

         \[\leadsto \frac{\sqrt[3]{{\left(b_2 \cdot b_2 - \color{blue}{c \cdot a}\right)}^{1.5}} - b_2}{a} \]

   7. Simplified62.1%

      \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}}} - b_2}{a} \]
   8. Step-by-step derivation

      1. flip--61.8%

         \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}} \cdot \sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}} - b_2 \cdot b_2}{\sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}} + b_2}}}{a} \]

      2. cbrt-unprod32.5%

         \[\leadsto \frac{\frac{\color{blue}{\sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5} \cdot {\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}}} - b_2 \cdot b_2}{\sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}} + b_2}}{a} \]

      3. pow-prod-up33.9%

         \[\leadsto \frac{\frac{\sqrt[3]{\color{blue}{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{\left(1.5 + 1.5\right)}}} - b_2 \cdot b_2}{\sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}} + b_2}}{a} \]

      4. metadata-eval33.9%

         \[\leadsto \frac{\frac{\sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{\color{blue}{3}}} - b_2 \cdot b_2}{\sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}} + b_2}}{a} \]

      5. pow333.9%

         \[\leadsto \frac{\frac{\sqrt[3]{\color{blue}{\left(\left(b_2 \cdot b_2 - c \cdot a\right) \cdot \left(b_2 \cdot b_2 - c \cdot a\right)\right) \cdot \left(b_2 \cdot b_2 - c \cdot a\right)}} - b_2 \cdot b_2}{\sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}} + b_2}}{a} \]

      6. add-cbrt-cube60.9%

         \[\leadsto \frac{\frac{\color{blue}{\left(b_2 \cdot b_2 - c \cdot a\right)} - b_2 \cdot b_2}{\sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}} + b_2}}{a} \]

      7. pow1/357.0%

         \[\leadsto \frac{\frac{\left(b_2 \cdot b_2 - c \cdot a\right) - b_2 \cdot b_2}{\color{blue}{{\left({\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}\right)}^{0.3333333333333333}} + b_2}}{a} \]

      8. pow-pow70.8%

         \[\leadsto \frac{\frac{\left(b_2 \cdot b_2 - c \cdot a\right) - b_2 \cdot b_2}{\color{blue}{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} + b_2}}{a} \]

      9. metadata-eval70.8%

         \[\leadsto \frac{\frac{\left(b_2 \cdot b_2 - c \cdot a\right) - b_2 \cdot b_2}{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{\color{blue}{0.5}} + b_2}}{a} \]

      10. pow1/270.8%

          \[\leadsto \frac{\frac{\left(b_2 \cdot b_2 - c \cdot a\right) - b_2 \cdot b_2}{\color{blue}{\sqrt{b_2 \cdot b_2 - c \cdot a}} + b_2}}{a} \]

   9. Applied egg-rr70.8%

      \[\leadsto \frac{\color{blue}{\frac{\left(b_2 \cdot b_2 - c \cdot a\right) - b_2 \cdot b_2}{\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2}}}{a} \]

   10. Taylor expanded in b_2 around 0 99.2%

       \[\leadsto \frac{\frac{\color{blue}{-1 \cdot \left(c \cdot a\right)}}{\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2}}{a} \]
   11. Step-by-step derivation

       1. associate-*r*99.2%

          \[\leadsto \frac{\frac{\color{blue}{\left(-1 \cdot c\right) \cdot a}}{\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2}}{a} \]

       2. neg-mul-199.2%

          \[\leadsto \frac{\frac{\color{blue}{\left(-c\right)} \cdot a}{\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2}}{a} \]

   12. Simplified99.2%

       \[\leadsto \frac{\frac{\color{blue}{\left(-c\right) \cdot a}}{\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2}}{a} \]
3. Recombined 4 regimes into one program.

