Cubic critical, wide range

Average Error: 52.5 → 6.2

Time: 14.1s

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 r99474 = b;
        double r99475 = -r99474;
        double r99476 = r99474 * r99474;
        double r99477 = 3.0;
        double r99478 = a;
        double r99479 = r99477 * r99478;
        double r99480 = c;
        double r99481 = r99479 * r99480;
        double r99482 = r99476 - r99481;
        double r99483 = sqrt(r99482);
        double r99484 = r99475 + r99483;
        double r99485 = r99484 / r99479;
        return r99485;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r99486 = -0.5;
        double r99487 = c;
        double r99488 = b;
        double r99489 = r99487 / r99488;
        double r99490 = r99486 * r99489;
        return r99490;
}

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

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

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 2019235 
(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)))