Error

Derivation

1. Initial program 28.7

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

3. Applied flip-+28.8

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

4. Applied associate-/l/28.8

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

5. Simplified0.4

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

7. Applied times-frac0.5

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

8. Simplified0.5

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

9. Simplified0.5

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

11. Applied pow10.5

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

12. Applied pow10.5

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

13. Applied pow-prod-down0.5

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

14. Simplified0.3

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

16. Applied div-inv0.4

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

17. Final simplification0.4

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

Reproduce

herbie shell --seed 2019007 +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)))

Details

Time bar (total: 43.5s)Debug log