Cubic critical, wide range

Average Error: 52.5 → 6.2

Time: 24.8s

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

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

↓

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

C

double f(double a, double b, double c) {
        double r4027954 = b;
        double r4027955 = -r4027954;
        double r4027956 = r4027954 * r4027954;
        double r4027957 = 3.0;
        double r4027958 = a;
        double r4027959 = r4027957 * r4027958;
        double r4027960 = c;
        double r4027961 = r4027959 * r4027960;
        double r4027962 = r4027956 - r4027961;
        double r4027963 = sqrt(r4027962);
        double r4027964 = r4027955 + r4027963;
        double r4027965 = r4027964 / r4027959;
        return r4027965;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r4027966 = c;
        double r4027967 = b;
        double r4027968 = r4027966 / r4027967;
        double r4027969 = -0.5;
        double r4027970 = r4027968 * r4027969;
        return r4027970;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.5

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

2. Simplified 52.5

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

3. Taylor expanded around inf 6.2

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

4. Final simplification 6.2

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :pre (and (< 4.930380657631324e-32 a 2.028240960365167e+31) (< 4.930380657631324e-32 b 2.028240960365167e+31) (< 4.930380657631324e-32 c 2.028240960365167e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))