jeff quadratic root 2

1. Split input into 3 regimes

2. if b < -1.22e112

   1. Initial program 50.3%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   2. Step-by-step derivation

   3. Simplified50.2%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ } \end{array}} \]

   4. Taylor expanded in b around inf 50.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\color{blue}{-2 \cdot \frac{c \cdot a}{b} + 2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]
   5. Step-by-step derivation

      1. fma-def50.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\color{blue}{\mathsf{fma}\left(-2, \frac{c \cdot a}{b}, 2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

      2. associate-/l*50.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \color{blue}{\frac{c}{\frac{b}{a}}}, 2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

      3. *-commutative50.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, \color{blue}{b \cdot 2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

   6. Simplified50.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\color{blue}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

   7. Taylor expanded in b around -inf 95.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(b - -1 \cdot b\right) \cdot \frac{-0.5}{a}\\ \end{array} \]
   8. Step-by-step derivation

      1. associate-*r/95.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b - -1 \cdot b\right) \cdot -0.5}{a}\\ \end{array} \]

      2. cancel-sign-sub-inv95.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + \left(--1\right) \cdot b\right) \cdot -0.5}{a}\\ \end{array} \]

      3. metadata-eval95.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + 1 \cdot b\right) \cdot -0.5}{a}\\ \end{array} \]

      4. *-un-lft-identity95.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + b\right) \cdot -0.5}{a}\\ \end{array} \]

   9. Applied egg-rr95.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + b\right) \cdot -0.5}{a}\\ \end{array} \]

   10. Taylor expanded in c around inf 95.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + b\right) \cdot -0.5}{a}\\ \end{array} \]

   if -1.22e112 < b < 1e104

   1. Initial program 85.3%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   if 1e104 < b

   1. Initial program 60.5%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   2. Step-by-step derivation

   3. Simplified60.5%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ } \end{array}} \]

   4. Taylor expanded in b around inf 90.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\color{blue}{-2 \cdot \frac{c \cdot a}{b} + 2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]
   5. Step-by-step derivation

      1. fma-def90.1%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\color{blue}{\mathsf{fma}\left(-2, \frac{c \cdot a}{b}, 2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

      2. associate-/l*98.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \color{blue}{\frac{c}{\frac{b}{a}}}, 2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

      3. *-commutative98.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, \color{blue}{b \cdot 2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

   6. Simplified98.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\color{blue}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

   7. Taylor expanded in b around -inf 98.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(b - -1 \cdot b\right) \cdot \frac{-0.5}{a}\\ \end{array} \]
   8. Step-by-step derivation

      1. div-inv98.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \color{blue}{c \cdot \frac{1}{\frac{b}{a}}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(b - -1 \cdot b\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

      2. clear-num98.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, c \cdot \color{blue}{\frac{a}{b}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(b - -1 \cdot b\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

   9. Applied egg-rr98.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \color{blue}{c \cdot \frac{a}{b}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(b - -1 \cdot b\right) \cdot \frac{-0.5}{a}\\ \end{array} \]
3. Recombined 3 regimes into one program.

4. Final simplification90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.22 \cdot 10^{+112}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + b\right) \cdot -0.5}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq 10^{+104}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, c \cdot \frac{a}{b}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(b + b\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

1. Split input into 3 regimes

2. if b < -1e114

   1. Initial program 50.3%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   2. Step-by-step derivation

   3. Simplified50.2%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ } \end{array}} \]

   4. Taylor expanded in b around inf 50.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\color{blue}{-2 \cdot \frac{c \cdot a}{b} + 2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]
   5. Step-by-step derivation

      1. fma-def50.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\color{blue}{\mathsf{fma}\left(-2, \frac{c \cdot a}{b}, 2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

      2. associate-/l*50.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \color{blue}{\frac{c}{\frac{b}{a}}}, 2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

      3. *-commutative50.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, \color{blue}{b \cdot 2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

   6. Simplified50.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\color{blue}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

   7. Taylor expanded in b around -inf 95.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(b - -1 \cdot b\right) \cdot \frac{-0.5}{a}\\ \end{array} \]
   8. Step-by-step derivation

      1. associate-*r/95.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b - -1 \cdot b\right) \cdot -0.5}{a}\\ \end{array} \]

      2. cancel-sign-sub-inv95.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + \left(--1\right) \cdot b\right) \cdot -0.5}{a}\\ \end{array} \]

      3. metadata-eval95.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + 1 \cdot b\right) \cdot -0.5}{a}\\ \end{array} \]

      4. *-un-lft-identity95.4%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + b\right) \cdot -0.5}{a}\\ \end{array} \]

