Error

Derivation

1. Split input into 2 regimes

2. if b < 1.4321471445306602e+153

   1. Initial program 14.3

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

   2. Initial simplification14.3

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

   4. Applied add-cube-cbrt14.5

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

   5. Applied sqrt-prod14.5

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

   6. Applied neg-mul-114.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\color{blue}{-1 \cdot b} - \sqrt{\sqrt[3]{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*} \cdot \sqrt[3]{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}} \cdot \sqrt{\sqrt[3]{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*} - b}\\ \end{array}\]

   7. Applied prod-diff14.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\color{blue}{(-1 \cdot b + \left(-\sqrt{\sqrt[3]{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}} \cdot \sqrt{\sqrt[3]{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*} \cdot \sqrt[3]{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}\right))_* + (\left(-\sqrt{\sqrt[3]{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt[3]{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*} \cdot \sqrt[3]{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}\right) + \left(\sqrt{\sqrt[3]{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}} \cdot \sqrt{\sqrt[3]{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*} \cdot \sqrt[3]{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}\right))_*}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*} - b}\\ \end{array}\]

   8. Simplified14.5

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

   9. Simplified14.5

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

   if 1.4321471445306602e+153 < b

   1. Initial program 60.3

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

   2. Initial simplification60.3

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

   3. Taylor expanded around 0 1.6

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

4. Final simplification13.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 1.4321471445306602 \cdot 10^{+153}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{-(\left(\sqrt{\sqrt[3]{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}\right) \cdot \left(\left|\sqrt[3]{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}\right|\right) + b)_*}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*} - b}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*} - b}\\ \end{array}\]

Runtime

Time bar (total: 25.8s)Debug logProfile

herbie shell --seed 2018252 +o rules:numerics
(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)))))))