Cubic critical, wide range

Average Error: 52.6 → 6.1

Time: 5.6s

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 r62141 = b;
        double r62142 = -r62141;
        double r62143 = r62141 * r62141;
        double r62144 = 3.0;
        double r62145 = a;
        double r62146 = r62144 * r62145;
        double r62147 = c;
        double r62148 = r62146 * r62147;
        double r62149 = r62143 - r62148;
        double r62150 = sqrt(r62149);
        double r62151 = r62142 + r62150;
        double r62152 = r62151 / r62146;
        return r62152;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r62153 = -0.5;
        double r62154 = c;
        double r62155 = b;
        double r62156 = r62154 / r62155;
        double r62157 = r62153 * r62156;
        return r62157;
}

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 6.1

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

3. Final simplification 6.1

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

Reproduce

herbie shell --seed 2020047 
(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)))