Cubic critical, wide range

Average Error: 52.6 → 6.2

Time: 14.7s

Precision: 64

\[4.93038 \cdot 10^{-32} \lt a \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt b \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt c \lt 2.02824 \cdot 10^{31}\]

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 r74981 = b;
        double r74982 = -r74981;
        double r74983 = r74981 * r74981;
        double r74984 = 3.0;
        double r74985 = a;
        double r74986 = r74984 * r74985;
        double r74987 = c;
        double r74988 = r74986 * r74987;
        double r74989 = r74983 - r74988;
        double r74990 = sqrt(r74989);
        double r74991 = r74982 + r74990;
        double r74992 = r74991 / r74986;
        return r74992;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r74993 = -0.5;
        double r74994 = c;
        double r74995 = b;
        double r74996 = r74994 / r74995;
        double r74997 = r74993 * r74996;
        return r74997;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.6

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

2. Simplified 52.6

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

3. Taylor expanded around inf 6.2

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

4. Final simplification 6.2

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

Reproduce

herbie shell --seed 2019195 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :pre (and (< 4.930380657631324e-32 a 2.028240960365167e+31) (< 4.930380657631324e-32 b 2.028240960365167e+31) (< 4.930380657631324e-32 c 2.028240960365167e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))