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 r85312 = b;
        double r85313 = -r85312;
        double r85314 = r85312 * r85312;
        double r85315 = 3.0;
        double r85316 = a;
        double r85317 = r85315 * r85316;
        double r85318 = c;
        double r85319 = r85317 * r85318;
        double r85320 = r85314 - r85319;
        double r85321 = sqrt(r85320);
        double r85322 = r85313 + r85321;
        double r85323 = r85322 / r85317;
        return r85323;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r85324 = -0.5;
        double r85325 = c;
        double r85326 = b;
        double r85327 = r85325 / r85326;
        double r85328 = r85324 * r85327;
        return r85328;
}

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)))