Cubic critical, wide range

Average Error: 52.5 → 6.2

Time: 13.2s

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 r52472 = b;
        double r52473 = -r52472;
        double r52474 = r52472 * r52472;
        double r52475 = 3.0;
        double r52476 = a;
        double r52477 = r52475 * r52476;
        double r52478 = c;
        double r52479 = r52477 * r52478;
        double r52480 = r52474 - r52479;
        double r52481 = sqrt(r52480);
        double r52482 = r52473 + r52481;
        double r52483 = r52482 / r52477;
        return r52483;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r52484 = -0.5;
        double r52485 = c;
        double r52486 = b;
        double r52487 = r52485 / r52486;
        double r52488 = r52484 * r52487;
        return r52488;
}

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)))