Quadratic roots, narrow range

Average Error: 28.5 → 16.5

Time: 13.6s

Precision: 64

\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le 2435.2513491695368:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} \cdot \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2 \cdot \left(c \cdot a\right)}{a \cdot b}}{2}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r1418860 = b;
        double r1418861 = -r1418860;
        double r1418862 = r1418860 * r1418860;
        double r1418863 = 4.0;
        double r1418864 = a;
        double r1418865 = r1418863 * r1418864;
        double r1418866 = c;
        double r1418867 = r1418865 * r1418866;
        double r1418868 = r1418862 - r1418867;
        double r1418869 = sqrt(r1418868);
        double r1418870 = r1418861 + r1418869;
        double r1418871 = 2.0;
        double r1418872 = r1418871 * r1418864;
        double r1418873 = r1418870 / r1418872;
        return r1418873;
}

↓

double f(double a, double b, double c) {
        double r1418874 = b;
        double r1418875 = 2435.2513491695368;
        bool r1418876 = r1418874 <= r1418875;
        double r1418877 = -4.0;
        double r1418878 = a;
        double r1418879 = r1418877 * r1418878;
        double r1418880 = c;
        double r1418881 = r1418874 * r1418874;
        double r1418882 = fma(r1418879, r1418880, r1418881);
        double r1418883 = sqrt(r1418882);
        double r1418884 = r1418883 * r1418882;
        double r1418885 = r1418881 * r1418874;
        double r1418886 = r1418884 - r1418885;
        double r1418887 = r1418874 + r1418883;
        double r1418888 = fma(r1418874, r1418887, r1418882);
        double r1418889 = r1418886 / r1418888;
        double r1418890 = r1418889 / r1418878;
        double r1418891 = 2.0;
        double r1418892 = r1418890 / r1418891;
        double r1418893 = -2.0;
        double r1418894 = r1418880 * r1418878;
        double r1418895 = r1418893 * r1418894;
        double r1418896 = r1418878 * r1418874;
        double r1418897 = r1418895 / r1418896;
        double r1418898 = r1418897 / r1418891;
        double r1418899 = r1418876 ? r1418892 : r1418898;
        return r1418899;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 2 regimes

2. if b < 2435.2513491695368

   1. Initial program 17.8

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

   2. Simplified 17.7

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

   4. Applied flip3-- 17.8

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

   5. Simplified 17.2

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

   6. Simplified 17.2

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

   if 2435.2513491695368 < b

   1. Initial program 37.0

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

   2. Simplified 36.9

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

   3. Taylor expanded around inf 15.9

      \[\leadsto \frac{\frac{\color{blue}{-2 \cdot \frac{a \cdot c}{b}}}{a}}{2}\]
   4. Using strategy rm

   5. Applied associate-*r/ 15.9

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

   6. Applied associate-/l/ 15.9

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

4. Final simplification 16.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 2435.2513491695368:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)} \cdot \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(b, b + \sqrt{\mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)}, \mathsf{fma}\left(-4 \cdot a, c, b \cdot b\right)\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2 \cdot \left(c \cdot a\right)}{a \cdot b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))