4. Final simplification88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -4.4 \cdot 10^{+101}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.95 \cdot 10^{-129}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 1.46 \cdot 10^{-72} \lor \neg \left(b_2 \leq 1.1 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{a \cdot c}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\\ \end{array} \]

1. Split input into 4 regimes

2. if b_2 < -6.99999999999999953e100

   1. Initial program 57.6%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative57.6%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg57.6%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified57.6%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around -inf 91.5%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]

   if -6.99999999999999953e100 < b_2 < 1.95000000000000005e-129

   1. Initial program 79.9%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative79.9%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg79.9%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified79.9%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   if 1.95000000000000005e-129 < b_2 < 3.3999999999999998e-72 or 1.35e-40 < b_2

   1. Initial program 14.3%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative14.3%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg14.3%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified14.3%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around inf 93.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]

   if 3.3999999999999998e-72 < b_2 < 1.35e-40

   1. Initial program 99.0%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative99.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg99.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified99.0%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
   4. Step-by-step derivation

      1. add-cbrt-cube83.1%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}} - b_2}{a} \]

      2. pow383.1%

         \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}^{3}}} - b_2}{a} \]

      3. pow1/378.0%

         \[\leadsto \frac{\color{blue}{{\left({\left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}^{3}\right)}^{0.3333333333333333}} - b_2}{a} \]

      4. sqrt-pow278.0%

         \[\leadsto \frac{{\color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(\frac{3}{2}\right)}\right)}}^{0.3333333333333333} - b_2}{a} \]

      5. fma-neg78.0%

         \[\leadsto \frac{{\left({\color{blue}{\left(\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)\right)}}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

      6. *-commutative78.0%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{c \cdot a}\right)\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

      7. distribute-rgt-neg-in78.0%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{c \cdot \left(-a\right)}\right)\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

      8. metadata-eval78.0%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} - b_2}{a} \]

   5. Applied egg-rr78.0%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b_2}{a} \]
   6. Step-by-step derivation

      1. unpow1/383.1%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{1.5}}} - b_2}{a} \]

      2. distribute-rgt-neg-out83.1%

         \[\leadsto \frac{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{-c \cdot a}\right)\right)}^{1.5}} - b_2}{a} \]

      3. *-commutative83.1%

         \[\leadsto \frac{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{a \cdot c}\right)\right)}^{1.5}} - b_2}{a} \]

      4. fma-neg83.1%

         \[\leadsto \frac{\sqrt[3]{{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right)}}^{1.5}} - b_2}{a} \]

      5. *-commutative83.1%

         \[\leadsto \frac{\sqrt[3]{{\left(b_2 \cdot b_2 - \color{blue}{c \cdot a}\right)}^{1.5}} - b_2}{a} \]

   7. Simplified83.1%

      \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}}} - b_2}{a} \]
   8. Step-by-step derivation

      1. clear-num83.1%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{\sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}} - b_2}}} \]

      2. inv-pow83.1%

         \[\leadsto \color{blue}{{\left(\frac{a}{\sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}} - b_2}\right)}^{-1}} \]

      3. pow1/378.0%

         \[\leadsto {\left(\frac{a}{\color{blue}{{\left({\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}\right)}^{0.3333333333333333}} - b_2}\right)}^{-1} \]

      4. pow-pow99.0%

         \[\leadsto {\left(\frac{a}{\color{blue}{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} - b_2}\right)}^{-1} \]

      5. metadata-eval99.0%

         \[\leadsto {\left(\frac{a}{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{\color{blue}{0.5}} - b_2}\right)}^{-1} \]

      6. pow1/299.0%

         \[\leadsto {\left(\frac{a}{\color{blue}{\sqrt{b_2 \cdot b_2 - c \cdot a}} - b_2}\right)}^{-1} \]

   9. Applied egg-rr99.0%

      \[\leadsto \color{blue}{{\left(\frac{a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}\right)}^{-1}} \]
   10. Step-by-step derivation

       1. unpow-199.0%

          \[\leadsto \color{blue}{\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}} \]

   11. Simplified99.0%

       \[\leadsto \color{blue}{\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}} \]
   12. Step-by-step derivation

