Average Error: 28.6 → 0.4
Time: 4.5s
Precision: binary64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[c \cdot \frac{1}{\left(-b\right) - \sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 28.6

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

  3. Applied flip-+28.6

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

  4. Simplified0.6

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

  5. Simplified0.6

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

  7. Applied *-un-lft-identity0.6

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

  8. Applied times-frac0.6

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

  9. Applied associate-/l*0.6

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

  10. Simplified0.4

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

  12. Applied associate-*r/0.4

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

  13. Applied associate-*r/0.5

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

  14. Applied associate-/r/0.5

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

  15. Simplified0.4

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

  16. Final simplification0.4

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

Reproduce

herbie shell --seed 2020185 
(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)))