Cubic critical, wide range

Average Error: 52.4 → 6.3

Time: 15.3s

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 r94699 = b;
        double r94700 = -r94699;
        double r94701 = r94699 * r94699;
        double r94702 = 3.0;
        double r94703 = a;
        double r94704 = r94702 * r94703;
        double r94705 = c;
        double r94706 = r94704 * r94705;
        double r94707 = r94701 - r94706;
        double r94708 = sqrt(r94707);
        double r94709 = r94700 + r94708;
        double r94710 = r94709 / r94704;
        return r94710;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r94711 = -0.5;
        double r94712 = c;
        double r94713 = b;
        double r94714 = r94712 / r94713;
        double r94715 = r94711 * r94714;
        return r94715;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.4

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

2. Simplified 52.4

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

3. Taylor expanded around inf 6.3

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

4. Final simplification 6.3

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

Reproduce

herbie shell --seed 2019326 
(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)))