Error

Target

Derivation

1. Split input into 3 regimes

2. if b < -2.8880468196997556e-80

   1. Initial program 51.8

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

   2. Taylor expanded around inf 51.8

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

   3. Simplified51.8

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

   5. Applied div-inv51.9

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

   6. Taylor expanded around -inf 10.1

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

   7. Simplified10.1

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

   if -2.8880468196997556e-80 < b < 2.743299671584071e+131

   1. Initial program 12.2

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

   2. Taylor expanded around inf 12.2

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

   3. Simplified12.3

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

   5. Applied *-un-lft-identity12.3

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

   6. Applied *-un-lft-identity12.3

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

   7. Applied distribute-lft-out--12.3

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

   8. Applied associate-/l*12.4

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

   if 2.743299671584071e+131 < b

   1. Initial program 53.3

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

   2. Taylor expanded around inf 2.4

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

4. Final simplification10.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.8880468196997556 \cdot 10^{-80}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.743299671584071 \cdot 10^{+131}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Runtime

Time bar (total: 41.7s)Debug logProfile

herbie shell --seed 2018353 +o rules:numerics
(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)))