Cubic critical, wide range

Average Error: 52.4 → 6.3

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 r100692 = b;
        double r100693 = -r100692;
        double r100694 = r100692 * r100692;
        double r100695 = 3.0;
        double r100696 = a;
        double r100697 = r100695 * r100696;
        double r100698 = c;
        double r100699 = r100697 * r100698;
        double r100700 = r100694 - r100699;
        double r100701 = sqrt(r100700);
        double r100702 = r100693 + r100701;
        double r100703 = r100702 / r100697;
        return r100703;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r100704 = -0.5;
        double r100705 = c;
        double r100706 = b;
        double r100707 = r100705 / r100706;
        double r100708 = r100704 * r100707;
        return r100708;
}

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 6.3

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

3. Final simplification 6.3

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

Reproduce

herbie shell --seed 2020046 +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)))