Quadratic roots, narrow range

Average Error: 28.6 → 16.9

Time: 11.7s

Precision: 64

\[1.0536712127723509 \cdot 10^{-08} \lt a \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt b \lt 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} \lt c \lt 94906265.62425156\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le 91.36334998724752:\\ \;\;\;\;\frac{\frac{\left(\left(c \cdot a\right) \cdot -4 + b \cdot b\right) \cdot \sqrt{\left(c \cdot a\right) \cdot -4 + b \cdot b} - b \cdot \left(b \cdot b\right)}{b \cdot \sqrt{\left(c \cdot a\right) \cdot -4 + b \cdot b} + \left(b \cdot b + \left(\left(c \cdot a\right) \cdot -4 + b \cdot b\right)\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b}{c} \cdot \left(-\frac{a}{a}\right)}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r459201 = b;
        double r459202 = -r459201;
        double r459203 = r459201 * r459201;
        double r459204 = 4.0;
        double r459205 = a;
        double r459206 = r459204 * r459205;
        double r459207 = c;
        double r459208 = r459206 * r459207;
        double r459209 = r459203 - r459208;
        double r459210 = sqrt(r459209);
        double r459211 = r459202 + r459210;
        double r459212 = 2.0;
        double r459213 = r459212 * r459205;
        double r459214 = r459211 / r459213;
        return r459214;
}

↓

double f(double a, double b, double c) {
        double r459215 = b;
        double r459216 = 91.36334998724752;
        bool r459217 = r459215 <= r459216;
        double r459218 = c;
        double r459219 = a;
        double r459220 = r459218 * r459219;
        double r459221 = -4.0;
        double r459222 = r459220 * r459221;
        double r459223 = r459215 * r459215;
        double r459224 = r459222 + r459223;
        double r459225 = sqrt(r459224);
        double r459226 = r459224 * r459225;
        double r459227 = r459215 * r459223;
        double r459228 = r459226 - r459227;
        double r459229 = r459215 * r459225;
        double r459230 = r459223 + r459224;
        double r459231 = r459229 + r459230;
        double r459232 = r459228 / r459231;
        double r459233 = 2.0;
        double r459234 = r459219 * r459233;
        double r459235 = r459232 / r459234;
        double r459236 = 1.0;
        double r459237 = r459215 / r459218;
        double r459238 = r459219 / r459219;
        double r459239 = -r459238;
        double r459240 = r459237 * r459239;
        double r459241 = r459236 / r459240;
        double r459242 = r459217 ? r459235 : r459241;
        return r459242;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 2 regimes

2. if b < 91.36334998724752

   1. Initial program 15.2

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

   2. Simplified 15.2

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

   4. Applied flip3-- 15.3

      \[\leadsto \frac{\color{blue}{\frac{{\left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}\right)}^{3} - {b}^{3}}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + \left(b \cdot b + \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot b\right)}}}{2 \cdot a}\]

   5. Simplified 14.6

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

   6. Simplified 14.6

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

   if 91.36334998724752 < b

   1. Initial program 34.2

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

   2. Simplified 34.2

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

   3. Taylor expanded around inf 17.9

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{a \cdot c}{b}}}{2 \cdot a}\]
   4. Using strategy rm

   5. Applied clear-num 17.9

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{-2 \cdot \frac{a \cdot c}{b}}}}\]

   6. Simplified 17.9

      \[\leadsto \frac{1}{\color{blue}{-\frac{a}{a} \cdot \frac{b}{c}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 16.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 91.36334998724752:\\ \;\;\;\;\frac{\frac{\left(\left(c \cdot a\right) \cdot -4 + b \cdot b\right) \cdot \sqrt{\left(c \cdot a\right) \cdot -4 + b \cdot b} - b \cdot \left(b \cdot b\right)}{b \cdot \sqrt{\left(c \cdot a\right) \cdot -4 + b \cdot b} + \left(b \cdot b + \left(\left(c \cdot a\right) \cdot -4 + b \cdot b\right)\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b}{c} \cdot \left(-\frac{a}{a}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))