Quadratic roots, wide range

Average Error: 52.7 → 6.1

Time: 15.1s

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

\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

↓

\[-1 \cdot \frac{c}{b}\]

C

double f(double a, double b, double c) {
        double r19077 = b;
        double r19078 = -r19077;
        double r19079 = r19077 * r19077;
        double r19080 = 4.0;
        double r19081 = a;
        double r19082 = r19080 * r19081;
        double r19083 = c;
        double r19084 = r19082 * r19083;
        double r19085 = r19079 - r19084;
        double r19086 = sqrt(r19085);
        double r19087 = r19078 + r19086;
        double r19088 = 2.0;
        double r19089 = r19088 * r19081;
        double r19090 = r19087 / r19089;
        return r19090;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r19091 = -1.0;
        double r19092 = c;
        double r19093 = b;
        double r19094 = r19092 / r19093;
        double r19095 = r19091 * r19094;
        return r19095;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.7

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

2. Simplified 52.7

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}}\]

3. Taylor expanded around inf 6.1

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]

4. Final simplification 6.1

   \[\leadsto -1 \cdot \frac{c}{b}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (< 4.93038e-32 a 2.02824e+31) (< 4.93038e-32 b 2.02824e+31) (< 4.93038e-32 c 2.02824e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))