Quadratic roots, medium range

Average Error: 43.9 → 12.0

Time: 20.3s

Precision: 64

\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]

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 r3082186 = b;
        double r3082187 = -r3082186;
        double r3082188 = r3082186 * r3082186;
        double r3082189 = 4.0;
        double r3082190 = a;
        double r3082191 = r3082189 * r3082190;
        double r3082192 = c;
        double r3082193 = r3082191 * r3082192;
        double r3082194 = r3082188 - r3082193;
        double r3082195 = sqrt(r3082194);
        double r3082196 = r3082187 + r3082195;
        double r3082197 = 2.0;
        double r3082198 = r3082197 * r3082190;
        double r3082199 = r3082196 / r3082198;
        return r3082199;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r3082200 = -1.0;
        double r3082201 = c;
        double r3082202 = b;
        double r3082203 = r3082201 / r3082202;
        double r3082204 = r3082200 * r3082203;
        return r3082204;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 43.9

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

2. Simplified 43.9

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

3. Taylor expanded around inf 12.0

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

4. Final simplification 12.0

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

Reproduce

herbie shell --seed 2019174 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))