Error

Target

Derivation

1. Split input into 3 regimes

2. if b < 1.8155453498736955e-116

   1. Initial program 20.6

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

   2. Applied simplify20.5

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

   if 1.8155453498736955e-116 < b < 7.199839052582607e+137

   1. Initial program 42.2

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

   2. Applied simplify42.2

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

   4. Applied flip--42.3

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

   5. Applied simplify15.5

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

   7. Applied add-cube-cbrt16.2

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\frac{\left(c \cdot a\right) \cdot \left(-4\right)}{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} + b}}{2 \cdot a}} \cdot \sqrt[3]{\frac{\frac{\left(c \cdot a\right) \cdot \left(-4\right)}{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} + b}}{2 \cdot a}}\right) \cdot \sqrt[3]{\frac{\frac{\left(c \cdot a\right) \cdot \left(-4\right)}{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} + b}}{2 \cdot a}}}\]

   8. Applied simplify16.3

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}} \cdot \sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}}\right)} \cdot \sqrt[3]{\frac{\frac{\left(c \cdot a\right) \cdot \left(-4\right)}{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} + b}}{2 \cdot a}}\]

   9. Applied simplify7.2

      \[\leadsto \left(\sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}} \cdot \sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}}\right) \cdot \color{blue}{\sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}}}\]
   10. Using strategy rm

   11. Applied add-cube-cbrt7.6

       \[\leadsto \left(\sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}} \cdot \sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}}} \cdot \sqrt[3]{\sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}}}\right)}\]

   12. Applied simplify7.6

       \[\leadsto \left(\sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}} \cdot \sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}}\right) \cdot \left(\color{blue}{\left(\sqrt[3]{\sqrt[3]{\frac{\left(-c\right) \cdot \frac{4}{2}}{b + \sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*}}}} \cdot \sqrt[3]{\sqrt[3]{\frac{\left(-c\right) \cdot \frac{4}{2}}{b + \sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*}}}}\right)} \cdot \sqrt[3]{\sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}}}\right)\]

   13. Applied simplify7.6

       \[\leadsto \left(\sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}} \cdot \sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\frac{\left(-c\right) \cdot \frac{4}{2}}{b + \sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*}}}} \cdot \sqrt[3]{\sqrt[3]{\frac{\left(-c\right) \cdot \frac{4}{2}}{b + \sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*}}}}\right) \cdot \color{blue}{\sqrt[3]{\sqrt[3]{\frac{\left(-c\right) \cdot \frac{4}{2}}{b + \sqrt{(\left(-4\right) \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}}}}\right)\]

   if 7.199839052582607e+137 < b

   1. Initial program 61.4

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

   2. Applied simplify61.4

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

   4. Applied flip--61.4

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

   5. Applied simplify37.7

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

   6. Taylor expanded around 0 14.7

      \[\leadsto \frac{\frac{\left(c \cdot a\right) \cdot \left(-4\right)}{\color{blue}{b} + b}}{2 \cdot a}\]
3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed 2018214 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

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

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