Average Error: 33.5 → 10.1
Time: 7.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.0323767944871679 \cdot 10^{127}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.17528679488360856 \cdot 10^{-69}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -4.0323767944871679 \cdot 10^{127}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 1.17528679488360856 \cdot 10^{-69}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\

\mathbf{else}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r62212 = b;
        double r62213 = -r62212;
        double r62214 = r62212 * r62212;
        double r62215 = 4.0;
        double r62216 = a;
        double r62217 = r62215 * r62216;
        double r62218 = c;
        double r62219 = r62217 * r62218;
        double r62220 = r62214 - r62219;
        double r62221 = sqrt(r62220);
        double r62222 = r62213 + r62221;
        double r62223 = 2.0;
        double r62224 = r62223 * r62216;
        double r62225 = r62222 / r62224;
        return r62225;
}
↓
double f(double a, double b, double c) {
        double r62226 = b;
        double r62227 = -4.032376794487168e+127;
        bool r62228 = r62226 <= r62227;
        double r62229 = 1.0;
        double r62230 = c;
        double r62231 = r62230 / r62226;
        double r62232 = a;
        double r62233 = r62226 / r62232;
        double r62234 = r62231 - r62233;
        double r62235 = r62229 * r62234;
        double r62236 = 1.1752867948836086e-69;
        bool r62237 = r62226 <= r62236;
        double r62238 = 1.0;
        double r62239 = 2.0;
        double r62240 = r62239 * r62232;
        double r62241 = -r62226;
        double r62242 = r62226 * r62226;
        double r62243 = 4.0;
        double r62244 = r62243 * r62232;
        double r62245 = r62244 * r62230;
        double r62246 = r62242 - r62245;
        double r62247 = sqrt(r62246);
        double r62248 = r62241 + r62247;
        double r62249 = r62240 / r62248;
        double r62250 = r62238 / r62249;
        double r62251 = -1.0;
        double r62252 = r62251 * r62231;
        double r62253 = r62237 ? r62250 : r62252;
        double r62254 = r62228 ? r62235 : r62253;
        return r62254;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if b < -4.032376794487168e+127

    1. Initial program 53.1

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

    2. Taylor expanded around -inf 3.1

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

    3. Simplified3.1

      \[\leadsto \color{blue}{1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\]

    if -4.032376794487168e+127 < b < 1.1752867948836086e-69

    1. Initial program 12.7

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

    3. Applied clear-num12.8

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}\]

    if 1.1752867948836086e-69 < b

    1. Initial program 53.9

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

    2. Taylor expanded around inf 8.8

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.0323767944871679 \cdot 10^{127}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.17528679488360856 \cdot 10^{-69}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))