Quadratic roots, wide range

Average Error: 52.6 → 0.1

Time: 6.3s

Precision: 64

\[4.93038 \cdot 10^{-32} \lt a \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt b \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt c \lt 2.02824 \cdot 10^{31}\]

Math

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

↓

\[\frac{4 \cdot c}{-\left(b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \cdot \frac{1}{2}\]

C

double f(double a, double b, double c) {
        double r45277 = b;
        double r45278 = -r45277;
        double r45279 = r45277 * r45277;
        double r45280 = 4.0;
        double r45281 = a;
        double r45282 = r45280 * r45281;
        double r45283 = c;
        double r45284 = r45282 * r45283;
        double r45285 = r45279 - r45284;
        double r45286 = sqrt(r45285);
        double r45287 = r45278 + r45286;
        double r45288 = 2.0;
        double r45289 = r45288 * r45281;
        double r45290 = r45287 / r45289;
        return r45290;
}

↓

double f(double a, double b, double c) {
        double r45291 = 4.0;
        double r45292 = c;
        double r45293 = r45291 * r45292;
        double r45294 = b;
        double r45295 = r45294 * r45294;
        double r45296 = a;
        double r45297 = r45291 * r45296;
        double r45298 = r45297 * r45292;
        double r45299 = r45295 - r45298;
        double r45300 = sqrt(r45299);
        double r45301 = r45294 + r45300;
        double r45302 = -r45301;
        double r45303 = r45293 / r45302;
        double r45304 = 1.0;
        double r45305 = 2.0;
        double r45306 = r45304 / r45305;
        double r45307 = r45303 * r45306;
        return r45307;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.6

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

3. Applied flip-+ 52.6

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

4. Simplified 0.4

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

6. Applied add-sqr-sqrt 0.5

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

7. Applied distribute-rgt-neg-in 0.5

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

8. Applied fma-neg 0.4

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

10. Applied *-un-lft-identity 0.4

    \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{1 \cdot \mathsf{fma}\left(\sqrt{b}, -\sqrt{b}, -\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}}{2 \cdot a}\]

11. Applied *-un-lft-identity 0.4

    \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(0 + 4 \cdot \left(a \cdot c\right)\right)}}{1 \cdot \mathsf{fma}\left(\sqrt{b}, -\sqrt{b}, -\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a}\]

12. Applied times-frac 0.4

    \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{0 + 4 \cdot \left(a \cdot c\right)}{\mathsf{fma}\left(\sqrt{b}, -\sqrt{b}, -\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}}{2 \cdot a}\]

13. Applied times-frac 0.4

    \[\leadsto \color{blue}{\frac{\frac{1}{1}}{2} \cdot \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\mathsf{fma}\left(\sqrt{b}, -\sqrt{b}, -\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{a}}\]

14. Simplified 0.4

    \[\leadsto \color{blue}{\frac{1}{2}} \cdot \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\mathsf{fma}\left(\sqrt{b}, -\sqrt{b}, -\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{a}\]

15. Simplified 0.2

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\frac{4 \cdot \left(a \cdot c\right)}{a}}{-\left(b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}\]
16. Using strategy rm

17. Applied *-un-lft-identity 0.2

    \[\leadsto \frac{1}{2} \cdot \frac{\frac{4 \cdot \left(a \cdot c\right)}{a}}{\color{blue}{1 \cdot \left(-\left(b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)\right)}}\]

18. Applied *-un-lft-identity 0.2

    \[\leadsto \frac{1}{2} \cdot \frac{\color{blue}{1 \cdot \frac{4 \cdot \left(a \cdot c\right)}{a}}}{1 \cdot \left(-\left(b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)\right)}\]

19. Applied times-frac 0.2

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\frac{1}{1} \cdot \frac{\frac{4 \cdot \left(a \cdot c\right)}{a}}{-\left(b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\right)}\]

20. Simplified 0.2

    \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{1} \cdot \frac{\frac{4 \cdot \left(a \cdot c\right)}{a}}{-\left(b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}\right)\]

21. Simplified 0.1

    \[\leadsto \frac{1}{2} \cdot \left(1 \cdot \color{blue}{\frac{4 \cdot c}{-\left(b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}\right)\]

22. Final simplification 0.1

    \[\leadsto \frac{4 \cdot c}{-\left(b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \cdot \frac{1}{2}\]

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (< 4.9303800000000003e-32 a 2.02824e+31) (< 4.9303800000000003e-32 b 2.02824e+31) (< 4.9303800000000003e-32 c 2.02824e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))