Cubic critical, narrow range

Average Error: 28.6 → 16.3

Time: 4.4s

Precision: 64

\[1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt a \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt b \lt 94906265.62425155937671661376953125 \land 1.053671212772350866701172186984739043147 \cdot 10^{-8} \lt c \lt 94906265.62425155937671661376953125\]

Math

\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

↓

\[\begin{array}{l} \mathbf{if}\;b \le 1395.437407789798953672288917005062103271:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, b, -\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r97902 = b;
        double r97903 = -r97902;
        double r97904 = r97902 * r97902;
        double r97905 = 3.0;
        double r97906 = a;
        double r97907 = r97905 * r97906;
        double r97908 = c;
        double r97909 = r97907 * r97908;
        double r97910 = r97904 - r97909;
        double r97911 = sqrt(r97910);
        double r97912 = r97903 + r97911;
        double r97913 = r97912 / r97907;
        return r97913;
}

↓

double f(double a, double b, double c) {
        double r97914 = b;
        double r97915 = 1395.437407789799;
        bool r97916 = r97914 <= r97915;
        double r97917 = r97914 * r97914;
        double r97918 = 3.0;
        double r97919 = a;
        double r97920 = r97918 * r97919;
        double r97921 = c;
        double r97922 = r97920 * r97921;
        double r97923 = r97917 - r97922;
        double r97924 = -r97923;
        double r97925 = fma(r97914, r97914, r97924);
        double r97926 = -r97914;
        double r97927 = sqrt(r97923);
        double r97928 = r97926 - r97927;
        double r97929 = r97925 / r97928;
        double r97930 = r97929 / r97920;
        double r97931 = -0.5;
        double r97932 = r97921 / r97914;
        double r97933 = r97931 * r97932;
        double r97934 = r97916 ? r97930 : r97933;
        return r97934;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 2 regimes

2. if b < 1395.437407789799

   1. Initial program 17.7

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
   2. Using strategy rm

   3. Applied flip-+ 17.7

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a}\]

   4. Simplified 16.8

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

   if 1395.437407789799 < b

   1. Initial program 36.8

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

   2. Taylor expanded around inf 15.9

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 16.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 1395.437407789798953672288917005062103271:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, b, -\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))