Cubic critical, wide range

Average Error: 52.4 → 6.2

Time: 4.1s

Precision: 64

\[4.93038 \cdot 10^{-32} \lt a \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt b \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt c \lt 2.02824 \cdot 10^{31}\]

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 r77647 = b;
        double r77648 = -r77647;
        double r77649 = r77647 * r77647;
        double r77650 = 3.0;
        double r77651 = a;
        double r77652 = r77650 * r77651;
        double r77653 = c;
        double r77654 = r77652 * r77653;
        double r77655 = r77649 - r77654;
        double r77656 = sqrt(r77655);
        double r77657 = r77648 + r77656;
        double r77658 = r77657 / r77652;
        return r77658;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r77659 = -0.5;
        double r77660 = c;
        double r77661 = b;
        double r77662 = r77660 / r77661;
        double r77663 = r77659 * r77662;
        return r77663;
}

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. Taylor expanded around inf 6.2

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

3. Final simplification 6.2

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

Reproduce

herbie shell --seed 2020039 +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)))