Cubic critical, wide range

Average Error: 52.6 → 6.1

Time: 6.1s

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 r149081 = b;
        double r149082 = -r149081;
        double r149083 = r149081 * r149081;
        double r149084 = 3.0;
        double r149085 = a;
        double r149086 = r149084 * r149085;
        double r149087 = c;
        double r149088 = r149086 * r149087;
        double r149089 = r149083 - r149088;
        double r149090 = sqrt(r149089);
        double r149091 = r149082 + r149090;
        double r149092 = r149091 / r149086;
        return r149092;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r149093 = -0.5;
        double r149094 = c;
        double r149095 = b;
        double r149096 = r149094 / r149095;
        double r149097 = r149093 * r149096;
        return r149097;
}

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. Simplified 52.6

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

3. Taylor expanded around inf 6.1

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

4. 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)))