Quadratic roots, wide range

Average Error: 52.5 → 6.2

Time: 12.8s

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 r17681 = b;
        double r17682 = -r17681;
        double r17683 = r17681 * r17681;
        double r17684 = 4.0;
        double r17685 = a;
        double r17686 = r17684 * r17685;
        double r17687 = c;
        double r17688 = r17686 * r17687;
        double r17689 = r17683 - r17688;
        double r17690 = sqrt(r17689);
        double r17691 = r17682 + r17690;
        double r17692 = 2.0;
        double r17693 = r17692 * r17685;
        double r17694 = r17691 / r17693;
        return r17694;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r17695 = -1.0;
        double r17696 = c;
        double r17697 = b;
        double r17698 = r17696 / r17697;
        double r17699 = r17695 * r17698;
        return r17699;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.5

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

2. Simplified 52.5

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

3. Taylor expanded around inf 6.2

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

4. Final simplification 6.2

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (< 4.93038e-32 a 2.02824e31) (< 4.93038e-32 b 2.02824e31) (< 4.93038e-32 c 2.02824e31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))