   9. Applied egg-rr95.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + b\right) \cdot -0.5}{a}\\ \end{array} \]

   10. Taylor expanded in c around inf 95.4%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + b\right) \cdot -0.5}{a}\\ \end{array} \]

   if -1e114 < b < 1.3600000000000001e-150

   1. Initial program 84.8%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   2. Step-by-step derivation

      1. expm1-log1p-u78.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. expm1-udef69.9%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{e^{\mathsf{log1p}\left(\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)} - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      3. *-commutative69.9%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{\color{blue}{c \cdot 2}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      4. add-sqr-sqrt65.6%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{c \cdot 2}{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      5. sqrt-unprod69.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{c \cdot 2}{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      6. sqr-neg69.8%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{c \cdot 2}{\sqrt{\color{blue}{b \cdot b}} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      7. sqrt-prod69.9%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{c \cdot 2}{\color{blue}{\sqrt{b} \cdot \sqrt{b}} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      8. add-sqr-sqrt69.9%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{c \cdot 2}{\color{blue}{b} - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      9. *-commutative69.9%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{c \cdot 2}{b - \sqrt{b \cdot b - \color{blue}{c \cdot \left(4 \cdot a\right)}}}\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      10. *-commutative69.9%

          \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;e^{\mathsf{log1p}\left(\frac{c \cdot 2}{b - \sqrt{b \cdot b - c \cdot \color{blue}{\left(a \cdot 4\right)}}}\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   3. Applied egg-rr69.9%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{e^{\mathsf{log1p}\left(\frac{c \cdot 2}{b - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\right)} - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   4. Step-by-step derivation

      1. expm1-def77.9%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c \cdot 2}{b - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      2. expm1-log1p84.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{c \cdot 2}{b - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

      3. associate-*r/84.5%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{c \cdot \frac{2}{b - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   5. Simplified84.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{c \cdot \frac{2}{b - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]

   if 1.3600000000000001e-150 < b

   1. Initial program 71.5%

      \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
   2. Step-by-step derivation

   3. Simplified71.5%

      \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ } \end{array}} \]

   4. Taylor expanded in b around inf 81.2%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\color{blue}{-2 \cdot \frac{c \cdot a}{b} + 2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]
   5. Step-by-step derivation

      1. fma-def81.2%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\color{blue}{\mathsf{fma}\left(-2, \frac{c \cdot a}{b}, 2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

      2. associate-/l*86.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \color{blue}{\frac{c}{\frac{b}{a}}}, 2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

      3. *-commutative86.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, \color{blue}{b \cdot 2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

   6. Simplified86.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\color{blue}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

   7. Taylor expanded in b around -inf 86.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(b - -1 \cdot b\right) \cdot \frac{-0.5}{a}\\ \end{array} \]
   8. Step-by-step derivation

      1. div-inv86.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \color{blue}{c \cdot \frac{1}{\frac{b}{a}}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(b - -1 \cdot b\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

      2. clear-num86.0%

         \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, c \cdot \color{blue}{\frac{a}{b}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(b - -1 \cdot b\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

   9. Applied egg-rr86.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \color{blue}{c \cdot \frac{a}{b}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(b - -1 \cdot b\right) \cdot \frac{-0.5}{a}\\ \end{array} \]
3. Recombined 3 regimes into one program.

4. Final simplification87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+114}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + b\right) \cdot -0.5}{a}\\ \end{array}\\ \mathbf{elif}\;b \leq 1.36 \cdot 10^{-150}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;c \cdot \frac{2}{b - \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, c \cdot \frac{a}{b}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(b + b\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

1. Initial program 71.5%

   \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
2. Step-by-step derivation

3. Simplified71.4%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ } \end{array}} \]

4. Taylor expanded in b around inf 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\color{blue}{-2 \cdot \frac{c \cdot a}{b} + 2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]
5. Step-by-step derivation

   1. fma-def68.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\color{blue}{\mathsf{fma}\left(-2, \frac{c \cdot a}{b}, 2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

   2. associate-/l*70.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \color{blue}{\frac{c}{\frac{b}{a}}}, 2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

   3. *-commutative70.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, \color{blue}{b \cdot 2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

6. Simplified70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\color{blue}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

7. Taylor expanded in b around -inf 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(b - -1 \cdot b\right) \cdot \frac{-0.5}{a}\\ \end{array} \]
8. Step-by-step derivation

   1. associate-*r/73.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b - -1 \cdot b\right) \cdot -0.5}{a}\\ \end{array} \]

   2. cancel-sign-sub-inv73.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + \left(--1\right) \cdot b\right) \cdot -0.5}{a}\\ \end{array} \]

   3. metadata-eval73.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + 1 \cdot b\right) \cdot -0.5}{a}\\ \end{array} \]

   4. *-un-lft-identity73.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + b\right) \cdot -0.5}{a}\\ \end{array} \]

9. Applied egg-rr73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + b\right) \cdot -0.5}{a}\\ \end{array} \]
10. Step-by-step derivation

