Cubic critical, wide range

Average Error: 52.8 → 6.0

Time: 13.1s

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 r64543 = b;
        double r64544 = -r64543;
        double r64545 = r64543 * r64543;
        double r64546 = 3.0;
        double r64547 = a;
        double r64548 = r64546 * r64547;
        double r64549 = c;
        double r64550 = r64548 * r64549;
        double r64551 = r64545 - r64550;
        double r64552 = sqrt(r64551);
        double r64553 = r64544 + r64552;
        double r64554 = r64553 / r64548;
        return r64554;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r64555 = -0.5;
        double r64556 = c;
        double r64557 = b;
        double r64558 = r64556 / r64557;
        double r64559 = r64555 * r64558;
        return r64559;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.8

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

2. Simplified 52.8

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

3. Taylor expanded around inf 6.0

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

4. Final simplification 6.0

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

Reproduce

herbie shell --seed 2019212 
(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)))