Cubic critical, wide range

Average Error: 52.6 → 6.2

Time: 3.9s

Precision: 64

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

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 r76606 = b;
        double r76607 = -r76606;
        double r76608 = r76606 * r76606;
        double r76609 = 3.0;
        double r76610 = a;
        double r76611 = r76609 * r76610;
        double r76612 = c;
        double r76613 = r76611 * r76612;
        double r76614 = r76608 - r76613;
        double r76615 = sqrt(r76614);
        double r76616 = r76607 + r76615;
        double r76617 = r76616 / r76611;
        return r76617;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r76618 = -0.5;
        double r76619 = c;
        double r76620 = b;
        double r76621 = r76619 / r76620;
        double r76622 = r76618 * r76621;
        return r76622;
}

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 6.2

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

3. Final simplification 6.2

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

Reproduce

herbie shell --seed 2020056 +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)))