Cubic critical, wide range

Average Error: 52.8 → 6.0

Time: 4.5s

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 r85457 = b;
        double r85458 = -r85457;
        double r85459 = r85457 * r85457;
        double r85460 = 3.0;
        double r85461 = a;
        double r85462 = r85460 * r85461;
        double r85463 = c;
        double r85464 = r85462 * r85463;
        double r85465 = r85459 - r85464;
        double r85466 = sqrt(r85465);
        double r85467 = r85458 + r85466;
        double r85468 = r85467 / r85462;
        return r85468;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r85469 = -0.5;
        double r85470 = c;
        double r85471 = b;
        double r85472 = r85470 / r85471;
        double r85473 = r85469 * r85472;
        return r85473;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.8

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

2. Taylor expanded around inf 6.0

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

3. Final simplification 6.0

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

Reproduce

herbie shell --seed 2020059 +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)))