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Target

Derivation

1. Split input into 4 regimes

2. if b < -6.215459245046153e+88

   1. Initial program 42.9

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

   2. Taylor expanded around -inf 3.6

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

   3. Applied simplify3.6

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

   if -6.215459245046153e+88 < b < 2.2207920706136116e-291

   1. Initial program 9.1

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

   3. Applied add-sqr-sqrt9.4

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

   4. Applied associate-/l*9.4

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

   5. Applied simplify9.4

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

   if 2.2207920706136116e-291 < b < 2.88215723413983e+143

   1. Initial program 35.1

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

   3. Applied flip-+35.2

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

   4. Applied simplify15.7

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

   6. Applied add-cube-cbrt16.4

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\frac{\left(a \cdot c\right) \cdot 4}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}} \cdot \sqrt[3]{\frac{\frac{\left(a \cdot c\right) \cdot 4}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}}\right) \cdot \sqrt[3]{\frac{\frac{\left(a \cdot c\right) \cdot 4}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}}}\]

   7. Applied simplify16.3

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\frac{4 \cdot c}{\frac{2}{1}}}{\left(-b\right) - \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}} \cdot \sqrt[3]{\frac{\frac{4 \cdot c}{\frac{2}{1}}}{\left(-b\right) - \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}}\right)} \cdot \sqrt[3]{\frac{\frac{\left(a \cdot c\right) \cdot 4}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}}\]

   8. Applied simplify9.0

      \[\leadsto \left(\sqrt[3]{\frac{\frac{4 \cdot c}{\frac{2}{1}}}{\left(-b\right) - \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}} \cdot \sqrt[3]{\frac{\frac{4 \cdot c}{\frac{2}{1}}}{\left(-b\right) - \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}}\right) \cdot \color{blue}{\sqrt[3]{\frac{\frac{1}{2} \cdot \left(4 \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}}}\]

   if 2.88215723413983e+143 < b

   1. Initial program 61.7

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

   3. Applied flip-+61.7

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

   4. Applied simplify35.7

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

   5. Taylor expanded around inf 13.1

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

   6. Applied simplify1.2

      \[\leadsto \color{blue}{\frac{c \cdot \left(1 \cdot \frac{4}{2}\right)}{\frac{c \cdot 2}{\frac{b}{a}} + \left(-\left(b + b\right)\right)}}\]
3. Recombined 4 regimes into one program.

4. Applied simplify6.7

   \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \le -6.215459245046153 \cdot 10^{+88}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{if}\;b \le 2.2207920706136116 \cdot 10^{-291}:\\ \;\;\;\;\frac{\sqrt{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} + \left(-b\right)}}{\frac{a \cdot 2}{\sqrt{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}}}\\ \mathbf{if}\;b \le 2.88215723413983 \cdot 10^{+143}:\\ \;\;\;\;\sqrt[3]{\frac{\frac{1}{2} \cdot \left(4 \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}} \cdot \left(\sqrt[3]{\frac{\frac{4 \cdot c}{2}}{\left(-b\right) - \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}} \cdot \sqrt[3]{\frac{\frac{4 \cdot c}{2}}{\left(-b\right) - \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4}{2} \cdot c}{\frac{c \cdot 2}{\frac{b}{a}} + \left(\left(-b\right) + \left(-b\right)\right)}\\ \end{array}}\]

Runtime

Time bar (total: 3.0m)Debug logProfile

herbie shell --seed 2019053 
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 0) (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))