Quadratic roots, wide range

Average Error: 52.7 → 0.1

Time: 6.1s

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

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

↓

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

C

double f(double a, double b, double c) {
        double r29610 = b;
        double r29611 = -r29610;
        double r29612 = r29610 * r29610;
        double r29613 = 4.0;
        double r29614 = a;
        double r29615 = r29613 * r29614;
        double r29616 = c;
        double r29617 = r29615 * r29616;
        double r29618 = r29612 - r29617;
        double r29619 = sqrt(r29618);
        double r29620 = r29611 + r29619;
        double r29621 = 2.0;
        double r29622 = r29621 * r29614;
        double r29623 = r29620 / r29622;
        return r29623;
}

↓

double f(double a, double b, double c) {
        double r29624 = 1.0;
        double r29625 = 2.0;
        double r29626 = r29624 / r29625;
        double r29627 = 4.0;
        double r29628 = c;
        double r29629 = b;
        double r29630 = -r29629;
        double r29631 = r29629 * r29629;
        double r29632 = a;
        double r29633 = r29627 * r29632;
        double r29634 = r29633 * r29628;
        double r29635 = r29631 - r29634;
        double r29636 = sqrt(r29635);
        double r29637 = r29630 - r29636;
        double r29638 = r29628 / r29637;
        double r29639 = r29627 * r29638;
        double r29640 = r29626 * r29639;
        return r29640;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.7

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

3. Applied flip-+ 52.7

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

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 *-un-lft-identity 0.4

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

7. Applied *-un-lft-identity 0.4

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

8. Applied times-frac 0.4

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

9. Applied times-frac 0.4

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

10. Simplified 0.4

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

11. Simplified 0.4

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

13. Applied associate-/l* 0.4

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

14. Simplified 0.3

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

16. Applied div-inv 0.3

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

17. Simplified 0.1

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

18. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020083 
(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)))