Error

Derivation

1. Split input into 2 regimes

2. if b < 1.0359194168322235e+82

   1. Initial program 15.1

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

   2. Initial simplification15.1

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

   4. Applied add-sqr-sqrt15.1

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

   5. Applied sqrt-prod15.2

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

   7. Applied add-cube-cbrt15.2

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

   8. Applied sqrt-prod15.2

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

   9. Simplified15.2

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

   11. Applied add-cube-cbrt15.2

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

   12. Applied cbrt-prod15.3

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

   if 1.0359194168322235e+82 < b

   1. Initial program 41.7

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

   2. Initial simplification41.7

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

   3. Taylor expanded around 0 5.1

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

4. Final simplification13.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 1.0359194168322235 \cdot 10^{+82}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}} \cdot \sqrt{\sqrt{\sqrt[3]{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}} \cdot \left|\sqrt[3]{\sqrt[3]{(\left(c \cdot -4\right) \cdot a + \left(b \cdot b\right))_*} \cdot \sqrt[3]{(\left(c \cdot -4\right) \cdot a + \left(b \cdot b\right))_*}} \cdot \sqrt[3]{\sqrt[3]{(\left(c \cdot -4\right) \cdot a + \left(b \cdot b\right))_*}}\right|}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*} - b}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*} - b}\\ \end{array}\]

Runtime

Time bar (total: 24.7s)Debug logProfile

herbie shell --seed 2018249 +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)))))))