quad2m (problem 3.2.1, negative)

Average Error: 33.6 → 10.5

Time: 4.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.3295118613703302 \cdot 10^{-13}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.29545095081340793 \cdot 10^{65}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

C

double f(double a, double b_2, double c) {
        double r12155 = b_2;
        double r12156 = -r12155;
        double r12157 = r12155 * r12155;
        double r12158 = a;
        double r12159 = c;
        double r12160 = r12158 * r12159;
        double r12161 = r12157 - r12160;
        double r12162 = sqrt(r12161);
        double r12163 = r12156 - r12162;
        double r12164 = r12163 / r12158;
        return r12164;
}

↓

double f(double a, double b_2, double c) {
        double r12165 = b_2;
        double r12166 = -1.3295118613703302e-13;
        bool r12167 = r12165 <= r12166;
        double r12168 = -0.5;
        double r12169 = c;
        double r12170 = r12169 / r12165;
        double r12171 = r12168 * r12170;
        double r12172 = 3.295450950813408e+65;
        bool r12173 = r12165 <= r12172;
        double r12174 = -r12165;
        double r12175 = r12165 * r12165;
        double r12176 = a;
        double r12177 = r12176 * r12169;
        double r12178 = r12175 - r12177;
        double r12179 = sqrt(r12178);
        double r12180 = r12174 - r12179;
        double r12181 = 1.0;
        double r12182 = r12181 / r12176;
        double r12183 = r12180 * r12182;
        double r12184 = 0.5;
        double r12185 = r12184 * r12170;
        double r12186 = 2.0;
        double r12187 = r12165 / r12176;
        double r12188 = r12186 * r12187;
        double r12189 = r12185 - r12188;
        double r12190 = r12173 ? r12183 : r12189;
        double r12191 = r12167 ? r12171 : r12190;
        return r12191;
}

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 < -1.3295118613703302e-13

   1. Initial program 55.0

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

   2. Taylor expanded around -inf 6.9

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

   if -1.3295118613703302e-13 < b_2 < 3.295450950813408e+65

   1. Initial program 15.4

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

   3. Applied div-inv 15.5

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

   if 3.295450950813408e+65 < b_2

   1. Initial program 40.0

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

   2. Taylor expanded around inf 4.7

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

4. Final simplification 10.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.3295118613703302 \cdot 10^{-13}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.29545095081340793 \cdot 10^{65}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +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))