Cubic critical, wide range

Average Error: 52.6 → 6.1

Time: 15.8s

Precision: 64

\[4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt a \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt b \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt c \lt 20282409603651670423947251286016\]

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 r60695 = b;
        double r60696 = -r60695;
        double r60697 = r60695 * r60695;
        double r60698 = 3.0;
        double r60699 = a;
        double r60700 = r60698 * r60699;
        double r60701 = c;
        double r60702 = r60700 * r60701;
        double r60703 = r60697 - r60702;
        double r60704 = sqrt(r60703);
        double r60705 = r60696 + r60704;
        double r60706 = r60705 / r60700;
        return r60706;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r60707 = -0.5;
        double r60708 = c;
        double r60709 = b;
        double r60710 = r60708 / r60709;
        double r60711 = r60707 * r60710;
        return r60711;
}

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

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

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 2019208 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (< 4.93038e-32 a 2.02824e31) (< 4.93038e-32 b 2.02824e31) (< 4.93038e-32 c 2.02824e31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))