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Target

Derivation

1. Split input into 4 regimes

2. if b < -1.0373992183972371e+38 or -7.587065739737278e-20 < b < -2.2869707272504606e-173

   1. Initial program 49.0

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

   2. Taylor expanded around -inf 47.4

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

   3. Applied simplify13.0

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

   if -1.0373992183972371e+38 < b < -7.587065739737278e-20

   1. Initial program 43.1

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

   3. Applied flip--43.2

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

   4. Applied simplify12.9

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

   5. Applied simplify12.9

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

   if -2.2869707272504606e-173 < b < 3.5962548250302795e+78

   1. Initial program 10.2

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

   3. Applied frac-2neg10.2

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

   4. Applied simplify10.2

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

   if 3.5962548250302795e+78 < b

   1. Initial program 40.5

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

   2. Taylor expanded around inf 3.9

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

   3. Applied simplify3.9

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

4. Applied simplify10.4

   \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \le -1.0373992183972371 \cdot 10^{+38}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{if}\;b \le -7.587065739737278 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{\left(a \cdot c\right) \cdot 4}{\sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)} - b}}{2 \cdot a}\\ \mathbf{if}\;b \le -2.2869707272504606 \cdot 10^{-173}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{if}\;b \le 3.5962548250302795 \cdot 10^{+78}:\\ \;\;\;\;\frac{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{-2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}}\]

Runtime

Time bar (total: 2.8m)Debug logProfile

herbie shell --seed 2018199 
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :herbie-target
  (if (< b 0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))