Cubic critical, wide range

Average Error: 52.1 → 0.1

Time: 4.8s

Precision: binary64

\[4.93038 \cdot 10^{-32} \lt a \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt b \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt c \lt 2.02824 \cdot 10^{31}\]

Math

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

↓

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

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.1

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

3. Applied flip-+ 52.1

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

4. Simplified 0.5

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

6. Applied *-un-lft-identity 0.5

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

7. Applied *-un-lft-identity 0.5

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

8. Applied times-frac 0.5

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

9. Applied times-frac 0.6

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

10. Simplified 0.6

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

11. Simplified 0.6

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

13. Applied pow1 0.6

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

14. Applied pow1 0.6

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

15. Applied pow-prod-down 0.6

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

16. Simplified 0.3

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

18. Applied *-un-lft-identity 0.3

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

19. Applied times-frac 0.3

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

20. Applied add-cube-cbrt 0.3

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

21. Applied times-frac 0.3

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

22. Simplified 0.3

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

23. Simplified 0.1

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

24. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020171 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (< 4.930380657631324e-32 a 2.028240960365167e+31) (< 4.930380657631324e-32 b 2.028240960365167e+31) (< 4.930380657631324e-32 c 2.028240960365167e+31))
  (/ (+ (neg b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))