Cubic critical, medium range

Average Error: 43.9 → 12.0

Time: 16.9s

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 r75164 = b;
        double r75165 = -r75164;
        double r75166 = r75164 * r75164;
        double r75167 = 3.0;
        double r75168 = a;
        double r75169 = r75167 * r75168;
        double r75170 = c;
        double r75171 = r75169 * r75170;
        double r75172 = r75166 - r75171;
        double r75173 = sqrt(r75172);
        double r75174 = r75165 + r75173;
        double r75175 = r75174 / r75169;
        return r75175;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r75176 = -0.5;
        double r75177 = c;
        double r75178 = b;
        double r75179 = r75177 / r75178;
        double r75180 = r75176 * r75179;
        return r75180;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 43.9

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

2. Simplified 43.9

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

3. Taylor expanded around inf 12.0

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

4. Final simplification 12.0

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

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e15) (< 1.11022e-16 b 9.0072e15) (< 1.11022e-16 c 9.0072e15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))