Cubic critical, wide range

Average Error: 52.6 → 6.2

Time: 6.4s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

\[4.93038 \cdot 10^{-32} \lt a \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt b \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt c \lt 2.02824 \cdot 10^{31}\]

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 r60016 = b;
        double r60017 = -r60016;
        double r60018 = r60016 * r60016;
        double r60019 = 3.0;
        double r60020 = a;
        double r60021 = r60019 * r60020;
        double r60022 = c;
        double r60023 = r60021 * r60022;
        double r60024 = r60018 - r60023;
        double r60025 = sqrt(r60024);
        double r60026 = r60017 + r60025;
        double r60027 = r60026 / r60021;
        return r60027;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r60028 = -0.5;
        double r60029 = c;
        double r60030 = b;
        double r60031 = r60029 / r60030;
        double r60032 = r60028 * r60031;
        return r60032;
}

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 2020042 
(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)))