Quadratic roots, wide range

Average Error: 52.2 → 6.4

Time: 14.7s

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 r17068 = b;
        double r17069 = -r17068;
        double r17070 = r17068 * r17068;
        double r17071 = 4.0;
        double r17072 = a;
        double r17073 = r17071 * r17072;
        double r17074 = c;
        double r17075 = r17073 * r17074;
        double r17076 = r17070 - r17075;
        double r17077 = sqrt(r17076);
        double r17078 = r17069 + r17077;
        double r17079 = 2.0;
        double r17080 = r17079 * r17072;
        double r17081 = r17078 / r17080;
        return r17081;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r17082 = -1.0;
        double r17083 = c;
        double r17084 = b;
        double r17085 = r17083 / r17084;
        double r17086 = r17082 * r17085;
        return r17086;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.2

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

2. Simplified 52.2

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

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 2019209 +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)))