Error

Derivation

1. Initial program 52.5

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

3. Applied flip-+52.6

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

4. Applied associate-/l/52.6

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

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 associate-/r*0.2

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

8. Simplified0.2

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

10. Applied *-un-lft-identity0.2

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

11. Applied *-un-lft-identity0.2

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

12. Applied distribute-lft-out--0.2

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

13. Applied times-frac0.2

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

14. Applied times-frac0.2

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

15. Simplified0.2

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

16. Simplified0.1

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

17. Final simplification0.1

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

Reproduce

herbie shell --seed 2019008 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :pre (and (< 4.930380657631324e-32 a 2.028240960365167e+31) (< 4.930380657631324e-32 b 2.028240960365167e+31) (< 4.930380657631324e-32 c 2.028240960365167e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))

Details

Time bar (total: 1.2m)Debug log