Error

Derivation

1. Initial program 43.9

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

3. Applied flip-+43.9

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

4. Applied associate-/l/43.9

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

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 sub-neg0.4

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

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

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

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

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

12. Applied associate-/r*0.4

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

13. Simplified0.4

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

15. Applied add-sqr-sqrt0.5

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

16. Final simplification0.5

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

Runtime

Time bar (total: 45.1s)Debug logProfile

herbie shell --seed 2018297 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))