Cubic critical, wide range

Average Error: 52.7 → 6.1

Time: 13.0s

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 r65772 = b;
        double r65773 = -r65772;
        double r65774 = r65772 * r65772;
        double r65775 = 3.0;
        double r65776 = a;
        double r65777 = r65775 * r65776;
        double r65778 = c;
        double r65779 = r65777 * r65778;
        double r65780 = r65774 - r65779;
        double r65781 = sqrt(r65780);
        double r65782 = r65773 + r65781;
        double r65783 = r65782 / r65777;
        return r65783;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r65784 = -0.5;
        double r65785 = c;
        double r65786 = b;
        double r65787 = r65785 / r65786;
        double r65788 = r65784 * r65787;
        return r65788;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.7

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

2. Simplified 52.7

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

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 2019325 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (< 4.93038e-32 a 2.02824e+31) (< 4.93038e-32 b 2.02824e+31) (< 4.93038e-32 c 2.02824e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))