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Target

Derivation

1. Split input into 4 regimes

2. if b < -6.193865508518113e+143

   1. Initial program 57.8

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

   2. Initial simplification57.8

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

   3. Taylor expanded around -inf 2.5

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

   if -6.193865508518113e+143 < b < 6.754842698368202e-188

   1. Initial program 9.5

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

   2. Initial simplification9.5

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

   if 6.754842698368202e-188 < b < 24.66056151892245

   1. Initial program 32.7

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

   2. Initial simplification32.7

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

   4. Applied flip--32.8

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

   5. Applied associate-/l/36.1

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

   6. Simplified23.7

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

   8. Applied associate-/l*17.9

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

   if 24.66056151892245 < b

   1. Initial program 55.0

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

   2. Initial simplification55.0

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

   3. Taylor expanded around inf 5.7

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

   4. Simplified5.7

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

4. Final simplification8.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.193865508518113 \cdot 10^{+143}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 6.754842698368202 \cdot 10^{-188}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 24.66056151892245:\\ \;\;\;\;\frac{c \cdot -4}{\frac{\left(\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + b\right) \cdot \left(a \cdot 2\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Runtime

Time bar (total: 42.3s)Debug logProfile

herbie shell --seed 2018278 
(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)))