quad2m (problem 3.2.1, negative)

Average Error: 34.5 → 10.5

Time: 8.2s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.78285893492843261 \cdot 10^{-126}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.1323104234385293 \cdot 10^{111}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, -2 \cdot \frac{b_2}{a}\right)\\ \end{array}\]

C

double f(double a, double b_2, double c) {
        double r15213 = b_2;
        double r15214 = -r15213;
        double r15215 = r15213 * r15213;
        double r15216 = a;
        double r15217 = c;
        double r15218 = r15216 * r15217;
        double r15219 = r15215 - r15218;
        double r15220 = sqrt(r15219);
        double r15221 = r15214 - r15220;
        double r15222 = r15221 / r15216;
        return r15222;
}

↓

double f(double a, double b_2, double c) {
        double r15223 = b_2;
        double r15224 = -4.7828589349284326e-126;
        bool r15225 = r15223 <= r15224;
        double r15226 = -0.5;
        double r15227 = c;
        double r15228 = r15227 / r15223;
        double r15229 = r15226 * r15228;
        double r15230 = 3.1323104234385293e+111;
        bool r15231 = r15223 <= r15230;
        double r15232 = -r15223;
        double r15233 = r15223 * r15223;
        double r15234 = a;
        double r15235 = r15234 * r15227;
        double r15236 = r15233 - r15235;
        double r15237 = sqrt(r15236);
        double r15238 = r15232 - r15237;
        double r15239 = 1.0;
        double r15240 = r15239 / r15234;
        double r15241 = r15238 * r15240;
        double r15242 = 0.5;
        double r15243 = -2.0;
        double r15244 = r15223 / r15234;
        double r15245 = r15243 * r15244;
        double r15246 = fma(r15242, r15228, r15245);
        double r15247 = r15231 ? r15241 : r15246;
        double r15248 = r15225 ? r15229 : r15247;
        return r15248;
}

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 < -4.7828589349284326e-126

   1. Initial program 51.3

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

   2. Taylor expanded around -inf 11.3

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

   if -4.7828589349284326e-126 < b_2 < 3.1323104234385293e+111

   1. Initial program 12.0

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

   3. Applied div-inv 12.1

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

   if 3.1323104234385293e+111 < b_2

   1. Initial program 49.7

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

   2. Taylor expanded around inf 3.4

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

   3. Simplified 3.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, -2 \cdot \frac{b_2}{a}\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 10.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.78285893492843261 \cdot 10^{-126}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 3.1323104234385293 \cdot 10^{111}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b_2}, -2 \cdot \frac{b_2}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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))