Quadratic roots, wide range

Average Error: 52.7 → 6.1

Time: 13.6s

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 r17720 = b;
        double r17721 = -r17720;
        double r17722 = r17720 * r17720;
        double r17723 = 4.0;
        double r17724 = a;
        double r17725 = r17723 * r17724;
        double r17726 = c;
        double r17727 = r17725 * r17726;
        double r17728 = r17722 - r17727;
        double r17729 = sqrt(r17728);
        double r17730 = r17721 + r17729;
        double r17731 = 2.0;
        double r17732 = r17731 * r17724;
        double r17733 = r17730 / r17732;
        return r17733;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r17734 = -1.0;
        double r17735 = c;
        double r17736 = b;
        double r17737 = r17735 / r17736;
        double r17738 = r17734 * r17737;
        return r17738;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.7

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

2. Simplified 52.7

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

3. Taylor expanded around inf 6.1

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

4. Final simplification 6.1

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

Reproduce

herbie shell --seed 2019304 +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)))