Cubic critical, narrow range

Average Error: 28.4 → 0.5

Time: 52.1s

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 r179107 = b;
        double r179108 = -r179107;
        double r179109 = r179107 * r179107;
        double r179110 = 3.0;
        double r179111 = a;
        double r179112 = r179110 * r179111;
        double r179113 = c;
        double r179114 = r179112 * r179113;
        double r179115 = r179109 - r179114;
        double r179116 = sqrt(r179115);
        double r179117 = r179108 + r179116;
        double r179118 = r179117 / r179112;
        return r179118;
}

↓

double f(double a, double b, double c) {
        double r179119 = 1.0;
        double r179120 = 3.0;
        double r179121 = r179119 / r179120;
        double r179122 = r179119 / r179121;
        double r179123 = a;
        double r179124 = b;
        double r179125 = -r179124;
        double r179126 = r179124 * r179124;
        double r179127 = r179120 * r179123;
        double r179128 = c;
        double r179129 = r179127 * r179128;
        double r179130 = r179126 - r179129;
        double r179131 = sqrt(r179130);
        double r179132 = r179125 - r179131;
        double r179133 = r179132 / r179128;
        double r179134 = r179123 / r179133;
        double r179135 = r179134 / r179123;
        double r179136 = r179120 / r179135;
        double r179137 = r179122 / r179136;
        return r179137;
}

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

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

16. Applied times-frac 0.6

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

17. Applied times-frac 0.5

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

18. Applied *-un-lft-identity 0.5

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

19. Applied times-frac 0.6

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

20. Applied associate-/r* 0.5

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

21. 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)))