Cubic critical, wide range

Average Error: 52.6 → 6.1

Time: 18.2s

Precision: 64

\[4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt a \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt b \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt c \lt 20282409603651670423947251286016\]

Math

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

↓

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

C

double f(double a, double b, double c) {
        double r3451912 = b;
        double r3451913 = -r3451912;
        double r3451914 = r3451912 * r3451912;
        double r3451915 = 3.0;
        double r3451916 = a;
        double r3451917 = r3451915 * r3451916;
        double r3451918 = c;
        double r3451919 = r3451917 * r3451918;
        double r3451920 = r3451914 - r3451919;
        double r3451921 = sqrt(r3451920);
        double r3451922 = r3451913 + r3451921;
        double r3451923 = r3451922 / r3451917;
        return r3451923;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r3451924 = c;
        double r3451925 = b;
        double r3451926 = r3451924 / r3451925;
        double r3451927 = -0.5;
        double r3451928 = r3451926 * r3451927;
        return r3451928;
}

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. Simplified 52.6

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

3. Taylor expanded around inf 6.1

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

4. Final simplification 6.1

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

Reproduce

herbie shell --seed 2019174 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :pre (and (< 4.930380657631324e-32 a 2.028240960365167e+31) (< 4.930380657631324e-32 b 2.028240960365167e+31) (< 4.930380657631324e-32 c 2.028240960365167e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))