Cubic critical, wide range

Average Error: 52.3 → 6.4

Time: 34.8s

Precision: 64

\[4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt a \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt b \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt c \lt 20282409603651670423947251286016\]

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 r95167 = b;
        double r95168 = -r95167;
        double r95169 = r95167 * r95167;
        double r95170 = 3.0;
        double r95171 = a;
        double r95172 = r95170 * r95171;
        double r95173 = c;
        double r95174 = r95172 * r95173;
        double r95175 = r95169 - r95174;
        double r95176 = sqrt(r95175);
        double r95177 = r95168 + r95176;
        double r95178 = r95177 / r95172;
        return r95178;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r95179 = -0.5;
        double r95180 = c;
        double r95181 = b;
        double r95182 = r95180 / r95181;
        double r95183 = r95179 * r95182;
        return r95183;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.3

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

2. Simplified 52.3

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

3. Taylor expanded around inf 6.4

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

4. Final simplification 6.4

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

Reproduce

herbie shell --seed 2019323 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (< 4.93038e-32 a 2.02824e+31) (< 4.93038e-32 b 2.02824e+31) (< 4.93038e-32 c 2.02824e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))