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 r76598 = b;
        double r76599 = -r76598;
        double r76600 = r76598 * r76598;
        double r76601 = 3.0;
        double r76602 = a;
        double r76603 = r76601 * r76602;
        double r76604 = c;
        double r76605 = r76603 * r76604;
        double r76606 = r76600 - r76605;
        double r76607 = sqrt(r76606);
        double r76608 = r76599 + r76607;
        double r76609 = r76608 / r76603;
        return r76609;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r76610 = -0.5;
        double r76611 = c;
        double r76612 = b;
        double r76613 = r76611 / r76612;
        double r76614 = r76610 * r76613;
        return r76614;
}

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)))