Cubic critical, wide range

Average Error: 52.7 → 6.1

Time: 4.3s

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 r82060 = b;
        double r82061 = -r82060;
        double r82062 = r82060 * r82060;
        double r82063 = 3.0;
        double r82064 = a;
        double r82065 = r82063 * r82064;
        double r82066 = c;
        double r82067 = r82065 * r82066;
        double r82068 = r82062 - r82067;
        double r82069 = sqrt(r82068);
        double r82070 = r82061 + r82069;
        double r82071 = r82070 / r82065;
        return r82071;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r82072 = -0.5;
        double r82073 = c;
        double r82074 = b;
        double r82075 = r82073 / r82074;
        double r82076 = r82072 * r82075;
        return r82076;
}

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. Taylor expanded around inf 6.1

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

3. Final simplification 6.1

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

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (< 4.9303800000000003e-32 a 2.02824e+31) (< 4.9303800000000003e-32 b 2.02824e+31) (< 4.9303800000000003e-32 c 2.02824e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))