Cubic critical, wide range

Average Error: 52.5 → 6.2

Time: 4.0s

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 r86445 = b;
        double r86446 = -r86445;
        double r86447 = r86445 * r86445;
        double r86448 = 3.0;
        double r86449 = a;
        double r86450 = r86448 * r86449;
        double r86451 = c;
        double r86452 = r86450 * r86451;
        double r86453 = r86447 - r86452;
        double r86454 = sqrt(r86453);
        double r86455 = r86446 + r86454;
        double r86456 = r86455 / r86450;
        return r86456;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r86457 = -0.5;
        double r86458 = c;
        double r86459 = b;
        double r86460 = r86458 / r86459;
        double r86461 = r86457 * r86460;
        return r86461;
}

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. 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 2020021 +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)))