Average Error: 28.3 → 0.3
Time: 1.3m
Precision: 64
Internal Precision: 576
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{-4}{2} \cdot \frac{c}{\sqrt{\frac{{b}^{4} - \left(4 \cdot \left(c \cdot a\right)\right) \cdot \left(4 \cdot \left(c \cdot a\right)\right)}{b \cdot b + 4 \cdot \left(c \cdot a\right)}} + b}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Initial simplification28.3

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

  4. Applied flip--28.3

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

  5. Applied associate-/l/28.3

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

  6. Simplified0.5

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

  8. Applied times-frac0.3

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

  9. Simplified0.3

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

  11. Applied flip--0.3

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

  12. Taylor expanded around 0 0.3

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

  13. Final simplification0.3

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

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018225 
(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)))