Average Error: 28.3 → 0.3
Time: 4.3s
Precision: binary64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{4}{2} \cdot \frac{c}{\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(4 \cdot \left(c \cdot a\right)\right)}^{3}}{{b}^{4} + 4 \cdot \left(a \cdot \left(c \cdot \left(4 \cdot \left(c \cdot a\right) + b \cdot b\right)\right)\right)}}}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

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. Simplified0.5

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

  5. Simplified0.5

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

  7. Applied *-un-lft-identity0.5

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

  8. Applied times-frac0.5

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

  9. Applied times-frac0.5

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

  10. Simplified0.5

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

  11. Simplified0.3

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

  13. Applied flip3--0.3

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

  14. Simplified0.3

    \[\leadsto \frac{4}{2} \cdot \frac{c}{\left(-b\right) - \sqrt{\frac{\color{blue}{{b}^{6} - {\left(4 \cdot \left(a \cdot c\right)\right)}^{3}}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(4 \cdot \left(a \cdot c\right)\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right) + \left(b \cdot b\right) \cdot \left(4 \cdot \left(a \cdot c\right)\right)\right)}}}\]

  15. Simplified0.3

    \[\leadsto \frac{4}{2} \cdot \frac{c}{\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(4 \cdot \left(a \cdot c\right)\right)}^{3}}{\color{blue}{{b}^{4} + 4 \cdot \left(a \cdot \left(c \cdot \left(b \cdot b + 4 \cdot \left(a \cdot c\right)\right)\right)\right)}}}}\]

  16. Final simplification0.3

    \[\leadsto \frac{4}{2} \cdot \frac{c}{\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(4 \cdot \left(c \cdot a\right)\right)}^{3}}{{b}^{4} + 4 \cdot \left(a \cdot \left(c \cdot \left(4 \cdot \left(c \cdot a\right) + b \cdot b\right)\right)\right)}}}\]

Reproduce

herbie shell --seed 2020179 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (neg b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))