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. Using strategy rm

3. Applied flip-+28.5

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

4. Applied associate-/l/28.5

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

5. Simplified0.5

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

7. Applied sub-neg0.5

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

8. Applied distribute-lft-in0.4

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

9. Simplified0.4

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

11. Applied *-un-lft-identity0.4

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

12. Applied times-frac0.4

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

13. Simplified0.4

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

14. Simplified0.3

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

16. Applied add-cbrt-cube0.3

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

17. Final simplification0.3

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

Reproduce

herbie shell --seed 2019068 +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)))