Quadratic roots, narrow range

Average Error: 28.4 → 0.6

Time: 19.1s

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

\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

↓

\[\frac{\frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot \left(-c\right)\right)\right)}^{2}} \cdot \left(\sqrt[3]{{\left(\sqrt[3]{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)}^{2}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot \left(-c\right)\right)}}\right)}}}{2 \cdot a}\]

C

double f(double a, double b, double c) {
        double r43264 = b;
        double r43265 = -r43264;
        double r43266 = r43264 * r43264;
        double r43267 = 4.0;
        double r43268 = a;
        double r43269 = r43267 * r43268;
        double r43270 = c;
        double r43271 = r43269 * r43270;
        double r43272 = r43266 - r43271;
        double r43273 = sqrt(r43272);
        double r43274 = r43265 + r43273;
        double r43275 = 2.0;
        double r43276 = r43275 * r43268;
        double r43277 = r43274 / r43276;
        return r43277;
}

↓

double f(double a, double b, double c) {
        double r43278 = 4.0;
        double r43279 = a;
        double r43280 = r43278 * r43279;
        double r43281 = c;
        double r43282 = r43280 * r43281;
        double r43283 = b;
        double r43284 = -r43283;
        double r43285 = -r43281;
        double r43286 = r43280 * r43285;
        double r43287 = fma(r43283, r43283, r43286);
        double r43288 = 2.0;
        double r43289 = pow(r43287, r43288);
        double r43290 = cbrt(r43289);
        double r43291 = pow(r43283, r43288);
        double r43292 = r43279 * r43281;
        double r43293 = r43278 * r43292;
        double r43294 = r43291 - r43293;
        double r43295 = cbrt(r43294);
        double r43296 = pow(r43295, r43288);
        double r43297 = cbrt(r43296);
        double r43298 = cbrt(r43287);
        double r43299 = cbrt(r43298);
        double r43300 = r43297 * r43299;
        double r43301 = r43290 * r43300;
        double r43302 = sqrt(r43301);
        double r43303 = r43284 - r43302;
        double r43304 = r43282 / r43303;
        double r43305 = 2.0;
        double r43306 = r43305 * r43279;
        double r43307 = r43304 / r43306;
        return r43307;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 28.4

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
2. Using strategy rm

3. Applied flip-+ 28.4

   \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

4. Simplified 0.5

   \[\leadsto \frac{\frac{\color{blue}{0 + \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
5. Using strategy rm

6. Applied add-cbrt-cube 0.5

   \[\leadsto \frac{\frac{0 + \left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right) \cdot \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right) \cdot \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}}}}}{2 \cdot a}\]

7. Simplified 0.5

   \[\leadsto \frac{\frac{0 + \left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{\sqrt[3]{\color{blue}{{\left(\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot \left(-c\right)\right)\right)}^{3}}}}}}{2 \cdot a}\]
8. Using strategy rm

9. Applied add-cube-cbrt 0.6

   \[\leadsto \frac{\frac{0 + \left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{\sqrt[3]{{\color{blue}{\left(\left(\sqrt[3]{\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot \left(-c\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot \left(-c\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot \left(-c\right)\right)}\right)}}^{3}}}}}{2 \cdot a}\]

10. Applied unpow-prod-down 0.6

    \[\leadsto \frac{\frac{0 + \left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{\sqrt[3]{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot \left(-c\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot \left(-c\right)\right)}\right)}^{3} \cdot {\left(\sqrt[3]{\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot \left(-c\right)\right)}\right)}^{3}}}}}}{2 \cdot a}\]

11. Applied cbrt-prod 0.7

    \[\leadsto \frac{\frac{0 + \left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{\color{blue}{\sqrt[3]{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot \left(-c\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot \left(-c\right)\right)}\right)}^{3}} \cdot \sqrt[3]{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot \left(-c\right)\right)}\right)}^{3}}}}}}{2 \cdot a}\]

12. Simplified 0.5

    \[\leadsto \frac{\frac{0 + \left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot \left(-c\right)\right)\right)}^{2}}} \cdot \sqrt[3]{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot \left(-c\right)\right)}\right)}^{3}}}}}{2 \cdot a}\]

13. Simplified 0.5

    \[\leadsto \frac{\frac{0 + \left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot \left(-c\right)\right)\right)}^{2}} \cdot \color{blue}{\sqrt[3]{\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot \left(-c\right)\right)}}}}}{2 \cdot a}\]
14. Using strategy rm

15. Applied add-cube-cbrt 0.5

    \[\leadsto \frac{\frac{0 + \left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot \left(-c\right)\right)\right)}^{2}} \cdot \sqrt[3]{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot \left(-c\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot \left(-c\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot \left(-c\right)\right)}}}}}}{2 \cdot a}\]

16. Applied cbrt-prod 0.6

    \[\leadsto \frac{\frac{0 + \left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot \left(-c\right)\right)\right)}^{2}} \cdot \color{blue}{\left(\sqrt[3]{\sqrt[3]{\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot \left(-c\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot \left(-c\right)\right)}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot \left(-c\right)\right)}}\right)}}}}{2 \cdot a}\]

17. Simplified 0.6

    \[\leadsto \frac{\frac{0 + \left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot \left(-c\right)\right)\right)}^{2}} \cdot \left(\color{blue}{\sqrt[3]{{\left(\sqrt[3]{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)}^{2}}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot \left(-c\right)\right)}}\right)}}}{2 \cdot a}\]

18. Final simplification 0.6

    \[\leadsto \frac{\frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot \left(-c\right)\right)\right)}^{2}} \cdot \left(\sqrt[3]{{\left(\sqrt[3]{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)}^{2}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(b, b, \left(4 \cdot a\right) \cdot \left(-c\right)\right)}}\right)}}}{2 \cdot a}\]

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4 a) c)))) (* 2 a)))