Error

Derivation

1. Initial program 28.3

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

2. Initial simplification28.3

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

4. Applied flip--28.3

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

5. Applied associate-/l/28.3

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

6. Simplified0.4

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

8. Applied associate-/l*0.4

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

10. Applied div-inv0.5

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

11. Applied times-frac0.5

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

12. Simplified0.4

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

13. Simplified0.4

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

14. Final simplification0.4

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

Runtime

Time bar (total: 26.9s)Debug logProfile

herbie shell --seed 2018354 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))