Cubic critical, wide range

Average Error: 52.6 → 6.1

Time: 6.3s

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 r150215 = b;
        double r150216 = -r150215;
        double r150217 = r150215 * r150215;
        double r150218 = 3.0;
        double r150219 = a;
        double r150220 = r150218 * r150219;
        double r150221 = c;
        double r150222 = r150220 * r150221;
        double r150223 = r150217 - r150222;
        double r150224 = sqrt(r150223);
        double r150225 = r150216 + r150224;
        double r150226 = r150225 / r150220;
        return r150226;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r150227 = -0.5;
        double r150228 = c;
        double r150229 = b;
        double r150230 = r150228 / r150229;
        double r150231 = r150227 * r150230;
        return r150231;
}

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 2020045 
(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)))