Cubic critical, wide range

Average Error: 52.5 → 6.2

Time: 14.7s

Precision: 64

\[4.930380657631324 \cdot 10^{-32} \lt a \lt 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} \lt b \lt 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} \lt c \lt 2.028240960365167 \cdot 10^{+31}\]

Math

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

↓

\[\frac{c}{b} \cdot \frac{-1}{2}\]

C

double f(double a, double b, double c) {
        double r1239970 = b;
        double r1239971 = -r1239970;
        double r1239972 = r1239970 * r1239970;
        double r1239973 = 3.0;
        double r1239974 = a;
        double r1239975 = r1239973 * r1239974;
        double r1239976 = c;
        double r1239977 = r1239975 * r1239976;
        double r1239978 = r1239972 - r1239977;
        double r1239979 = sqrt(r1239978);
        double r1239980 = r1239971 + r1239979;
        double r1239981 = r1239980 / r1239975;
        return r1239981;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r1239982 = c;
        double r1239983 = b;
        double r1239984 = r1239982 / r1239983;
        double r1239985 = -0.5;
        double r1239986 = r1239984 * r1239985;
        return r1239986;
}

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. Taylor expanded around inf 6.2

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}}\]

3. Final simplification 6.2

   \[\leadsto \frac{c}{b} \cdot \frac{-1}{2}\]

Reproduce

herbie shell --seed 2019154 +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 a) c)))) (* 3 a)))