Cubic critical, narrow range

Average Error: 28.6 → 0.3

Time: 20.5s

Precision: 64

\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]

Math

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

↓

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

C

double f(double a, double b, double c) {
        double r100374 = b;
        double r100375 = -r100374;
        double r100376 = r100374 * r100374;
        double r100377 = 3.0;
        double r100378 = a;
        double r100379 = r100377 * r100378;
        double r100380 = c;
        double r100381 = r100379 * r100380;
        double r100382 = r100376 - r100381;
        double r100383 = sqrt(r100382);
        double r100384 = r100375 + r100383;
        double r100385 = r100384 / r100379;
        return r100385;
}

↓

double f(double a, double b, double c) {
        double r100386 = c;
        double r100387 = b;
        double r100388 = -r100387;
        double r100389 = r100387 * r100387;
        double r100390 = 3.0;
        double r100391 = a;
        double r100392 = r100390 * r100391;
        double r100393 = r100392 * r100386;
        double r100394 = r100389 - r100393;
        double r100395 = sqrt(r100394);
        double r100396 = r100388 - r100395;
        double r100397 = r100386 / r100396;
        return r100397;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 28.6

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

3. Applied flip-+ 28.6

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

4. Simplified 0.6

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

6. Applied *-un-lft-identity 0.6

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

7. Applied *-un-lft-identity 0.6

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

8. Applied times-frac 0.6

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

9. Applied associate-/l* 0.6

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

10. Simplified 0.5

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

12. Applied *-un-lft-identity 0.5

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

13. Applied times-frac 0.4

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

14. Applied associate-/r* 0.4

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

15. Simplified 0.4

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

17. Applied *-un-lft-identity 0.4

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

18. Applied *-un-lft-identity 0.4

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

19. Applied times-frac 0.4

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

20. Applied add-sqr-sqrt 0.4

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

21. Applied times-frac 0.4

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

22. Applied add-cube-cbrt 0.4

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

23. Applied add-cube-cbrt 0.4

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

24. Applied times-frac 0.4

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

25. Applied times-frac 0.4

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

26. Simplified 0.4

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

27. Simplified 0.3

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

28. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019326 
(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))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))