Error

Derivation

1. Split input into 4 regimes

2. if b < -2.687654240788987e+144

   1. Initial program 57.4

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

   2. Taylor expanded around -inf 3.1

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b} - \frac{2}{3} \cdot \frac{b}{a}}\]

   3. Simplified3.1

      \[\leadsto \color{blue}{(\frac{-2}{3} \cdot \left(\frac{b}{a}\right) + \left(\frac{c}{\frac{b}{\frac{1}{2}}}\right))_*}\]

   if -2.687654240788987e+144 < b < 1.7618946534600747e-308

   1. Initial program 8.4

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

   3. Applied associate-/r*8.4

      \[\leadsto \color{blue}{\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}}\]

   if 1.7618946534600747e-308 < b < 1.7958087442557617e+106

   1. Initial program 32.4

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

   3. Applied flip-+32.5

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a}\]

   4. Applied associate-/l/36.8

      \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(3 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}\]

   5. Simplified20.1

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

   7. Applied times-frac16.1

      \[\leadsto \color{blue}{\frac{3}{3 \cdot a} \cdot \frac{c \cdot a}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}\]

   8. Simplified16.0

      \[\leadsto \color{blue}{\frac{1}{a}} \cdot \frac{c \cdot a}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}\]

   9. Simplified16.0

      \[\leadsto \frac{1}{a} \cdot \color{blue}{\frac{c \cdot a}{\left(-b\right) - \sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}}\]
   10. Using strategy rm

   11. Applied pow116.0

       \[\leadsto \frac{1}{a} \cdot \color{blue}{{\left(\frac{c \cdot a}{\left(-b\right) - \sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}\right)}^{1}}\]

   12. Applied pow116.0

       \[\leadsto \color{blue}{{\left(\frac{1}{a}\right)}^{1}} \cdot {\left(\frac{c \cdot a}{\left(-b\right) - \sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}\right)}^{1}\]

   13. Applied pow-prod-down16.0

       \[\leadsto \color{blue}{{\left(\frac{1}{a} \cdot \frac{c \cdot a}{\left(-b\right) - \sqrt{(-3 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}\right)}^{1}}\]

   14. Simplified8.2

       \[\leadsto {\color{blue}{\left(\frac{c}{\left(-b\right) - \sqrt{(\left(a \cdot -3\right) \cdot c + \left(b \cdot b\right))_*}}\right)}}^{1}\]

   if 1.7958087442557617e+106 < b

   1. Initial program 59.0

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

   2. Taylor expanded around inf 2.3

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}}\]
3. Recombined 4 regimes into one program.

4. Final simplification6.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.687654240788987 \cdot 10^{+144}:\\ \;\;\;\;(\frac{-2}{3} \cdot \left(\frac{b}{a}\right) + \left(\frac{c}{\frac{b}{\frac{1}{2}}}\right))_*\\ \mathbf{elif}\;b \le 1.7618946534600747 \cdot 10^{-308}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} + \left(-b\right)}{3}}{a}\\ \mathbf{elif}\;b \le 1.7958087442557617 \cdot 10^{+106}:\\ \;\;\;\;\frac{c}{\left(-b\right) - \sqrt{(\left(a \cdot -3\right) \cdot c + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2018360 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))

Details

Time bar (total: 25.8s)Debug log