Cubic critical, wide range

Average Error: 52.6 → 6.1

Time: 6.3s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

\[4.93038 \cdot 10^{-32} \lt a \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt b \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt c \lt 2.02824 \cdot 10^{31}\]

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 r52606 = b;
        double r52607 = -r52606;
        double r52608 = r52606 * r52606;
        double r52609 = 3.0;
        double r52610 = a;
        double r52611 = r52609 * r52610;
        double r52612 = c;
        double r52613 = r52611 * r52612;
        double r52614 = r52608 - r52613;
        double r52615 = sqrt(r52614);
        double r52616 = r52607 + r52615;
        double r52617 = r52616 / r52611;
        return r52617;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r52618 = -0.5;
        double r52619 = c;
        double r52620 = b;
        double r52621 = r52619 / r52620;
        double r52622 = r52618 * r52621;
        return r52622;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.6

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

2. Taylor expanded around inf 6.1

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

3. Final simplification 6.1

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (< 4.9303800000000003e-32 a 2.02824e+31) (< 4.9303800000000003e-32 b 2.02824e+31) (< 4.9303800000000003e-32 c 2.02824e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))