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Derivation

1. Split input into 4 regimes

2. if b < -8.852140361364619e+83

   1. Initial program 41.0

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

   2. Taylor expanded around -inf 4.2

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

   if -8.852140361364619e+83 < b < 2.2718361572016638e-107

   1. Initial program 11.5

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

   3. Applied *-un-lft-identity11.5

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

   4. Applied *-un-lft-identity11.5

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

   5. Applied distribute-lft-out11.5

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

   6. Applied associate-/l*11.6

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

   if 2.2718361572016638e-107 < b < 1.0536625090564815e+86

   1. Initial program 41.0

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

   3. Applied flip-+41.1

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

   4. Applied associate-/l/44.1

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

   5. Simplified19.6

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

   if 1.0536625090564815e+86 < b

   1. Initial program 57.9

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

   2. Taylor expanded around inf 2.6

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

   3. Simplified2.6

      \[\leadsto \color{blue}{\frac{-c}{b}}\]
3. Recombined 4 regimes into one program.

4. Final simplification9.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.852140361364619 \cdot 10^{+83}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 2.2718361572016638 \cdot 10^{-107}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} + \left(-b\right)}}\\ \mathbf{elif}\;b \le 1.0536625090564815 \cdot 10^{+86}:\\ \;\;\;\;\frac{4 \cdot \left(c \cdot a\right)}{\left(\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}\right) \cdot \left(a \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Runtime

Time bar (total: 24.8s)Debug logProfile

herbie shell --seed 2018286 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))