Cubic critical, wide range

Average Error: 52.3 → 6.4

Time: 5.0s

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 r107697 = b;
        double r107698 = -r107697;
        double r107699 = r107697 * r107697;
        double r107700 = 3.0;
        double r107701 = a;
        double r107702 = r107700 * r107701;
        double r107703 = c;
        double r107704 = r107702 * r107703;
        double r107705 = r107699 - r107704;
        double r107706 = sqrt(r107705);
        double r107707 = r107698 + r107706;
        double r107708 = r107707 / r107702;
        return r107708;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r107709 = -0.5;
        double r107710 = c;
        double r107711 = b;
        double r107712 = r107710 / r107711;
        double r107713 = r107709 * r107712;
        return r107713;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.3

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

2. Taylor expanded around inf 6.4

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

3. Final simplification 6.4

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

Reproduce

herbie shell --seed 2020033 +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)))