Cubic critical, wide range

Average Error: 52.5 → 6.2

Time: 13.3s

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 r58241 = b;
        double r58242 = -r58241;
        double r58243 = r58241 * r58241;
        double r58244 = 3.0;
        double r58245 = a;
        double r58246 = r58244 * r58245;
        double r58247 = c;
        double r58248 = r58246 * r58247;
        double r58249 = r58243 - r58248;
        double r58250 = sqrt(r58249);
        double r58251 = r58242 + r58250;
        double r58252 = r58251 / r58246;
        return r58252;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r58253 = -0.5;
        double r58254 = c;
        double r58255 = b;
        double r58256 = r58254 / r58255;
        double r58257 = r58253 * r58256;
        return r58257;
}

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. Simplified 52.5

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

3. Taylor expanded around inf 6.2

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

4. Final simplification 6.2

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (< 4.93038e-32 a 2.02824e31) (< 4.93038e-32 b 2.02824e31) (< 4.93038e-32 c 2.02824e31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))