Cubic critical, wide range

Average Error: 52.5 → 6.2

Time: 13.4s

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

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

↓

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

C

double f(double a, double b, double c) {
        double r67159 = b;
        double r67160 = -r67159;
        double r67161 = r67159 * r67159;
        double r67162 = 3.0;
        double r67163 = a;
        double r67164 = r67162 * r67163;
        double r67165 = c;
        double r67166 = r67164 * r67165;
        double r67167 = r67161 - r67166;
        double r67168 = sqrt(r67167);
        double r67169 = r67160 + r67168;
        double r67170 = r67169 / r67164;
        return r67170;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r67171 = -0.5;
        double r67172 = c;
        double r67173 = b;
        double r67174 = r67172 / r67173;
        double r67175 = r67171 * r67174;
        return r67175;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.5

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

2. Simplified 52.5

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

3. Taylor expanded around inf 6.2

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

4. Final simplification 6.2

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

Reproduce

herbie shell --seed 2019303 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (< 4.93038e-32 a 2.02824e31) (< 4.93038e-32 b 2.02824e31) (< 4.93038e-32 c 2.02824e31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))