Cubic critical, medium range

Average Error: 44.2 → 11.8

Time: 54.0s

Precision: 64

\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]

Math

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

↓

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

C

double f(double a, double b, double c) {
        double r4604879 = b;
        double r4604880 = -r4604879;
        double r4604881 = r4604879 * r4604879;
        double r4604882 = 3.0;
        double r4604883 = a;
        double r4604884 = r4604882 * r4604883;
        double r4604885 = c;
        double r4604886 = r4604884 * r4604885;
        double r4604887 = r4604881 - r4604886;
        double r4604888 = sqrt(r4604887);
        double r4604889 = r4604880 + r4604888;
        double r4604890 = r4604889 / r4604884;
        return r4604890;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r4604891 = c;
        double r4604892 = b;
        double r4604893 = r4604891 / r4604892;
        double r4604894 = -0.5;
        double r4604895 = r4604893 * r4604894;
        return r4604895;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 44.2

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

2. Simplified 44.2

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

3. Taylor expanded around inf 11.8

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

4. Final simplification 11.8

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

Reproduce

herbie shell --seed 2019170 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))