Average Error: 28.9 → 0.4
Time: 6.0s
Precision: binary64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{1}{{\left(\frac{-b}{c} - \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{c}\right)}^{1}}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 28.9

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

  3. Applied flip-+29.0

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

  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

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

  7. Simplified0.6

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

  9. Applied pow10.6

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

  10. Applied pow10.6

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

  11. Applied pow-prod-down0.6

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

  12. Simplified0.4

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

  14. Applied div-sub0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2020164 
(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)))