Quadratic roots, wide range

Average Error: 52.4 → 6.2

Time: 14.2s

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 r14982 = b;
        double r14983 = -r14982;
        double r14984 = r14982 * r14982;
        double r14985 = 4.0;
        double r14986 = a;
        double r14987 = r14985 * r14986;
        double r14988 = c;
        double r14989 = r14987 * r14988;
        double r14990 = r14984 - r14989;
        double r14991 = sqrt(r14990);
        double r14992 = r14983 + r14991;
        double r14993 = 2.0;
        double r14994 = r14993 * r14986;
        double r14995 = r14992 / r14994;
        return r14995;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r14996 = -1.0;
        double r14997 = c;
        double r14998 = b;
        double r14999 = r14997 / r14998;
        double r15000 = r14996 * r14999;
        return r15000;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.4

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

2. Simplified 52.4

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

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 2019326 +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)))