Cubic critical, medium range

Average Error: 43.7 → 12.1

Time: 12.3s

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

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

↓

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

C

double f(double a, double b, double c) {
        double r47833 = b;
        double r47834 = -r47833;
        double r47835 = r47833 * r47833;
        double r47836 = 3.0;
        double r47837 = a;
        double r47838 = r47836 * r47837;
        double r47839 = c;
        double r47840 = r47838 * r47839;
        double r47841 = r47835 - r47840;
        double r47842 = sqrt(r47841);
        double r47843 = r47834 + r47842;
        double r47844 = r47843 / r47838;
        return r47844;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r47845 = -0.5;
        double r47846 = c;
        double r47847 = b;
        double r47848 = r47846 / r47847;
        double r47849 = r47845 * r47848;
        return r47849;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 43.7

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

2. Simplified 43.7

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

3. Taylor expanded around inf 12.1

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

4. Final simplification 12.1

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

Reproduce

herbie shell --seed 2019325 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e+15) (< 1.11022e-16 b 9.0072e+15) (< 1.11022e-16 c 9.0072e+15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))