Cubic critical, wide range

Average Error: 52.7 → 6.1

Time: 4.6s

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 r65403 = b;
        double r65404 = -r65403;
        double r65405 = r65403 * r65403;
        double r65406 = 3.0;
        double r65407 = a;
        double r65408 = r65406 * r65407;
        double r65409 = c;
        double r65410 = r65408 * r65409;
        double r65411 = r65405 - r65410;
        double r65412 = sqrt(r65411);
        double r65413 = r65404 + r65412;
        double r65414 = r65413 / r65408;
        return r65414;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r65415 = -0.5;
        double r65416 = c;
        double r65417 = b;
        double r65418 = r65416 / r65417;
        double r65419 = r65415 * r65418;
        return r65419;
}

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 2020083 +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)))