Quadratic roots, wide range

Average Error: 52.3 → 6.4

Time: 13.2s

Precision: 64

\[4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt a \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt b \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt c \lt 20282409603651670423947251286016\]

Math

\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

↓

\[-1 \cdot \frac{c}{b}\]

C

double f(double a, double b, double c) {
        double r17609 = b;
        double r17610 = -r17609;
        double r17611 = r17609 * r17609;
        double r17612 = 4.0;
        double r17613 = a;
        double r17614 = r17612 * r17613;
        double r17615 = c;
        double r17616 = r17614 * r17615;
        double r17617 = r17611 - r17616;
        double r17618 = sqrt(r17617);
        double r17619 = r17610 + r17618;
        double r17620 = 2.0;
        double r17621 = r17620 * r17613;
        double r17622 = r17619 / r17621;
        return r17622;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r17623 = -1.0;
        double r17624 = c;
        double r17625 = b;
        double r17626 = r17624 / r17625;
        double r17627 = r17623 * r17626;
        return r17627;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.3

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

2. Simplified 52.3

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}}\]

3. Taylor expanded around inf 6.4

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

4. Final simplification 6.4

   \[\leadsto -1 \cdot \frac{c}{b}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (< 4.93038e-32 a 2.02824e+31) (< 4.93038e-32 b 2.02824e+31) (< 4.93038e-32 c 2.02824e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))