Cubic critical, wide range

Average Error: 52.5 → 6.2

Time: 4.5s

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 r75869 = b;
        double r75870 = -r75869;
        double r75871 = r75869 * r75869;
        double r75872 = 3.0;
        double r75873 = a;
        double r75874 = r75872 * r75873;
        double r75875 = c;
        double r75876 = r75874 * r75875;
        double r75877 = r75871 - r75876;
        double r75878 = sqrt(r75877);
        double r75879 = r75870 + r75878;
        double r75880 = r75879 / r75874;
        return r75880;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r75881 = -0.5;
        double r75882 = c;
        double r75883 = b;
        double r75884 = r75882 / r75883;
        double r75885 = r75881 * r75884;
        return r75885;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.5

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

2. Taylor expanded around inf 6.2

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

3. Final simplification 6.2

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

Reproduce

herbie shell --seed 2020003 +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)))