Cubic critical, narrow range

Average Error: 28.9 → 0.3

Time: 19.0s

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

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

↓

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

C

double f(double a, double b, double c) {
        double r102183 = b;
        double r102184 = -r102183;
        double r102185 = r102183 * r102183;
        double r102186 = 3.0;
        double r102187 = a;
        double r102188 = r102186 * r102187;
        double r102189 = c;
        double r102190 = r102188 * r102189;
        double r102191 = r102185 - r102190;
        double r102192 = sqrt(r102191);
        double r102193 = r102184 + r102192;
        double r102194 = r102193 / r102188;
        return r102194;
}

↓

double f(double a, double b, double c) {
        double r102195 = 3.0;
        double r102196 = a;
        double r102197 = r102195 * r102196;
        double r102198 = c;
        double r102199 = b;
        double r102200 = -r102199;
        double r102201 = -r102198;
        double r102202 = r102199 * r102199;
        double r102203 = fma(r102201, r102197, r102202);
        double r102204 = sqrt(r102203);
        double r102205 = r102200 - r102204;
        double r102206 = r102198 / r102205;
        double r102207 = r102206 / r102197;
        double r102208 = r102197 * r102207;
        return r102208;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 28.9

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

3. Applied flip-+ 28.9

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

4. Simplified 0.6

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

5. Simplified 0.6

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

7. Applied associate-/r* 0.6

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

8. Simplified 0.6

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

10. Applied *-un-lft-identity 0.6

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

11. Applied *-un-lft-identity 0.6

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

12. Applied *-un-lft-identity 0.6

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

13. Applied times-frac 0.5

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

14. Applied times-frac 0.5

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

15. Applied times-frac 0.5

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

16. Simplified 0.5

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

17. Simplified 0.3

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

18. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))