Cubic critical, wide range

Average Error: 52.3 → 6.4

Time: 30.1s

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 r59041 = b;
        double r59042 = -r59041;
        double r59043 = r59041 * r59041;
        double r59044 = 3.0;
        double r59045 = a;
        double r59046 = r59044 * r59045;
        double r59047 = c;
        double r59048 = r59046 * r59047;
        double r59049 = r59043 - r59048;
        double r59050 = sqrt(r59049);
        double r59051 = r59042 + r59050;
        double r59052 = r59051 / r59046;
        return r59052;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r59053 = -0.5;
        double r59054 = c;
        double r59055 = b;
        double r59056 = r59054 / r59055;
        double r59057 = r59053 * r59056;
        return r59057;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.3

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

2. Simplified 52.3

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

3. Taylor expanded around inf 6.4

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

4. Final simplification 6.4

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

Reproduce

herbie shell --seed 2019323 +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)))