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Derivation

1. Split input into 2 regimes

2. if b < 4.50657349639539e+128

   1. Initial program 14.9

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

   2. Initial simplification14.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\left(-4 \cdot a\right) \cdot c + b \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\left(-4 \cdot a\right) \cdot c + b \cdot b} - b}\\ \end{array}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt15.0

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

   6. Applied add-cube-cbrt15.1

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\left(-4 \cdot a\right) \cdot c + b \cdot b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\left(\sqrt[3]{\sqrt{\left(-4 \cdot a\right) \cdot c + b \cdot b}} \cdot \sqrt[3]{\sqrt{\left(-4 \cdot a\right) \cdot c + b \cdot b}}\right) \cdot \sqrt[3]{\sqrt{\left(-4 \cdot a\right) \cdot c + b \cdot b}}} \cdot \sqrt{\sqrt{\left(-4 \cdot a\right) \cdot c + b \cdot b}} - b}\\ \end{array}\]

   if 4.50657349639539e+128 < b

   1. Initial program 53.6

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

   2. Initial simplification53.6

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

   3. Taylor expanded around 0 3.2

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{b}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{\left(-4 \cdot a\right) \cdot c + b \cdot b} - b}\\ \end{array}\]
3. Recombined 2 regimes into one program.

4. Final simplification13.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 4.50657349639539 \cdot 10^{+128}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}} \cdot \sqrt{\sqrt[3]{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}} \cdot \left(\sqrt[3]{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}} \cdot \sqrt[3]{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array}\]

Runtime

Time bar (total: 26.2s)Debug logProfile

herbie shell --seed 2018351 
(FPCore (a b c)
  :name "jeff quadratic root 1"
  (if (>= b 0) (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ (* 2 c) (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))))))