Quadratic roots, wide range

Average Error: 52.4 → 6.3

Time: 7.2s

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 r22941 = b;
        double r22942 = -r22941;
        double r22943 = r22941 * r22941;
        double r22944 = 4.0;
        double r22945 = a;
        double r22946 = r22944 * r22945;
        double r22947 = c;
        double r22948 = r22946 * r22947;
        double r22949 = r22943 - r22948;
        double r22950 = sqrt(r22949);
        double r22951 = r22942 + r22950;
        double r22952 = 2.0;
        double r22953 = r22952 * r22945;
        double r22954 = r22951 / r22953;
        return r22954;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r22955 = -1.0;
        double r22956 = c;
        double r22957 = b;
        double r22958 = r22956 / r22957;
        double r22959 = r22955 * r22958;
        return r22959;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.4

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

2. Simplified 52.4

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

3. Taylor expanded around inf 6.3

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

4. Final simplification 6.3

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

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(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)))