Cubic critical, narrow range

Average Error: 28.4 → 0.5

Time: 52.6s

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

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

↓

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

C

double f(double a, double b, double c) {
        double r170453 = b;
        double r170454 = -r170453;
        double r170455 = r170453 * r170453;
        double r170456 = 3.0;
        double r170457 = a;
        double r170458 = r170456 * r170457;
        double r170459 = c;
        double r170460 = r170458 * r170459;
        double r170461 = r170455 - r170460;
        double r170462 = sqrt(r170461);
        double r170463 = r170454 + r170462;
        double r170464 = r170463 / r170458;
        return r170464;
}

↓

double f(double a, double b, double c) {
        double r170465 = 1.0;
        double r170466 = 3.0;
        double r170467 = r170465 / r170466;
        double r170468 = r170465 / r170467;
        double r170469 = a;
        double r170470 = b;
        double r170471 = -r170470;
        double r170472 = r170470 * r170470;
        double r170473 = r170466 * r170469;
        double r170474 = c;
        double r170475 = r170473 * r170474;
        double r170476 = r170472 - r170475;
        double r170477 = sqrt(r170476);
        double r170478 = r170471 - r170477;
        double r170479 = r170478 / r170474;
        double r170480 = r170469 / r170479;
        double r170481 = r170480 / r170469;
        double r170482 = r170466 / r170481;
        double r170483 = r170468 / r170482;
        return r170483;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 28.4

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

3. Applied flip-+ 28.4

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

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 0.6

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

12. Applied associate-/l* 0.6

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

14. Applied *-un-lft-identity 0.6

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

15. Applied *-un-lft-identity 0.6

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

16. Applied *-un-lft-identity 0.6

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

17. Applied times-frac 0.6

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

18. Applied times-frac 0.6

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

19. Applied times-frac 0.5

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

20. Applied *-un-lft-identity 0.5

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

21. Applied times-frac 0.6

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

22. Applied associate-/r* 0.5

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

23. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(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)))