Cubic critical, wide range

Average Error: 52.7 → 0.1

Time: 21.1s

Precision: 64

\[4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt a \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt b \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt c \lt 20282409603651670423947251286016\]

Math

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

↓

\[\frac{3 \cdot a}{3 \cdot a} \cdot \frac{c}{\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}\]

C

double f(double a, double b, double c) {
        double r84263 = b;
        double r84264 = -r84263;
        double r84265 = r84263 * r84263;
        double r84266 = 3.0;
        double r84267 = a;
        double r84268 = r84266 * r84267;
        double r84269 = c;
        double r84270 = r84268 * r84269;
        double r84271 = r84265 - r84270;
        double r84272 = sqrt(r84271);
        double r84273 = r84264 + r84272;
        double r84274 = r84273 / r84268;
        return r84274;
}

↓

double f(double a, double b, double c) {
        double r84275 = 3.0;
        double r84276 = a;
        double r84277 = r84275 * r84276;
        double r84278 = r84277 / r84277;
        double r84279 = c;
        double r84280 = b;
        double r84281 = -r84280;
        double r84282 = 2.0;
        double r84283 = pow(r84280, r84282);
        double r84284 = r84276 * r84279;
        double r84285 = r84275 * r84284;
        double r84286 = r84283 - r84285;
        double r84287 = sqrt(r84286);
        double r84288 = r84281 - r84287;
        double r84289 = r84279 / r84288;
        double r84290 = r84278 * r84289;
        return r84290;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.7

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

3. Applied flip-+ 52.7

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

4. Simplified 0.5

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

5. Simplified 0.5

   \[\leadsto \frac{\frac{0 + \left(a \cdot c\right) \cdot 3}{\color{blue}{\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}}}{3 \cdot a}\]
6. Using strategy rm

7. Applied *-un-lft-identity 0.5

   \[\leadsto \frac{\frac{0 + \left(a \cdot c\right) \cdot 3}{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}\right)}}}{3 \cdot a}\]

8. Applied *-un-lft-identity 0.5

   \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(0 + \left(a \cdot c\right) \cdot 3\right)}}{1 \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}\right)}}{3 \cdot a}\]

9. Applied times-frac 0.5

   \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{0 + \left(a \cdot c\right) \cdot 3}{\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}}}{3 \cdot a}\]

10. Applied associate-/l* 0.6

    \[\leadsto \color{blue}{\frac{\frac{1}{1}}{\frac{3 \cdot a}{\frac{0 + \left(a \cdot c\right) \cdot 3}{\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}}}}\]

11. Simplified 0.5

    \[\leadsto \frac{\frac{1}{1}}{\color{blue}{\frac{3 \cdot a}{\frac{\left(3 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}}}}\]
12. Using strategy rm

13. Applied *-un-lft-identity 0.5

    \[\leadsto \frac{\frac{1}{1}}{\color{blue}{1 \cdot \frac{3 \cdot a}{\frac{\left(3 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}}}}\]

14. Applied add-cube-cbrt 0.5

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{1}{1}} \cdot \sqrt[3]{\frac{1}{1}}\right) \cdot \sqrt[3]{\frac{1}{1}}}}{1 \cdot \frac{3 \cdot a}{\frac{\left(3 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}}}\]

15. Applied times-frac 0.5

    \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{1}{1}} \cdot \sqrt[3]{\frac{1}{1}}}{1} \cdot \frac{\sqrt[3]{\frac{1}{1}}}{\frac{3 \cdot a}{\frac{\left(3 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}}}}\]

16. Simplified 0.5

    \[\leadsto \color{blue}{1} \cdot \frac{\sqrt[3]{\frac{1}{1}}}{\frac{3 \cdot a}{\frac{\left(3 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}}}\]

17. Simplified 0.1

    \[\leadsto 1 \cdot \color{blue}{\left(\frac{3 \cdot a}{3 \cdot a} \cdot \frac{c}{\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}\right)}\]

18. Final simplification 0.1

    \[\leadsto \frac{3 \cdot a}{3 \cdot a} \cdot \frac{c}{\left(-b\right) - \sqrt{{b}^{2} - 3 \cdot \left(a \cdot c\right)}}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (< 4.93038e-32 a 2.02824e+31) (< 4.93038e-32 b 2.02824e+31) (< 4.93038e-32 c 2.02824e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))