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Target

Derivation

1. Split input into 4 regimes

2. if b < -2.835340391058544e+148

   1. Initial program 58.7

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

   2. Simplified58.7

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

   3. Taylor expanded around -inf 2.1

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

   if -2.835340391058544e+148 < b < 3.587545867948427e-80

   1. Initial program 12.5

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

   2. Simplified12.5

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

   if 3.587545867948427e-80 < b < 8.686813508697628e-23 or 6.168799619006917e+62 < b

   1. Initial program 54.1

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

   2. Simplified54.1

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

   3. Taylor expanded around inf 6.7

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

   4. Simplified6.7

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

   if 8.686813508697628e-23 < b < 6.168799619006917e+62

   1. Initial program 46.2

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

   2. Simplified46.2

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

   4. Applied clear-num46.2

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

   6. Applied *-un-lft-identity46.2

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

   7. Applied times-frac46.2

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

   8. Applied associate-/r*46.2

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

   10. Applied flip--46.2

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

   11. Applied associate-/r/46.2

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

   12. Applied div-inv46.2

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

   13. Applied times-frac46.2

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

   14. Simplified12.1

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

   15. Simplified12.1

       \[\leadsto \frac{\left(-4 \cdot a\right) \cdot c + 0}{a} \cdot \color{blue}{\frac{\frac{1}{2}}{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} + b}}\]
3. Recombined 4 regimes into one program.

4. Final simplification9.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.835340391058544 \cdot 10^{+148}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 3.587545867948427 \cdot 10^{-80}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 8.686813508697628 \cdot 10^{-23} \lor \neg \left(b \le 6.168799619006917 \cdot 10^{+62}\right):\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2}}{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} + b} \cdot \frac{c \cdot \left(-4 \cdot a\right)}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 
(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)))

Details

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