       1. div-inv99.2%

          \[\leadsto \frac{1}{\color{blue}{a \cdot \frac{1}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}} \]

   13. Applied egg-rr99.2%

       \[\leadsto \frac{1}{\color{blue}{a \cdot \frac{1}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -7 \cdot 10^{+100}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.95 \cdot 10^{-129}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 3.4 \cdot 10^{-72} \lor \neg \left(b_2 \leq 1.35 \cdot 10^{-40}\right):\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a \cdot \frac{1}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \end{array} \]

1. Split input into 3 regimes

2. if b_2 < -7.9000000000000001e37

   1. Initial program 63.3%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative63.3%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg63.3%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified63.3%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around -inf 90.2%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]

   if -7.9000000000000001e37 < b_2 < 1.95000000000000005e-129 or 2.1000000000000001e-71 < b_2 < 1.22e-40

   1. Initial program 79.8%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative79.8%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg79.8%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified79.8%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around 0 73.0%

      \[\leadsto \frac{\sqrt{\color{blue}{-1 \cdot \left(c \cdot a\right)}} - b_2}{a} \]
   5. Step-by-step derivation

      1. mul-1-neg73.0%

         \[\leadsto \frac{\sqrt{\color{blue}{-c \cdot a}} - b_2}{a} \]

      2. distribute-rgt-neg-out73.0%

         \[\leadsto \frac{\sqrt{\color{blue}{c \cdot \left(-a\right)}} - b_2}{a} \]

   6. Simplified73.0%

      \[\leadsto \frac{\sqrt{\color{blue}{c \cdot \left(-a\right)}} - b_2}{a} \]

   if 1.95000000000000005e-129 < b_2 < 2.1000000000000001e-71 or 1.22e-40 < b_2

   1. Initial program 14.3%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative14.3%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg14.3%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified14.3%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around inf 93.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -7.9 \cdot 10^{+37}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.95 \cdot 10^{-129} \lor \neg \left(b_2 \leq 2.1 \cdot 10^{-71}\right) \land b_2 \leq 1.22 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

1. Split input into 4 regimes

2. if b_2 < -1.55e101

   1. Initial program 57.6%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative57.6%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg57.6%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified57.6%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around -inf 91.5%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]

   if -1.55e101 < b_2 < 1.95000000000000005e-129

   1. Initial program 79.9%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative79.9%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg79.9%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified79.9%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   if 1.95000000000000005e-129 < b_2 < 3.3999999999999998e-72 or 1.22e-40 < b_2

   1. Initial program 14.3%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative14.3%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg14.3%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified14.3%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around inf 93.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]

   if 3.3999999999999998e-72 < b_2 < 1.22e-40

   1. Initial program 99.0%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative99.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg99.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified99.0%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around 0 99.0%

      \[\leadsto \frac{\sqrt{\color{blue}{-1 \cdot \left(c \cdot a\right)}} - b_2}{a} \]
   5. Step-by-step derivation

      1. mul-1-neg99.0%

         \[\leadsto \frac{\sqrt{\color{blue}{-c \cdot a}} - b_2}{a} \]

      2. distribute-rgt-neg-out99.0%

         \[\leadsto \frac{\sqrt{\color{blue}{c \cdot \left(-a\right)}} - b_2}{a} \]

   6. Simplified99.0%

      \[\leadsto \frac{\sqrt{\color{blue}{c \cdot \left(-a\right)}} - b_2}{a} \]
3. Recombined 4 regimes into one program.

