NMSE problem 3.2.1

Average Error: 32.8 → 10.2

Time: 29.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.031575300615258 \cdot 10^{-39}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.9089378078751267 \cdot 10^{+122}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

C

double f(double a, double b_2, double c) {
        double r1638836 = b_2;
        double r1638837 = -r1638836;
        double r1638838 = r1638836 * r1638836;
        double r1638839 = a;
        double r1638840 = c;
        double r1638841 = r1638839 * r1638840;
        double r1638842 = r1638838 - r1638841;
        double r1638843 = sqrt(r1638842);
        double r1638844 = r1638837 - r1638843;
        double r1638845 = r1638844 / r1638839;
        return r1638845;
}

↓

double f(double a, double b_2, double c) {
        double r1638846 = b_2;
        double r1638847 = -3.031575300615258e-39;
        bool r1638848 = r1638846 <= r1638847;
        double r1638849 = -0.5;
        double r1638850 = c;
        double r1638851 = r1638850 / r1638846;
        double r1638852 = r1638849 * r1638851;
        double r1638853 = 1.9089378078751267e+122;
        bool r1638854 = r1638846 <= r1638853;
        double r1638855 = -r1638846;
        double r1638856 = a;
        double r1638857 = r1638855 / r1638856;
        double r1638858 = r1638846 * r1638846;
        double r1638859 = r1638850 * r1638856;
        double r1638860 = r1638858 - r1638859;
        double r1638861 = sqrt(r1638860);
        double r1638862 = r1638861 / r1638856;
        double r1638863 = r1638857 - r1638862;
        double r1638864 = 0.5;
        double r1638865 = r1638851 * r1638864;
        double r1638866 = 2.0;
        double r1638867 = r1638846 / r1638856;
        double r1638868 = r1638866 * r1638867;
        double r1638869 = r1638865 - r1638868;
        double r1638870 = r1638854 ? r1638863 : r1638869;
        double r1638871 = r1638848 ? r1638852 : r1638870;
        return r1638871;
}

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 < -3.031575300615258e-39

   1. Initial program 53.3

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

   2. Taylor expanded around -inf 7.8

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

   if -3.031575300615258e-39 < b_2 < 1.9089378078751267e+122

   1. Initial program 13.7

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

   3. Applied div-sub 13.7

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

   if 1.9089378078751267e+122 < b_2

   1. Initial program 50.0

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

   3. Applied div-sub 50.0

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

   4. Taylor expanded around inf 3.4

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 10.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.031575300615258 \cdot 10^{-39}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.9089378078751267 \cdot 10^{+122}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{1}{2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))