quad2m (problem 3.2.1, negative)

Average Error: 34.5 → 10.6

Time: 4.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.279587145681289 \cdot 10^{-136}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 6.8526453862578789 \cdot 10^{139}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

C

double f(double a, double b_2, double c) {
        double r11712 = b_2;
        double r11713 = -r11712;
        double r11714 = r11712 * r11712;
        double r11715 = a;
        double r11716 = c;
        double r11717 = r11715 * r11716;
        double r11718 = r11714 - r11717;
        double r11719 = sqrt(r11718);
        double r11720 = r11713 - r11719;
        double r11721 = r11720 / r11715;
        return r11721;
}

↓

double f(double a, double b_2, double c) {
        double r11722 = b_2;
        double r11723 = -1.279587145681289e-136;
        bool r11724 = r11722 <= r11723;
        double r11725 = -0.5;
        double r11726 = c;
        double r11727 = r11726 / r11722;
        double r11728 = r11725 * r11727;
        double r11729 = 6.852645386257879e+139;
        bool r11730 = r11722 <= r11729;
        double r11731 = -r11722;
        double r11732 = r11722 * r11722;
        double r11733 = a;
        double r11734 = r11733 * r11726;
        double r11735 = r11732 - r11734;
        double r11736 = sqrt(r11735);
        double r11737 = r11731 - r11736;
        double r11738 = 1.0;
        double r11739 = r11738 / r11733;
        double r11740 = r11737 * r11739;
        double r11741 = 0.5;
        double r11742 = r11741 * r11727;
        double r11743 = 2.0;
        double r11744 = r11722 / r11733;
        double r11745 = r11743 * r11744;
        double r11746 = r11742 - r11745;
        double r11747 = r11730 ? r11740 : r11746;
        double r11748 = r11724 ? r11728 : r11747;
        return r11748;
}

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 < -1.279587145681289e-136

   1. Initial program 51.1

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

   2. Taylor expanded around -inf 12.2

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

   if -1.279587145681289e-136 < b_2 < 6.852645386257879e+139

   1. Initial program 11.1

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

   3. Applied div-inv 11.2

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

   if 6.852645386257879e+139 < b_2

   1. Initial program 57.7

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

   2. Taylor expanded around inf 2.1

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

4. Final simplification 10.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.279587145681289 \cdot 10^{-136}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 6.8526453862578789 \cdot 10^{139}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 +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))