Error

Derivation

1. Split input into 2 regimes

2. if b < 7.145891006680855e-06

   1. Initial program 17.7

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

   2. Simplified17.7

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

   4. Applied add-cube-cbrt17.9

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

   5. Applied sqrt-prod17.9

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

   if 7.145891006680855e-06 < b

   1. Initial program 22.4

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

   2. Simplified22.4

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

   3. Taylor expanded around 0 5.6

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

   5. Applied flip3--5.6

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

   6. Applied associate-/l/5.6

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

   7. Simplified5.6

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

4. Final simplification13.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 7.145891006680855 \cdot 10^{-06}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\sqrt[3]{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*} \cdot \sqrt[3]{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}} \cdot \sqrt{\sqrt[3]{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{(b \cdot b + \left(\left(a \cdot c\right) \cdot -4\right))_* \cdot \sqrt{(b \cdot b + \left(\left(a \cdot c\right) \cdot -4\right))_*} - \left(b \cdot b\right) \cdot b}{\left(\left(b \cdot b + \sqrt{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*} \cdot b\right) + \sqrt{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*} \cdot \sqrt{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}\right) \cdot \left(2 \cdot a\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019089 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 2"
  (if (>= b 0) (/ (* 2 c) (- (- b) (sqrt (- (* b b) (* (* 4 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))))