Cubic critical, wide range

Average Error: 52.3 → 6.4

Time: 33.5s

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 r128393 = b;
        double r128394 = -r128393;
        double r128395 = r128393 * r128393;
        double r128396 = 3.0;
        double r128397 = a;
        double r128398 = r128396 * r128397;
        double r128399 = c;
        double r128400 = r128398 * r128399;
        double r128401 = r128395 - r128400;
        double r128402 = sqrt(r128401);
        double r128403 = r128394 + r128402;
        double r128404 = r128403 / r128398;
        return r128404;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r128405 = -0.5;
        double r128406 = c;
        double r128407 = b;
        double r128408 = r128406 / r128407;
        double r128409 = r128405 * r128408;
        return r128409;
}

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)))