NMSE problem 3.2.1

Average Error: 33.6 → 10.5

Time: 3.8s

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 r92177 = b_2;
        double r92178 = -r92177;
        double r92179 = r92177 * r92177;
        double r92180 = a;
        double r92181 = c;
        double r92182 = r92180 * r92181;
        double r92183 = r92179 - r92182;
        double r92184 = sqrt(r92183);
        double r92185 = r92178 - r92184;
        double r92186 = r92185 / r92180;
        return r92186;
}

↓

double f(double a, double b_2, double c) {
        double r92187 = b_2;
        double r92188 = -1.3295118613703302e-13;
        bool r92189 = r92187 <= r92188;
        double r92190 = -0.5;
        double r92191 = c;
        double r92192 = r92191 / r92187;
        double r92193 = r92190 * r92192;
        double r92194 = 3.295450950813408e+65;
        bool r92195 = r92187 <= r92194;
        double r92196 = -r92187;
        double r92197 = r92187 * r92187;
        double r92198 = a;
        double r92199 = r92198 * r92191;
        double r92200 = r92197 - r92199;
        double r92201 = sqrt(r92200);
        double r92202 = r92196 - r92201;
        double r92203 = 1.0;
        double r92204 = r92203 / r92198;
        double r92205 = r92202 * r92204;
        double r92206 = 0.5;
        double r92207 = r92206 * r92192;
        double r92208 = 2.0;
        double r92209 = r92187 / r92198;
        double r92210 = r92208 * r92209;
        double r92211 = r92207 - r92210;
        double r92212 = r92195 ? r92205 : r92211;
        double r92213 = r92189 ? r92193 : r92212;
        return r92213;
}

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 "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))