Error

Derivation

1. Initial program 28.5

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

2. Simplified28.5

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

4. Applied flip--28.5

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

6. Applied *-un-lft-identity28.5

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

7. Applied *-un-lft-identity28.5

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

8. Applied div-inv28.5

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

9. Applied times-frac28.5

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

10. Applied times-frac28.5

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

11. Simplified0.5

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

12. Simplified0.5

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

14. Applied pow10.5

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

15. Applied pow10.5

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

16. Applied pow10.5

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

17. Applied pow10.5

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

18. Applied pow-prod-down0.5

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

19. Applied pow-prod-down0.5

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

20. Applied pow-prod-down0.5

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

21. Simplified0.3

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

22. Final simplification0.3

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

Reproduce

herbie shell --seed 2019094 +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)))