Cubic critical, wide range

Average Error: 52.3 → 6.3

Time: 4.5s

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

\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

↓

\[-0.5 \cdot \frac{c}{b}\]

C

double f(double a, double b, double c) {
        double r79846 = b;
        double r79847 = -r79846;
        double r79848 = r79846 * r79846;
        double r79849 = 3.0;
        double r79850 = a;
        double r79851 = r79849 * r79850;
        double r79852 = c;
        double r79853 = r79851 * r79852;
        double r79854 = r79848 - r79853;
        double r79855 = sqrt(r79854);
        double r79856 = r79847 + r79855;
        double r79857 = r79856 / r79851;
        return r79857;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r79858 = -0.5;
        double r79859 = c;
        double r79860 = b;
        double r79861 = r79859 / r79860;
        double r79862 = r79858 * r79861;
        return r79862;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.3

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

2. Taylor expanded around inf 6.3

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}}\]

3. Final simplification 6.3

   \[\leadsto -0.5 \cdot \frac{c}{b}\]

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (< 4.9303800000000003e-32 a 2.02824e+31) (< 4.9303800000000003e-32 b 2.02824e+31) (< 4.9303800000000003e-32 c 2.02824e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))