Cubic critical, wide range

Average Error: 52.3 → 6.4

Time: 4.0s

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 r88551 = b;
        double r88552 = -r88551;
        double r88553 = r88551 * r88551;
        double r88554 = 3.0;
        double r88555 = a;
        double r88556 = r88554 * r88555;
        double r88557 = c;
        double r88558 = r88556 * r88557;
        double r88559 = r88553 - r88558;
        double r88560 = sqrt(r88559);
        double r88561 = r88552 + r88560;
        double r88562 = r88561 / r88556;
        return r88562;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r88563 = -0.5;
        double r88564 = c;
        double r88565 = b;
        double r88566 = r88564 / r88565;
        double r88567 = r88563 * r88566;
        return r88567;
}

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. Taylor expanded around inf 6.4

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

3. Final simplification 6.4

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

Reproduce

herbie shell --seed 2020060 +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)))