Cubic critical, wide range

Average Error: 52.3 → 6.4

Time: 13.8s

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 r62496 = b;
        double r62497 = -r62496;
        double r62498 = r62496 * r62496;
        double r62499 = 3.0;
        double r62500 = a;
        double r62501 = r62499 * r62500;
        double r62502 = c;
        double r62503 = r62501 * r62502;
        double r62504 = r62498 - r62503;
        double r62505 = sqrt(r62504);
        double r62506 = r62497 + r62505;
        double r62507 = r62506 / r62501;
        return r62507;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r62508 = -0.5;
        double r62509 = c;
        double r62510 = b;
        double r62511 = r62509 / r62510;
        double r62512 = r62508 * r62511;
        return r62512;
}

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

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

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 +o rules:numerics
(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)))