Average Error: 29.2 → 0.3
Time: 5.2s
Precision: binary64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{c}{\left(-b\right) - \sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)}}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 29.2

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

  3. Applied flip-+29.1

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

  4. Simplified0.6

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

  5. Simplified0.6

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

  7. Applied frac-2neg0.6

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

  8. Simplified0.5

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

  9. Simplified0.5

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

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

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

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

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

  13. Applied times-frac0.5

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

  14. Applied times-frac0.6

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

  15. Applied times-frac0.5

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

  16. Simplified0.5

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

  17. Simplified0.3

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

  18. Final simplification0.3

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

Reproduce

herbie shell --seed 2020182 
(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)))