Average Error: 34.1 → 10.7
Time: 5.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.8371925747446876 \cdot 10^{53}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 8.67970785211126629 \cdot 10^{-40}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -4.8371925747446876 \cdot 10^{53}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 8.67970785211126629 \cdot 10^{-40}:\\
\;\;\;\;\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 r56187 = b;
        double r56188 = -r56187;
        double r56189 = r56187 * r56187;
        double r56190 = 4.0;
        double r56191 = a;
        double r56192 = r56190 * r56191;
        double r56193 = c;
        double r56194 = r56192 * r56193;
        double r56195 = r56189 - r56194;
        double r56196 = sqrt(r56195);
        double r56197 = r56188 + r56196;
        double r56198 = 2.0;
        double r56199 = r56198 * r56191;
        double r56200 = r56197 / r56199;
        return r56200;
}
↓
double f(double a, double b, double c) {
        double r56201 = b;
        double r56202 = -4.837192574744688e+53;
        bool r56203 = r56201 <= r56202;
        double r56204 = 1.0;
        double r56205 = c;
        double r56206 = r56205 / r56201;
        double r56207 = a;
        double r56208 = r56201 / r56207;
        double r56209 = r56206 - r56208;
        double r56210 = r56204 * r56209;
        double r56211 = 8.679707852111266e-40;
        bool r56212 = r56201 <= r56211;
        double r56213 = 1.0;
        double r56214 = 2.0;
        double r56215 = r56214 * r56207;
        double r56216 = -r56201;
        double r56217 = r56201 * r56201;
        double r56218 = 4.0;
        double r56219 = r56218 * r56207;
        double r56220 = r56219 * r56205;
        double r56221 = r56217 - r56220;
        double r56222 = sqrt(r56221);
        double r56223 = r56216 + r56222;
        double r56224 = r56215 / r56223;
        double r56225 = r56213 / r56224;
        double r56226 = -1.0;
        double r56227 = r56226 * r56206;
        double r56228 = r56212 ? r56225 : r56227;
        double r56229 = r56203 ? r56210 : r56228;
        return r56229;
}

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 < -4.837192574744688e+53

    1. Initial program 37.6

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

    2. Taylor expanded around -inf 5.7

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

    3. Simplified5.7

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

    if -4.837192574744688e+53 < b < 8.679707852111266e-40

    1. Initial program 15.4

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

    3. Applied clear-num15.5

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

    if 8.679707852111266e-40 < b

    1. Initial program 55.1

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

    2. Taylor expanded around inf 7.5

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

  4. Final simplification10.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.8371925747446876 \cdot 10^{53}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 8.67970785211126629 \cdot 10^{-40}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))