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Derivation

1. Split input into 4 regimes

2. if b < -8.422430181155895e+146

   1. Initial program 57.7

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

   2. Initial simplification57.7

      \[\leadsto \frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2 \cdot a}\]

   3. Taylor expanded around -inf 2.3

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}}\]

   if -8.422430181155895e+146 < b < -1.347277676039659e-308

   1. Initial program 8.4

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

   2. Initial simplification8.4

      \[\leadsto \frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2 \cdot a}\]

   3. Taylor expanded around -inf 8.4

      \[\leadsto \frac{\sqrt{\color{blue}{{b}^{2} - 4 \cdot \left(a \cdot c\right)}} - b}{2 \cdot a}\]

   if -1.347277676039659e-308 < b < 4.331216038455128e+110

   1. Initial program 33.8

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

   2. Initial simplification33.8

      \[\leadsto \frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2 \cdot a}\]
   3. Using strategy rm

   4. Applied flip--33.9

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b \cdot b}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}}}{2 \cdot a}\]

   5. Applied associate-/l/38.2

      \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b \cdot b}{\left(2 \cdot a\right) \cdot \left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b\right)}}\]

   6. Simplified21.5

      \[\leadsto \frac{\color{blue}{\left(c \cdot -4\right) \cdot a}}{\left(2 \cdot a\right) \cdot \left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b\right)}\]
   7. Using strategy rm

   8. Applied associate-/r*16.1

      \[\leadsto \color{blue}{\frac{\frac{\left(c \cdot -4\right) \cdot a}{2 \cdot a}}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}}\]

   9. Taylor expanded around 0 9.0

      \[\leadsto \frac{\color{blue}{-2 \cdot c}}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}\]

   if 4.331216038455128e+110 < b

   1. Initial program 59.2

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

   2. Initial simplification59.2

      \[\leadsto \frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2 \cdot a}\]

   3. Taylor expanded around inf 1.8

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]

   4. Simplified1.8

      \[\leadsto \color{blue}{\frac{-c}{b}}\]
3. Recombined 4 regimes into one program.

4. Final simplification6.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.422430181155895 \cdot 10^{+146}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le -1.347277676039659 \cdot 10^{-308}:\\ \;\;\;\;\frac{\sqrt{{b}^{2} - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 4.331216038455128 \cdot 10^{+110}:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2018360 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))

Details

Time bar (total: 15.6s)Debug log