quad2m (problem 3.2.1, negative)

Average Error: 34.0 → 10.5

Time: 12.8s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.913471920057083513053706311365148123626 \cdot 10^{-110}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 106106913250787377152:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, \frac{b_2}{a} \cdot -2\right)\\ \end{array}\]

C

double f(double a, double b_2, double c) {
        double r23922 = b_2;
        double r23923 = -r23922;
        double r23924 = r23922 * r23922;
        double r23925 = a;
        double r23926 = c;
        double r23927 = r23925 * r23926;
        double r23928 = r23924 - r23927;
        double r23929 = sqrt(r23928);
        double r23930 = r23923 - r23929;
        double r23931 = r23930 / r23925;
        return r23931;
}

↓

double f(double a, double b_2, double c) {
        double r23932 = b_2;
        double r23933 = -1.9134719200570835e-110;
        bool r23934 = r23932 <= r23933;
        double r23935 = -0.5;
        double r23936 = c;
        double r23937 = r23936 / r23932;
        double r23938 = r23935 * r23937;
        double r23939 = 1.0610691325078738e+20;
        bool r23940 = r23932 <= r23939;
        double r23941 = -r23932;
        double r23942 = r23932 * r23932;
        double r23943 = a;
        double r23944 = r23943 * r23936;
        double r23945 = r23942 - r23944;
        double r23946 = sqrt(r23945);
        double r23947 = r23941 - r23946;
        double r23948 = r23947 / r23943;
        double r23949 = 0.5;
        double r23950 = r23932 / r23943;
        double r23951 = -2.0;
        double r23952 = r23950 * r23951;
        double r23953 = fma(r23949, r23937, r23952);
        double r23954 = r23940 ? r23948 : r23953;
        double r23955 = r23934 ? r23938 : r23954;
        return r23955;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.9134719200570835e-110

   1. Initial program 51.3

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

   2. Taylor expanded around -inf 10.7

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]

   if -1.9134719200570835e-110 < b_2 < 1.0610691325078738e+20

   1. Initial program 12.7

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
   2. Using strategy rm

   3. Applied pow1 12.7

      \[\leadsto \color{blue}{{\left(\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\right)}^{1}}\]

   if 1.0610691325078738e+20 < b_2

   1. Initial program 34.5

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

   2. Taylor expanded around inf 6.8

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]

   3. Simplified 6.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, \frac{b_2}{a} \cdot -2\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 10.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.913471920057083513053706311365148123626 \cdot 10^{-110}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 106106913250787377152:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, \frac{b_2}{a} \cdot -2\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))