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Target

Derivation

1. Split input into 4 regimes

2. if b < -3.1121915926751862e+81 or -6.495235596303068e-31 < b < -5.2599359043084235e-101

   1. Initial program 53.6

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

   2. Taylor expanded around -inf 44.2

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

   3. Applied simplify8.0

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

   if -3.1121915926751862e+81 < b < -6.495235596303068e-31

   1. Initial program 45.7

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

   3. Applied div-inv45.7

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

   5. Applied flip--45.8

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

   6. Applied simplify15.9

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

   7. Applied simplify16.0

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

   if -5.2599359043084235e-101 < b < 6.045122565443192e+91

   1. Initial program 11.8

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

   3. Applied div-inv11.9

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

   if 6.045122565443192e+91 < b

   1. Initial program 42.4

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

   2. Taylor expanded around inf 4.5

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

   3. Applied simplify4.5

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

4. Applied simplify10.0

   \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \le -3.1121915926751862 \cdot 10^{+81}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{if}\;b \le -6.495235596303068 \cdot 10^{-31}:\\ \;\;\;\;\frac{\left(a \cdot c\right) \cdot 4}{\sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)} - b} \cdot \frac{1}{2 \cdot a}\\ \mathbf{if}\;b \le -5.2599359043084235 \cdot 10^{-101}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{if}\;b \le 6.045122565443192 \cdot 10^{+91}:\\ \;\;\;\;\frac{1}{2 \cdot a} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}}\]

Runtime

Time bar (total: 2.6m)Debug logProfile

herbie shell --seed 2018166 
(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)))