    1. div-inv72.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \color{blue}{c \cdot \frac{1}{\frac{b}{a}}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(b - -1 \cdot b\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

    2. clear-num72.9%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, c \cdot \color{blue}{\frac{a}{b}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(b - -1 \cdot b\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

11. Applied egg-rr73.1%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \color{blue}{c \cdot \frac{a}{b}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + b\right) \cdot -0.5}{a}\\ \end{array} \]

12. Final simplification73.1%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, c \cdot \frac{a}{b}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + b\right) \cdot -0.5}{a}\\ \end{array} \]

1. Initial program 71.5%

   \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
2. Step-by-step derivation

3. Simplified71.4%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ } \end{array}} \]

4. Taylor expanded in b around inf 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\color{blue}{-2 \cdot \frac{c \cdot a}{b} + 2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]
5. Step-by-step derivation

   1. fma-def68.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\color{blue}{\mathsf{fma}\left(-2, \frac{c \cdot a}{b}, 2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

   2. associate-/l*70.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \color{blue}{\frac{c}{\frac{b}{a}}}, 2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

   3. *-commutative70.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, \color{blue}{b \cdot 2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

6. Simplified70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\color{blue}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

7. Taylor expanded in b around -inf 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(b - -1 \cdot b\right) \cdot \frac{-0.5}{a}\\ \end{array} \]
8. Step-by-step derivation

   1. associate-*r/73.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b - -1 \cdot b\right) \cdot -0.5}{a}\\ \end{array} \]

   2. cancel-sign-sub-inv73.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + \left(--1\right) \cdot b\right) \cdot -0.5}{a}\\ \end{array} \]

   3. metadata-eval73.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + 1 \cdot b\right) \cdot -0.5}{a}\\ \end{array} \]

   4. *-un-lft-identity73.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + b\right) \cdot -0.5}{a}\\ \end{array} \]

9. Applied egg-rr73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + b\right) \cdot -0.5}{a}\\ \end{array} \]

10. Taylor expanded in c around inf 40.1%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{b}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + b\right) \cdot -0.5}{a}\\ \end{array} \]

11. Final simplification40.1%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + b\right) \cdot -0.5}{a}\\ \end{array} \]

1. Initial program 71.5%

   \[\begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array} \]
2. Step-by-step derivation

3. Simplified71.4%

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ } \end{array}} \]

4. Taylor expanded in b around inf 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\color{blue}{-2 \cdot \frac{c \cdot a}{b} + 2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]
5. Step-by-step derivation

   1. fma-def68.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\color{blue}{\mathsf{fma}\left(-2, \frac{c \cdot a}{b}, 2 \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

   2. associate-/l*70.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \color{blue}{\frac{c}{\frac{b}{a}}}, 2 \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

   3. *-commutative70.7%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, \color{blue}{b \cdot 2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

6. Simplified70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\color{blue}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(b - \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -4\right)\right)}\right) \cdot \frac{-0.5}{a}\\ \end{array} \]

7. Taylor expanded in b around -inf 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(b - -1 \cdot b\right) \cdot \frac{-0.5}{a}\\ \end{array} \]
8. Step-by-step derivation

   1. associate-*r/73.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b - -1 \cdot b\right) \cdot -0.5}{a}\\ \end{array} \]

   2. cancel-sign-sub-inv73.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + \left(--1\right) \cdot b\right) \cdot -0.5}{a}\\ \end{array} \]

   3. metadata-eval73.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + 1 \cdot b\right) \cdot -0.5}{a}\\ \end{array} \]

   4. *-un-lft-identity73.1%

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + b\right) \cdot -0.5}{a}\\ \end{array} \]

9. Applied egg-rr73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;-2 \cdot \frac{c}{\mathsf{fma}\left(-2, \frac{c}{\frac{b}{a}}, b \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + b\right) \cdot -0.5}{a}\\ \end{array} \]

10. Taylor expanded in c around 0 73.0%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{-1 \cdot \frac{c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + b\right) \cdot -0.5}{a}\\ \end{array} \]
11. Step-by-step derivation

    1. associate-*r/73.0%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-1 \cdot c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + b\right) \cdot -0.5}{a}\\ \end{array} \]

    2. neg-mul-173.0%

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{\color{blue}{-c}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + b\right) \cdot -0.5}{a}\\ \end{array} \]

12. Simplified73.0%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\color{blue}{\frac{-c}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + b\right) \cdot -0.5}{a}\\ \end{array} \]

13. Final simplification73.0%

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \geq 0:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + b\right) \cdot -0.5}{a}\\ \end{array} \]

herbie shell --seed 2023167 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  :precision binary64
  (if (>= b 0.0) (/ (* 2.0 c) (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a))))