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 r69999 = b;
        double r70000 = -r69999;
        double r70001 = r69999 * r69999;
        double r70002 = 3.0;
        double r70003 = a;
        double r70004 = r70002 * r70003;
        double r70005 = c;
        double r70006 = r70004 * r70005;
        double r70007 = r70001 - r70006;
        double r70008 = sqrt(r70007);
        double r70009 = r70000 + r70008;
        double r70010 = r70009 / r70004;
        return r70010;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r70011 = -0.5;
        double r70012 = c;
        double r70013 = b;
        double r70014 = r70012 / r70013;
        double r70015 = r70011 * r70014;
        return r70015;
}

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 6.2

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

4. 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)))