4. Final simplification88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.55 \cdot 10^{+101}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.95 \cdot 10^{-129}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq 3.4 \cdot 10^{-72} \lor \neg \left(b_2 \leq 1.22 \cdot 10^{-40}\right):\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(-c\right)} - b_2}{a}\\ \end{array} \]

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 68.5%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative68.5%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg68.5%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified68.5%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around -inf 64.5%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}} \]

   if -4.999999999999985e-310 < b_2

   1. Initial program 34.0%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative34.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg34.0%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified34.0%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around inf 68.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a} + 0.5 \cdot \frac{c}{b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

1. Split input into 2 regimes

2. if b_2 < 8.2000000000000005e-305

   1. Initial program 69.1%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative69.1%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg69.1%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified69.1%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around 0 41.5%

      \[\leadsto \frac{\sqrt{\color{blue}{-1 \cdot \left(c \cdot a\right)}} - b_2}{a} \]
   5. Step-by-step derivation

      1. mul-1-neg41.5%

         \[\leadsto \frac{\sqrt{\color{blue}{-c \cdot a}} - b_2}{a} \]

      2. distribute-rgt-neg-out41.5%

         \[\leadsto \frac{\sqrt{\color{blue}{c \cdot \left(-a\right)}} - b_2}{a} \]

   6. Simplified41.5%

      \[\leadsto \frac{\sqrt{\color{blue}{c \cdot \left(-a\right)}} - b_2}{a} \]

   7. Taylor expanded in c around 0 28.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{b_2}{a}} \]
   8. Step-by-step derivation

      1. associate-*r/28.3%

         \[\leadsto \color{blue}{\frac{-1 \cdot b_2}{a}} \]

      2. neg-mul-128.3%

         \[\leadsto \frac{\color{blue}{-b_2}}{a} \]

   9. Simplified28.3%

      \[\leadsto \color{blue}{\frac{-b_2}{a}} \]

   if 8.2000000000000005e-305 < b_2

   1. Initial program 33.1%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative33.1%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg33.1%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified33.1%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around inf 69.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq 8.2 \cdot 10^{-305}:\\ \;\;\;\;\frac{-b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

1. Split input into 2 regimes

2. if b_2 < 8.2000000000000005e-305

   1. Initial program 69.1%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative69.1%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg69.1%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified69.1%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]
   4. Step-by-step derivation

      1. add-cbrt-cube53.7%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}} - b_2}{a} \]

      2. pow353.6%

         \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}^{3}}} - b_2}{a} \]

      3. pow1/351.5%

         \[\leadsto \frac{\color{blue}{{\left({\left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right)}^{3}\right)}^{0.3333333333333333}} - b_2}{a} \]

      4. sqrt-pow251.5%

         \[\leadsto \frac{{\color{blue}{\left({\left(b_2 \cdot b_2 - a \cdot c\right)}^{\left(\frac{3}{2}\right)}\right)}}^{0.3333333333333333} - b_2}{a} \]

      5. fma-neg51.5%

         \[\leadsto \frac{{\left({\color{blue}{\left(\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)\right)}}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

      6. *-commutative51.5%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{c \cdot a}\right)\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

      7. distribute-rgt-neg-in51.5%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{c \cdot \left(-a\right)}\right)\right)}^{\left(\frac{3}{2}\right)}\right)}^{0.3333333333333333} - b_2}{a} \]

      8. metadata-eval51.5%

         \[\leadsto \frac{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} - b_2}{a} \]

   5. Applied egg-rr51.5%

      \[\leadsto \frac{\color{blue}{{\left({\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} - b_2}{a} \]
   6. Step-by-step derivation

      1. unpow1/353.6%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)\right)}^{1.5}}} - b_2}{a} \]

      2. distribute-rgt-neg-out53.6%

         \[\leadsto \frac{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, \color{blue}{-c \cdot a}\right)\right)}^{1.5}} - b_2}{a} \]

      3. *-commutative53.6%

         \[\leadsto \frac{\sqrt[3]{{\left(\mathsf{fma}\left(b_2, b_2, -\color{blue}{a \cdot c}\right)\right)}^{1.5}} - b_2}{a} \]

      4. fma-neg53.6%

         \[\leadsto \frac{\sqrt[3]{{\color{blue}{\left(b_2 \cdot b_2 - a \cdot c\right)}}^{1.5}} - b_2}{a} \]

      5. *-commutative53.6%

         \[\leadsto \frac{\sqrt[3]{{\left(b_2 \cdot b_2 - \color{blue}{c \cdot a}\right)}^{1.5}} - b_2}{a} \]

