Cubic critical, wide range

Average Error: 52.6 → 6.1

Time: 6.2s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

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

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 r66280 = b;
        double r66281 = -r66280;
        double r66282 = r66280 * r66280;
        double r66283 = 3.0;
        double r66284 = a;
        double r66285 = r66283 * r66284;
        double r66286 = c;
        double r66287 = r66285 * r66286;
        double r66288 = r66282 - r66287;
        double r66289 = sqrt(r66288);
        double r66290 = r66281 + r66289;
        double r66291 = r66290 / r66285;
        return r66291;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r66292 = -0.5;
        double r66293 = c;
        double r66294 = b;
        double r66295 = r66293 / r66294;
        double r66296 = r66292 * r66295;
        return r66296;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.6

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

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 2020047 
(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)))