Quadratic roots, narrow range

Average Error: 28.2 → 16.3

Time: 25.5s

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

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le 1984.600261148631:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} \cdot \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}, b, b \cdot b + \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(a \cdot c\right) \cdot -2}{a \cdot b}}{2}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r1457901 = b;
        double r1457902 = -r1457901;
        double r1457903 = r1457901 * r1457901;
        double r1457904 = 4.0;
        double r1457905 = a;
        double r1457906 = r1457904 * r1457905;
        double r1457907 = c;
        double r1457908 = r1457906 * r1457907;
        double r1457909 = r1457903 - r1457908;
        double r1457910 = sqrt(r1457909);
        double r1457911 = r1457902 + r1457910;
        double r1457912 = 2.0;
        double r1457913 = r1457912 * r1457905;
        double r1457914 = r1457911 / r1457913;
        return r1457914;
}

↓

double f(double a, double b, double c) {
        double r1457915 = b;
        double r1457916 = 1984.600261148631;
        bool r1457917 = r1457915 <= r1457916;
        double r1457918 = a;
        double r1457919 = c;
        double r1457920 = r1457918 * r1457919;
        double r1457921 = -4.0;
        double r1457922 = r1457915 * r1457915;
        double r1457923 = fma(r1457920, r1457921, r1457922);
        double r1457924 = sqrt(r1457923);
        double r1457925 = r1457924 * r1457923;
        double r1457926 = r1457922 * r1457915;
        double r1457927 = r1457925 - r1457926;
        double r1457928 = r1457922 + r1457923;
        double r1457929 = fma(r1457924, r1457915, r1457928);
        double r1457930 = r1457927 / r1457929;
        double r1457931 = r1457930 / r1457918;
        double r1457932 = 2.0;
        double r1457933 = r1457931 / r1457932;
        double r1457934 = -2.0;
        double r1457935 = r1457920 * r1457934;
        double r1457936 = r1457918 * r1457915;
        double r1457937 = r1457935 / r1457936;
        double r1457938 = r1457937 / r1457932;
        double r1457939 = r1457917 ? r1457933 : r1457938;
        return r1457939;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 2 regimes

2. if b < 1984.600261148631

   1. Initial program 17.2

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

   2. Simplified 17.1

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

   4. Applied flip3-- 17.1

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

   5. Simplified 16.6

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

   6. Simplified 16.6

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

   if 1984.600261148631 < b

   1. Initial program 36.7

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

   2. Simplified 36.6

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

   3. Taylor expanded around inf 16.0

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

   5. Applied associate-*r/ 16.0

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

   6. Applied associate-/l/ 16.0

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

4. Final simplification 16.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 1984.600261148631:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} \cdot \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right) - \left(b \cdot b\right) \cdot b}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}, b, b \cdot b + \mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(a \cdot c\right) \cdot -2}{a \cdot b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 +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)))