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Derivation

1. Split input into 5 regimes

2. if b < -2.8671071746705253e+127

   1. Initial program 51.2

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

   2. Simplified51.2

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

   3. Taylor expanded around -inf 2.8

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

   if -2.8671071746705253e+127 < b < 1.354454736393541e-170

   1. Initial program 11.4

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

   2. Simplified11.3

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

   4. Applied clear-num11.5

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}}}\]

   if 1.354454736393541e-170 < b < 9.22274670563189e-75

   1. Initial program 24.9

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

   2. Simplified24.9

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

   4. Applied flip--25.0

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

   5. Applied associate-/l/31.2

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

   6. Simplified25.9

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

   8. Applied times-frac18.2

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

   9. Simplified18.2

      \[\leadsto \color{blue}{\frac{\frac{c}{\frac{-1}{2}}}{a}} \cdot \frac{a}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}\]

   if 9.22274670563189e-75 < b < 1.849572225033813e-22

   1. Initial program 37.0

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

   2. Simplified37.0

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

   4. Applied add-sqr-sqrt37.5

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

   5. Applied times-frac37.5

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

   if 1.849572225033813e-22 < b

   1. Initial program 54.3

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

   2. Simplified54.3

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

   3. Taylor expanded around inf 6.8

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

   4. Simplified6.8

      \[\leadsto \color{blue}{\frac{-c}{b}}\]
3. Recombined 5 regimes into one program.

4. Final simplification10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.8671071746705253 \cdot 10^{+127}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.354454736393541 \cdot 10^{-170}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}}\\ \mathbf{elif}\;b \le 9.22274670563189 \cdot 10^{-75}:\\ \;\;\;\;\frac{a}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} + b} \cdot \frac{\frac{c}{\frac{-1}{2}}}{a}\\ \mathbf{elif}\;b \le 1.849572225033813 \cdot 10^{-22}:\\ \;\;\;\;\frac{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}}{a} \cdot \frac{\sqrt{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}}{2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))

Details

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