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Target

Derivation

1. Split input into 4 regimes

2. if b < -4.1796980969696804e+30

   1. Initial program 56.6

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

   2. Initial simplification56.6

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

   4. Applied associate-/r*56.6

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

   6. Applied div-inv56.6

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

   7. Taylor expanded around -inf 4.7

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

   8. Simplified4.7

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

   if -4.1796980969696804e+30 < b < -2.5235826594530557e-108

   1. Initial program 37.7

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

   2. Initial simplification37.7

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

   4. Applied flip--37.7

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

   5. Applied associate-/l/40.9

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

   6. Simplified19.1

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

   if -2.5235826594530557e-108 < b < 5.2381578698044655e+62

   1. Initial program 12.3

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

   2. Initial simplification12.3

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

   4. Applied clear-num12.4

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

   if 5.2381578698044655e+62 < b

   1. Initial program 37.6

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

   2. Initial simplification37.6

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

   3. Taylor expanded around inf 5.3

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

4. Final simplification9.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.1796980969696804 \cdot 10^{+30}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -2.5235826594530557 \cdot 10^{-108}:\\ \;\;\;\;\frac{4 \cdot \left(c \cdot a\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) + \sqrt{-4 \cdot \left(c \cdot a\right) + b \cdot b}\right)}\\ \mathbf{elif}\;b \le 5.2381578698044655 \cdot 10^{+62}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\left(-b\right) - \sqrt{-4 \cdot \left(c \cdot a\right) + b \cdot b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Runtime

Time bar (total: 34.8s)Debug logProfile

herbie shell --seed 2018273 
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :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)))