jeff quadratic root 1

Average Error: 20.1 → 7.8

Time: 19.0s

Precision: 64

Math

\[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.434128173054205306503791529718852543763 \cdot 10^{153}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \left(a \cdot \left(\sqrt[3]{\frac{c}{b}} \cdot \sqrt[3]{\frac{c}{b}}\right)\right) \cdot \sqrt[3]{\frac{c}{b}}, b \cdot -2\right)}\\ \end{array}\\ \mathbf{elif}\;b \le -5.066042609614386619694957941213132915138 \cdot 10^{-310} \lor \neg \left(b \le 5.031608061939102936286074782173578716838 \cdot 10^{53}\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b - 2 \cdot \frac{a \cdot c}{b}\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\frac{\left(a \cdot c\right) \cdot 4}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r32978 = b;
        double r32979 = 0.0;
        bool r32980 = r32978 >= r32979;
        double r32981 = -r32978;
        double r32982 = r32978 * r32978;
        double r32983 = 4.0;
        double r32984 = a;
        double r32985 = r32983 * r32984;
        double r32986 = c;
        double r32987 = r32985 * r32986;
        double r32988 = r32982 - r32987;
        double r32989 = sqrt(r32988);
        double r32990 = r32981 - r32989;
        double r32991 = 2.0;
        double r32992 = r32991 * r32984;
        double r32993 = r32990 / r32992;
        double r32994 = r32991 * r32986;
        double r32995 = r32981 + r32989;
        double r32996 = r32994 / r32995;
        double r32997 = r32980 ? r32993 : r32996;
        return r32997;
}

↓

double f(double a, double b, double c) {
        double r32998 = b;
        double r32999 = -1.4341281730542053e+153;
        bool r33000 = r32998 <= r32999;
        double r33001 = 0.0;
        bool r33002 = r32998 >= r33001;
        double r33003 = -r32998;
        double r33004 = r32998 * r32998;
        double r33005 = 4.0;
        double r33006 = a;
        double r33007 = r33005 * r33006;
        double r33008 = c;
        double r33009 = r33007 * r33008;
        double r33010 = r33004 - r33009;
        double r33011 = sqrt(r33010);
        double r33012 = r33003 - r33011;
        double r33013 = 2.0;
        double r33014 = r33013 * r33006;
        double r33015 = r33012 / r33014;
        double r33016 = r33013 * r33008;
        double r33017 = r33008 / r32998;
        double r33018 = cbrt(r33017);
        double r33019 = r33018 * r33018;
        double r33020 = r33006 * r33019;
        double r33021 = r33020 * r33018;
        double r33022 = -2.0;
        double r33023 = r32998 * r33022;
        double r33024 = fma(r33013, r33021, r33023);
        double r33025 = r33016 / r33024;
        double r33026 = r33002 ? r33015 : r33025;
        double r33027 = -5.0660426096144e-310;
        bool r33028 = r32998 <= r33027;
        double r33029 = 5.031608061939103e+53;
        bool r33030 = r32998 <= r33029;
        double r33031 = !r33030;
        bool r33032 = r33028 || r33031;
        double r33033 = r33006 * r33008;
        double r33034 = r33033 / r32998;
        double r33035 = r33013 * r33034;
        double r33036 = r32998 - r33035;
        double r33037 = r33003 - r33036;
        double r33038 = r33037 / r33014;
        double r33039 = r33003 + r33011;
        double r33040 = r33016 / r33039;
        double r33041 = r33002 ? r33038 : r33040;
        double r33042 = r33033 * r33005;
        double r33043 = r33042 / r33012;
        double r33044 = r33016 / r33043;
        double r33045 = r33002 ? r33015 : r33044;
        double r33046 = r33032 ? r33041 : r33045;
        double r33047 = r33000 ? r33026 : r33046;
        return r33047;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 3 regimes

2. if b < -1.4341281730542053e+153

   1. Initial program 39.5

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 39.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \end{array}\]

   4. Applied sqrt-prod 39.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\\ \end{array}\]

   5. Taylor expanded around -inf 6.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}\\ \end{array}\]

   6. Simplified 6.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{a \cdot c}{b}, b \cdot -2\right)}}\\ \end{array}\]
   7. Using strategy rm

   8. Applied *-un-lft-identity 6.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \color{blue}{\frac{a \cdot c}{1 \cdot b}}, b \cdot -2\right)}\\ \end{array}\]

   9. Applied times-frac 1.4

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\mathsf{fma}\left(2, \frac{a}{1} \cdot \frac{c}{b}, b \cdot -2\right)}}\\ \end{array}\]

   10. Simplified 1.4

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(\color{blue}{2}, a \cdot \frac{c}{b}, b \cdot -2\right)}\\ \end{array}\]
   11. Using strategy rm

   12. Applied add-cube-cbrt 1.4

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \color{blue}{a \cdot \left(\left(\sqrt[3]{\frac{c}{b}} \cdot \sqrt[3]{\frac{c}{b}}\right) \cdot \sqrt[3]{\frac{c}{b}}\right)}, b \cdot -2\right)}\\ \end{array}\]

   13. Applied associate-*r* 1.4

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\mathsf{fma}\left(2, \left(a \cdot \left(\sqrt[3]{\frac{c}{b}} \cdot \sqrt[3]{\frac{c}{b}}\right)\right) \cdot \sqrt[3]{\frac{c}{b}}, b \cdot -2\right)}}\\ \end{array}\]

   if -1.4341281730542053e+153 < b < -5.0660426096144e-310 or 5.031608061939103e+53 < b

   1. Initial program 18.7

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

   2. Taylor expanded around inf 8.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \color{blue}{\left(b - 2 \cdot \frac{a \cdot c}{b}\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]

   if -5.0660426096144e-310 < b < 5.031608061939103e+53

   1. Initial program 10.0

      \[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 10.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right)} + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \end{array}\]

   4. Applied sqrt-prod 10.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\left(-b\right) + \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\\ \end{array}\]
   5. Using strategy rm

   6. Applied flip-+ 10.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\color{blue}{\frac{2 \cdot c}{\frac{\left(-b\right) \cdot \left(-b\right) - \left(\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right) \cdot \left(\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\right)}{\left(-b\right) - \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}}\\ \end{array}\]

   7. Simplified 10.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\color{blue}{2 \cdot c}}{\frac{\left(a \cdot c\right) \cdot 4}{\left(-b\right) - \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\\ \end{array}\]

   8. Simplified 10.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{\frac{\left(a \cdot c\right) \cdot 4}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\\ \end{array}\]
3. Recombined 3 regimes into one program.

4. Final simplification 7.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.434128173054205306503791529718852543763 \cdot 10^{153}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \left(a \cdot \left(\sqrt[3]{\frac{c}{b}} \cdot \sqrt[3]{\frac{c}{b}}\right)\right) \cdot \sqrt[3]{\frac{c}{b}}, b \cdot -2\right)}\\ \end{array}\\ \mathbf{elif}\;b \le -5.066042609614386619694957941213132915138 \cdot 10^{-310} \lor \neg \left(b \le 5.031608061939102936286074782173578716838 \cdot 10^{53}\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b - 2 \cdot \frac{a \cdot c}{b}\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\frac{\left(a \cdot c\right) \cdot 4}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 1"
  :precision binary64
  (if (>= b 0.0) (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ (* 2 c) (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))))))