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Target

Derivation

1. Split input into 4 regimes

2. if b < -8.3719584206639e+49

   1. Initial program 35.5

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

   2. Initial simplification35.5

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

   3. Taylor expanded around -inf 6.5

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

   if -8.3719584206639e+49 < b < 6.389550873438033e-306

   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

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

   4. Applied div-inv9.7

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

   if 6.389550873438033e-306 < b < 8.385120305158761e+80

   1. Initial program 31.1

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

   2. Initial simplification31.1

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

   4. Applied flip--31.2

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

   5. Applied associate-/l/36.2

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

   6. Simplified22.4

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

   8. Applied associate-/r*16.4

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

   9. Taylor expanded around 0 9.4

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

   if 8.385120305158761e+80 < b

   1. Initial program 57.7

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

   2. Initial simplification57.7

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

   3. Taylor expanded around inf 2.9

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

   4. Simplified2.9

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

4. Final simplification7.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.3719584206639 \cdot 10^{+49}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 6.389550873438033 \cdot 10^{-306}:\\ \;\;\;\;\left(\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{elif}\;b \le 8.385120305158761 \cdot 10^{+80}:\\ \;\;\;\;\frac{-2 \cdot c}{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Runtime

Time bar (total: 41.8s)Debug logProfile

herbie shell --seed 2018339 
(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)))