Quadratic roots, wide range

Average Error: 52.2 → 6.4

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 r18032 = b;
        double r18033 = -r18032;
        double r18034 = r18032 * r18032;
        double r18035 = 4.0;
        double r18036 = a;
        double r18037 = r18035 * r18036;
        double r18038 = c;
        double r18039 = r18037 * r18038;
        double r18040 = r18034 - r18039;
        double r18041 = sqrt(r18040);
        double r18042 = r18033 + r18041;
        double r18043 = 2.0;
        double r18044 = r18043 * r18036;
        double r18045 = r18042 / r18044;
        return r18045;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r18046 = -1.0;
        double r18047 = c;
        double r18048 = b;
        double r18049 = r18047 / r18048;
        double r18050 = r18046 * r18049;
        return r18050;
}

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 2019305 +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)))