Cubic critical, wide range

Average Error: 52.5 → 6.2

Time: 6.7s

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 r67998 = b;
        double r67999 = -r67998;
        double r68000 = r67998 * r67998;
        double r68001 = 3.0;
        double r68002 = a;
        double r68003 = r68001 * r68002;
        double r68004 = c;
        double r68005 = r68003 * r68004;
        double r68006 = r68000 - r68005;
        double r68007 = sqrt(r68006);
        double r68008 = r67999 + r68007;
        double r68009 = r68008 / r68003;
        return r68009;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r68010 = -0.5;
        double r68011 = c;
        double r68012 = b;
        double r68013 = r68011 / r68012;
        double r68014 = r68010 * r68013;
        return r68014;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.5

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

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 2020043 
(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)))