Quadratic roots, full range

Average Error: 33.0 → 8.7

Time: 18.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.6519381339788066 \cdot 10^{+37}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le -1.3761661522305357 \cdot 10^{-153}:\\ \;\;\;\;\frac{\frac{\sqrt{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}{\frac{a}{\sqrt{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}}}{2}\\ \mathbf{elif}\;b \le 6.555431533807236 \cdot 10^{+28}:\\ \;\;\;\;\frac{\frac{a \cdot \frac{-4 \cdot c}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r1838583 = b;
        double r1838584 = -r1838583;
        double r1838585 = r1838583 * r1838583;
        double r1838586 = 4.0;
        double r1838587 = a;
        double r1838588 = r1838586 * r1838587;
        double r1838589 = c;
        double r1838590 = r1838588 * r1838589;
        double r1838591 = r1838585 - r1838590;
        double r1838592 = sqrt(r1838591);
        double r1838593 = r1838584 + r1838592;
        double r1838594 = 2.0;
        double r1838595 = r1838594 * r1838587;
        double r1838596 = r1838593 / r1838595;
        return r1838596;
}

↓

double f(double a, double b, double c) {
        double r1838597 = b;
        double r1838598 = -1.6519381339788066e+37;
        bool r1838599 = r1838597 <= r1838598;
        double r1838600 = c;
        double r1838601 = r1838600 / r1838597;
        double r1838602 = a;
        double r1838603 = r1838597 / r1838602;
        double r1838604 = r1838601 - r1838603;
        double r1838605 = 2.0;
        double r1838606 = r1838604 * r1838605;
        double r1838607 = r1838606 / r1838605;
        double r1838608 = -1.3761661522305357e-153;
        bool r1838609 = r1838597 <= r1838608;
        double r1838610 = -4.0;
        double r1838611 = r1838610 * r1838600;
        double r1838612 = r1838597 * r1838597;
        double r1838613 = fma(r1838611, r1838602, r1838612);
        double r1838614 = sqrt(r1838613);
        double r1838615 = r1838614 - r1838597;
        double r1838616 = sqrt(r1838615);
        double r1838617 = r1838602 / r1838616;
        double r1838618 = r1838616 / r1838617;
        double r1838619 = r1838618 / r1838605;
        double r1838620 = 6.555431533807236e+28;
        bool r1838621 = r1838597 <= r1838620;
        double r1838622 = r1838614 + r1838597;
        double r1838623 = r1838611 / r1838622;
        double r1838624 = r1838602 * r1838623;
        double r1838625 = r1838624 / r1838602;
        double r1838626 = r1838625 / r1838605;
        double r1838627 = -2.0;
        double r1838628 = r1838627 * r1838601;
        double r1838629 = r1838628 / r1838605;
        double r1838630 = r1838621 ? r1838626 : r1838629;
        double r1838631 = r1838609 ? r1838619 : r1838630;
        double r1838632 = r1838599 ? r1838607 : r1838631;
        return r1838632;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 4 regimes

2. if b < -1.6519381339788066e+37

   1. Initial program 33.6

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

   2. Simplified 33.6

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

   3. Taylor expanded around -inf 6.5

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}}}{2}\]

   4. Simplified 6.5

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

   if -1.6519381339788066e+37 < b < -1.3761661522305357e-153

   1. Initial program 5.2

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

   2. Simplified 5.2

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

   4. Applied clear-num 5.3

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

   6. Applied *-un-lft-identity 5.3

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

   7. Applied *-un-lft-identity 5.3

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

   8. Applied times-frac 5.3

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

   9. Applied add-cube-cbrt 5.3

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

   10. Applied times-frac 5.3

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

   11. Simplified 5.3

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

   12. Simplified 5.2

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

   14. Applied add-sqr-sqrt 5.6

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

   15. Applied associate-/l* 5.6

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

   if -1.3761661522305357e-153 < b < 6.555431533807236e+28

   1. Initial program 24.5

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

   2. Simplified 24.5

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

   4. Applied clear-num 24.5

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

   6. Applied *-un-lft-identity 24.5

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

   7. Applied *-un-lft-identity 24.5

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

   8. Applied times-frac 24.5

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

   9. Applied add-cube-cbrt 24.5

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

   10. Applied times-frac 24.5

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

   11. Simplified 24.5

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

   12. Simplified 24.5

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

   14. Applied flip-- 25.0

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

   15. Simplified 17.1

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

   17. Applied *-un-lft-identity 17.1

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

   18. Applied times-frac 14.9

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

   19. Simplified 14.9

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

   if 6.555431533807236e+28 < b

   1. Initial program 56.3

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

   2. Simplified 56.3

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

   3. Taylor expanded around inf 4.5

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]
3. Recombined 4 regimes into one program.

4. Final simplification 8.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.6519381339788066 \cdot 10^{+37}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le -1.3761661522305357 \cdot 10^{-153}:\\ \;\;\;\;\frac{\frac{\sqrt{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}{\frac{a}{\sqrt{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}}}{2}\\ \mathbf{elif}\;b \le 6.555431533807236 \cdot 10^{+28}:\\ \;\;\;\;\frac{\frac{a \cdot \frac{-4 \cdot c}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} + b}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))