Cubic critical, wide range

Average Error: 52.4 → 6.2

Time: 4.1s

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 r95331 = b;
        double r95332 = -r95331;
        double r95333 = r95331 * r95331;
        double r95334 = 3.0;
        double r95335 = a;
        double r95336 = r95334 * r95335;
        double r95337 = c;
        double r95338 = r95336 * r95337;
        double r95339 = r95333 - r95338;
        double r95340 = sqrt(r95339);
        double r95341 = r95332 + r95340;
        double r95342 = r95341 / r95336;
        return r95342;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r95343 = -0.5;
        double r95344 = c;
        double r95345 = b;
        double r95346 = r95344 / r95345;
        double r95347 = r95343 * r95346;
        return r95347;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.4

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

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 2020062 +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)))