quad2m (problem 3.2.1, negative)

Average Error: 33.8 → 10.1

Time: 4.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.369694371126339229257094016308893237032 \cdot 10^{-83}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.678238127073728805877873599258558355989 \cdot 10^{53}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

C

double f(double a, double b_2, double c) {
        double r12043 = b_2;
        double r12044 = -r12043;
        double r12045 = r12043 * r12043;
        double r12046 = a;
        double r12047 = c;
        double r12048 = r12046 * r12047;
        double r12049 = r12045 - r12048;
        double r12050 = sqrt(r12049);
        double r12051 = r12044 - r12050;
        double r12052 = r12051 / r12046;
        return r12052;
}

↓

double f(double a, double b_2, double c) {
        double r12053 = b_2;
        double r12054 = -1.3696943711263392e-83;
        bool r12055 = r12053 <= r12054;
        double r12056 = -0.5;
        double r12057 = c;
        double r12058 = r12057 / r12053;
        double r12059 = r12056 * r12058;
        double r12060 = 2.678238127073729e+53;
        bool r12061 = r12053 <= r12060;
        double r12062 = 1.0;
        double r12063 = a;
        double r12064 = -r12053;
        double r12065 = r12053 * r12053;
        double r12066 = r12063 * r12057;
        double r12067 = r12065 - r12066;
        double r12068 = sqrt(r12067);
        double r12069 = r12064 - r12068;
        double r12070 = r12063 / r12069;
        double r12071 = r12062 / r12070;
        double r12072 = 0.5;
        double r12073 = r12072 * r12058;
        double r12074 = 2.0;
        double r12075 = r12053 / r12063;
        double r12076 = r12074 * r12075;
        double r12077 = r12073 - r12076;
        double r12078 = r12061 ? r12071 : r12077;
        double r12079 = r12055 ? r12059 : r12078;
        return r12079;
}

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 < -1.3696943711263392e-83

   1. Initial program 53.2

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

   2. Taylor expanded around -inf 9.4

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

   if -1.3696943711263392e-83 < b_2 < 2.678238127073729e+53

   1. Initial program 12.8

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

   3. Applied clear-num 12.9

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

   if 2.678238127073729e+53 < b_2

   1. Initial program 38.1

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

   2. Taylor expanded around inf 5.7

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

4. Final simplification 10.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.369694371126339229257094016308893237032 \cdot 10^{-83}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.678238127073728805877873599258558355989 \cdot 10^{53}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 +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))