Cubic critical, wide range

Average Error: 52.2 → 6.4

Time: 13.8s

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 r60688 = b;
        double r60689 = -r60688;
        double r60690 = r60688 * r60688;
        double r60691 = 3.0;
        double r60692 = a;
        double r60693 = r60691 * r60692;
        double r60694 = c;
        double r60695 = r60693 * r60694;
        double r60696 = r60690 - r60695;
        double r60697 = sqrt(r60696);
        double r60698 = r60689 + r60697;
        double r60699 = r60698 / r60693;
        return r60699;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r60700 = -0.5;
        double r60701 = c;
        double r60702 = b;
        double r60703 = r60701 / r60702;
        double r60704 = r60700 * r60703;
        return r60704;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.2

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

2. Taylor expanded around inf 6.4

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

3. Final simplification 6.4

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

Reproduce

herbie shell --seed 2019305 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3 a) c)))) (* 3 a)))