Average Error: 28.8 → 0.5
Time: 6.0s
Precision: binary64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\sqrt{3}}{\frac{\sqrt{3}}{a}} \cdot \frac{c \cdot 1}{a \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 28.8

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

  3. Applied flip-+28.8

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

  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 div-inv0.6

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

  7. Applied times-frac0.6

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

  8. Simplified0.5

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

  10. Applied add-sqr-sqrt0.9

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

  11. Applied times-frac0.8

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

  12. Applied add-sqr-sqrt0.6

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

  13. Applied times-frac0.6

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

  14. Applied associate-*l*0.6

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

  15. Simplified0.5

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

  16. Final simplification0.5

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

Reproduce

herbie shell --seed 2020169 
(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)))