Error

Derivation

1. Split input into 3 regimes

2. if b_2 < -2.8671071746705253e+127

   1. Initial program 51.2

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

   2. Simplified51.2

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]

   3. Taylor expanded around -inf 2.8

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]

   4. Simplified2.8

      \[\leadsto \color{blue}{(-2 \cdot \left(\frac{b_2}{a}\right) + \left(\frac{c}{\frac{b_2}{\frac{1}{2}}}\right))_*}\]

   if -2.8671071746705253e+127 < b_2 < 9.637680726124706e-24

   1. Initial program 15.0

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

   2. Simplified15.0

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]

   3. Taylor expanded around 0 15.0

      \[\leadsto \frac{\sqrt{\color{blue}{{b_2}^{2} - a \cdot c}} - b_2}{a}\]
   4. Using strategy rm

   5. Applied *-un-lft-identity15.0

      \[\leadsto \frac{\color{blue}{1 \cdot \left(\sqrt{{b_2}^{2} - a \cdot c} - b_2\right)}}{a}\]

   6. Applied associate-/l*15.2

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\sqrt{{b_2}^{2} - a \cdot c} - b_2}}}\]

   if 9.637680726124706e-24 < b_2

   1. Initial program 54.3

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

   2. Simplified54.3

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]

   3. Taylor expanded around 0 54.3

      \[\leadsto \frac{\sqrt{\color{blue}{{b_2}^{2} - a \cdot c}} - b_2}{a}\]

   4. Taylor expanded around inf 6.8

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]
3. Recombined 3 regimes into one program.

4. Final simplification10.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.8671071746705253 \cdot 10^{+127}:\\ \;\;\;\;(-2 \cdot \left(\frac{b_2}{a}\right) + \left(\frac{c}{\frac{b_2}{\frac{1}{2}}}\right))_*\\ \mathbf{elif}\;b_2 \le 9.637680726124706 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{{b_2}^{2} - c \cdot a} - b_2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

Details

Time bar (total: 20.4s)Debug log