   7. Simplified53.6%

      \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(b_2 \cdot b_2 - c \cdot a\right)}^{1.5}}} - b_2}{a} \]

   8. Taylor expanded in b_2 around -inf 63.2%

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}} \]
   9. Step-by-step derivation

      1. metadata-eval63.2%

         \[\leadsto \color{blue}{\frac{-2}{1}} \cdot \frac{b_2}{a} \]

      2. times-frac63.2%

         \[\leadsto \color{blue}{\frac{-2 \cdot b_2}{1 \cdot a}} \]

      3. *-commutative63.2%

         \[\leadsto \frac{\color{blue}{b_2 \cdot -2}}{1 \cdot a} \]

      4. times-frac63.0%

         \[\leadsto \color{blue}{\frac{b_2}{1} \cdot \frac{-2}{a}} \]

      5. /-rgt-identity63.0%

         \[\leadsto \color{blue}{b_2} \cdot \frac{-2}{a} \]

   10. Simplified63.0%

       \[\leadsto \color{blue}{b_2 \cdot \frac{-2}{a}} \]

   if 8.2000000000000005e-305 < b_2

   1. Initial program 33.1%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative33.1%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg33.1%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified33.1%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around inf 69.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq 8.2 \cdot 10^{-305}:\\ \;\;\;\;b_2 \cdot \frac{-2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

1. Split input into 2 regimes

2. if b_2 < 8.2000000000000005e-305

   1. Initial program 69.1%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative69.1%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg69.1%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified69.1%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around -inf 63.2%

      \[\leadsto \frac{\color{blue}{-2 \cdot b_2}}{a} \]
   5. Step-by-step derivation

      1. *-commutative63.2%

         \[\leadsto \frac{\color{blue}{b_2 \cdot -2}}{a} \]

   6. Simplified63.2%

      \[\leadsto \frac{\color{blue}{b_2 \cdot -2}}{a} \]

   if 8.2000000000000005e-305 < b_2

   1. Initial program 33.1%

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
   2. Step-by-step derivation

      1. +-commutative33.1%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

      2. unsub-neg33.1%

         \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

   3. Simplified33.1%

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

   4. Taylor expanded in b_2 around inf 69.2%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq 8.2 \cdot 10^{-305}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

1. Initial program 48.7%

   \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
2. Step-by-step derivation

   1. +-commutative48.7%

      \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}{a} \]

   2. unsub-neg48.7%

      \[\leadsto \frac{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a} \]

3. Simplified48.7%

   \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}} \]

4. Taylor expanded in b_2 around 0 33.3%

   \[\leadsto \frac{\sqrt{\color{blue}{-1 \cdot \left(c \cdot a\right)}} - b_2}{a} \]
5. Step-by-step derivation

   1. mul-1-neg33.3%

      \[\leadsto \frac{\sqrt{\color{blue}{-c \cdot a}} - b_2}{a} \]

   2. distribute-rgt-neg-out33.3%

      \[\leadsto \frac{\sqrt{\color{blue}{c \cdot \left(-a\right)}} - b_2}{a} \]

6. Simplified33.3%

   \[\leadsto \frac{\sqrt{\color{blue}{c \cdot \left(-a\right)}} - b_2}{a} \]

7. Taylor expanded in c around 0 13.9%

   \[\leadsto \color{blue}{-1 \cdot \frac{b_2}{a}} \]
8. Step-by-step derivation

   1. associate-*r/13.9%

      \[\leadsto \color{blue}{\frac{-1 \cdot b_2}{a}} \]

   2. neg-mul-113.9%

      \[\leadsto \frac{\color{blue}{-b_2}}{a} \]

9. Simplified13.9%

   \[\leadsto \color{blue}{\frac{-b_2}{a}} \]

10. Final simplification13.9%

    \[\leadsto \frac{-b_2}{a} \]

herbie shell --seed 2023166 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))