Cubic critical, narrow range

Average Error: 28.7 → 0.4

Time: 54.7min

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]

Math

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

↓

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

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 28.7

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

2. Simplified 28.7

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

4. Applied flip-- 28.7

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

5. Simplified 0.4

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

7. Applied *-un-lft-identity 0.4

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

8. Applied distribute-lft-neg-in 0.4

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

9. Applied times-frac 0.3

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

10. Applied associate-/l* 0.3

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

12. Applied div-inv 0.4

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

13. Applied times-frac 0.6

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

14. Applied add-cube-cbrt 0.6

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

15. Applied distribute-rgt-neg-in 0.6

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

16. Applied times-frac 0.6

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

17. Applied times-frac 0.5

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

18. Simplified 0.4

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

19. Simplified 0.4

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

21. Applied associate-*l* 0.4

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

22. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020181 
(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)))