Cubic critical, wide range

Average Error: 52.4 → 6.3

Time: 5.8s

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 r100447 = b;
        double r100448 = -r100447;
        double r100449 = r100447 * r100447;
        double r100450 = 3.0;
        double r100451 = a;
        double r100452 = r100450 * r100451;
        double r100453 = c;
        double r100454 = r100452 * r100453;
        double r100455 = r100449 - r100454;
        double r100456 = sqrt(r100455);
        double r100457 = r100448 + r100456;
        double r100458 = r100457 / r100452;
        return r100458;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r100459 = -0.5;
        double r100460 = c;
        double r100461 = b;
        double r100462 = r100460 / r100461;
        double r100463 = r100459 * r100462;
        return r100463;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.4

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

2. Taylor expanded around inf 6.3

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

3. Final simplification 6.3

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

Reproduce

herbie shell --seed 2020046 +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)))