Average Error: 28.2 → 0.5
Time: 8.2s
Precision: binary64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}}{3} \cdot \frac{a \cdot c}{a}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 28.2

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

  3. Applied flip-+28.2

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

  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 clear-num0.6

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

  7. Simplified0.6

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

  9. Applied div-inv0.6

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

  10. Applied add-sqr-sqrt0.6

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

  11. Applied times-frac0.6

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

  12. Applied times-frac0.6

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

  13. Simplified0.6

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

  14. Simplified0.5

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

  15. Final simplification0.5

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

Reproduce

herbie shell --seed 2020162 
(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)))