Quadratic roots, wide range

Average Error: 52.1 → 6.5

Time: 7.3s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

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

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 r19075 = b;
        double r19076 = -r19075;
        double r19077 = r19075 * r19075;
        double r19078 = 4.0;
        double r19079 = a;
        double r19080 = r19078 * r19079;
        double r19081 = c;
        double r19082 = r19080 * r19081;
        double r19083 = r19077 - r19082;
        double r19084 = sqrt(r19083);
        double r19085 = r19076 + r19084;
        double r19086 = 2.0;
        double r19087 = r19086 * r19079;
        double r19088 = r19085 / r19087;
        return r19088;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r19089 = -1.0;
        double r19090 = c;
        double r19091 = b;
        double r19092 = r19090 / r19091;
        double r19093 = r19089 * r19092;
        return r19093;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.1

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

2. Simplified 52.1

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

3. Taylor expanded around inf 6.5

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

4. Final simplification 6.5

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

Reproduce

herbie shell --seed 2019350 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4 a) c)))) (* 2 a)))