Average Error: 28.5 → 0.4
Time: 5.0s
Precision: binary64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{1}{\frac{-b}{c} + \frac{-\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{c}}\]

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. 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

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

  6. Applied associate-*r*0.5

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

  8. Applied clear-num0.5

    \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{\frac{0 + \left(3 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}}\]

  9. Simplified0.5

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

  11. Applied sub-neg0.5

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

  12. Applied distribute-lft-in0.5

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

  13. Simplified0.4

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

  14. Simplified0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2020175 
(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)))