Error

Target

Derivation

1. Split input into 3 regimes

2. if b < 2.101736454867319e-250

   1. Initial program 20.6

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

   2. Initial simplification20.6

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

   3. Taylor expanded around 0 20.6

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

   4. Simplified20.6

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

   if 2.101736454867319e-250 < b < 5.539836417000253e+151

   1. Initial program 36.0

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

   2. Initial simplification36.0

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

   3. Taylor expanded around 0 36.0

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

   4. Simplified36.0

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

   6. Applied div-inv36.1

      \[\leadsto \color{blue}{\left(\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b\right) \cdot \frac{1}{2 \cdot a}}\]
   7. Using strategy rm

   8. Applied flip--36.1

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

   9. Applied associate-*l/36.2

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

   10. Simplified14.1

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

   11. Taylor expanded around inf 7.7

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

   if 5.539836417000253e+151 < b

   1. Initial program 62.7

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

   2. Initial simplification62.7

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

   3. Taylor expanded around 0 62.7

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

   4. Simplified62.8

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

   6. Applied div-inv62.8

      \[\leadsto \color{blue}{\left(\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b\right) \cdot \frac{1}{2 \cdot a}}\]
   7. Using strategy rm

   8. Applied flip--62.8

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

   9. Applied associate-*l/62.8

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

   10. Simplified37.0

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

   11. Taylor expanded around 0 7.8

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

4. Final simplification14.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 2.101736454867319 \cdot 10^{-250}:\\ \;\;\;\;\frac{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 5.539836417000253 \cdot 10^{+151}:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{2}}{a} \cdot \left(\left(c \cdot -4\right) \cdot a\right)}{b + b}\\ \end{array}\]

Runtime

Time bar (total: 18.3s)Debug logProfile

herbie shell --seed 2018290 +o rules:numerics
(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)))