Quadratic roots, full range

Average Error: 33.8 → 15.8

Time: 29.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.802032183745334768196101346563616909151 \cdot 10^{-307}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)} - b}}}{2}\\ \mathbf{elif}\;b \le 4.033968672442674816629306589314441243721 \cdot 10^{160}:\\ \;\;\;\;\frac{\left(\frac{4 \cdot c}{\sqrt{b + \sqrt{\mathsf{fma}\left(4, c \cdot \left(-a\right), b \cdot b\right)}}} \cdot \frac{a}{a}\right) \cdot \frac{-1}{\sqrt{\sqrt{\sqrt[3]{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)}} \cdot \left|\sqrt[3]{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)}\right| + b}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(c \cdot \left(-a\right), 4, 0\right)}{b \cdot 2}}{a}}{2}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r60506 = b;
        double r60507 = -r60506;
        double r60508 = r60506 * r60506;
        double r60509 = 4.0;
        double r60510 = a;
        double r60511 = r60509 * r60510;
        double r60512 = c;
        double r60513 = r60511 * r60512;
        double r60514 = r60508 - r60513;
        double r60515 = sqrt(r60514);
        double r60516 = r60507 + r60515;
        double r60517 = 2.0;
        double r60518 = r60517 * r60510;
        double r60519 = r60516 / r60518;
        return r60519;
}

↓

double f(double a, double b, double c) {
        double r60520 = b;
        double r60521 = -1.8020321837453348e-307;
        bool r60522 = r60520 <= r60521;
        double r60523 = 1.0;
        double r60524 = a;
        double r60525 = -r60524;
        double r60526 = 4.0;
        double r60527 = r60525 * r60526;
        double r60528 = c;
        double r60529 = r60520 * r60520;
        double r60530 = fma(r60527, r60528, r60529);
        double r60531 = sqrt(r60530);
        double r60532 = r60531 - r60520;
        double r60533 = r60524 / r60532;
        double r60534 = r60523 / r60533;
        double r60535 = 2.0;
        double r60536 = r60534 / r60535;
        double r60537 = 4.033968672442675e+160;
        bool r60538 = r60520 <= r60537;
        double r60539 = r60526 * r60528;
        double r60540 = r60528 * r60525;
        double r60541 = fma(r60526, r60540, r60529);
        double r60542 = sqrt(r60541);
        double r60543 = r60520 + r60542;
        double r60544 = sqrt(r60543);
        double r60545 = r60539 / r60544;
        double r60546 = r60524 / r60524;
        double r60547 = r60545 * r60546;
        double r60548 = -1.0;
        double r60549 = -r60539;
        double r60550 = fma(r60524, r60549, r60529);
        double r60551 = cbrt(r60550);
        double r60552 = sqrt(r60551);
        double r60553 = fabs(r60551);
        double r60554 = r60552 * r60553;
        double r60555 = r60554 + r60520;
        double r60556 = sqrt(r60555);
        double r60557 = r60548 / r60556;
        double r60558 = r60547 * r60557;
        double r60559 = r60558 / r60535;
        double r60560 = 0.0;
        double r60561 = fma(r60540, r60526, r60560);
        double r60562 = 2.0;
        double r60563 = r60520 * r60562;
        double r60564 = r60561 / r60563;
        double r60565 = r60564 / r60524;
        double r60566 = r60565 / r60535;
        double r60567 = r60538 ? r60559 : r60566;
        double r60568 = r60522 ? r60536 : r60567;
        return r60568;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 3 regimes

2. if b < -1.8020321837453348e-307

   1. Initial program 21.4

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

   2. Simplified 21.5

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

   4. Applied clear-num 21.6

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

   5. Simplified 21.5

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

   if -1.8020321837453348e-307 < b < 4.033968672442675e+160

   1. Initial program 35.3

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

   2. Simplified 35.3

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

   4. Applied flip-- 35.4

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

   5. Simplified 16.5

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

   6. Simplified 16.5

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

   8. Applied *-un-lft-identity 16.5

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

   9. Applied add-sqr-sqrt 16.7

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

   10. Applied *-un-lft-identity 16.7

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

   11. Applied times-frac 16.7

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

   12. Applied times-frac 15.9

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

   13. Simplified 15.9

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

   14. Simplified 9.3

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

   16. Applied add-cube-cbrt 9.4

       \[\leadsto \frac{\frac{1}{\sqrt{b + \sqrt{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)} \cdot \sqrt[3]{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)}}}}} \cdot \left(\frac{-a}{a} \cdot \frac{4 \cdot c}{\sqrt{b + \sqrt{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)}}}\right)}{2}\]

   17. Applied sqrt-prod 9.4

       \[\leadsto \frac{\frac{1}{\sqrt{b + \color{blue}{\sqrt{\sqrt[3]{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)} \cdot \sqrt[3]{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)}} \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)}}}}} \cdot \left(\frac{-a}{a} \cdot \frac{4 \cdot c}{\sqrt{b + \sqrt{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)}}}\right)}{2}\]

   18. Simplified 9.4

       \[\leadsto \frac{\frac{1}{\sqrt{b + \color{blue}{\left|\sqrt[3]{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)}\right|} \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)}}}} \cdot \left(\frac{-a}{a} \cdot \frac{4 \cdot c}{\sqrt{b + \sqrt{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)}}}\right)}{2}\]

   19. Simplified 9.4

       \[\leadsto \frac{\frac{1}{\sqrt{b + \left|\sqrt[3]{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)}\right| \cdot \color{blue}{\sqrt{\sqrt[3]{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)}}}}} \cdot \left(\frac{-a}{a} \cdot \frac{4 \cdot c}{\sqrt{b + \sqrt{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)}}}\right)}{2}\]

   if 4.033968672442675e+160 < b

   1. Initial program 64.0

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

   2. Simplified 64.0

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

   4. Applied flip-- 64.0

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

   5. Simplified 38.6

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

   6. Simplified 38.6

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

   7. Taylor expanded around 0 14.6

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\left(-a\right) \cdot c, 4, 0\right)}{\color{blue}{2 \cdot b}}}{a}}{2}\]
3. Recombined 3 regimes into one program.

4. Final simplification 15.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.802032183745334768196101346563616909151 \cdot 10^{-307}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)} - b}}}{2}\\ \mathbf{elif}\;b \le 4.033968672442674816629306589314441243721 \cdot 10^{160}:\\ \;\;\;\;\frac{\left(\frac{4 \cdot c}{\sqrt{b + \sqrt{\mathsf{fma}\left(4, c \cdot \left(-a\right), b \cdot b\right)}}} \cdot \frac{a}{a}\right) \cdot \frac{-1}{\sqrt{\sqrt{\sqrt[3]{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)}} \cdot \left|\sqrt[3]{\mathsf{fma}\left(a, -4 \cdot c, b \cdot b\right)}\right| + b}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(c \cdot \left(-a\right), 4, 0\right)}{b \cdot 2}}{a}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))