Average Error: 28.5 → 0.3
Time: 4.7s
Precision: binary64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{-c}{b + \sqrt{\frac{{b}^{4} - 3 \cdot \left(3 \cdot \left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)\right)}{3 \cdot \left(c \cdot a\right) + b \cdot b}}}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 28.5

    \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

  2. Simplified28.5

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

  4. Applied flip--28.5

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

  5. Simplified0.6

    \[\leadsto \frac{\frac{\color{blue}{0 - 3 \cdot \left(a \cdot c\right)}}{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} + b}}{3 \cdot a}\]

  6. Simplified0.6

    \[\leadsto \frac{\frac{0 - 3 \cdot \left(a \cdot c\right)}{\color{blue}{b + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}}}{3 \cdot a}\]
  7. Using strategy rm

  8. Applied sub0-neg0.6

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

  9. Applied distribute-frac-neg0.6

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

  10. Applied distribute-frac-neg0.6

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

  11. Simplified0.3

    \[\leadsto -\color{blue}{1 \cdot \frac{c}{b + \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}}\]
  12. Using strategy rm

  13. Applied flip--0.3

    \[\leadsto -1 \cdot \frac{c}{b + \sqrt{\color{blue}{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(3 \cdot \left(a \cdot c\right)\right) \cdot \left(3 \cdot \left(a \cdot c\right)\right)}{b \cdot b + 3 \cdot \left(a \cdot c\right)}}}}\]

  14. Simplified0.3

    \[\leadsto -1 \cdot \frac{c}{b + \sqrt{\frac{\color{blue}{{b}^{4} - 3 \cdot \left(3 \cdot \left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)\right)}}{b \cdot b + 3 \cdot \left(a \cdot c\right)}}}\]

  15. Simplified0.3

    \[\leadsto -1 \cdot \frac{c}{b + \sqrt{\frac{{b}^{4} - 3 \cdot \left(3 \cdot \left(a \cdot \left(a \cdot \left(c \cdot c\right)\right)\right)\right)}{\color{blue}{3 \cdot \left(a \cdot c\right) + b \cdot b}}}}\]

  16. Final simplification0.3

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

Reproduce

herbie shell --seed 2020184 
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3.0 a) c)))) (* 3.0 a)))