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. Using strategy rm

3. Applied flip-+28.3

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

4. Applied associate-/l/28.3

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

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-rgt-in0.4

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

9. Simplified0.4

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

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

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

12. Applied times-frac0.4

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

13. Simplified0.4

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

14. Simplified0.3

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

16. Applied add-sqr-sqrt0.3

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

17. Applied sqrt-prod0.4

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

18. Applied fma-def0.3

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

19. Final simplification0.3

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018304 +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)))