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Target

Derivation

1. Split input into 4 regimes

2. if b < -1.3565616138219603e+114

   1. Initial program 48.8

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

   2. Initial simplification48.8

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

   4. Applied div-inv48.8

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

   5. Taylor expanded around -inf 3.8

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

   if -1.3565616138219603e+114 < b < -2.3557420274181203e-276

   1. Initial program 8.8

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

   2. Initial simplification8.8

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

   if -2.3557420274181203e-276 < b < 2.1917466413699358e+71

   1. Initial program 29.3

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

   2. Initial simplification29.3

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

   4. Applied div-inv29.3

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

   6. Applied flip--29.4

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

   7. Applied associate-*l/29.5

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

   8. Simplified16.0

      \[\leadsto \frac{\color{blue}{\frac{-4 \cdot \left(a \cdot c\right) + 0}{\frac{a}{\frac{1}{2}}}}}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}\]

   9. Taylor expanded around inf 9.5

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

   if 2.1917466413699358e+71 < b

   1. Initial program 57.4

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

   2. Initial simplification57.4

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

   4. Applied div-inv57.4

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

   5. Taylor expanded around inf 3.3

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

   6. Simplified3.3

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

4. Final simplification6.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.3565616138219603 \cdot 10^{+114}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le -2.3557420274181203 \cdot 10^{-276}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 2.1917466413699358 \cdot 10^{+71}:\\ \;\;\;\;\frac{c \cdot -2}{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018263 
(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)))